No Prices No Games!: Four Economic Models
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Richter, Michael; Rubinstein, Ariel Book No Prices No Games!: Four Economic Models Provided in Cooperation with: Open Book Publishers Suggested Citation: Richter, Michael; Rubinstein, Ariel (2024) : No Prices No Games!: Four Economic Models, ISBN 978-1-80511-310-2, Open Book Publishers, Cambridge, https://doi.org/10.11647/OBP.0404 This Version is available at: https://hdl.handle.net/10419/305343 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
NO PRICES NO GAMES! FOUR ECONOMIC MODELS 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!. Cambridge, UK: Open Book Publishers, 2024, https://doi.org/10.11647/OBP.0404 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. Digital material and resources associated with this volume are available at https:// doi.org/10.11647/OBP.0404#resources. 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 Cover image: Ariel Rubinstein Cover concept: Michael Richter and Ariel Rubinstein Cover design: Jeevanjot Kaur Nagpal
Contents Personal Note vii Notation and Terminology ix 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 46 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 76 3.5 A First Welfare Theorem 79 3.6 A Second Welfare Theorem 81 3.7 Primitive Equilibrium – Examples 83 3.8 Initial Status Equilibrium 85 4 Biased Preferences Equilibrium 91 4.1 The Economy and the Equilibrium Concept 92 4.2 The Give-and-Take Economy 98 4.3 The Fixed-Prices Exchange Economy 100 4.4 Housing-Type Economies 104 5 A Comparison to Game Theory 111 5.1 The Matching Economy 112 5.2 The Jungle Equilibrium 115 5.3 Restricting Partnerships: Pairwise Y-equilibrium 120 5.4 Prestige by Partner: Status Equilibrium 122 5.5 Prestige by Self: Initial Status Equilibrium 124 5.6 A Comparison of Approaches 127 5.7 The Majority Voting Economy 127 5.8 Convex Y-equilibrium 128 5.9 Biased Preferences Equilibrium 130 5.10 The Majority Voting Game and Nash Equilibrium 131 5.11 Comparing our Approaches with Nash Equilibrium 133 References 134
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. This project began just over a decade ago and developed from a series of papers, most of which we wrote jointly. 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. We wish to acknowledge generous assistance from two outstanding individuals: Martin Osborne, who was kind, as always, and shared with us the format of the book which he originally designed (so, if you are happy with the format, you should thank him) and Áron Tóbiás, who contributed so much of his time to carefully reviewing a draft of the book and saved us (and you) from a large number of errors. We are also grateful to Tuval Danenberg for his comments. MR: I am grateful to the support of my wife, Emel Yildirim-Richter, without whom this project would never have been completed. URLs: https://mrichter.co and https://arielrubinstein.tau.ac.il
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).
1Equilibrium in the Jungle The standard economic approach treats economic activity as voluntary: all involved parties are doing whatever they do of their own free will. When analyzed using the competitive equilibrium approach, economic agents operate within bounds set by a price system that they take as given, but their decisions are free — no one forces them to act. When analyzed using the gametheoretical approach, agents behave strategically, and in equilibrium they best respond to correct predictions about the other agents’ behavior, and, again, no one can force anyone to take a particular action. However, life is not just a series of voluntary actions. An agent (or a group of agents) might use power to seize assets from others or to force others to do things against their will. Resources are often transferred from one agent to another based on the exercise of power, rather than due to the satisfaction of mutual wants. While an agent can use power to force another to behave against his best interests, there is often no need to actually use power since the mere threat of doing so can be sufficient to persuade a weaker agent to give in. Economic Theory typically ignores the use of power as a driver of social activity. In the words of Hirshleifer (1994) (see also Bowles and Gintis (1992) and Grossman (1995) who express similar sentiments): ... the mainline Marshallian tradition has ... almost entirely overlooked what I will call the dark side of the force — to wit, crime, war, and politics. ... Appropriating, grabbing, confiscating what you want — and, on the flip side, defending, protecting, sequestering what you already have — that’s economic activity too. As the title of the book promises, we consider economic interactions that are harmonized without the emergence of a price system or the use of strategic ©2024 Michael Richter and Ariel Rubinstein, CC BY-NC-ND 4.0 https://doi.org/10.11647/OBP.0404.01
20 Chapter 1. Equilibrium in the Jungle 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 since if (yi)∈F Pareto dominates (xi)then pyi≥pxifor all iwith strict inequality for any agent ifor whom yiixiand thus Σi∈Npyi>Σi∈Npxiwhich contradicts the fact that the two sums must be 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
1.4 Comments on the Jungle Equilibrium 25 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 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.
