No Prices No Games! Four Economic Models
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Richter, Michael Book No Prices No Games! Four Economic Models Provided in Cooperation with: Open Book Publishers Suggested Citation: Richter, Michael (2024) : No Prices No Games! Four Economic Models, ISBN 9781805114581, Open Book Publishers, Cambridge, https://doi.org/10.11647/OBP.0438 This Version is available at: https://hdl.handle.net/10419/312707 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/
NO PRICES NO GAMES! FOUR ECONOMIC MODELS SECOND EDITION
NO PRICES NO GAMES! FOUR ECONOMIC MODELS SECOND EDITION Michael Richter Baruch College Royal Holloway, University of London Ariel Rubinstein Tel Aviv University New York University
c2024 Michael Richter and Ariel Rubinstein This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs license (CC BY-NC-ND 4.0). This license allows you to share, copy, distribute, and transmit the work providing you do not modify the work, you do not use the work for commercial purposes, you attribute the work to the authors, and you provide a link to the license. Attribution should not in any way suggest that the authors endorse you or your use of the work and should include the following information. Michael Richter and Ariel Rubinstein, No Prices No Games!. Second Edition. Cambridge, UK: Open Book Publishers, 2024, https://doi.org/10.11647/OBP.0438 Further details about CC BY-NC-ND licenses are available at https://creativecommons. org/licenses/by-nc-nd/4.0/. All external links were active at the time of publication unless otherwise stated and have been archived via the Internet Archive Wayback Machine at https://archive. org/web. ISBN Paperback: 978-1-80511-308-9 ISBN Hardback: 978-1-80511-309-6 ISBN Digital (PDF): 978-1-80511-310-2 DOI: 10.11647/OBP.0404 Information about any revised version of this work will be provided at https://doi. org/10.11647/OBP.0438. Cover image: Ariel Rubinstein Cover concept: Michael Richter and Ariel Rubinstein Cover design: Jeevanjot Kaur Nagpal Version: 2024.11.12
Contents Personal Note vii Notation and Terminology xi 0 Introduction 1 0.1 The Book 1 0.2 The Notion of an Economy 3 0.3 Examples of Economies 4 0.4 Equilibrium Concepts 8 1 Equilibrium in the Jungle 13 1.1 The Housing Jungle: Model and Equilibrium 15 1.2 The Jungle Equilibrium: Welfare 18 1.3 Comparison to the Competitive Equilibrium 21 1.4 Comments on the Jungle Equilibrium 25 1.5 The Division Jungle 28 1.6 The Division Jungle: Comments on Welfare 33 1.7 A Didactic Perspective 34 2 The Permissible and the Forbidden 37 2.1 The Y-Equilibrium Concept 39 2.2 Y-Equilibrium, Pareto Optimality, and Envy-Freeness 43 2.3 Euclidean Economies 45 2.4 The “Kosher” Economy 47 2.5 Convex Y-Equilibrium 49 2.6 Pareto Optimality and Existence of Convex Y-Equilibrium 51 2.7 A Structure Theorem for Convex Y-equilibrium 53 2.8 The Division Economy 55 2.9 The Give-and-Take Economy 59 2.10 The Stay Close Economy 61
vi Contents 3 Status and Indoctrination 65 3.1 Status Equilibrium 67 3.2 Status Equilibrium – Examples 68 3.3 A Detour: Convex Preferences 71 3.4 Primitive Equilibrium 77 3.5 A First Welfare Theorem 81 3.6 A Second Welfare Theorem 83 3.7 Primitive Equilibrium – More Examples 84 3.8 Initial Status Equilibrium 87 4 Biased Preferences Equilibrium 93 4.1 The Economy and the Equilibrium Concept 95 4.2 The Give-and-Take Economy 100 4.3 The Fixed-Prices Exchange Economy 102 4.4 Housing-Type Economies 106 5 A Comparison to Game Theory 113 5.1 The Matching Economy 114 5.2 The Jungle Equilibrium 118 5.3 Restricting Partnerships: Pairwise Y-equilibrium 123 5.4 Prestige by Partner: Status Equilibrium 125 5.5 Prestige by Self: Initial Status Equilibrium 127 5.6 Comparing the Approaches 130 5.7 The Majority Voting Economy 132 5.8 Convex Y-equilibrium 133 5.9 Biased Preferences Equilibrium 134 5.10 The Majority Voting Game and Nash Equilibrium 135 5.11 Comparing our Approaches with Nash Equilibrium 137 References 139
Personal Note We feel some dissatisfaction with current trends in Economic Theory. Novelty has fallen by the wayside. Models have become overly complicated and excessively sophisticated mathematically. Papers are too long and contain few new fundamental ideas. Authors go to great lengths to masquerade theoretical work as being applied. This book contains a collection of models in Economic Theory that are simple in their approach and straightforward mathematically. They include new concepts and are presented concisely, without any pretend claims regarding their direct applied usefulness. At best, the models have helped us to understand various social institutions, such as power, status, social norms, and preference biases as a means to achieve harmony in economic environments. Needless to say, we do not advocate for the adoption of any of these institutions but, rather, investigate their rationales. Our main objective is to disrupt the convention that every economic model should be either a market with prices or a strategic game. The book brings the papers together in a unified language and an accessible style. It can be used to teach a unit in an advanced Economic Theory course or as a source for independent study. This project began just over a decade ago and developed from a series of papers, most of which we wrote jointly. We are still working on the project. This version of the book is the first revision of the original manuscript and includes both minor corrections as well as a rewriting of Chapter 5. We intend to continue developing the book in the coming years. MR: https://mrichter.co AR: https://arielrubinstein.tau.ac.il or https://arielrubinstein.org
2 Chapter 0. Introduction rationality assumption with explicit reference to decision procedures, while the Behavioral Economics literature added realistic psychological motives to purely materialistic considerations. However, these developments left in place the standard view of economic interactions as being resolved through prices or games. This short book is aimed primarily at young economists. It is intended to demonstrate models of interaction between agents with NO PRICES and NO GAMES. We do not claim that these models are any more (or less) “true”, “realistic”, or “useful” than others. In fact, we do not believe that these adjectives are even relevant to models in Economic Theory. As mentioned, we view these models as economic stories: they are interesting; they capture some aspect of reality; they are elegant; they are novel; or ... not. In the models we study, agents are purely self-interested, and equilibrium reflects a social institution that systematically alters either the agents’ choice sets or their preferences. In this respect, the models are closer to market models than to game-theoretical models, and, as in the case of market models, an equilibrium will not be just a profile of choices made by individuals, but will also specify an additional price-like element that uniformly affects all agents. While we do not have any applied message, working on these topics has brought us to the realization that economic harmony can be achieved by institutions other than prices or games. Of course, this realization could have happened even without any models, but they illuminate how such institutions may function in bringing harmony to economic situations. We focus on four institutions: Power (Chapter 1), Social Norms governing what is permissible and what is forbidden (Chapter 2), Status (Chapter 3), and Preference Biases (Chapter 4). In the last part of the book (Chapter 5), we compare our approach to other more established ones. We refrain from any normative assessments of the institutions. Such judgements are left to the reader.
0.2 The Notion of an Economy 3 0.2 The Notion of an Economy The stage on which this book’s plots will be performed is a formal model called an economy. The model is intended to abstractly capture situations in which each agent in a society chooses an alternative and there exists a fundamental tension between the agents’ personal desires and society-wide feasibility constraints. (For example, in a Walrasian economy, consumers have unquenchable desires, but overall resources are limited.) The model’s abstraction allows us to consider examples that are “economics” in the conventional sense of the term, but also others that are not. Nevertheless, the term “economy” will be used throughout since all of the models feature the fundamental economic conundrum: individuals’ desires cannot all be satisfied due to feasibility constraints on the profiles of choices that can be made in the society. Definition: Economy An economy is a tuple ‹N,X,(%i)i∈N,F›where: •N={1,...,n}is the set of agents. •Xis a set of personal alternatives. Each agent chooses an element from X. In its most general form, no structure is imposed on X; however, we sometimes consider the special case where Xis a subset of a Euclidean space. •%iis agent i’s preference relation over the set X. The fact that preferences are defined over Xrather than over the set of choice profiles embodies the assumption that there are no externalities: each agent cares only about his chosen alternative irrespective of what other agents choose (as in the case of markets, but unlike in the case of games).
4 Chapter 0. Introduction •F⊂XNis a non-empty set of feasible profiles. A choice profile (xi)i∈Nspecifies an element xi∈Xfor each agent i∈N. The set XNis comprised of all choice profiles. Not all profiles are feasible, and the feasibility constraint is given by a set F⊂XN. Unless stated otherwise, we assume that Fis closed under all permutations (i.e. the feasibility constraint is anonymous and does not discriminate between agents). We usually abbreviate (xi)i∈Nas (xi). An economy without preferences, ‹N,X,F›, is called an environment. Sometimes, we consider an extended version of an economy which specifies for each agent ian element eiin X, with the interpretation that i always has the right to choose ei. The vector (ei)is required to be in F, namely the allocation of these initial rights is feasible. The role of the vector (ei)is analogous to that of the profile of initial endowments in the standard exchange economy. Definition: Extended Economy An extended economy is a tuple ‹N,X,(%i)i∈N,F,(ei)i∈N›where: • ‹N,X,(%i)i∈N,F›is an economy. •(ei)i∈Nis a feasible initial profile. 0.3 Examples of Economies We now introduce some economies which appear throughout the book. As mentioned, some of the examples are traditional economic settings while others demonstrate the framework’s ability to model a variety of alternative social situations.
0.3 Examples of Economies 5 Example: The Housing Economy The set Xcontains ndistinct elements called houses (recall that nis the number of agents) and each agent ihas preferences %iover the houses. Each agent chooses a house, but no two agents can occupy the same one. That is, Fis the set of profiles that assigns a distinct house to every agent. This economy is the iconic model of Shapley and Scarf (1974). The model is attractive due to its simplicity and its usefulness as a platform for introducing a rich variety of concepts. If each agent’s ideal is distinct, then the situation is “bliss”, there are no conflicting desires, and so there is no need for a social institution to achieve harmony in the society. However, bliss does not usually exist, and, therefore, we need social institutions to resolve the conflict between agents’ desires and societal feasibility. Example: The Division Economy There are Kcommodities, and the set of alternatives X=RK +consists of the non-negative bundles of those commodities. Preference relations are monotonic, continuous, and convex. As in standard market settings, there are limited resources, and the set of feasible profiles F={(xi)|Σixi=e}is the set of all partitions of a total endowment e∈RK +among the agents. If we would add initial endowments to the model, then we would obtain the classical framework used by economists since Edgeworth (1881) to discuss voluntary exchange and competitive equilibrium. Bliss is always impossible, unlimited wants must be constrained in the face of limited resources, and achieving social harmony requires some social institution.
6 Chapter 0. Introduction Example: The Give-and-Take Economy There are situations in life in which redistribution is imposed by an authority that forces individuals to comply, and there are others in which redistribution is accomplished by means of voluntary exchange between individuals. There are further situations (e.g. a soup kitchen) in which exchange is carried out by unilateral actions: some individuals give while others take without any exercise of power, commitments to “return the favour”, or coercion by an authority. These actions are self-motivated: some people like to give, while others like to take. But typically, such motives will not balance each other out, and social norms are needed to achieve harmony. Formally, we consider the following give-and-take economy, which was first studied by Sprumont (1991). Let X= [−1,1], where a positive xrepresents a withdrawal of xfrom a social fund (i.e. taking) and a negative xrepresents a contribution of |x|to the social fund (i.e. giving). Preferences are assumed to be continuous and strictly convex (that is, single-peaked) but need not be monotonic. Feasibility requires that the social fund is balanced, that is, F={(xi)|Σixi=0}. Example: The Clubs Economy The set Xconsists of a finite set of clubs (see Buchanan (1965)). Each agent chooses a single club to become a member of. Agents have preferences over the clubs and not over the clubs’ members. The feasibility constraint is defined by the limits on how many people can belong to each club. Specifically, there is a vector of positive integers (qx)x∈Xwhere qxis the quota for club x(for non-triviality, we require that the sum of the quotas is at least n). The set of feasible profiles are those for which no club is chosen by more people than allowed by its capacity.
0.3 Examples of Economies 7 Example: The Stay Close Economy This example illustrates the potential of our abstract concept to expand the scope of classical economic analysis. It does not involve goods but nonetheless fits squarely into our concept of an economy. In this example, Xis a set of locations in some geographical area. Each agent chooses a location in Xand has preferences over the locations. Not every profile of locations is feasible because the society is under threat and its survival depends upon the ability of its members to quickly reach one another in the case of danger. Therefore, all members need to live close enough to each other so that whenever one of them is attacked the others can quickly come to his defence. Formally, the feasibility constraint Frequires that the distance between any two agents does not exceed some constant d. When dis very large, every agent can choose his ideal location, but when dis small, this is no longer feasible. We refer to the special case when d=0 as the consensus economy. This fits, for example, the situation of a political party whose members need to present a united front. That is, in order to maintain cohesion, all members of the party need to express the same position. Example: The Matching Economy Matching problems are classics of Cooperative Game Theory. Agents have to find a match, and each agent has a preference relation over his potential partners. This situation fits our framework by letting the set of alternatives Xbe the set of agents N. That is, each agent chooses a partner, which can be himself. Each has a preference relation on X that places himself at the bottom. The feasibility constraint Fstipulates that for any iand j, if ichooses j, then jmust choose i. Note that this feasibility constraint differs from those in the previous examples in that Fis not closed under all permutations.
8 Chapter 0. Introduction Example: The Sequential Production Economy A group of nagents works in nshifts to transform an initial product x0 into a different product. Each works one shift, and the agents may work in any order. An agent’s ability to produce a product, which might be just an intermediate product, depends on the output of the previous shift. The group possesses a technology that enables certain transformations of one product into another. More precisely, Xis a set of products that includes x0. Each agent has preferences for the product that he produces (rather than for the final product). The common production technology is a correspondence T from Xto Xwhere T(x)is the set of outputs which xcan be transformed into. Any agent can choose to be “idle” and not transform the product produced in the previous shift, that is x∈T(x). Thus, Fis the set of all permutations of profiles (x1,...,xn)such that xm∈T(xm−1)for m=1,...,n. 0.4 Equilibrium Concepts This book introduces and analyzes several solution concepts and applies them to a variety of economic environments. In general, a solution concept relates to some domain of economic environments and determines for each environment a set of harmonious outcomes. These outcomes are harmonious in the sense that the assumed forces that may disturb harmony are neutralized. In our setting, the domain of a solution concept is a class of economies and a candidate for equilibrium typically includes two components: (i) A profile of choices — one choice for each agent. (ii) A specification of certain parameters that systematically influence either agents’ choice problems or their preference relations. Harmony is achieved in equilibrium as follows: agents make individually optimal choices, and the parameters restrict their choice sets (or, in one case,
0.4 Equilibrium Concepts 9 biases their preferences) to be compatible in the sense that the resulting profile of choices is feasible. The concepts will differ in the parameters and in how they restrict agents’ choice sets. The solution concepts discussed in the book can be divided into two groups. In the choice group, each agent’s choice set depends on a price-like equilibrium parameter but not on the equilibrium profile of choices. Such choices must be individually optimal and compatible. These concepts are similar in structure to the notion of competitive equilibrium whose parameters are prices and each agent’s choice set (budget set) is determined solely by his initial endowment and the prices. Three of our solution concepts belong to this group: Y-equilibrium (Chapter 2). The price-like parameter in a Y-equilibrium is a set of alternatives which is interpreted as the set of “permissible” alternatives that uniformly binds all agents. When making a choice, an agent only needs to know the set of permissible alternatives and nothing else. In equilibrium, the permissible set is a maximal set of alternatives from among those which satisfy the following property: if every agent chooses a preference-maximizing alternative from this set, then the resulting choice profile is feasible. Initial Status Equilibrium (Chapter 3). This concept relates to an extended economy wherein the notion of an economy is enriched with an additional element: a feasible profile of alternatives, one for each agent, in which the alternative designated to an agent is interpreted as one that he always has the right to choose. The price-like parameter in an initial status equilibrium is an ordering of the alternatives that can be interpreted as “status” or “value”. An agent’s choice set is comprised of all alternatives which have a weakly lower status than his endowment. In equilibrium, a status ordering prevails such that each agent’s designated alternative is his most preferred from among his choice set, namely the set of all alternatives that are of weakly lower status than his initial alternative. As always, an equilibrium profile of choices has to be feasible.
10 Chapter 0. Introduction Biased Preferences Equilibrium (Chapter 4). The price-like parameter in a biased preferences equilibrium is a vector that systematically biases agents’ preferences. In this model, agents’ choice sets are fixed and unaffected by the parameters. Rather, in an equilibrium, a systematic bias prevails such that each agent chooses a most-preferred alternative from his choice set, according to his biased preferences, and the profile of choices is feasible. In the deviation group of solution concepts, an equilibrium is a profile of choices that is immune to any single agent’s deviation from his prescribed alternative to any alternative in a set determined by the equilibrium parameters. This is the approach taken in Game Theory. For example, a Nash equilibrium is a profile of actions such that, for each agent, the outcome of that profile is not worse for him than any other outcome he can achieve given the other players’ choices in the profile. Two of our solution concepts fall into this group: Jungle Equilibrium (Chapter 1). In this case, the economy is extended with an exogenous power ranking of the agents; but, in an equilibrium, there are no additional parameters. In the jungle, an agent can steal from those that are weaker than himself; therefore, his choice set is determined by his equilibrium choice as well as the choices of those who are weaker than him. A jungle equilibrium is a profile of choices such that each agent’s assigned choice is preference-maximal from among the set of the alternatives he can obtain by stealing resources from weaker agents. Status Equilibrium (Chapter 3). Again, the price-like equilibrium parameter is an ordering over the alternatives that connotes status (or value). However, in this case, an agent’s choice set depends not only on this parameter but also on his own equilibrium alternative. In detail, his choice set is the set of all alternatives which are weakly lower-ranked than his equilibrium alternative (rather than his initial alternative). An equilibrium is a status ordering and a profile of optimal choices such that the profile of choices is feasible.
0.4 Equilibrium Concepts 11 The book analyzes each of these solution concepts both in the abstract, by means of general propositions, and more concretely, by applying the solution concepts to a variety of economic environments (some familiar and some novel).
18 Chapter 1. Equilibrium in the Jungle that an equilibrium exists. Proposition 1.1 above leaves open the possibility that other equilibria may exist. However, we will now show that, given the assumption that all preference relations are strict, the equilibrium is unique. Proposition 1.2: Uniqueness Every jungle housing economy has a unique equilibrium. Proof: Consider the jungle housing economy ‹N,X,(%i)i∈N,F,B›. Assume, contrary to the claim, that (ai)and (bi)are two different equilibria of the jungle. Denote by i∗the strongest individual ifor whom ai6=bi. Suppose that ai∗i∗bi∗. Since the set of houses allocated to individuals 1 through i∗−1 is the same in both (ai)and (bi), it must be that in (bi), the house ai∗is held by an agent jwho is weaker than i∗. Thus, i∗Bj and bj=ai∗i∗bi∗which contradicts (bi)being an equilibrium. 1.2 The Jungle Equilibrium: Welfare We move to discuss two fundamental welfare theorems. In abstract, the first states that, for any initial conditions, equilibrium outcomes are Pareto-optimal profiles; while the second states that Pareto-optimal profiles are equilibrium outcomes for some initial conditions. In the jungle housing economy, the initial condition is the power relation. We now bring proofs of the fundamental welfare theorems for the jungle housing economy (Abdulkadıro˘ glu and Sönmez (1998) show equivalent results that the set of allocations obtained by a serial dictatorship of some order is equal to the set of Pareto-optimal allocations). Proposition 1.3: The First Welfare Theorem The jungle equilibrium is Pareto optimal.
1.2 The Jungle Equilibrium: Welfare 19 Proof: Recall the assumption that preferences are strict. Let (xi)be the jungle equilibrium. Assume that, contrary to the claim, there is a feasible profile (yi)that Pareto dominates (xi). Let ibe the strongest agent for whom yi6=xi. Then, yiixiand xj=yifor some agent jweaker than i, contradicting the fact that (xi)is a jungle equilibrium. The above proof relies on the strictness of the agents’ preferences. If some individuals have indifferences in their preferences, then a jungle equilibrium might not be Pareto optimal. For example, in the case of two agents and two houses aand b, if a∼1band a2b, then both (x1,x2)=(a,b)and (b,a)are jungle equilibria, but the profile (a,b)is not Pareto optimal. We now move to the second welfare theorem. Since the initial conditions for the jungle housing economy are a power relation, the appropriate second welfare theorem states that for any Pareto-optimal profile, there is a power relation for which the jungle equilibrium is precisely that profile. Recall that in the standard exchange model, the social planner assigns initial endowments to the agents with the expectation that trade between them will yield the desired allocation of the total endowment. Analogously, in the jungle housing economy, the social planner assigns the power relation with the expectation that the law of the jungle will yield the desired allocation of the houses. Proposition 1.4: The Second Welfare Theorem Given any housing economy ‹N,X,(%i)i∈N,F›and Pareto-optimal profile (xi), there exists a power relation Bsuch that (xi)is the unique jungle equilibrium of the jungle housing economy ‹N,X,(%i)i∈N,F,B›.
20 Chapter 1. Equilibrium in the Jungle Proof: First, note that in every Pareto-optimal profile (xi), at least one individual is allocated his favourite house: Otherwise, start with some agent i0, and define ik+1to be the agent who holds ik’s favourite house (ik+16=ik because no agent’s favourite house is his current house). Since Nis finite, there will eventually be some lsuch that k>l≥0 and ik+1=il. Then, assigning yij=xij+1for each l≤j≤k, and keeping yj=xjfor all other agents, we obtain a feasible allocation (yi)which Pareto-dominates (xi). We construct a power relation Bas follows: Let i1be an agent for whom xi1is his first-best house and make him the most powerful agent. Now remove i1from the set of individuals and xi1from the set of houses. The inductive process continues as follows: at the beginning of the k+1st stage, kagents have been assigned power. The allocation of the remaining houses among the remaining agents is Pareto optimal; therefore, identify an agent ik+1for whom xik+1is his favourite house from among X−{xi1,...,xik}and make him the (k+1)st-most powerful individual. By construction, for each agent i, the house xiis preferred by iover every house that is allocated to an individual weaker than him according to B. Thus, (xi)is a jungle equilibrium of ‹N,X,(%i)i∈N,F,B›. Externalities: To incorporate externalities, we modify the model by defining the agents’ preferences over the set of feasible profiles (rather than the set of houses) and by allowing indifferences. The definition of a jungle equilibrium also needs to be modified. When deciding whether to confiscate a house, an agent compares the current profile to the one that would result if he does so. One way to proceed is by interpreting iBjto mean that agent ican force jto exchange houses: itakes over the house occupied by j and forces jto accept the house ipreviously occupied. Thus, an equilibrium of the jungle with externalities ‹N,X,(%i)i∈N,F,B›is a feasible profile (ai)such that for no two agents j,j0∈Nis it the case that jBj0and (bi)j(ai), where (bi)is the allocation that differs from (ai)only in the fact that bj=aj0and bj0=aj.
1.3 Comparison to the Competitive Equilibrium 21 In the model with externalities, a jungle equilibrium does not necessarily exist. For example, consider a case with 3 agents where 1 B2B3 and X={a,b,c}. Think of the houses as being located clockwise on a circle: a→b→c→a. Suppose that agent 1 top-ranks the three profiles where he is the clockwise neighbour of 2. Likewise, agent 2 top-ranks the three profiles where he is the clockwise neighbour of 1. There is no equilibrium because in any profile, agent 3 is the clockwise neighbour of either agent 1 or 2, in which case the other agent desires agent 3’s position and is stronger than him. It is also easy to find an example with three individuals in which a jungle equilibrium exists but is not Pareto optimal. 1.3 Comparison to the Competitive Equilibrium Shapley and Scarf (1974) used the extended housing economy for studying the notion of competitive equilibrium in a simple setting with discrete goods. Recall that the extended housing economy is a tuple ‹N,X,(%i)i∈N,F,(ei)i∈N› where ‹N,X,(%i)i∈N,F›is a housing economy and (ei)is a feasible profile which is interpreted as an initial allocation of the houses. Thus, instead of a power relation, the housing economy model is enriched with the specification of an initial endowment for each agent. Shapley and Scarf (1974) define a competitive equilibrium for this extended economy to be a profile of prices (one real number to each house) and a profile of houses such that: (i) each agent prefers his assigned house to any that is not more expensive than his initial endowment and (ii) the housing assignment is feasible. Formally: Definition: Competitive Equilibrium Acompetitive equilibrium for an extended housing economy is a tuple ‹(px)x∈X,(xi)i∈N›where (px)x∈Xis a profile of prices and (xi)is a profile of houses such that: (i) For every individual i, the house xiis %i-maximal in {x|pei≥px}. (ii) The profile (xi)is in F.
22 Chapter 1. Equilibrium in the Jungle The following proposition, due to Shapley and Scarf (1974), shows that a competitive equilibrium exists. The proof, due to David Gale, uses an algorithm which is based on the notion of a top-trading cycle. Given any group of agents with initial endowments, a top-trading cycle is a cycle of agents all of whom most prefer the house of the next agent in the cycle from among those that the group members are endowed with. If an agent prefers his own house to all others then he makes a cycle of length one. We will see that a top-trading cycle always exists. The top-trading cycle algorithm proceeds as follows: at each stage, a toptrading cycle is identified. Each agent in the cycle is exclusively assigned the house of the next agent in the cycle (which he prefers from among the houses that were not assigned previously). All houses in the cycle are assigned the same price, which is lower than the prices of all previously assigned houses, and both the assigned agents and the assigned houses are removed. Proposition 1.5: Existence of Competitive Equilibrium For any extended housing economy, a competitive equilibrium exists. Proof: Let ‹N,X,(%i)i∈N,F,(ei)i∈N›be an extended housing economy. We first show that a top-trading cycle exists for every group of agents G. Start arbitrarily with an agent i0∈G, and define ik+1∈Gas the initial holder of ik’s favourite house from the set of houses belonging to G. Since the group is finite, there will eventually be some lsuch that k≥l≥0 and ik+1=il. Then, the sequence (il,...,ik)constitutes a top-trading cycle. See Figure 1.1 for an illustration of the argument where l=2 and k=5. i0i1i2=i6i3i4i5 wants wants wants wants wants wants Figure 1.1 The Top–Trading Cycle algorithm.
1.3 Comparison to the Competitive Equilibrium 23 The algorithm constructs a partition {I1,...,Il,...,IL}of Nas follows: First, find a top-trading cycle from the group of all agents. Set I1to be the set of members of this cycle and assign to each of them the house he most prefers. Continue inductively: at stage l+1, find a top-trading cycle from among the group N−I1−...−Iland for each member of the cycle assign the house which he most prefers from among those initially held by the group. Set Il+1to be the set of members in the cycle. Continue in this fashion until a partition is completed. Choose a sequence of numbers p1>p2>... >pL>0 and, for each x∈X, define px=plwhere the agent who initially occupies xis in Il. The assigned profile (xi)together with the price vector (px)constitutes a competitive equilibrium because (xi)∈F, and every agent iin Ilchooses his favourite house from within his “budget set”, namely the set of houses initially held by the members of Il∪...∪IL. Comparing the above construction to that of the jungle equilibrium clarifies the source of power in the market vs. the source of power in the jungle. In Gale’s construction, in each round some agents obtain their favourite house from among those not allocated in previous rounds. So too in the jungle equilibrium. However, in the case of competitive equilibrium, the order is determined by the existence of a “top-trading cycle” which indicates the parties’ joint interest in making an exchange, whereas in the jungle the order is determined by power, independently of the agents’ preferences. Given that the preference relations are assumed to be strict, there is a unique competitive equilibrium allocation (for a proof, see Osborne and Rubinstein (2023)). However, this allocation can be supported by many price systems, and it can even be that one house is more expensive than another in one equilibrium price system but less expensive in another. The two fundamental welfare theorems hold for the competitive equilibrium in this model:
24 Chapter 1. Equilibrium in the Jungle (a) Any competitive equilibrium ‹(px),(xi)›is Pareto-optimal: if (yi)∈FPareto dominates (xi)then pyi≥pxifor all iwith strict inequality for any agent ifor whom yiixiand thus Σi∈Npyi>Σi∈Npxialthough the two sums are equal. (b) For any Pareto-optimal allocation (xi)there is a price vector (px)such that ‹(px),(xi)›is a competitive equilibrium. By Proposition 1.5 a competitive equilibrium exists for the extended economy with the initial allocation (xi). Its allocation (yi)is weakly Pareto superior to (xi)and since (xi)is Pareto-optimal it must coincide with (xi). Therefore, if we start with (ei)=(xi)the proof constructs a competitive equilibrium in which each agent ikeeps xi. Power and Wealth: Since the jungle equilibrium is Pareto optimal, it can be supported by prices as a competitive equilibrium. This invites a natural question: what is the relationship between power and wealth? First, there is always a price system in which “stronger” in the jungle economy means “richer” in the competitive equilibrium of the extended housing economy with the initial endowment profile being the jungle equilibrium of the jungle economy. Formally, let (xi)be the jungle equilibrium in the housing economy jungle ‹N,X,(%i)i∈N,F,B›. The extended housing economy ‹N,X,(%i)i∈N,F,(ei=xi)i∈N›has a competitive equilibrium ‹(px),(xi)›where pxi>pxjwhenever ij. However, other equilibrium price vectors may exist. For example, if the strongest agent top-ranks his own house while all other agents bottom-rank it, then there also exists a competitive price vector in which the strongest agent is the poorest. In fact, if we modify the economy somewhat, then there may be no jungle equilibrium in which the statement “stronger =richer” holds. For example, recall the clubs economy where each agent chooses one club from the set X, and no more than qxagents can choose club x. Consider the economy with 4 agents, where X={a,b}and qa=qb=2. If the preferences are such that agent 1 prefers aand all other agents prefer b, then the unique jungle equilibrium is (a,b,b,a). However, in this equilibrium, every agent obtains his first-best club except for agent 4 and to prevent agent 4 from getting what he wants it must be that pb>pa. Thus, any price vector which supports the jungle equilibrium allocation must have the property that the strongest agent is the poorest.
1.4 Comments on the Jungle Equilibrium 25 1.4 Comments on the Jungle Equilibrium Comparative statics: The jungle equilibrium satisfies the expected comparative statics property that advancing an agent in the power ranking cannot hurt the agent. To see this, recall that there is a unique jungle equilibrium and it can be calculated via a serial dictatorship procedure. When an individual agent becomes stronger, all agents who are still stronger than him will continue to make the same choices, while the individual now gets to choose earlier and, therefore, has a strictly larger set of houses to choose from. On the other hand, in the case of competitive equilibrium, improving an agent’s initial house endowment, according to his own preferences, might make him worse off in equilibrium. Although the new house is better for him, it might be unattractive to other agents. Thus, when applying the top-trading cycle algorithm, it could be that he initially appeared in the first cycle and, after the “improvement”, he now appears in the last cycle and, therefore, ends up worse off in the new equilibrium than in the old one. Manipulability: The jungle equilibrium is immune to preference misrepresentations by an agent. Again, the unique jungle equilibrium can be calculated by the serial dictatorship algorithm. When it is an agent’s turn to choose, the set of alternatives that he chooses from is unaffected by his declared preferences, and, thus, he can do no better by misrepresenting his preferences. This nonmanipulability property also holds for competitive equilibria. Indifferences: Even if some of the agents’ preferences are not strict, the serial dictatorship procedure still produces a jungle equilibrium. However, it is not necessarily unique since, when an agent has to make a choice, he might have more than one maximal option and each produces a different equilibrium. Note that indifferences can also create a multiplicity of competitive equilibrium profiles in the housing economy market. Equilibrium and Dynamics: The jungle equilibrium concept is static, like most solution concepts in Economic Theory. The following is an example of dynamics that lead to a jungle equilibrium: At the beginning, all agents are
26 Chapter 1. Equilibrium in the Jungle assigned to be “homeless”. At stage t+1, given the assignment of the agents at stage tto X∪{homeless}, every homeless agent chooses his favourite house from among those that, at the end of stage t, are either: i) vacant or ii) assigned to an agent weaker than him. Every agent who currently occupies a house chooses to stay there. At the end of stage t+1, if a house is chosen by only one agent, then he settles there. If more than one agent chooses the same house, then the strongest among them settles there and all the rest remain homeless. Proposition 1.6: Equilibrium Dynamics The above dynamics converges in at most nstages to the jungle equilibrium. Proof: Let Htbe the set of homeless agents at the beginning of stage tand it be the most powerful among them. If there are any homeless agents at stage t+1, then itit+1: To see why, note that at stage t,itwill obtain a home because all homeless agents are weaker than him and so he will win at any home which he approaches. Furthermore, all agents stronger than itremain in their homes as no one challenges them. Thus, in the beginning of stage t+1, all homeless agents must be weaker than it. Therefore, after at most nstages, all agents have a home and the process terminates at a profile (xi). Suppose that (xi)is different than the jungle equilibrium profile (yi). Take ito be the strongest agent for whom xi6=yi. Thus, iBjwhere jis the agent who holds yi, i.e. xj=yi. By Proposition 1.2, yiis i-maximum in X− {y1,...,yi−1}=X− {x1,...,xi−1}and therefore yiixi. At the stage in the algorithm where i selected xiit must be that yiwas being held by someone stronger than i. But, in the algorithm, when a house changes hands, it can only go to someone stronger so as it eventually reaches jit must be that jBi, a contradiction.
1.4 Comments on the Jungle Equilibrium 27 A different power relation for each house: A key assumption in the jungle model is the uniformity of the power relation: if an agent iis able to evict agent jfrom one house, then he is able to evict him from any house. An extension of the model allows for dependence of the power relation on the house in dispute. Suppose that, for each house x∈X, there is a strict power ordering Bxwhere iBxjmeans that agent iis stronger than agent jin a fight over house x. That is, if agent joccupies xand iBxj, then agent ican confiscate x. An equilibrium in the economy with house-dependent power relations ‹N,X,(%i)i∈N,F,(Bx)x∈X›is a profile (xi)such that there are no two agents iand jsuch that iprefers the house occupied by jto the house he occupies (xjixi) and iis stronger than jregarding xj(iBxjj). As commented on in Rubinstein and Yıldız (2022), the notion of a jungle equilibrium in ‹N,X,(%i)i∈N,F,(Bx)x∈X›is equivalent to pairwise stability in an auxiliary two-sided matching problem between Nand Xwhere each agent i∈Nhas the preference %iover Xand each house x∈Xhas the preference relation Bxover N. A profile (xi)is pairwise stable if there is no pair iand xjsuch that iprefers xjover xi(xjixi) and xj“prefers” iover j(iBxjj). Therefore, a profile is pairwise stable in the auxiliary matching problem if and only if it is a jungle equilibrium with house-dependent power relations. Gale and Shapley (1962) showed, using the deferred acceptance algorithm, that a pairwise stable matching exists in any two-sided matching problem. Thus, in the jungle with house-dependent power relations, a jungle equilibrium also exists. Since the pairwise stable matching need not be unique, neither is the jungle equilibrium when the power relation is house-dependent. Finally, Gale and Sotomayor (1985)’s analysis implies that there is always a jungle equilibrium (xi)which is weakly Pareto optimal, in the sense that there is no assignment (zi)such that ziixifor every i∈N.