26 Chapter 1. Equilibrium in the Jungle 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 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
1.4 Comments on the Jungle Equilibrium 27 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. 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 xif he wishes to do so. 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 the two-sided matching problem between Nand Xwhere each agent i∈Nhas the preference %iover Xand each house x∈Xhas the preference relation Bxover N. An assignment (xi)is pairwise stable if there is no pair iand xjsuch that iprefers xjover xi(xjixi) and xj“prefers” iover j(iBxjj). Therefore, an assignment 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.
28 Chapter 1. Equilibrium in the Jungle 1.5 The Division Jungle We now apply the jungle concept to a version of the division economy. To the definition of a division economy from Chapter 0, we add a profile (Xi)i∈Nof personal consumption sets, which represent bounds on each agent’s ability to consume. These sets can be thought of as either physical limits on what a person can consume or what possessions he can protect. Note that, in the housing economy, there is an implicit assumption of a similar nature, namely that an agent can hold only one house. The following is the formal definition of a jungle division economy (throughout, when comparing bundles, the notation x≤ymeans that xk≤ykfor every commodity k): Definition: Jungle Division Economy Ajungle division economy is a tuple ‹N,(Xi)i∈N,(%i)i∈N,F,B›where: •N={1,...,n}is the set of agents. •Xi⊆RK +is agent i’s personal consumption set in a K-commodity world. The sets Xiare assumed to be compact, convex, and satisfy free disposal (that is, if xi∈Xi,y∈RK +and y≤xi, then y∈Xi). •%iare preferences over Xiand assumed to satisfy continuity, strict monotonicity, and strict convexity. •Fis the set of all profiles of bundles (xi)such that: (i) xi∈Xifor all i, and (ii) Σi∈Nxi≤ewhere e∈RK +is an aggregate bundle available for distribution among the agents. •is a strict power ordering over N. Given a profile (xi), denote the “leftover” bundle e−Σi∈Nxias x0. We now turn to modifying the definition of a jungle equilibrium to fit the division jungle. There are (at least) two possible definitions that coincide with that of the housing economy. The first is a strong jungle equilibrium which
1.5 The Division Jungle 29 is a feasible profile such that no agent can assemble a preferable bundle by combining his own bundle with all bundles held by weaker agents and the leftover bundle. By this definition, the stability of a profile is disturbed by the possibility that an agent can attack more than one weaker agent. The second definition is a weak jungle equilibrium, which is a feasible profile such that no agent can assemble a preferable bundle by combining his own bundle with one other that is either held by a weaker agent or is the leftover bundle. Formally: Definition: Strong Jungle Equilibrium Astrong jungle equilibrium is a feasible profile (xi)with the property that there is no agent iand bundle yi∈Xisuch that: (i) yiixi. (ii) yi≤xi+ ΣiBjxj+x0(the agent takes from weaker agents and from the leftover bundle and potentially disposes of some of his possessions). Definition: Weak Jungle Equilibrium Aweak jungle equilibrium is a feasible profile (xi)with the property that there is no agent iand bundle yi∈Xisuch that: (i) yiixi. (ii) Either (a) or (b) holds. (a) yi≤xi+xjfor some jfor whom iBj(the agent steals from a single weaker agent and then may dispose of some of his possessions); or (b) yi≤xi+x0(the agent takes from the leftover bundle and then may dispose of some of his possessions). Note that the above definitions use inequalities rather than equalities. This is because, when a stronger agent seizes other resources, he might be put outside of his consumption set and, thus, needs either to take less or to dispose of some goods in order to remain in his consumption set. Obviously, any strong jungle equilibrium is also a weak jungle equilibrium.
36 Chapter 1. Equilibrium in the Jungle 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, ©2024 Michael Richter and Ariel Rubinstein, CC BY-NC-ND 4.0 https://doi.org/10.11647/OBP.0404.02
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. We now proceed to the formal definition of the equilibrium notion.
2.1 The Y-Equilibrium Concept 39 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. 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.