34 Chapter 1. Equilibrium in the Jungle mismatch implies that for a division economy, not every Pareto-optimal profile can be obtained as a jungle equilibrium by appending some power relation to that economy. Power and wealth: Making a statement about the relation between power and wealth in the division economy is more involved than in the housing economy since the existence of a competitive equilibrium price vector that supports the jungle equilibrium is not guaranteed, even if all of the agents’ consumption sets are the same. For a discussion of this issue, see Piccione and Rubinstein (2007). 1.7 A Didactic Perspective The discussion in this chapter also has a didactic purpose, as expressed in the personal concluding remarks made by one of us in Piccione and Rubinstein (2007), which are essentially quoted here (with some small changes): When I present the model in public lectures, I ask the audience to imagine that they are attending the first lecture of a course at the University of the Jungle, entitled Introduction to the Principles of Economics. The analogy of such a presentation to the way we introduce the market equilibrium in a standard Microeconomics course serves as a device to shed light on the implicit message that Microeconomics students receive from us. Being faithful to the classical economic tradition, the jungle model does not stray far from the standard exchange economy. We use terminology that is familiar to any economics student. After having defined the notion of jungle equilibrium, we conduct the same type of analysis that can be found in any microeconomics textbook on competitive equilibrium. We show existence and then discuss the first and second fundamental welfare theorems. We emphasise the analogy between the initial endowments in an exchange economy and the initial distribution of power in the jungle: both are used to determine
1.7 A Didactic Perspective 35 the equilibrium distribution of commodities among the agents. Were I teaching this model, I would also add the standard comments regarding externalities and the place for government intervention. There are arguments which attempt to dismiss the comparison between markets and jungles: One might argue that the market has the virtue of providing incentives to “produce” and to enlarge the size of the “pie” to be distributed among the agents. On the other hand, one could also argue that the jungle provides incentives to develop power. In the market economy, agents invest effort in producing more goods. In the jungle economy, agents invest effort in becoming stronger, an asset for a society that needs to defend itself against invaders or invade others in order to accumulate resources. One might argue that market mechanisms preserve resources that would otherwise have been wasted in conflict. Note, however, that under complete information a stronger agent can persuade a weaker one to part with his goods using only the threat of force. Societies often create rituals that help individuals gauge the power of others and thereby avoid the costs of conflict. Under incomplete information, the market also wastes resources. And finally, I have not mentioned the obvious transaction costs that are also associated with market institutions. One might argue that labour is a good that should be treated differently. However, the long history of slavery shows this to be inaccurate.
36 Chapter 1. Equilibrium in the Jungle One might also argue that the virtue of the market system is that it exploits people’s natural desire to acquire wealth. In contrast, the jungle just uses people’s natural willingness to exercise power and to dominate. Obviously, I am not arguing in favour of adopting the jungle system. The comparison between the jungle and market mechanisms depends on our assessment of the characteristics with which agents enter the model. If the distribution of the initial holdings in the market reflects social values that we wish to promote, we might regard the market outcome as agreeable. However, if the initial wealth is allocated unfairly, dishonestly or arbitrarily, then we might not favour the market system. Similarly, if power is desirable then we might advocate for the jungle system, but if the distribution of power reflects brute force that threatens lives then we would clearly not be in favour.
2The Permissible and the Forbidden Picture in your mind a family consisting of nmembers. The grandparents have prepared a holiday feast and all are sitting happily around a long table. When the main dish is served, the grandparents act as dictators, putting a portion of it on each family member’s plate and making sure they eat it to the last bite. And then, dessert arrives and with it a dramatic turn of events. Grandma and Grandpa enter the room with their famous homemade pie. Everyone loves their pie and gazes eagerly at its entrance. Given the chance, each family member would gladly eat more than 1/nof the pie. At this point, the grandparents declare that they will not interfere in the division of the pie and will let the younger generation use their academic knowledge to decide how the pie is divided. One member of the family, an economist, suggests that each family member should be endowed with 1/nof the pie and — since some perhaps appreciate the pie more, while others perhaps less — a market should operate under the table where members can exchange slices of the pie for money. Another member of the family, a game theorist, suggests that the grandparents conduct an auction. He claims that this might be fun and, more importantly, the pie will be divided optimally. Hopefully, in your family, neither markets nor auctions are used to resolve such a conflict and, instead, harmony is achieved by means of a social norm: each family member does not dare to even consider taking more than the socially acceptable amount, say q, of the pie. Obviously, not every qwill bring harmony to the family. If q>1/n, then a family crisis would erupt since there would not be enough pie to satisfy the family members. All family members would race to get their slice, and some will be disappointed because they are unable to realize their anticipation of eating qof the pie. If q<1/n, then no conflict arises, but the members of the family would feel uneasy looking at the leftovers on the table and, next year, c2024, Michael Richter and Ariel Rubinstein, CC BY-NC-ND 4.0 https://doi.org/10.11647/OBP.0438.02
38 Chapter 2. The Permissible and the Forbidden would feel justified in taking a bit more. If q=1/n, then harmony prevails. It is optimal for each family member to take q, and any loosening of the norm will lead to demands which cannot be satisfied. We think of a bound on the portion that one can take as an example of a natural social norm that specifies what is considered permissible (“done”) and forbidden (“not done”). Such a norm resolves the family’s allocation problem but not with prices or games. Following Richter and Rubinstein (2020), we analyze the Y-equilibrium concept. It is defined as a set of permissible alternatives (which is the same for all agents) combined with a profile of choices (one for each agent) such that: (i) each agent’s choice is optimal from among the permissible alternatives; (ii) the profile of choices is feasible; and (iii) the set of permissible alternatives is maximal in the sense that there is no superset of permissible alternatives from which a profile satisfying (i) and (ii) can be found. By this definition, two forces make a permissible set unstable: the first modifies the permissible set in the case that the profile of (intended) choices is not feasible, while the second loosens restrictions on the permissible set as long as a new profile of optimal choices is feasible. The Y-equilibrium concept reflects a decentralized institution for achieving harmony in a society. We envision that, without a central authority, the same invisible hand that calculates equilibrium prices so “effectively” is also able to determine a maximal set of permissible alternatives that are compatible with self-maximizing behavior. The above forces adjust the social norm until harmony is achieved. While we do not provide a general dynamic process that converges to Y-equilibrium, in Richter and Rubinstein (2020), for several examples, we demonstrated natural tâtonnement-like processes that lead to a Y-equilibrium.
2.1 The Y-Equilibrium Concept 39 In standard economic settings, equilibrium prices can also be thought of as being determined by a central authority (or platform) that wishes to ensure that trade is viable. Similarly, one can think of the permissible set in a Y-equilibrium as a norm dictated by an authority that wishes to achieve harmony in society without imposing any unnecessary restrictions on the individuals. We now proceed to the formal definition of the equilibrium notion. 2.1 The Y-Equilibrium Concept Recall that an economy is a tuple ‹N,X,(%i)i∈N,F›where Nis the set of agents, Xis the set of alternatives that each agent chooses from, %iis agent i’s preferences on X, and F⊆XNis the set of feasible choice profiles. A candidate for an equilibrium is a configuration which consists of a subset of X, called a permissible set, together with a profile of choices: Definition: Configuration Aconfiguration is a pair ‹Y,(yi)i∈N›where Y⊆Xand (yi)i∈Nis a profile of elements in Y. We refer to Yas a permissible set and to (yi)i∈Nas an outcome. As explained in Chapter 0, a candidate for a solution in this book has a structure analogous to that of a competitive equilibrium. It is comprised of a profile of choices (one for each agent) and an additional parameter. In a configuration, the additional parameter is a permissible set, that is taken by all agents as given and uniformly binds the choices of all agents. Analogously, in a competitive equilibrium, the additional parameter is a price system, that is taken by all agents as given and uniformly binds the exchanges of all agents. Before defining the equilibrium concept, we need an additional concept: a para-equilibrium is a configuration where each individual maximizes his interests given the permissible set and the resulting choice profile is feasible.
40 Chapter 2. The Permissible and the Forbidden Definition: Para-equilibrium Apara-equilibrium is a configuration ‹Y,(yi)›satisfying: (i) For all i,yiis a %i-maximal alternative in Y. (ii) The profile (yi)is in F. A Y-equilibrium is a para-equilibrium such that any expansion of the permissible set will lead to a violation of feasibility if agents self-maximize with respect to the expanded permissible set. Definition: Y-equilibrium AY-equilibrium is a para-equilibrium ‹Y,(yi)›such that there is no paraequilibrium ‹Z,(zi)›for which Zis a strict superset of Y. As mentioned earlier, we view the permissible set not as being determined by an authority but, rather, as evolving through an invisible-hand-like process with two forces: First, if the profile of intended choices from the permissible set is not feasible, then alternatives are removed or added to the permissible set. Second, when the profile of chosen alternatives is feasible, additional alternatives are added to the permissible set as long as harmony is not disturbed. Note that (yi)can differ from (zi), that is, when assessing the existence of a larger permissible set, choices can adapt to the loosening. We take the permissible set to be uniform for all agents, although we are aware that there are situations in life where norms are nonuniform, such as allowing handicapped drivers to park in places where others are not permitted. The uniformity of the permissible set in our model is analogous to the uniformity of the price system in models of competitive equilibrium (although prices are often not uniform in real life). In some circumstances, uniformity can be viewed as an expression of equality of opportunity. It also is a simplicity property: in order to be followed, norms must be simple and clear, and norms are simpler when they do not distinguish between agents based on their names or preferences.
2.1 The Y-Equilibrium Concept 41 Example: A Housing Economy Consider the housing economy with N={1,2},X={a,b,c,d,e}, and preferences a1b1c1d1eand a2c2b2e2d. One para-equilibrium is Y={d,e},y1=d,y2=e. This is not a Y-equilibrium since Y={b,c,d,e}with y1=b,y2=cis also a para-equilibrium with a larger permissible set. The latter is the unique Y-equilibrium since the alternative acannot be a member of any para-equilibrium permissible set as it is the top-ranked for both agents. Incidentally, the Y-equilibrium outcome is not Pareto-optimal because ais left unassigned. Existence: Not every economy has a Y-equilibrium. In any housing economy, if at least two agents have the same strict preferences over the houses, then no Y-equilibrium exists. This is because, whatever the permissible set is, those two agents will pick the same house, which violates feasibility. This demonstrates that social norms regarding “the permissible and the forbidden” do not resolve conflicts when agents have similar preferences yet feasibility requires them to make different choices. Example: A Single Pie Consider the grandparents’ pie economy discussed in the beginning of the chapter. There are nfamily members, and a pie of size 1 is to be divided among them. The set of alternatives is X= [0,1]where x∈Xis a share of the pie. Each agent prefers to get as large a share as possible. The feasibility constraint states that the sum of their choices cannot exceed 1 (though some pie can be left over). To see that this economy has a unique Y-equilibrium, notice first that the pair ‹Y= [0,1/n],(yi≡1/n)›is a para-equilibrium. There is no para-equilibrium with a point above 1/nin the permissible set since, then, every agent would choose a point above 1/n, which is not feasible. Therefore, the above pair is a Y-equilibrium. There is no other Y- equilibrium since the permissible set in any para-equilibrium is a subset of [0,1/n].
42 Chapter 2. The Permissible and the Forbidden Example: The Quorum Economy Consider an economy with a finite set of clubs, X. Agents have preferences over the clubs (without regard to the clubs’ memberships). In order to operate, each club xneeds a minimal quorum of mx≤n (rather than having a maximal capacity as in the clubs economy). That is, feasibility requires that each club xis either empty or chosen by at least mxmembers. A special case is the consensus economy where mx=n for all x, that is, feasibility requires that all agents make the same choice. In general, if every agent were to choose his favourite club, then there would be non-empty clubs with less than a quorum. The role of the permissible set is to help the agents to coordinate their choices while imposing minimal restrictions on the permissible clubs. A Y-equilibrium always exists: First, a para-equilibrium exists because any configuration Y={x}combined with all agents choosing xis a para-equilibrium. Second, since the set of subsets of Xis finite, there is a para-equilibrium with a permissible set that cannot be expanded. However, Pareto optimality is not guaranteed, as illustrated by the following example. Let n=6, X={a,b,c}, and mx=3 for all x. Two agents have the preferences abc, two have the preferences bca, and two have the preferences cab. Obviously, there is no para-equilibrium with Y=X. Furthermore, there is no paraequilibrium with exactly two permissible clubs since four of the agents would choose one club and only two would choose the other, violating feasibility. As above, having a single club open is a para-equilibrium and since there are no multi-club para-equilibria, it is a Y-equilibrium. Thus, there are three Y-equilibria, each with a single different club open. Each Y-equilibrium outcome is not Pareto-optimal since there is an unopened club that is strictly preferred by four agents and, therefore, there is a Pareto improvement where exactly three of those four agents switch to that more-preferred club.
2.2 Y-Equilibrium, Pareto Optimality, and Envy-Freeness 43 The Y-equilibrium concept is not meant to be normative in any sense. However, it has two fairness properties: (i) All agents face the same choice set. Analogously, in the standard competitive equilibrium, all agents face the same trading opportunities. (ii) It is envy-free (see Foley (1966) and Varian (1974)). Envy-freeness ensures that no agent can complain that someone else is assigned an alternative that he prefers. Definition: Envy-freeness A profile (yi)i∈Nis envy-free if, for all i6=j,yi%iyj. The concepts of para-equilibrium and envy-freeness are closely related. A profile is envy-free if and only if it is the outcome of some para-equilibrium: First, any para-equilibrium outcome is envy-free (no agent can envy another’s choice since all agents choose from the same set). Second, if a profile (yi)is envy-free, then ‹{y1,...,yn},(yi)›is a para-equilibrium. 2.2 Y-Equilibrium, Pareto Optimality, and Envy-Freeness We have seen that Y-equilibrium profiles need not be overall Pareto-optimal. Nonetheless, they still satisfy some efficiency criterion. We now show that the Y-equilibrium profiles are precisely those which are Pareto optimal from among the set of feasible envy-free profiles. Proposition 2.1: Y-equilibrium Outcome Characterization A profile is a Y-equilibrium outcome if and only if it is Pareto-optimal among all feasible envy-free profiles. Proof: Let ‹Y,(yi)›be a Y-equilibrium. The profile (yi)is feasible and envyfree. If it is not Pareto-optimal among the feasible envy-free profiles, then
50 Chapter 2. The Permissible and the Forbidden There are two motivations for requiring a permissible set to be convex: (i) Suppose that on a certain highway, you are told that it is permitted to drive at 20 mph and at 80 mph. Naturally, you conclude that it is also permitted to drive at 50 mph. In contrast, if you are told that it is forbidden to drive on that highway both at 20 mph and at 80 mph, you wouldn’t instinctively conclude that 50 mph is also forbidden. This highlights an asymmetry between the permissible and the forbidden. Forbidden actions are usually “extreme”, while permissible actions are generally a sort of “middle ground”. (As always, exceptions exist: on an ice road in Estonia, it is only permitted to drive at speeds in the intervals 10–25 kph and 40–70 kph.) (ii) As mentioned earlier, for a norm to be accepted and internalized, simplicity is a virtue. In this vein, the restriction of attention to convex permissible sets can also be viewed as a simplicity requirement. In the one-dimensional case described above, a convex permissible set is simply a minimum and maximum speed. We will demonstrate later that, in higher-dimensional spaces, the equilibrium convex permissible sets are simple in the sense that they can be described by a small number of linear inequalities. The requirement that the permissible set is convex is similar in spirit to the standard assumption that agents choose from budget sets that are determined by common linear prices. The linearity of prices is a form of simplicity and is an attractive assumption even if in reality prices are often not linear. Definition: Convex Y-equilibrium A para-equilibrium ‹Y,(yi)›of a convex Euclidean economy is convex if Yis convex. A convex Y-equilibrium is a convex para-equilibrium ‹Y,(yi)›such that there is no other convex para-equilibrium ‹Z,(zi)›with a larger permissible set Z)Y. As in the Y-equilibrium case for Euclidean economies, any convex Y- equilibrium has a closed permissible set. If not, then the closure of its permissible set, which is also convex, together with the same profile
2.6 Pareto Optimality and Existence of Convex Y-Equilibrium 51 of alternatives, would constitute a convex para-equilibrium with a larger permissible set. A Y-equilibrium with a convex permissible set is a convex Y- equilibrium. However, a convex Y-equilibrium need not be a Y-equilibrium (it might be that there is no larger convex para-equilibrium permissible set, but there is a larger non-convex para-equilibrium permissible set). 2.6 Pareto Optimality and Existence of Convex Y-Equilibrium Proposition 2.1 states that the Y-equilibrium profiles are exactly those which are Pareto-optimal among the para-equilibrium profiles (which are the feasible envy-free profiles). For convex Y-equilibria, there is a partial analogue: profiles which are Pareto-optimal among the convex para-equilibrium profiles are convex Y-equilibria profiles. However, when discussing the exchange economy, we will see that there can be convex Y-equilibrium profiles that are not Paretooptimal among the convex para-equilibrium profiles. Proposition 2.4: A Sufficient Condition for a Profile to be a Convex Y-equilibrium Outcome For convex Euclidean economies, any profile which is Pareto-optimal among the convex para-equilibrium outcomes is a convex Y-equilibrium outcome. Proof: Given a convex Euclidean economy, let (yi)be a convex para-equilibrium outcome that is Pareto-optimal among the convex para-equilibrium outcomes. Let Pbe the collection of all convex sets Yfor which ‹Y,(yi)› is a convex para-equilibrium. Endow Pwith the partial order ⊇. We will use Zorn’s Lemma to show that Phas a maximal element. (A reminder of Zorn’s Lemma: Given a partially ordered set P, if every chain — a completely ordered subset of P— has an upper bound in P, then the set Phas at least one maximal element.)
52 Chapter 2. The Permissible and the Forbidden Given a chain Cof elements in P, let Ube the union of the sets in C. Clearly, Uis an upper bound on C, and we now show that Uis in P. The set Uis convex since for any two points x,y∈U, there is some Y∈ C such that x,y∈Yand, since any convex combination of xand yis in Y, it is also in U. To show that the tuple ‹U,(yi)›is a para-equilibrium, it suffices to show that, for each i, the element yiis %i-maximal in U. If there is an x∈Usuch that xiyifor some i, then there is Y∈C such that x∈Y, contradicting that ‹Y,(yi)›is a para-equilibrium. Let Y∗be a maximal element of P. It is left to show that ‹Y∗,(yi)›is a Y-equilibrium. Suppose that there is a convex para-equilibrium ‹Z,(zi)› such that Z)Y∗. It must be that zi%iyifor all i. Since (yi)is Paretooptimal from among the convex para-equilibrium outcomes, it must be that zi∼iyifor all i. Then, ‹Z,(yi)›is also a convex para-equilibrium, contradicting the maximality of Y∗. For Euclidean economies, we have already shown that a Y-equilibrium always exists (Proposition 2.3). The following proposition demonstrates that a convex Y-equilibrium also exists. Proposition 2.5: Existence of a Convex Y-equilibrium Every convex Euclidean economy has a convex Y-equilibrium. Proof: Let Obe the set of convex para-equilibrium outcomes. The set Ois not empty since Fcontains a constant profile (yi≡y∗)and the pair ‹{y∗},(yi≡y∗)›is trivially a convex para-equilibrium.
2.7 A Structure Theorem for Convex Y-equilibrium 53 The set Ois compact. To see this, since O⊆Fand Fis compact, it suffices to show that Ois closed. Take a sequence ‹Yt,(yi t)›of paraequilibria such that (yi t)converges to (zi)as t→ ∞. Let Z⊆Xbe the convex hull of the limit allocations {z1,...,zn}. The configuration ‹Z,(zi)›is a convex para-equilibrium since if there is an agent jand a convex combination of the {z1,...,zn}such that Σi∈Nλizijzj, then by continuity, for some large enough t,Σi∈Nλiyi tjyj t. Since Ytis convex, it holds that Σi∈Nλiyi t∈Yt, but this violates ‹Yt,(yi t)›being a convex para-equilibrium. Since Ois compact, the same argument as in Proposition 2.3 implies the existence of a profile that is Pareto-optimal in Oand, by Proposition 2.4, it is a convex Y-equilibrium outcome. 2.7 A Structure Theorem for Convex Y-equilibrium Much of Economic Theory deals with establishing conditions that guarantee the existence of a solution concept. Theorems about the structure of equilibrium are less common, although, in our opinion, are more interesting. We now show that our assumptions on the economy, together with a differentiability condition, guarantee that the permissible set of convex equilibria is an intersection of at most nhalf-spaces (recall that nis the number of agents). Thus, the requirement that the permissible set is convex implies that the convex Y-equilibrium permissible set takes a relatively simple form. Proposition 2.6: The Structure of Convex Y-equilibria Let ‹Y,(yi)›be a convex Y-equilibrium in a differentiable Euclidean economy. Let J={i|yiis not the %i-global maximum in X}. Then, there is a profile of closed half-spaces (Hj)j∈J, such that Y=∩j∈JHj.
54 Chapter 2. The Permissible and the Forbidden y1 y4 y5 y2 y3 %4 %5 %1 Y Figure 2.1 An illustration of Proposition 2.6 (note that J={1,4,5}) Proof: First, note that if J=;, that is, every agent is assigned his first-best, then Y=X(which is the degenerate case where Yis the intersection of an empty set of half-spaces). Otherwise, for every j∈J, let Hjbe the unique half-space of alternatives containing yjsuch that yjis strictly preferred to all other elements in Hj. Its existence is guaranteed by the assumptions of differentiability and strict convexity of the agents’ preference relations. We first show that Yis a subset of ∩j∈JHj: Suppose that for some j∈Jthere is an alternative wj∈Y−Hj. By the differentiability and strict convexity of j’s preferences, and for small " > 0, it holds that "wj+(1−")yjjyj. By convexity of Yit holds that "wj+(1−")yj∈Y. Therefore, yjis not %j-maximal in Y, a contradiction. To show that the permissible set Yis equal to ∩j∈JHj, it remains to be shown that ‹∩j∈JHj,(yi)›is a convex para-equilibrium. This follows from:
2.8 The Division Economy 55 (i) The set ∩j∈JHjis convex. (ii) For each agent i,yi∈Y⊆∩j∈JHj. (iii)For each j∈J,yjis the %j-maximum in Hjand, thus, also in ∩j∈JHj. (iv) For each i/∈J,yiis the %i-global maximum and, thus, also in ∩j∈JHj. 2.8 The Division Economy A leading economic problem is the division of a bundle among the members of a society. The grandparents single pie economy is its simplest version. The only convex Y-equilibrium is the intuitively appealing norm that forbids taking more than 1/nth of the pie. For the multi-good division economy, the analogous norm which allows an agent to take up to 1/nth of the total bundle is typically not a Y-equilibrium permissible set because it does not allow any trades. We proceed by exploring the properties of convex Y-equilibria in a differentiable division economy, formally defined as: Definition: Differentiable Division Economy Adifferentiable division economy ‹N,X,(%i)i∈N,F›is a differentiable Euclidean economy such that: (i) The set of alternatives is all bundles with mcommodities, i.e. X=Rm +. (ii) Every preference relation %iis strictly monotonic (besides being continuous, strictly convex, and differentiable). (iii) There is a bundle e∈Rm ++ such that (xi)∈Fif and only if Σixi≤e. The following claim draws a connection between convex Y-equilibrium and egalitarian competitive equilibrium (see Foley (1966) and Varian (1974)) which is a competitive equilibrium of the exchange economy in which each agent is initially endowed with 1/nof the total bundle. We will see that every egalitarian competitive equilibrium outcome is a convex Y-equilibrium outcome and, if at least one agent selects an interior bundle, then its permissible set is identical to the egalitarian competitive equilibrium’s common budget set.
56 Chapter 2. The Permissible and the Forbidden Claim: Egalitarian Competitive Equilibria and Convex Y-equilibria Let ‹p,(yi)›be an egalitarian competitive equilibrium in a differentiable division economy. Then, there is a convex Y-equilibrium with the same allocation ‹Y,(yi)›. Furthermore, if at least one of the bundles yjis strictly positive, then Ymust be B={y|p∙y≤p∙e/n}. Proof: The pair ‹B,(yi)›is a convex para-equilibrium and (yi)is overall Paretooptimal by the standard first welfare theorem. Thus, by Proposition 2.4, (yi)is a convex Y-equilibrium outcome. If ‹Y,(yi)›is a convex Y-equilibrium, then by Proposition 2.6, Y=∩i∈NHi, where Hiis the lower half-space of %iat yi(since no agent has his first-best, it holds that J=N). For all i,B⊆Hi, since otherwise there exists zi∈B\Hiand, by differentiability and strict convexity, yi would not be %i-optimal in B. If for some jthe bundle yjhas a zero coordinate, then it can be that B(Hj, but if for any jthe bundle yjis strictly positive, then Hj=Band, therefore, Y=∩iHi=B. Comments: Every overall Pareto-optimal interior convex Y-equilibrium profile is an egalitarian competitive equilibrium allocation: Let ‹Y,(yi)›be a convex Y-equilibrium such that each bundle yiis interior. By monotonicity, the alternative yiis never %i-globally maximal and thus, by Proposition 2.6, Y=∩i∈NHiwhere Hiis the lower half-space of %iat yi and, by monotonicity, there is a positive vector piand a positive number wi such that Hi={x|pi∙x≤wi}. Since every yiis interior and the allocation is Pareto optimal, the half-spaces must be parallel (otherwise, any two agents on non-parallel half-spaces could make a Pareto-improving local exchange) that is, there is a positive vector psuch that pi=pfor all i. It follows that Y={x|p∙x≤w}for some positive vector pand a positive number w. By
2.8 The Division Economy 57 monotonicity, p∙yi=wfor all i. Since p∙e=p∙Σi∈Nyi=nw , we have p∙yi=w=p∙(e/n). Thus, (yi)is a competitive egalitarian equilibrium allocation with price vector p. There can exist a non-interior Pareto-optimal convex equilibrium outcome that is not an egalitarian competitive equilibrium allocation: Here is a simple example: Let n=3, m=2, e= (5,5)and the agents’ preferences be represented by the utility functions specified in Figure 2.2, panel (a) (a slight modification of the preferences will make the preference relations strictly convex): u1(x1,x2) = x1 u2(x1,x2) = x1+x2 u3(x1,x2) = x2 (a) Utility functions 1 2 3y3 y2 y1 Y 123 u3 u2 u1 (b) Illustration Figure 2.2 A convex Y-equilibrium with a non-egalitarian Pareto-optimal outcome. Let y1= (3,0),y2= (2,2)and y3= (0,3)(Figure 2.2., panel (b)). The allocation (yi)is Pareto-optimal: If (zi)Pareto-dominates (yi), then zi 1+zi 2≥ yi 1+yi 2for all iwith at least one inequality. Thus, Σi(zi 1+zi 2)>Σ(yi 1+yi 2) = 10, which is not feasible. The set Yis the intersection of (Hi)where each Hiis a half-space of bundles below i’s indifference curve, which includes yi. The pair ‹Y,(yi)›is a convex para-equilibrium and, by Proposition 2.4, (yi) is a convex Y-equilibrium outcome. To see this directly, note that if there were a larger convex para-equilibrium, ‹Z,(zi)›, thenZwould contain an element that is not in Y. Any such element is strictly preferred to yifor at least one agent i. Thus, (zi) would Pareto-dominate (yi).
58 Chapter 2. The Permissible and the Forbidden There can exist a non Pareto-optimal interior convex equilibrium outcome: Consider the economy (depicted in Figure 2.3) with two agents, two goods, total bundle e= (3,3), and kinked utility functions as depicted (a small deviation could make them strictly convex). Agent 1’s indifference curve has slope −1.25, and agent 2’s indifference curve has slope −0.8. The depicted allocation y1= (2,1)and y2= (1,2)is not Paretooptimal since it is mutually beneficial to have agent 1 get one additional unit of good 1 and one unit fewer of good 2. 1 2 y1= (2,1) y2= (1,2) Y 1 2 u2 u1 Expansion Path2 Expansion Path1 Figure 2.3 A non Pareto-optimal convex equilibrium. The configuration ‹Y,(yi)›is a convex para-equilibrium. In any larger convex para-equilibrium, ‹Z,(zi)›, the convex set Zincludes a bundle that is strictly better for at least one of the agents and, therefore, z16=y1 and z26=y2. Given the agents’ indifference curves, it must be that in z1agent 1 receives more than 2 units of good 1 . Then, it must be that z1 1+z1 2>3 since if z1 1+z1 2≤3 then the intersection point of the line between zand (2,2)(which has a slope of at most −2 which is smaller than −1.25) and the expansion ray of agent 1 is strictly preferred by agent 1 to z. Likewise z2 1+z2 2>3, a contradiction. Initial Endowments: Recall that a division economy differs from the standard exchange economy as it does not specify an initial distribution of the goods. One way to incorporate initial endowments into our framework is by the following notion of a trade economy. Let (ei)be an initial endowment profile. Let X=Rmwhere a member of Xis interpreted as a trade (and thus includes negative components as well). Set Fto include all profiles of trades (ti)such that Σiti=0 and for every agent i, the post-trade bundle ti+ei≥0. As to the preferences, assume that each agent ihas a basic preference relation %i cover the set of bundles (satisfying the standard division economy assumptions).
2.9 The Give-and-Take Economy 59 Among trades that give an agent a non-negative amount of every good, agent i’s preferences %ion Xare induced from their basic preferences by ti%isiif ti+ei%i csi+ei. Every agent prefers the no-exchange option 0 to any trade which leaves them with a negative amount of any good. Analogous results to the previous claims for the division economy also hold for the trade economy: (i) the profile of trades in any competitive equilibrium in the standard exchange economy is a convex Y-equilibrium outcome in this trade economy, and (ii) any Pareto-optimal convex Y-equilibrium outcome in the trade economy, where at least one agent has a strictly positive post-trade allocation, is a profile of trades in a competitive equilibrium of the standard exchange economy. 2.9 The Give-and-Take Economy Recall that in the give-and-take economy, the set of alternatives is X= [−1,1], where a positive xrepresents a withdrawal of xfrom a social fund while a negative xrepresents a contribution of −x. Feasibility requires that the social fund be balanced, that is, (xi)∈Fiff Σixi=0. All agents have strictly convex preferences over Xwith agent i’s ideal denoted by peaki. As mentioned earlier, the give-and-take economy is an economic situation in which the market plays no role. We will see that norms regarding what is permissible and what is forbidden can serve as an effective non-market tool for achieving harmony. The case Σipeaki=0 is “bliss”: everything is permitted and ‹X,(peaki)› is a convex Y-equilibrium. However, in general, there is tension between feasibility and the agents’ desires. The following claim characterizes the convex Y-equilibrium for the case where the sum of what people ideally want to take is greater than what people ideally want to give. We will see now that, in this case, there is a unique convex Y-equilibrium. In it, people are allowed to give as much as they want but there is a bound on the maximum that can be taken, and its outcome is Pareto optimal.
66 Chapter 3. Status and Indoctrination ordering). The indoctrination does not affect the agent’s basic preferences (in contrast to the biased preferences model discussed in Chapter 4) but, rather, modifies his choice set. An agent only considers moving from one alternative to another if it benefits society (and will only make such a move if he also personally benefits). With the above interpretations in mind, we will discuss two types of equilibria, which fit into the two categories of equilibrium discussed in Section 0.4. The first is the status equilibrium: it is a feasible profile of choices and a public ordering such that no agent strictly prefers any alternative that is weakly lower-ranked by the public ordering than the one assigned to him. Thus, the public ordering limits the agents’ deviations from the equilibrium profile: an agent who is assigned an alternative only considers deviating to alternatives that are weakly lower-ranked (by the public ordering) than the one he is assigned. Deviations are purely self-serving and contemplated without regard to feasibility. In the taxonomy of Section 0.4, the status equilibrium belongs to the deviation group (like Nash equilibrium). In the last section of the chapter, we will study the initial status equilibrium concept, which fits into the choice group (like competitive equilibrium). This concept operates on an extended economy (in which the model of an economy is extended to include an initial profile). In the initial profile, each agent is assigned an alternative that he can always choose and which, together with the public ordering, determine the agent’s choice set. An initial status equilibrium consists of a feasible profile and a public ordering, but this time the profile of choices must be such that no agent strictly prefers any alternative that is weakly lower-ranked by the public ordering than the one initially assigned to him. Thus, the public ordering limits an agent’s choice set: he only considers alternatives that are weakly lowerranked than his initial alternative.
3.1 Status Equilibrium 67 3.1 Status Equilibrium Definition: Status Equilibrium Given an economy ‹N,X,(%i)i∈N,F›, a status equilibrium is a pair ‹P,(xi)i∈N›where (xi)i∈Nis a profile and Pis an ordering (a complete, reflexive, and transitive binary relation) on Xsatisfying: Feasibility: the profile (xi)is in F. Personal optimality: for every agent i, the element xiis %i-maximal in {z∈X|xiPz}. The ordering Pis referred to as a public ordering. As mentioned, under the first two interpretations of a public ordering, it ranks the alternatives by value or prestige. The term aPb means that a is more expensive than b or that a is more prestigious than b. An equilibrium public ordering stabilizes the equilibrium profile in the sense that every agent is satisfied with his assignment given that he is bounded by the worth (or prestige) of his assigned alternative. Under the third interpretation, Pis a social motive that systematically affects an agent’s willingness to exchange his assigned alternative. The term aPb means that “ais less socially desirable than b” (this is not a mistake... less and not more). If an agent iis assigned xi, then he cannot bear the idea of exchanging it for an alternative that is less socially desirable and, therefore, he only considers more socially desirable alternatives (i.e. those which are lower-ranked by P). Under this interpretation, an equilibrium consists of a public ordering and a feasible profile in which no agent both: i) wishes to exchange his assigned alternative, according to his personal preferences and ii) is able to justify the exchange as furthering society’s interests, according to the public ordering. Proposition 3.1: A Second Welfare Theorem Any Pareto-optimal profile is a status equilibrium profile.
68 Chapter 3. Status and Indoctrination Proof: Let (ai)be a Pareto-optimal profile. Define the binary relation Don A={a1,...,an}by xDy if xis desired by a holder of y, that is, there are iand jsuch that x=aijaj=y. If Dhas a cycle, then there is a set of agents who can permute their alternatives among themselves (recall that Fis closed under permutations) so that all of them are strictly better off, contradicting (ai)being Pareto-optimal. Since Dhas no cycles, it can be extended to a complete ordering over A. Then, Dcan be extended to a strict ordering Pon the entire set Xby putting all elements in X−Aabove all elements in A(making all unassigned elements “unaffordable”) and arbitrarily ranking the elements in X−Aamong themselves. Personal optimality holds since, for every agent i, the alternative aiis optimal in {x|aiPx}(if aiPx, then x=ajfor some j, and if iwere to prefer it, then xDai, which contradicts aiPx since Pextends D). By the same proof, any feasible profile (Pareto-optimal or not) for which the relation Ddoes not have cycles is a status equilibrium profile. In particular, in the consensus economy, where all agents have to make the same choice, any profile that assigns the same element x∗to all agents is supported by any public ordering that ranks x∗as the unique lowest element in Xand thus all other alternatives are “blocked”. Such a profile might be not Pareto-optimal. Thus, any Pareto-optimal profile is a status equilibrium profile, but a status equilibrium profile does not have to be Pareto-optimal. 3.2 Status Equilibrium – Examples Example: The Jobs Economy Let Xbe a non-singleton set of types of jobs. Each agent holds strict preferences on X. Feasibility is given by a vector (nx)x∈Xwhere nxis the number of available jobs of type x(non-emptiness of Frequires that Σx∈Xnx≥n).
3.2 Status Equilibrium – Examples 69 A public ordering in this example has a natural interpretation of social status, which is often associated with a job. Once an agent is assigned to a job, he cannot switch to a higher-status job but he can switch to any job of equal or lower status (e.g. a professor can move to a lowerranked university but not to a higher-ranked one). The housing model of Shapley and Scarf (1974) is the special case where nx≡1 and |X|=n. Claim: The following holds for the jobs economy: (i) If Σxnx=n, then the First Welfare Theorem holds: every status equilibrium profile is Pareto-optimal. (ii) If Σxnx>n, then the First Welfare Theorem fails: there is always a status equilibrium profile that is not Pareto-optimal. Proof: (i) Let ‹P,(xi)›be a status equilibrium. Assume by contradiction that the feasible profile (yi)Pareto-dominates (xi). Let jbe an agent for whom xjis P-maximal from among {xi|yi6=xi}. Since preferences are strict, it must be that yjjxjand, therefore, yjPx j. Since Σxnx=n, it must be that in any feasible profile, all jobs are filled. Therefore, there is another agent whose original job is yjand whose new job is not, contradicting the P-maximality of xjfrom among {xi|yi6=xi}. (ii) Let (xi)be a Pareto-optimal profile. By Proposition 3.1, there is a public ordering such that ‹P,(xi)›is a status equilibrium. Let zdenote a job with spare capacity, and let jbe an agent who does not have job z(which exists since Σxnx>nand |X|>1). Let (yi)be the feasible profile obtained from (xi)by moving jfrom xjto z. Since (xi)is Paretooptimal, every agent who does not have job zstrictly prefers his assigned job to zand thus, (yi)is not Pareto-optimal. Let P0be the public ordering obtained from Pby moving zto the bottom rank. The pair ‹P0,(yi)›is clearly a status equilibrium.
70 Chapter 3. Status and Indoctrination Example: R-Monotonic Preferences Let Rbe a strict partial ordering (irreflexive, transitive, and antisymmetric but not necessarily complete) on X. A preference relation % is R-monotonic if abwhenever aRb. For example, let Xbe a set of bundles and Rbe defined by xRy if the bundle xcontains weakly more than yof every good and strictly more of at least one. In this case, R-monotonicity is the standard notion of strong monotonicity. It will now be shown that, for any economy with R-monotonic preferences, any status equilibrium profile can also be supported as a status equilibrium with an R-monotonic public ordering. Thus, a stronger assumption on agents’ preferences (R-monotonicity) leads to stronger conclusions about the equilibrium public ordering (being R- monotonic). Claim: Let Rbe a strict partial ordering and let ‹N,X,(%i)i∈N,F›be an economy where every preference %iis R-monotonic. If ‹P,(xi)i∈N›is a status equilibrium, then there is an R-monotonic ordering Qsuch that ‹Q,(xi)i∈N›is also a status equilibrium. Proof: Define the desire binary relation Das y Dz if there is an agent who is assigned zand strictly prefers y(and thus, it must be that y Pz strictly). Let S=R∪D. The relation Sis acyclic: if not, let z1S1z2S2z3S3...zmSmz1 be a minimal cycle where each Siis either Ror D. •It cannot be that all Siare Rbecause Ris acyclic. •It cannot be that all Siare Dsince zi−1Dziimplies zi−1Pzistrictly and thus, a D-cycle implies a strict P-cycle, which is impossible. •It cannot be that the cycle S1,...,Smcontains both Dand R. This is because, if it did, then it would contain an Rfollowed by a D. However, if aRbDc, then there is a jsuch that c=xjand bjc. Since j
3.3 A Detour: Convex Preferences 71 extends R, it follows that ajband therefore, ajcand so aDc. Therefore, the cycle can be shortened. Thus, Sis a strict partial ordering. Extend Sto an ordering Q. Since S extends R, so does Qand thus, Qis R-monotonic. Since Sextends D, so doesQand thus, ‹Q,(xi)›is a status equilibrium. 3.3 A Detour: Convex Preferences In Section 3.4, we will refine the notion of a status equilibrium by imposing some structure to the public ordering. In preparation, we make a detour to the concept of convex preferences. One conventional definition of convex preferences for Euclidean spaces requires that if ais weakly preferred to b, then any convex combination of a and bis also weakly preferred to b. This definition is equivalent to requiring that all upper contours (sets of the type {x|xa}) are convex sets. Both of these definitions refer to the term “convex combination”, which itself uses an algebraic structure on the space of alternatives and so does not apply to economies where the set Xlacks such a structure. Following Richter and Rubinstein (2019), we suggest an alternative definition of convex preferences which generalizes the standard Euclidean notion and is also applicable to spaces without algebraic structure. A cornerstone of this approach is the view that preferences are built from primitive building blocks. Here, we take the building blocks to be the members of a set of orderings Λ, which we call primitive orderings. Each primitive ordering is a complete, reflexive, and transitive binary relation over the set X(indifferences are allowed). We interpret the primitive orderings as expressions of objective attributes of the alternatives that are in the vocabulary of all agents.
72 Chapter 3. Status and Indoctrination The assumption behind this definition is that, when thinking about replacing an alternative b∈X, an agent has in mind a necessary criterion (primitive ordering) that is critical, in the sense that, for an alternative to be better than b, it must be better by this criterion. Note that the critical criterion can depend on b. For example, imagine a department chair who is contemplating replacing b, who is a weak teacher. In this case, the critical consideration may be pedagogical ability, and any teacher who is pedagogically worse than bwill be rejected. However, this does not mean that any candidate who is pedagogically better than bwill be preferred. Again, the critical criterion can vary from one alternative to another: when the department chair considers replacing c, who is a great teacher and a poor researcher, he may feel that research ability is now critical, and thus, any candidate who is a worse researcher than cwill be judged to be a worse candidate than c. Definition: Λ-convex Preferences Let Xbe a set of objects and Λbe a set of orderings on Xreferred to as primitive orderings. The symbol Drepresents a generic member of Λ. A preference relation %on Xis Λ-convex if: ∀b∈X,∃D∈Λsuch that for x6=bit is necessary for xbthat xBb. A preference relation %on Xis Λ-strictly convex if: ∀b∈X,∃D∈Λsuch that for x6=bit is necessary for x%bthat xBb. In both definitions, the ordering Dis called a critical direction at b; (there can be multiple critical directions). Three comments: (i) Every (strict) primitive ordering in Λis Λ-(strictly) convex: for each alternative, the primitive ordering itself is a critical direction. (ii) A “Pareto” property holds: If band care distinct, bDcfor every D∈Λ, and %is Λ-convex, then b%c. This is because there is a critical ordering D attached to band bDcand therefore, ccannot be strictly preferred to b. For Λ-strictly convex preferences, the conclusion is stronger, namely, bc.
3.3 A Detour: Convex Preferences 73 (iii) In Richter and Rubinstein (2019), we also suggested other similar definitions of convex preferences and discussed their connection to Edelman and Jamison (1985)’s notion of “abstract convexity”. Underpinning our convexity notion is the abstraction of a concept that plays a fundamental role in economic analysis when we talk about convex preferences on a Euclidean space: for each alternative, there is a hyperplane which contains it, such that all weakly preferred alternatives lie on one side of the hyperplane. In the same spirit, our notion of convex preferences requires that for every alternative there is a primitive ordering that puts all preferred alternatives on one side of the ordering. % D Figure 3.1 A supporting hyperplane and its corresponding critical direction The definition of convex preferences is attractive for several reasons: (a) It is compelling as a procedural assumption of preference formation. (b) It emphasizes and allows for the dependence of the convexity property on the specification of the considerations used to construct preferences. (c) It does not require any algebraic structure. (d) For continuous preferences on Euclidean spaces, it generalizes the standard convexity notion as shown in the following example: Example: Euclidean Space with Algebraic Linear Orderings Let Xbe an open convex subset of a Euclidean space. For any vector v6=0, define the algebraic linear ordering ≥vby x≥vyif v∙x≥v∙y. Let Ψbe the set of all algebraic linear orderings. Claim: Let %be a continuous preference relation on X. Then: %is convex by the standard definition if and only if %is Ψ-convex.
74 Chapter 3. Status and Indoctrination Proof: Assume %is convex by the standard definition, i.e. for every b∈X, the set U(b) = {z|zb}is convex. It is also open, since %is continuous. By the separating hyperplane theorem, there exists ≥v∈Ψ such that x>vbfor every x∈U(b). That is, ≥vis a critical direction. Assume %is Ψ-convex. Let a,c∈Xsuch that a,cb, and let z be an element on the line between aand c. By Ψ-convexity, there is a critical direction ≥vat z. Then, z≥vaor z≥vcor both, and since %is Ψ-convex, it follows that z%aor z%cor both and thus zb. The above example can be applied to the one-dimensional case. There, the set Ψconsists of two orderings DL(which ranks left alternatives higher) and DR(which ranks right alternatives higher). The claim demonstrated that for the one-dimensional case, Ψ-convexity of continuous preferences is equivalent to standard convexity of preferences, namely any such preferences have a maximal region (which may be an interval or a single element), the preferences are weakly increasing up to this region, and weakly decreasing beyond it. The following example shows that strict Ψ-convexity is equivalent to singlepeakedness of preferences. Example: Left and Right Let X= [0,1], and suppose that Λcontains two orderings: the rightist DR (which ranks elements to the right higher) and the leftist DL(which ranks elements to the left higher). A preference relation is single-peaked if: (i) it has a unique maximum point (peak) in X; and (ii) it is strictly increasing below the peak and strictly decreasing above it. Claim: Let Λ = {DL,DR}and X= [0,1]. A continuous preference relation is Λ-strictly convex if and only if it is single-peaked. Proof: Suppose %is singled-peaked. At any b>peak, the ordering DL is critical, while at any b<peak the ordering DRis critical. At the peak, both orderings are critical.
3.3 A Detour: Convex Preferences 75 Suppose %is Λ-strictly convex. Since the preferences are continuous and Xis compact, there is a %-maximal element. Suppose that there are two %-maximal elements x<z, and let y∈(x,z). The ordering DLis not a critical direction at ybecause z%yand z6BLy. Likewise, DRis not a critical direction at xbecause x%yand x6BRy. Therefore, there is no critical direction at y, a contradiction to the Λ-strictly convexity of %. Let Mbe the %-maximal element. For every y<x<M, the critical ordering at xmust be DRand thus y≺x. Therefore, %is strictly increasing to the left of M. Likewise, for every x>M, the critical ordering must be DL, and %is strictly decreasing to the right of M. Thus, the preferences are single-peaked with the peak at M. We now illustrate the richness of the Λ-convexity notion by considering its application to the case of preferences over collections of distinct goods. Example: Collections Let Zbe a set of distinct indivisible goods and Xbe the set of all collections of items from Z. Here, unlike in the housing economy, an agent can have more than one good or none at all. We will use the Greek symbols Θand Φfor collections of goods. For every v, a non-negative function on Z, let Dvbe the ordering of X represented by the utility function v(Θ) = Σz∈Θv(z). That is, v(Θ) is the sum of the v-values attached to the individual items in the set Θ. Let Λ be the set of such orderings. An interpretation of these orderings is that a value is attached to each good and the value of a collection is the sum of the values of the goods in the collection. Claim: A preference is Λ-convex if and only if it is weakly monotonic with respect to the inclusion relation.
82 Chapter 3. Status and Indoctrination Proposition 3.2: A First Welfare Theorem Let ‹N,X,F,Λ›be a convex environment. (i) If the convex environment satisfies condition D, then for any profile of Λ-convex preferences (%i)i∈Nany primitive equilibrium profile (ai)of the convex economy ‹N,X,(%i)i∈N,F,Λ›is weakly Pareto-optimal (there is no other feasible (bi)such that for all ieither biiaior bi=ai). (ii) If the convex environment fails condition D, then there are Λ-convex preferences (%i)i∈Nsuch that the convex economy ‹N,X,(%i)i∈N,F,Λ› has a primitive equilibrium profile that is not weakly Pareto-optimal. Proof: (i) Consider a primitive equilibrium ‹D,(ai)›. If (ai)is not weakly Paretooptimal, then there is another feasible profile (bi)such that for all ieither biiaior bi=ai. Then, for all i, either biBaior bi=ai, contradicting condition D. (ii) Since condition Dfails, there exist two distinct feasible profiles, (ai) and (bi), and a primitive ordering Dsuch that for all i,aiBbior ai=bi. Extend the convex environment to a convex economy by endowing each agent with the same Λ-convex preference relation D. Then, ‹D,(bi)›is a primitive equilibrium that is not weakly Pareto-optimal. Note that when condition D is satisfied, every primitive equilibrium profile is weakly Pareto-optimal, but it might be not Pareto optimal. For example, for the housing economy with two houses, two agents, preferences a1band a∼2b, and any set of primitive orderings, condition D holds, but ‹ab,(b,a)› is a primitive equilibrium with a Pareto-nonoptimal profile.
3.6 A Second Welfare Theorem 83 3.6 A Second Welfare Theorem We have seen that the Second Welfare Theorem does not generally hold. Essentially, it requires the following Richness property: Definition: Richness The convex economy ‹N,X,(%i),F,Λ›satisfies Richness if the following holds: Let (ai)be a feasible profile and Diand Djbe two different primitive orderings such that (recall the notation B(D,ai) = {x|aiDx}): (i) aiis %i-maximal in B(Di,ai)but not in B(Dj,ai); and (ii) ajis %j-maximal in B(Dj,aj)but not in B(Di,aj). Then, there is a pair of alternatives (bi,bj)6= (ai,aj)such that: (I) (bi,bj,a−i,j)∈Fand (II) (bi,bj)Pareto-dominates (ai,aj) (That is, bi%iaiand bj%jajwith at least one strict preference.) The Richness property is illustrated in Figure 3.3 using an Edgeworth box. It states that, for any feasible profile (ai), if D1is a critical direction for agent 1 at a1and D2is a critical direction for agent 2 at a2, and the directions are not identical, then there is a feasible mutually beneficial reconfiguration of their bundles (b1,b2), which leaves all other agents unchanged. • • %2 %1 D2 D1 (a1,a2) (b1,b2) Figure 3.3 Richness Proposition 3.3: A Second Welfare Theorem Let ‹N,X,(%i)i∈N,F,Λ›be a convex economy that satisfies Richness. Then, any Pareto-optimal profile is a primitive equilibrium profile.
84 Chapter 3. Status and Indoctrination Proof: Let (xi)i∈Nbe a Pareto-optimal profile. Let Oibe the set of all critical directions of %iat xi(that is, the set of all D∈Λsatisfying that for any z, if zixi, then zBxi). By the Λ-convexity of the preferences, Oi6=;. If ∩iOiwere empty, then there would be two agents iand jsuch that Oiand Ojare non-nested sets. Take Di∈Oi\Ojand Dj∈Oj\Oi. The element xiis %i-maximal in B(Di,xi)but not in B(Dj,xi), and analogously for agent j. By the Richness property, there is a pair of elements (bi,bj)such that the modified profile obtained by replacing the pair (xi,xj)with (bi,bj)is feasible and Pareto-dominates (xi)i∈N, which contradicts the Pareto-optimality of (xi). Thus, there exists D∈∩iOi, and therefore, ‹D,(xi)i∈N›is a primitive equilibrium. In Richter and Rubinstein (2015), it is shown that under the following two additional assumptions the Richness property is also necessary for the Second Welfare Theorem to hold: (i) differentiability of preferences and (ii) there are no two alternatives, xand x0, such that xDx0for all primitive orderings D. 3.7 Primitive Equilibrium – More Examples Example: The Division Economy Let X=RL +be the set of bundles in an L-commodity world. Let z= (zl) be the vector of total endowment which has to be fully divided, that is, (xi)is feasible if Σn i=1xi=z. Let Ψ+be the set of all positive algebraic orderings, namely all ≥vwhere v∈RL +\{0}. Let the set of primitive orderings Λbe a non-empty (finite or infinite) subset of Ψ+. All agents hold monotonic Λ-convex preference relations. Two simple cases: When Λcontains a single ordering ≥v, all agents hold the same preference relation, %=≥v. When Λ = {≥(1,0),≥(0,1)}, every indifference curve is “right-angled”.
3.7 Primitive Equilibrium – More Examples 85 The First Welfare Theorem holds since the economy satisfies condition Dand thus, by Proposition 3.2, any primitive equilibrium profile is weakly Pareto-optimal. The Richness property used in Proposition 3.3 holds and thus any Pareto-optimal allocation is a primitive equilibrium profile. This is somewhat stronger than the textbook Second Welfare Theorem which states that any Pareto-optimal allocation is an equilibrium allocation supported by some linear ordering while Proposition 3.3 states that the equilibrium public ordering can be drawn from Λ. Example: The Collections Economy A non-empty finite set of distinct indivisible goods Zis to be partitioned among the agents. Unlike in the housing economy, each agent chooses acollection of goods, which can have more than one good or none at all. Let Xbe the set of all subsets of Z. We will use lower-case letters for goods and the Greek symbols Θand Φfor collections of goods. The set Fcontains all profiles that allocate each item in Zto exactly one agent. For every v, a positive-valued function on Z, let Dvbe the ordering of Xrepresented by the utility function v(Θ) = Σz∈Θv(z). That is, v(Θ) is the sum of the v-values attached to the individual items in the set Θ. Let Λbe the set of such strict orderings. As shown earlier, the Λ-convex preferences are exactly all preferences that are weakly monotonic with respect to the inclusion relation. We assume that all agents’ preference relations are strict and Λ-convex. In this economy, a primitive equilibrium has the interpretation that a price is attached to each good and the price of a collection is the sum of the prices of the goods in the collection. In contrast, a status equilibrium has the interpretation that there is a price for each collection. Claim: For the collections economy: the set of primitive equilibrium profiles ⊆the set of Pareto-optimal profiles ⊆the set of status equilibrium profiles, and these inclusions can be strict.
86 Chapter 3. Status and Indoctrination Proof: To establish the first inclusion, by Proposition 3.2 it suffices to verify that condition Dholds. Take a primitive ordering Dv. For any two distinct feasible profiles, (Θi)and (Φi), it holds that Σiv(Θi) = Σiv(Φi) = v(Z). Thus, it cannot be that v(Θi)≥v(Φi)for all iwith at least one strict inequality. However, there can be Pareto-optimal profiles that are not primitive equilibrium profiles. For example, let Z={a,b,c,d}and n=2. Both agents have preferences that rank any cardinally larger set higher and are therefore, Λ-convex. To simplify notation, denote the set of goods {x,y} as xy . Table 3.1 depicts the agents’ preferences over two-element sets: %1%2 ac,bd ad ,bc ab cd ad ,bc,cd ab,ac,bd Table 3.1 Preferences with a Pareto-optimal profile that is not a primitive equilibrium profile (highlighted). The profile (x1,x2)=(ab,cd )is Pareto-optimal. However, there is no public ordering ≥vthat supports this profile as a primitive equilibrium. If there were, then ac >vab (to ensure that ab is optimal for agent 1), which implies that v(c)>v(b). Similarly, we can conclude that v(b)> v(d)>v(a)>v(c), a contradiction. For the second inclusion, recall that by Proposition 3.1, any Paretooptimal profile is a status equilibrium profile. However, there are collection economies with status equilibrium profiles that are not Pareto-optimal. For example, suppose that Z={a,b,c,d},n=2, and both agents have identical Λ-convex preferences %∗satisfying that the sets ac and bd are %∗-superior to ab and cd . Then, ‹%∗,(ab,cd )›is a status equilibrium that is not Pareto-optimal.
3.8 Initial Status Equilibrium 87 This example demonstrates a stark contrast between equilibria with item-pricing (where the price of a bundle is the sum of the individual items’ prices) and those with bundle-pricing (where a price is attached to each bundle). The following table summarizes the above claim: Item-pricing Bundle-pricing equilibria equilibria First Welfare Theorem ØX Second Welfare Theorem XØ Table 3.2 Depiction of the Claim 3.8 Initial Status Equilibrium In this section, we extend the definition of a status equilibrium to cover extended economies. To remind the reader, an extended economy is an economy with the specification of an additional feasible profile (ei)i∈N interpreted as an “initial profile”. It specifies an alternative for each agent which he has the absolute right to choose, independently of other agents’ choices and of the equilibrium parameters. When the alternatives are assets, the initial profile can be thought of as specifying initial ownership. Definition: Initial Status Equilibrium Given an extended economy ‹N,X,(%i)i∈N,F,(ei)i∈N›: An initial status equilibrium is a pair ‹P,(xi)i∈N›where Pis an ordering on Xand (xi)is a feasible profile such that every agent i’s assigned alternative xiis %i-optimal in his “budget set” B(P,ei) = {x∈X|eiPx}. In an initial status equilibrium, an agent’s choice set consists of all alternatives that are weakly P-inferior to his initial alternative. In contrast, in a status equilibrium, an agent’s choice set consists of all alternatives that are weakly P-inferior to his equilibrium alternative.
88 Chapter 3. Status and Indoctrination Two comments: (i) If xiis %i-maximal in B(P,ei), then xiis also %i-maximal in B(P,xi). Thus, any initial status equilibrium of an extended economy is also a status equilibrium of the underlying economy. (ii) If ‹P,(xi)›is a status equilibrium, then for every strict ordering P0which is a tiebreaking of P, the pair ‹P0,(xi)›is also a status equilibrium. This is not the case for an initial status equilibrium: Consider the extended housing economy with two houses aand b, two agents, initial profile (e1,e2) = (a,b), and preference relations b1aand a2b. The public ordering that equally ranks aand band the profile (b,a)constitute an initial status equilibrium for the extended economy. However, breaking this indifference will invalidate the equilibrium since one agent will not be able to “afford” the other house. Even though a status equilibrium exists when a Pareto-optimal profile does (Proposition 3.1), the following example demonstrates that the existence of an initial status equilibrium is not guaranteed even for finite extended economies. Example: An Extended Jobs Economy Consider the jobs economy of Section 3.2 with 3 agents, two jobs aand b, and capacities na=2 and nb=1. Assume that agents 1 and 2 prefer b and agent 3 prefers a. There are two Pareto-optimal profiles (b,a,a)and (a,b,a), both of which are status equilibrium profiles (with the public ordering bPa). If either of those profiles is the initial profile, then it is also an initial status equilibrium profile. However, if the initial profile is (a,a,b), where each agent starts with the alternative he dislikes, then an initial status equilibrium does not exist. To see why, note that an equilibrium public ordering cannot rank aweakly above b, because then agents 1 and 2 would both choose b, violating feasibility. Nor can it be that bis ranked strictly above a, because then all three agents would choose a, again violating feasibility.
3.8 Initial Status Equilibrium 89 The “problem” is that the initial status equilibrium concept does not allow for the exchange of aand bbetween 1 and 3 (or between 2 and 3) due to the equilibrium concept’s inability in allowing different budget sets for two agents with the same initial alternative. The reader may wonder why no equilibrium exists in this extended economy whereas an equilibrium does exist in the standard competitive market model. The reason is that, in the standard competitive market model there is also money in the economy and a monetary amount can be attached to the transaction of exchanging afor bso that at least one of the two agents who prefer bto awould be indifferent between conducting the transaction or refraining from it. Then, the public ordering in the standard market is not merely ordinal but cardinal, indicating the monetary amount required to exchange a lower-ranked good for a higher-ranked one. Thus, the existence of an initial status equilibrium is not guaranteed when the initial profile assigns identical elements to different agents. However, whenever every agent has a distinct initial alternative, the following proposition establishes the existence of an initial status equilibrium. Furthermore, it shows that if in addition there is a strict partial ordering Rsuch that all individual preferences are R-monotonic (that is, if aRb then aibfor all i), then there is an R-monotonic equilibrium public ordering. Taking Rto be the empty binary relation gives the baseline result of Shapley and Scarf (1974) (presented in Section 1.3). Proposition 3.4: Existence of an Initial Status Equilibrium Any extended economy ‹N,X,(%i)i∈N,F,(ei)i∈N›where all initial alternatives are distinct has an initial status equilibrium. If, in addition, all preference relations are R-monotonic with respect to a strict partial ordering R, then the public ordering can be taken to be R-monotonic as well.
90 Chapter 3. Status and Indoctrination Proof: Let Ybe the set of alternatives in the initial profile (ei). For any Z⊆Y, define M(Z) = {i|ei=zfor some z∈Z}to be the set of agents initially assigned to alternatives in Z. Following the construction in Proposition 1.5, select a sequence of top trading cycles, B1,...,BT. Define a partial ordering Pon Yby aPb if a∈Bt,b∈Bs, and t≤s(all elements in the same Btare P-indifferent). We need to extend Pto all of X. Partition X\Yinto sets A1,...,AT+1as follows: For any x∈X\Y, let x∈Atwhere tis the smallest index such that there is an agent i∈M(Bt)who strictly prefers x over all elements in Bt. If there is no such t, then let x∈AT+1. Place the elements in any Atbelow Bt−1and above Bt. Define Pon Atas any arbitrary expansion of R. To see that Pexpands Reverywhere, consider aand bsuch that aRb (and thus, all agents prefer ato b). AT+1 BT AT B2 A2 B1 A1 Figure 3.4 The Construction •If b∈Y, then b∈Btfor some t. If a∈Y, then it must belong to an earlier trading cycle because no agent would top-rank bwhen ais present, and thus, aPb. If a/∈Y, then the agent who top-ranks b∈Bt prefers ato all elements of Bt. Therefore, a∈Aswith s≤t, thus, aPb. •If b∈Atfor some t≤T, then for some i∈M(Bt)it holds that biy for all y∈Bt. Thus, ialso prefers aover all y∈Bt. If a∈Y, then a belongs to a previous trading cycle and if a/∈Y, then it belongs to As with s≤t. In either case aPb (for the case that a,b∈At, recall that P expands Ron At). •If b∈AT+1and a/∈AT+1, then aPb and if a∈AT+1, then aPb because Pexpands Ron AT+1.
3.8 Initial Status Equilibrium 91 The profile (yi), which assigns to each agent i∈M(Bt)the element yi that eipoints to in the top trading cycle Bt, together with P, constitutes an initial status equilibrium: First, (yi)is feasible since it is a permutation of the initial profile. Second, for every i,eiPy ibecause yiand eiare in the same cycle. Third, suppose that ziyifor some agent i. Then, if z∈Y, it belongs to an earlier cycle. If z/∈Y, then iprefers zto all elements of Bt, and so z∈Asfor s≤t. In either event, z Py i. Example: The Extended Give-and-Take Economy Extend the give-and-take economy by adding an initial profile (ei)i∈N that is feasible, Σei=0. Each agent ifor whom ei>0 has the right to take eifrom the public fund, while each agent ifor whom ei<0 has the right to contribute −ei. Remember that every agent ihas continuous and strictly convex (and thus single-peaked) preferences with a peak at peaki. As before, we focus on the case where Σpeaki>0. Here, we adopt the interpretation that aPb means that bis more socially beneficial than a. Each agent chooses how much to give or take from the alternatives that are more socially beneficial than his initial assignment. The existence of an initial status equilibrium for this extended economy is guaranteed by Proposition 3.4 only for the case that all ei are distinct. Here, we construct a simple initial status equilibrium with an attractive structure which also demonstrates existence even when the eiare not distinct. Let Pzbe the ordering that places all alternatives between −1 and zequally at the bottom and is strictly increasing from zto 1. Every agent ifaces the interval budget set [−1,max{z,ei}]and so has a unique optimal choice which is continuous in z, weakly increasing, and strictly increasing for z∈[ei,peaki](in the case that ei<peaki). Given the total indifference ordering P1, every agent would choose peaki, and the sum of their chosen actions would be Σipeaki>0. Given the strictly increasing
98 Chapter 4. Biased Preferences Equilibrium Example: The Difficulty in Extending the Bias Notion Let K={1,2,3}and %be represented by uwhere u(x1,x2,0) = 2x1and u(x1,x2,1) = x1+x2. Let the bias vector be β= (βk)=(1,2,1). Suppose that there are biased preferences %βsuch that for any two goods kand l,MRSk,l(x,%β) = βk βlMRSk,l(x,%)at every alternative x. Then: (i) The MRS1,2 is unchanged (and remains ∞) at every (x1,x2,0). (ii) The MRS1,2 is changed from 1 to 1/2 at every (x1,x2,1). (iii) The MRS1,3 is unchanged at any alternative. By (i) and (iii), it holds that (3,5,1)∼β(4,5,0)∼β(4,6,0)∼β(2,6,1) because u(3,5,1) = u(4,5,0) = u(4,6,0) = u(2,6,1). But then, (3,5,1)∼β (2,6,1), contradicting (ii). Thus, there is no preference relation %βfor which all biased marginal rates of substitution are β-scaled versions of the original. We now define the solution concept. As explained above, harmony will be achieved by a uniform social shift of the weights placed on the consideration functions. An equilibrium consists of a profile (xi)of alternatives and a bias β such that: (i) for every agent i, the alternative xiis optimal in Xiaccording to his biased preferences; and (ii) the profile is feasible. Definition: Biased Preferences Equilibrium Abiased preferences equilibrium is a tuple ‹β,(xi)›where β∈Band (xi) is a profile of choices, such that: (i) For every agent i, the alternative xiis optimal in Xiaccording to the preferences induced by T(ui,β). (ii) The profile (xi)is in F. We will refer to a profile of alternatives that is Pareto-optimal according to the agents’ initial preferences as pre-Pareto optimal. Obviously, since agents choose optimally given their biased preferences, a profile of biased preferences equilibrium choices is always ex-post Pareto optimal.
4.1 The Economy and the Equilibrium Concept 99 Example: The Service Economy Let X={a,b}where x∈Xis a charging station. The amount of time that agent ineeds to charge his vehicle is ti≤1. The charging stations vary along two aspects aand bwith each aspect possessed by only one of the two stations. That is, the utility function of agent isatisfies vi a(a)>0, vi a(b) = 0 and vi b(a) = 0 , vi b(b)>0. Each station can serve agents as long as the total time required to serve them is not more than 1, thus F includes all the profiles (xi)such that P{i|xi=x}ti≤1 for both x∈X. Let αi=vi a(a)/vi b(b)and assume that α1> α2>∙∙∙> αn. Given a bias vector β, denote μ=βb/βa. An agent iwill choose aif αi> μ, he will choose bif αi< μ, and will be indifferent between the options if αi=μ. The following service economy has a unique biased preferences equilibrium profile. Agents 1,...,4 initially prefer to be served in awhile agent 5 initially prefers to be served at b. Agent 1 2 3 4 5 αi54321/3 ti0.3 0.6 0.7 0.1 0.1 Table 4.1 An equilibrium in the Service economy The biased preferences equilibrium profile is unique: agents 1 and 2 choose awhile the others choose band is supported by any bias for which 3 ≤μ≤4. This is not a pre-Pareto optimal profile: moving agent 4 from bto ais a Pareto improvement that is impossible in the biased preferences equilibrium because blocking agent 3 from attending aforces the bias to be such that agent 4 is also biased towards b. Of course, an equilibrium may not exist. For example, if the above table is modified so that t5=0.25 (instead of 0.1), then no biased preferences equilibrium exists. In the rest of the chapter, we analyze the biased preferences equilibrium for several examples: the give-and-take economy, the exchange economy with fixed prices, and a few housing-type economies.
100 Chapter 4. Biased Preferences Equilibrium 4.2 The Give-and-Take Economy We return to an old friend: the give-and-take economy. Recall that, in this economy, each agent decides how much to contribute to or withdraw from a social fund, and feasibility requires that the total contributions equal the total withdrawals. All agents face the same choice set [−1,1], where, as usual, a positive number represents the amount taken from (and a negative number the amount given to) the social fund. To fit it into the current framework, let the two considerations be g(giving) and t(taking). The consideration gis generous (people like to give), while the consideration tis selfish (people like to take). Initially, an agent i’s choice balances between these two considerations by choosing the maximizer of ui g(x) + ui t(x), where ui gis a strictly decreasing and strictly concave function while ui tis a strictly increasing and strictly concave function. It follows that both are continuous. Denote by peakithe unique maximizer of ui g(x) + ui t(x), which is agent i’s most-preferred choice in [−1,1], and assume that it is interior. Any bias βpushes all agents’ preferences in the same direction by systematically altering the tradeoff between generosity and selfishness. Denote μ=βt/βg. A μabove 1 biases the agents’ preferences towards selfishness, while aμbelow 1 biases the agents’ preferences towards generosity. Under the above assumptions, there is a unique equilibrium profile, and it is pre-Pareto optimal: Proposition 4.1: Uniqueness and Pre-Pareto Optimality (i) The give-and-take economy has a unique biased preferences equilibrium (up to a rescaling of the bias vector). (ii) The equilibrium profile is pre-Pareto optimal.
4.2 The Give-and-Take Economy 101 Proof: (i) Given the bias (1,μ), each agent ioptimizes ui g(x) + μ∙ui t(x)and has a unique optimal choice denoted by xi(μ). Since all of the xifunctions are increasing and continuous, so is the net overall “demand” from the social fund, Σixi(μ). This sum is positive when μis sufficiently large and negative when μis sufficiently small. Therefore, there is a μ∗for which the sum is zero, and ‹(1,μ∗),(xi(μ∗))›is a biased preferences equilibrium. The parameter μ∗is unique since xi(μ)is strictly increasing when −1<xi(μ)<1, and it must be that xi(μ∗)is interior for some i. (It is impossible that all agents choose 1 or all choose −1. It is also impossible that in equilibrium some choose 1 and the other −1 since if μ∗≥1, then for every i,xi(μ∗)≥peaki>−1 and if μ∗≤1, then for every i, xi(μ∗)≤peaki<1.) (ii) If μ∗=1, then every agent’s equilibrium choice is his unbiased firstbest. If μ∗>1, then every agent ichooses xi(μ∗)≥peaki, and any exante Pareto improvement (yi)must satisfy yi≤xi(μ∗)with at least one strict inequality, but that contradicts feasibility since 0 = Σxi(μ∗)>Σyi. Thus, the equilibrium profile is pre-Pareto optimal. The case μ∗<1 is analogous. Note that in the biased preferences equilibrium, balancing the social fund is a shared responsibility of all agents: when the sum of the agents’ peaks is positive (i.e. there is an overall preference for taking), the equilibrium bias (μ∗<1) overweighs generosity and every agent chooses a point below his peak. This is essentially true for primitive equilibria (in every equilibrium profile all agents are assigned to alternatives weakly below their peaks). In contrast, in a Y-equilibrium, there is a uniform cap on withdrawals and only the greediest agents are impacted, and in a jungle equilibrium, only the weakest agents are restricted.
102 Chapter 4. Biased Preferences Equilibrium 4.3 The Fixed-Prices Exchange Economy The next example is related to the literature on economies with fixed prices (see Benassy (1986) and the references therein). Let X=RK ++ be the set of bundles in a world with a set of goods K. Every agent ihas an initial endowment ei∈X, and exchange takes place according to a fixed price vector p= (pk). Accordingly, Xi={x∈X|p∙x=p∙ei}, and the set of feasible profiles is F={(xi)∈ΠiXi|Σixi= Σiei}. All agents share the same considerations, one for each good. Each consideration function ui k(x)is a function of only xk, which is assumed to be increasing, twice-differentiable, and strictly concave. In economies with fixed prices, rationing is typically the mechanism used to achieve harmony. That is, upper bounds are established on the consumable quantity of each good. In contrast, in a biased preferences equilibrium, economic harmony is achieved by means of a systematic adjustment of preferences. Proposition 4.2: Biased Preferences Equilibria in Exchange Economies with Fixed Prices In any exchange economy with fixed prices: (i) A biased preferences equilibrium exists. (ii) All biased preferences equilibrium outcomes are pre-Pareto optimal. Proof: (i) To illustrate, consider the two-good two-agent case, which can be depicted using an Edgeworth Box (see Figure 4.1). Assume that agents do not like consuming on the boundary, i.e. the derivative of every consideration function ui kat 0 is infinity. Let (x1,x2)∈X1×X2be an overall Pareto-optimal allocation of e1+e2, which always exists. Then, at (x1,x2), both agents have the same marginal rate of substitution μ. If μ=p1/p2, then no bias is needed, that is, ‹β= (1,1),(xi)›is a biased preferences equilibrium. If μ6=p1/p2, then the bias β= (p1,p2μ)
4.3 The Fixed-Prices Exchange Economy 103 modifies both preferences so that the MRS1,2 of the biased preferences of each iat xiis μβ1/β2=p1/p2. Thus, ‹β,(xi)›is a biased preferences equilibrium. (x1,x2) p1 p2 u1u1 u2 u2 (e1,e2) β(x1,x2) p1 p2 T(u1,β) T(u2,β)(e1,e2) Figure 4.1 Equilibrium in an Edgeworth Box For the case of any number of agents and more than two goods (and even without the boundary assumptions), Keiding (1981) (following Balasko (1979)) showed that there is a vector q= (qk)and an allocation (xi)such that p∙xi=p∙eifor all i, and if yiixi, then q∙yi>q∙xi. Therefore, for every agent i, any good kthat he consumes and any other good l, it holds that MRSk,l(xi)≥qk/ql. Consequently, by setting β= (pk/qk)k∈K, it holds that biasedMRSk,l(xi) = βk βlMRSk,l(xi)≥βkqk βlql=pk pl(the bound is an equality if xi l>0). Therefore, for every agent i, given the price vector pand the initial bundle ei, the bundle xiis optimal for i’s biased preferences. Thus, ‹β,(xi)›is a biased preferences equilibrium. (ii) Let ‹β,(xi)›be a biased preferences equilibrium. Then, for each agent iand any good lthat he consumes, the MRSk,lof the biased preferences T(β,ui)at xiis bounded from above by pk/pl. Therefore, the MRSk,lof his initial preferences at xiis bounded from above by pk/βk pl/βl. Thus, (xi) is a Walrasian equilibrium outcome in the unbiased economy with price vector pk/βkand initial endowment (xi). Therefore, by the standard First Welfare Theorem, (xi)is pre-Pareto optimal.
104 Chapter 4. Biased Preferences Equilibrium We now consider an example with two goods and linear preferences where the biased preferences equilibrium can easily be calculated. Example: Linear Preferences Suppose that there are two agents, two goods, and that for every agent i, the two consideration functions are linear, that is, ui 1(x1) = x1and ui 2(x2) = αix2where every αiis a positive number. Consider the configuration depicted in Figure 4.2: T(β,u1)T(β,u2)u1u2u1u2T(β,u1)T(β,u2) p1 p2 u1 u1 u2 u2 T(u1,β) T(β,u1) T(u2,β) T(β,u2) (e1,e2) contract curve biased preferences equilibrium Figure 4.2 Biased linear preferences in an Edgeworth Box (dashed line =the budget lines; black solid lines =initial preferences; red solid lines =biased preferences; blue line =the contract curve for the initial preferences) In this example (other configurations can be analyzed similarly): (i) Agent 2 likes good 2 more than agent 1 does, that is, α2> α1. (ii) The ratio p1/p2is greater than both agents’ (constant) personal marginal rates of substitution, that is, p1/p2>1/α1>1/α2. (iii) In any feasible allocation, both agents must consume positive amounts of good 1. The economy is not in harmony because, given (ii), both agents wish to purchase only good 2.
4.3 The Fixed-Prices Exchange Economy 105 In any biased preferences equilibrium, there is an agent iwho consumes good 2, and by (iii), he also consumes good 1. By the linearity of the preferences, agent imust be indifferent between all alternatives in Xi, that is, p1/p2=β1/(αiβ2). If i=1, then by (i), agent 2 does not consume good 1, violating (iii). Thus, it must be that i=2, and the bias satisfies p1/p2=β1/(α2β2). Such a bias is part of the equilibrium depicted in Figure 4.2, where agent 1 consumes only good 1, and agent 2 (who is indifferent between all bundles in his budget set) consumes all of good 2 and the remainder of good 1. It follows that this is the unique biased preferences equilibrium. Failure of Individual Rationality: An interesting feature of a biased preferences equilibrium is that, even though it is pre-Pareto optimal, “Individual Rationality” can fail: in an equilibrium, an agent might choose a bundle that is inferior to his endowment bundle when judged by his initial preferences, as in the previous example. By his original preferences, agent 1 is worse off in the equilibrium than he was with his initial endowment, since he trades some of his good 2 endowment for good 1, but ex-ante he would prefer to do the opposite. Example: Non-Convex Preferences In the standard exchange economy with non-convex preferences, a competitive equilibrium may not exist: there may be no price vector for which the sum of the demands equals the total bundle. Nevertheless, there may be a price vector for which a biased preferences equilibrium exists. Thus, prices and biased preferences together may achieve harmony when the standard competitive equilibrium tools fail to do so. To illustrate, consider the division economy where both agents have the non-convex preferences represented by (x1)2+2(x2)2and the initial endowments are e1= (1,1)and e2= (2,2). There is no standard competitive equilibrium. Given any price vector, each agent will
106 Chapter 4. Biased Preferences Equilibrium consume only one of the two goods, and since the agents have the same preferences, any equilibrium price vector must make each agent indifferent between the two goods, i.e. p= (1,p2). But, then agent 2 will demand more than 3 units of one of the goods. In contrast, a biased preferences equilibrium exists. Let p= (2,1)and β= (8,1). Each agent’s biased utility function is 4(x1)2+ (x2)2. Agent 1’s optimal bundles are (1.5,0)and (0,3), and agent 2’s optimal bundles are (3,0)and (0,6). Thus, the bias β, together with the allocation x1= (0,3) and x2= (3,0), is a biased preferences equilibrium in the exchange economy with fixed prices p. Note that agent 1 is initially poorer than agent 2, but in the equilibrium, agent 1 is better off according to the initial preferences! 4.4 Housing-Type Economies We return to the classic housing economy of Shapley and Scarf (1974), in which there is a set Nof agents and an equally-sized set Hof houses. Each agent i chooses a single house, that is, Xi=H. Let vi(h)>0 be agent i’s valuation of house h. The model can be enriched to fit our framework by taking the set of considerations to be Hand setting ui h(xi) = vi(h)if xi=hand 0 otherwise. Given a bias vector (βh), an agent iderives utility βhvi(h)from house h. Example: The following table presents the consideration function values in a housing economy with two agents. h1h2 v1(h)4 3 v2(h)3 1 Table 4.2 House utilities
4.4 Housing-Type Economies 107 Both agents initially prefer house h1. To achieve harmony, the bias must boost h2so that one agent will choose it, but not to the extent that both will. For example, a biased preferences equilibrium is obtained by the bias (1,2), which results in agent 1 choosing h2and agent 2 choosing h1. Of course, other biases are possible but, in all biased preferences equilibria, agent 1 gets h2and agent 2 gets h1. Note that, in the biased preferences equilibrium profile, the product of the ex-ante values (3∙3=9) is larger than that in the other assignment (4 ∙1=4). We will see below that this is not a coincidence. We say that a feasible profile (xi)is Nash maximal if it maximizes Πi∈Nvi(xi) over all feasible profiles. We now show that the set of biased preferences equilibrium profiles is precisely the set of Nash-maximal profiles and thus, any biased preferences equilibrium profile is pre-Pareto optimal. The proof is a direct application of Shapley and Shubik (1971) (see also Gale (1984) for a proof using the KKM Lemma). Proposition 4.3: Biased Preferences Equilibrium =Nash Maximality In the housing economy, the set of biased preferences equilibrium profiles is the set of Nash-maximal profiles. Proof: Let (hi)i∈Nbe a Nash-maximal profile, that is, it maximizes Σi∈Nln(vi(xi)) over all feasible assignments. By Shapley and Shubik (1971), there exists a price vector (ph)so that for each agent i, the house hiis a maximizer of ln(vi(xi)) −pxi, and therefore, it is also a maximizer of vi(xi)/epxi. Thus, ‹(βh=1/eph)h∈H,(hi)i∈N›constitutes a biased preferences equilibrium. In the other direction, let ‹β,(hi)›be a biased preferences equilibrium and (xi)be any other assignment. For each i,βhivi(hi)≥βxivi(xi)and therefore, Πiβhivi(hi)≥Πiβxivi(xi). Since Πiβhi= Πiβxi, it follows that Πivi(hi)≥Πivi(xi), that is, (hi)is Nash maximal.
114 Chapter 5. A Comparison to Game Theory The second battleground is a “political economy” setting. There is a group of agents with views on a political issue. Each agent chooses a position and cares only about the position he himself chooses (and not about the outcome of the process). There is a need that a majority of agents choose the same position; otherwise, a crisis ensues. Traditionally, such a situation is modelled as a non-cooperative game, and its Nash equilibria are calculated. Extending Richter and Rubinstein (2021), we compare this approach to two of the approaches discussed in this book: •The convex Y-equilibrium in which society is governed by (convex) norms that specify what is permissible and what is forbidden. •The biased preferences equilibrium in which where preferences are systematically biased. On both battlegrounds, the matching problem and the political economy setting, we will see that the new approaches lead to very different outcomes than the traditional ones. 5.1 The Matching Economy In the matching economy, Nis an even-numbered population of nagents. The set of alternatives is taken to be the set of agents, i.e. X=N. A pairing is a profile (xi)i∈Nthat specifies, for every i, a partner xi6=i, such that if iis paired with j, then jis paired with i. The feasibility set Fis the set of all pairings. Note that F is not closed under permutations (if iand jexchange partners, then feasibility requires that xiand xjalso do). A match between two agents iand jis denoted as i↔j. We assume that agents prefer to have any partner over being alone; in other words, every agent bottom-ranks himself. Therefore, it will be sufficient to specify, for each agent i, a strict preference relation iover X\{i}, the set of all other agents. As mentioned, the standard solution concept for this economy is pairwise stability, which is a pairing such that there are no two agents in different pairs who prefer each other over their respective current partners. Formally, a pairing (xi)is pairwise stable if there is no iand jsuch that jixiand ijxj.
5.1 The Matching Economy 115 The two-sided matching economy is a special case of the matching economy. The set of agents Nis partitioned into two equally-sized groups, N1and N2, and every agent prefers any agent from the other group over any agent from his own group. We say that a pairing is mixed if every couple has one member from each group. For two-sided matching economies, a pairwisestable pairing always exists and can be calculated using Gale and Shapley (1962)’s deferred acceptance algorithm. However, in the general matching economy, a pairwise-stable pairing often does not exist. The following is Gale and Shapley (1962)’s canonical example of a matching economy with no pairwise-stable pairing: Agent 1 2 3 4 1st Preference 2 3 1 1 2nd Preference 3 1 2 2 3rd Preference 4 4 4 3 Table 5.1 A matching economy with no pairwise-stable pairing. To see that there is no pairwise-stable pairing, consider a candidate pairing. Let ibe the agent matched with 4. Agent iprefers every other agent over 4, and there is an agent j∈{1,2,3}\{i}who top-ranks i. Thus, the couple i↔jblocks the candidate pairing from being pairwise stable. As discussed in Section 0.4, we distinguish between two types of equilibrium concepts: the choice type (such as competitive equilibrium) and the deviation type (such as Nash equilibrium). Two of the concepts which we will apply, the Y-equilibrium and the initial status equilibrium, belong to the choice type. In these solution concepts, some internally determined parameter will restrict every agent’s choice set, such that there is a pairing in which every agent’s partner is one he most prefers from his choice set. The other solution concepts that we apply to the matching problem belong to the deviation type. An equilibrium concept of this type captures immunity to certain threats that would “rock the boat”. Such a solution concept consists of a pairing and an internally determined parameter that restricts every agent’s deviation possibilities, such that no agent can profitably deviate.
116 Chapter 5. A Comparison to Game Theory Pairwise stability belongs to the deviation class. Given a pairing, the threat to pairwise stability is a potential deviation by two agents who prefer each other over their current partners. We have something different in mind — unilateral threats: for harmony to be disturbed, it is sufficient that even one agent is willing and able to approach another. The threat is merely the approach of one agent to another who is not his partner, whether or not his approach is reciprocated. The perspective of life which we model is that a pairing can be destabilized not by coalitions. It is sufficient that an agent Aapproaches an agent B(who is not matched with A) and expresses his desire that Babandons his current partner and matches with him instead. This destabilizes society regardless of whether or not Breciprocates A’s affections. Why does Aapproach B? Actually, why not? He may know that Balso prefers him over B’s current partner (this is the premise of pairwise stability). Even if he knows that Bdoes not prefer him, Amight hope that if he approaches B, then Bwill feel flattered and change his mind. Finally, Amight not know B’s preferences and simply tries his luck. Generally, it is impossible that every agent is paired with his top choice since agents’ desires are not perfectly reciprocated. Therefore, achieving stability when agents can make unilateral approaches requires restrictions on which approaches are allowed. Given such restrictions, we say that a pairing is unilaterally stable if there is no agent who wishes to approach another and is able to do so. A familiar and very restrictive social norm forbids anyone from approaching any matched individual. Such a norm achieves harmony in a society, but at the cost of drastically curtailing personal freedom. The restrictions described in the following three sections involve social institutions (power, taboos, and status) that limit an agent’s ability to act, but in a less draconian manner. The following is a running example which will be used to illustrate the different solution concepts:
5.1 The Matching Economy 117 Example: The Common-ranking Two-sided Matching Economy Acommon-ranking two-sided matching economy is a two-sided matching economy where every agent in N1ranks members of N2according to a common ranking j12j22...2jn/2, and every agent in N2ranks his potential partners in N1according to i11i21...1in/2. In such an economy, the set of Pareto-optimal pairings is the set of all mixed pairings (recall, a pairing is mixed if every agent is paired with an agent from the other side). To see this, any pairing that matches two members of the same side also matches two members on the other side. However, in that case, all four agents can be beneficially re-paired with members of the other side, which is a Pareto improvement. On the other hand, in any mixed matching, improving one agent’s situation requires moving another agent down the common preference ladder; thus, no Pareto improvements exist. Before proceeding, let us recall the classic serial dictatorship algorithm: There is an equal number of agents and objects and the agents are strictly ranked. The agents each choose an object according to their rank. That is, the highest-ranked agent selects his most-preferred object; then the secondhighest agent selects his most-preferred object from those remaining, and so on until all agents have made a selection. We modify this algorithm as follows: Again, the agents are strictly ordered, but this time, the highest-ranked agent selects a partner from among the other agents. Both he and his partner are removed, and the highest-ranked remaining agent selects a partner from among the remaining agents, and they are also removed. This algorithm proceeds until every agent either “chooses” or is “chosen”. Formally, a pairing (xi)is the outcome of the modified serial dictatorship procedure with the ordering if there is a sequence of n/2 agents i1,...,in/2 such that i1is the -maximal and xi1is his most-preferred agent from N, agent i2is the -maximal and xi1is his most-preferred from N−{i1,xi1}, and so on.
118 Chapter 5. A Comparison to Game Theory Denote by MSD, the set of pairings that result from the modified serial dictatorship procedure for some ordering . Obviously, MSD is non-empty and any pairing in MSD is Pareto-optimal. 5.2 The Jungle Equilibrium The solution concepts we apply here are related to the jungle model described in Chapter 1. In this section, an equilibrium candidate is a tuple ‹B,(xi)›where Bis a strict ordering on Nand (xi)is a pairing. The statement iBjmeans that “iis more powerful than j”. We consider several variants of the jungle equilibrium that differ in which circumstances power prevents an agent from approaching another. In a J1-equilibrium, an agent is only able to approach agents who are weaker than himself. In a J2-equilibrium, an agent needs to be stronger than both the agent he is approaching and that agent’s current partner. In a J2*-equilibrium, he has to be stronger than the agent he is approaching and his own partner. In a J3-equilibrium, an agent needs to be stronger than all three of these agents: the agent he is approaching, his partner and his own partner. The following table summarizes the situations in which an agent ican approach agent j: i j xi xj Concept Power Requirement J1 ij J2 ijand ixj J2* ijand ixi J3 ijand ixjand ixi Figure 5.1 A potential approach by ito j(left panel) and the conditions under which such an approach can be made for each solution concept (right panel). We now arrive to the formal definition of the J1-equilibrium concept. Definition: J1-Equilibrium AJ1-equilibrium is a tuple ‹B,(xi)›in which there are no two agents i and jsuch that iprefers jover his current partner (that is, jixi) and i is more powerful than j(that is, iBj).
5.2 The Jungle Equilibrium 119 In a J1-equilibrium pairing, it is possible that an agent Aprefers another agent Bto his current partner. But, agent Ais prohibited from approaching Bbecause Bis stronger than him. In contrast, according to pairwise stability, what prevents agent Afrom approaching agent Bis that Bwill reject him. The following proposition states that the J1-equilibrium concept is stricter than both pairwise stability and MSD. Since pairwise-stable pairings do not always exist, neither will J1-equilibria. Proposition 5.1: J1-equilibrium Properties (i) Every J1-equilibrium pairing is both pairwise stable and an MSD outcome (and thus Pareto optimal). (ii) A pairwise-stable pairing might not be a J1-equilibrium pairing. (iii) An MSD pairing might not be a J1-equilibrium pairing. Proof: (i) Let ‹B,(xi)›be a J1-equilibrium. The profile (xi)is clearly the outcome of the MSD procedure with the ordering . In order to show that (xi)is pairwise stable, suppose that there are two agents iand jwho strictly prefer each other to their current partners. One of them must be B- stronger than the other, and he prefers the weaker agent over his current partner, thus violating the J1-equilibrium condition. (ii) In a J1-equilibrium, the strongest agent is matched with his first-best choice. In the following matching economy, the (red) pairing 1↔2 and 3↔4 is pairwise stable, but no agent is matched with his first best: Agent 1 2 3 4 1st Preference 4 3 1 2 2nd Preference 2 1 4 3 3rd Preference 3 4 2 1 Table 5.2 Preferences with a pairwise-stable pairing (in red) and an MSD pairing (in blue), neither of which is a J1-equilibrium outcome.
120 Chapter 5. A Comparison to Game Theory (iii) In the economy presented in Table 5.2, the (blue) pairing 1↔4 and 2↔3 is an MSD outcome (with the ordering 1 234), but both 3 and 4 are matched with their worst partner, which cannot occur in a J1-equilibrium. Example: The Common-ranking Two-sided Matching Economy The pairing {i1↔j1,i2↔j2,∙∙∙,in/2↔jn/2}combined with any power ordering that satisfies i1,j1i2,j2∙∙∙in/2,jn/2is a J1-equilibrium. There is no other J1-equilibrium pairing: Since every agent in N2top- ranks i1, agent i1must be stronger than everyone in N2except perhaps his partner. This means that, in any equilibrium, i1has to be matched with his first-best, namely j1. Similarly, j1must be stronger than all members of N1, except possibly i1. This pattern continues down the ranking. Among the remaining agents, i2and j2are matched, and i2 must be more powerful than {j3,...,jn/2}while j2must be stronger than {i3,...,in/2}and so on. In a J1-equilibrium, the ability of one agent to approach another depends solely on the power relationship between them. However, in the context of the matching economy, any approach involves not only the agent who initiates the approach and the approached agent but also their partners. The following solution concepts take this into account. Definition: J2-Equilibrium, J2*-equilibrium and J3-equilibrium AJ2-equilibrium is a tuple ‹B,(xi)›for which there are no iand jsuch that jixiand iBj,xj. AJ2*-equilibrium is a tuple ‹B,(xi)›for which there are no iand jsuch that jixiand iBj,xi. AJ3-equilibrium is a tuple ‹B,(xi)›for which there are no iand jsuch that jixiand iBj,xi,xj.
5.2 The Jungle Equilibrium 121 Obviously, every J1-equilibrium is also a J2-equilibrium as well as a J2*- equilibrium and every J2-equilibrium or J2*-equilibrium is a J3-equilibrium. Part (i) of the following proposition proves that in terms of sets of outcomes, the three concepts are identical and equal to MSD. Thus, these equilibria always exist and their outcomes are Pareto optimal. Part (ii) shows that the concepts are neither weaker nor stronger than pairwise stability. Since pairwise-stable pairings are Pareto optimal, there can be a Pareto-optimal pairing which is not a J3-equilibrium outcome. Part (iii) shows that any ordering is a J3-equilibrium ordering (which is not the case for the J2- and J2*-equilibrium concepts). Proposition 5.2: J2, J2* and J3-equilibrium Properties (i) The sets of J2-equilibrium pairings, J2*-equilibrium pairings, and J3- equilibrium pairings are identical and equal to MSD (and thus such pairings always exist and are Pareto-optimal). (ii) The set of pairwise stable outcomes and MSD do not include one another. (iii) For any strict ordering , there is a J3-equilibrium with the power relation . Proof: (i) Choose an arbitrary strict ordering of the agents and apply the modified serial dictatorship procedure. Let i1,...,in/2be the agents who make a choice according the procedure and j1,...,jn/2be the agents who are chosen where ikchooses jkfor each k. Any power relation satisfying i1Bi2B∙∙∙Bin/2Bj1,...,jn/2supports this matching as a J2-equilibrium. Any agent who might be preferred by ikover jkmust have been paired earlier, and thus, is either stronger than ikor has a partner who is. No jlcan approach another agent because every other couple, ik↔jk, has at least one member who is stronger than him, namely ik.
122 Chapter 5. A Comparison to Game Theory The power relation i1B∗j1B∗i2B∗j2∙∙∙B∗in/2B∗jn/2supports the the matching as a J2*-equilibrium. No jlcan approach another agent because his partner ilis stronger than he is. Any agent who might be preferred by ikover jkmust have been paired earlier, and thus, is stronger than ik. To complete the proof it is sufficient to show that if ‹,(xi)›is a J3-equilibrium, then (xi)is the outcome of the MSD procedure with the ordering B. Order the matches in this J3-equilibrium as follows: a1↔b1,...,an/2↔bn/2where akBbkfor all kand a1Ba2B∙∙∙BaK. Thus, the pairing (xi)will be the result of the MSD with the ordering B. (ii) For the matching economy depicted in Table 5.2, the red highlighted pairing is pairwise stable but is not in MSD since no agent gets his firstbest. The blue highlighted pairing is in MSD but is not pairwise stable. (iii) Let be a strict ordering and (xi)be the MSD outcome with this ordering. Then, ‹,(xi)›is a J3-equilibrium. By the MSD procedure, half of the agents “make a choice” while the rest “are chosen”. Any agent who “makes a choice” can only prefer agents who are matched before him, i.e. those who are stronger than him or are paired with a stronger partner. Any “chosen” agent is neutralized by being matched with a stronger partner. Example: The Common-ranking Two-sided Matching Economy Every mixed pairing (xi)is a J2-equilibrium pairing (and thus also a J2*- and a J3-equilibrium outcome) supported by assigning the power relations of agents in each side by the rank of their partners (that is, for every two members iand jfrom the same side assign iBjif xiis higher-ranked than xj).
5.3 Restricting Partnerships: Pairwise Y-equilibrium 123 While every mixed pairing is part of a J2-equilibrium, not every power relation is. For example, in the case of four agents, there is no J2- equilibrium with the power relation j2i1i2j1. This is because j2is the most powerful, and must be matched with i1. Thus, the only candidate pairing is {i1↔j2,i2↔j1}. But this is not a J2-equilibrium because i1prefers j1over j2, and is stronger than both i2and j1. 5.3 Restricting Partnerships: Pairwise Y-equilibrium We now adjust the Y-equilibrium concept (Chapter 2) to fit the matching economy. Since every agent needs a partner, uniformly restricting the set of permitted partners will leave some agents without a partner. Instead, we model social norms that determine which pairs are permitted and which are forbidden. Definition: Y-Equilibrium Let Mbe the set of all pairs. A para-Y-equilibrium is a tuple ‹Y,(xi)› where Y⊆Mand (xi)is a pairing such that, for every agent i,xiis imaximal in {j|i↔j∈Y}. A Y-equilibrium is a para-Y-equilibrium such that there is no other para-Y-equilibrium ‹Z,(yi)›with Y⊂Z. Any set of permissible pairs Yinduces, for each agent i, a choice set of permissible partners {j|i↔j∈Y}. Thus, unlike in Chapter 2, here the Y- equilibrium notion treats agents asymmetrically in the sense that different agents face different choice sets with the restriction that if jis permissible for i, then iis permissible for j. The adapted Y-equilibrium notion requires that, for any larger permissible set, there is an iand jsuch that iwould choose jbut jwould not choose i. The following proposition shows that the set of Y-equilibrium pairings is the set of all Pareto-optimal pairings and thus, always exists.
130 Chapter 5. A Comparison to Game Theory (ii) Given a pair-rankable matching economy with a partition I1,...,In/2 of N, the pairing in which every agent is matched with his doubleton’s partner is the only initial status equilibrium pairing. This is because the two agents in I1must be matched in any initial status equilibrium since they top-rank each other and, for any equilibrium ranking, one of them can “afford” the other. The same argument then applies to the agents in I2and so on. Obviously, this pairing with the ranking Pfrom part (i) is a J1-equilibrium. Example: The Common-ranking Two-sided Matching Economy The economy is uniquely pair-rankable with Iq={iq,jq}. By Proposition 5.6, the unique initial status equilibrium is the pairing {i1↔j1,i2↔j2,...} with the status ranking Pthat ranks xPy if x∈Ip,y∈Iqand p≤q. 5.6 Comparing the Approaches Figure 5.2 summarizes the relationship between the different equilibrium concepts, pairwise stability (PS), Pareto optimality (Pareto) and the modified serial dictatorship procedure outcomes (MSD). The lines in the diagram stand for inclusions (each inclusion can be strict). In particular, all equilibrium pairings are Pareto optimal (the First Welfare Theorem); only the Y-equilibrium satisfies the Second Welfare Theorem; and only the J2-, J3-, and Y-equilibria are guaranteed to exist. The symbols S and IS stand for status equilibrium and initial status equilibrium, respectively. IS S J1 J2=J2*=J3=MSD PS Y=Pareto Exists Figure 5.2 Relationship between the concepts.
5.6 Comparing the Approaches 131 One can imagine solution concepts other than those analysed in this chapter. An example is due to Herings and Zhou (2024) (see also Herings (2024)). They model a social restriction on the approaches that an agent can make. They require that for any pair of agents, the norm allows at least one of them to approach the other. In an equilibrium, every agent approaches his optimal partner from among those he is allowed to approach. Definition: EE-equilibrium An EE-equilibrium is a pairing (xi)i∈Nand a profile of permissible sets of agents (Ai)i∈Nsuch that: (i) For every two agents iand j, either i∈Ajor j∈Ai(or both). (ii) For every i, agent xiis i’s most-preferred partner from Ai. It is shown now that the EE-equilibrium outcomes are identical to the pairwisestable pairings. Proposition 5.6: EE-equilibrium =Pairwise Stable The set of EE outcomes is equal to the set of pairwise-stable pairings. Proof Let (xi)be a pairwise-stable pairing. Define Ai={z:xi%iz}. By definition, for every i,xiis %i-maximal within Ai. In addition, for any two unmatched agents iand j, by pairwise stability it must be that xi%ijor xj%ji, and therefore j∈Aior i∈Aj. Thus, ‹(Ai),(xi)›is an EE-equilibrium. Let ‹(Ai),(xi)›be an EE-equilibrium. Then, for every iand jwho are not matched, it must be that i∈Ajand so xj%ji, or j∈Aiand so xi%ij. Therefore, (xi)is pairwise stable.
132 Chapter 5. A Comparison to Game Theory 5.7 The Majority Voting Economy We now turn to compare our approach to that of Non-Cooperative Game Theory. As mentioned, our battleground is the following majority voting economy: There is an odd number of agents (n≥3), each of whom chooses a position in X= [−1,1]. Each agent ihas continuous and strictly convex preferences over Xwith a unique peak, denoted by peaki. All peaks are distinct and we can assume that −1<peak1<peak2<∙∙∙<peakn<1. To simplify notation, denote the leftmost peak, the median peak, and the rightmost peak as L,M, and R, respectively. The set F⊆XNconsists of all profiles for which at least τ= (n+1)/2 members choose the same position. If a profile (xi)has a point shared by at least τmembers, we denote it by O((xi)) and refer to it as the overall position (clearly, it is not possible to have two such positions). The situation we have in mind is one where societal harmony requires that a majority of members declare the same position and if there is no such majority, then a crisis bursts. An example is a committee or a jury who, according to their procedure, must come out with a position that is supported by a majority of members. Another example is of a political party whose leaders must take positions on an issue. To prevent the public from becoming confused and abandoning the party, at least a majority of them need to take the same position. Note that in Hotelling (1929) and its many extensions, an agent cares only about the group’s position, while in Downs (1957), an agent also cares about his chosen position. We go a step further and assume that an agent cares only about his chosen position and does not care at all about the group’s majority position.
5.8 Convex Y-equilibrium 133 5.8 Convex Y-equilibrium First, we analyze the economy’s convex Y-equilibria (Chapter 2). Recall that a convex Y-equilibrium is a configuration ‹Y,(yi)›(where Yis a convex subset of Xand (yi)is a profile of choices from Y) that satisfies the three conditions: (i) Rationality: for all i,yiis a %i-maximal position in Y. (ii) Feasibility: (yi)∈F. (iii) Set maximality: there is no convex set Z⊃Yand profile (zi)∈Fsuch that ziis a %i-maximal alternative in Zfor all i. Recall that, in any convex Y-equilibrium, the permissible set is closed (if not, then the closure of the permissible set with the same profile is a larger convex para-equilibrium). Since Yis convex and closed and all preference relations are strictly convex, every agent’s maximal position is unique. We now show that there are exactly two convex Y-equilibria in this economy and both have the overall position M(the median peak): a “rightist” equilibrium in which Mand all positions to its right are permissible and a “leftist” equilibrium in which Mand all positions to its left are permissible (see Figure 5.2). −11 peak1peak2peak3=Mpeak4peak5 x1x2x3=x4=x5 Figure 5.3 A leftist equilibrium Proposition 5.7: Convex Y-equilibria in the Majority Voting Economy In the majority voting economy, there are two convex Y-equilibria. Their permissible sets are [−1,M]and [M,1]. Both have the overall position M.
134 Chapter 5. A Comparison to Game Theory Proof: The set [−1,M]is a convex para-equilibrium permissible set because all of the rightist agents and the median voter vote M, thereby constituting a majority. Likewise, [M,1]is a convex para-equilibrium permissible set. To show that [−1,M]and [M,1]are convex Y-equilibrium permissible sets and that there are no others, it suffices to show that any convex paraequilibrium permissible set is a subset of either [−1,M]or [M,1]. To see this, note that if a convex permissible set contains points to both the left and right of M, then no position attracts majority support: Mmust be in the permissible set since it is convex and the median agent selects it, all leftist agents (a minority) choose positions to the left of Mand all rightist agents (also a minority) choose positions to the right of M. Thus, every convex Y-para-equilibrium’s permissible set is contained in [−1,M]or [M,1]. At first glance, Proposition 5.7 appears to be a kind of “median voter theorem” since the only convex Y-equilibrium overall position is the median. Thus, in terms of outcomes, the convex Y-equilibrium involves a compromise; however, the price of this “happy ending” is that only positions to one side of Mare permitted. 5.9 Biased Preferences Equilibrium To fit the model into Chapter 4’s definition of an economy, we modify the specification of the agents’ preferences as we did for the give-and-take economy. Assume that each agent ihas two considerations in mind, labelled land r, and maximizes the utility function ui(x) = ui l(x) + ui r(x)where ui lis strictly decreasing and represents an argument for leftist positions, while ui ris strictly increasing and represents an argument for rightist ones. The functions ui land ui rare differentiable with non-zero and finite derivatives at each point. They are also strictly concave, which implies that their sum induces convex preferences with a unique peak, denoted by peaki, and we assume that all peaks are different.
5.10 The Majority Voting Game and Nash Equilibrium 135 A bias (λl,λr)transforms an agent i’s utility function into λlui l(x)+λrui r(x). Consequently, a leftist bias (where λl> λr) moves the peak of every agent to the left, while a rightist bias (where λl< λr) moves all of the peaks to the right. There always exist extreme biased preferences equilibria where a majority of agents agree on the rightmost position supported by a rightist bias, which increases the weight of the rightist consideration strongly enough that at least a majority of individuals become extreme rightists. Likewise, there are also extreme leftist biased preferences equilibria. It is possible that there are biased preferences equilibria where a majority of individuals move in one direction, and it just so happens that a majority of the biased peaks coincide but this would be a fluke occurrence. However, there is never a biased preferences equilibrium with a majority at the median position (since the peaks are distinct, a bias is necessary for agreement, but if there is a rightist bias, then any equilibrium must have a right-of-median overall position, and vice-versa if there is a leftist bias). To summarize, extreme positions are always biased preferences equilibrium overall positions and the median never is, whereas the unique Y- equilibrium overall position is the median. 5.10 The Majority Voting Game and Nash Equilibrium To model the majority voting economy ‹N,X,(%i)i∈N,F›as a strategic game, let the set of players be N, and let each player’s set of actions be the set of positions X. Each player jhas a preference relation %j ∗on the set of all choice profiles, defined by x= (xi)%j ∗y= (yi)if either: (i) x∈Fand y/∈F; or (ii) both x,y∈For both x,y/∈Fand xj%jyj. In other words, every agent’s lexicographical first priority is harmony, and his second priority is his own position. We apply the standard Nash equilibrium to this game. We distinguish between non-crisis Nash equilibria (in which a majority of players agree on a position) and crisis Nash equilibria (in which no overall position exists). Recall
136 Chapter 5. A Comparison to Game Theory that nis odd. If n>3, then there is a unique crisis equilibrium in which every agent chooses his own peak. No other crisis equilibria exist since, if the outcome of the game is a crisis, then every agent chooses his peak, because otherwise any agent could profitably deviate to his peak, whether or not that results in harmony. If n=3, then a crisis equilibrium does not exist since any agent can deviate to one of the other two positions and thus, avoid a crisis. In this game, the notion of a non-crisis Nash equilibrium is identical to that of the social equilibrium in Debreu (1952)’s model of generalized games (see Tóbiás (2022) for a review of conditions guaranteeing its existence). A social equilibrium is a profile of actions in Fsuch that every player’s action is a best response from among the set of actions that are available to him given the other players’ actions. In other words, the profile after the deviation must be in F. Formally, (xi)∈Fis a social equilibrium if for each i, the action xiis optimal for ifrom among all the actions tisuch that (ti,x−i)∈F. The difference between the Nash and Debreu formulations is purely semantic: every player is either not interested in moving from a non-crisis profile to a crisis profile (in the Nash formulation) or is not even allowed to do so (in Debreu’s formulation). All profiles in which a bare majority of exactly τ= (n+1)/2 agents choose the same position (whatever it is), while the rest choose their peaks, are noncrisis Nash equilibria. These equilibria can be extremely unnatural in that the coalition which supports them does not have anything to do with the position being supported. In particular, there are non-crisis Nash equilibria for any overall position, even extreme ones that are outside of [L,R], and the agents supporting the overall position need not be those whose peaks are closest to it. We will now see that there are no other non-crisis Nash equilibria. Proposition 5.8: Nash Equilibrium in the Voting Game If n≥5, then the set of non-crisis Nash equilibria in the voting game consists of all profiles for which there is a position chosen by exactly τ agents while the rest choose their peaks.
5.11 Comparing our Approaches with Nash Equilibrium 137 Proof: These are Nash equilibria: No agent at the majority position can deviate profitably since, if he did so, then a crisis would ensue because his former position would no longer be a majority position and neither would his new position (all other agents are choosing their peaks which are distinct, so any new position would have at most two agents, but n≥5). All other agents are at their first-best, they choose their peak, and no crisis occurs. Therefore, they do not want to deviate. To see that there are no other non-crisis Nash equilibria, consider a Nash equilibrium in which at least τagents choose a common position t. An agent who does not choose tis not critical in maintaining harmony and therefore, must be at his peak. If strictly more than τagents choose t, then at least one of them is not at his peak and could deviate profitably. Comment: In Richter and Rubinstein (2021), we conducted similar comparisons and reached similar conclusions regarding other conditions for “holding the group together”: (i) a consensus among a super majority of agents, (ii) all positions are sufficiently close to the median position, or (iii) all positions are sufficiently close to the average position. 5.11 Comparing our Approaches with Nash Equilibrium The above analysis clarifies the significant differences between the convex Y- equilibrium, the biased preferences equilibrium, and the Nash equilibrium of the above political game. For the convex Y-equilibrium concept, Mis the only overall position. For the biased preferences equilibrium concept, typically only the extreme positions, −1 and 1, are overall positions. In contrast, for the Nash equilibrium concept, all positions, even those outside the range [L,R], are overall positions. Furthermore, a convex Y-equilibrium is “monotonic” in the sense that if agent i’s ideal position is to the left of j’s then his chosen position
138 Chapter 5. A Comparison to Game Theory is weakly to the left of j’s. In contrast, there are always non-monotonic Nash equilibria. The biased preferences equilibrium case is less clear: the existence of a non-monotonic equilibrium depends on the underlying utility functions. Notice that the Nash equilibria require a high degree of coordination between the agents. In contrast, the Y-equilibrium and biased preferences equilibrium concepts only require that agents know either the social restrictions or biases, but not the behaviour of others. This is like the marketplace where individuals only need to know prices, but not other agents’ actions. Let us emphasise: we are not saying that the standard game-theoretical approach is “wrong”, nor do we insist that the Y-equilibrium or biased preferences equilibrium approaches are “right”. Rather, and as already mentioned, we are suggesting that the reader not automatically apply Nashequilibrium-like concepts but instead considers alternative solution concepts in the spirit of those described in this book.
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