40 Chapter 2. The Permissible and the Forbidden 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. 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
2.1 The Y-Equilibrium Concept 41 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 Yequilibrium since the permissible set in any para-equilibrium is a subset of [0,1/n]. 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
42 Chapter 2. The Permissible and the Forbidden (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. 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.
2.2 Y-Equilibrium, Pareto Optimality, and Envy-Freeness 43 (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 there is a feasible envy-free profile (zi)that Pareto-dominates (yi). The configuration ‹Y∪{z1,...,zn},(zi)›is a para-equilibrium (since zi%izj for all i,j, and zi%iyi%iyfor all iand y∈Y). By Pareto dominance, ziiyifor at least one agent iand therefore zi/∈Y. Thus, Y∪{z1,...,zn} is a strict superset of Y, contradicting the definition of Y-equilibrium.
44 Chapter 2. The Permissible and the Forbidden In the other direction, let (yi)be Pareto-optimal among the feasible envy-free profiles. Let Ybe the set of all elements in this profile plus any element which is weakly inferior to yifor every agent i, namely, Y=Si{yi}∪{x|for all i,yi%ix}. The configuration ‹Y,(yi)›is a para-equilibrium. In order to show that it is also a Y-equilibrium, we need to invalidate the existence of a para-equilibrium ‹Z,(zi)›for which Z)Y. If it exists, then, zi%iyifor all iand (zi)is envy-free. Let x∈Z−Y. By the definition of Y, there is an agent jfor whom xjyj and, consequently, zj%jxjyj. Therefore, (zi)is a feasible envyfree profile that Pareto-dominates (yi), contradicting (yi) being Paretooptimal among the feasible envy-free profiles. Thus, no such paraequilibrium ‹Z,(zi)›exists and, therefore, ‹Y,(yi)›is a Y-equilibrium. We do not take overall Pareto optimality as a necessary condition for the plausibility or desirability of a solution concept. Still, a natural question is: What condition guarantees that any Y-equilibrium outcome is overall Paretooptimal (and not just among the envy-free profiles)? One such condition is the imitation property:Fsatisfies the imitation property if, whenever a profile is in F, so is any profile for which one agent adopts the alternative chosen by another agent instead of his own. That is, for any (ai)∈Fand any i,j∈N, the profile where aiis replaced with ajis also in F. An example where the imitation property holds is the stay close economy (described in Chapter 0) since, if one agent adopts another’s position, the maximal distance between any two agents does not increase. Proposition 2.2: The Imitation Property and Pareto Optimality Assume that Fsatisfies the imitation property. Then, a profile is a Yequilibrium outcome if and only if it is overall Pareto optimal.
2.3 Euclidean Economies 45 Proof: Let ‹Y,(yi)›be a Y-equilibrium. Assume by contradiction that there is a feasible profile (zi)which Pareto-dominates (yi). We construct a profile (xi)as follows: Assign x1, a %1-maximal alternative from {z1,...,zN}, to agent 1. Assign x2, a %2-maximal alternative from {x1,z2,...,zN}, to agent 2, and so on. In this construction, the profile selected at each stage is feasible (due to the imitation property) and xi%izifor all i. Furthermore, for every agent i, the alternative xiis %i-maximal from {x1,...,xi−1,zi,...,zN}⊇{x1,...,xN}. Thus, (xi)is feasible and envyfree. It weakly Pareto-dominates (zi)and thus Pareto-dominates (yi), contradicting Proposition 2.1. The other direction follows immediately from Proposition 2.1 because, under the imitation condition on F, every Pareto-optimal profile is envy-free and, therefore, is also Pareto-optimal among the envy-free allocations. 2.3 Euclidean Economies In many common economic models, such as Walrasian economies, the set of alternatives is taken to be a subset of a Euclidean space with standard closedness, convexity, and differentiability restrictions on the alternatives, the preference relations, and the feasibility set. We now consider our framework in a Euclidean setting. Definition: Euclidean Economy AEuclidean economy is an economy ‹N,X,(%i)i∈N,F›such that: (i) The set Xis a closed subset of some Euclidean space. (ii) For each i, the preferences %iare continuous. (iii) The feasibility set Fis anonymous (closed under permutations), compact, and contains at least one constant profile.
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=y1and z26=y2. Given the agents’ indifference curves, in z1agent 1 receives more of good 1 and less of good 2 than in y1. Furthermore, agent 1 must not prefer another bundle on the line segment between z1and the corner (2,2). This means that the slope between these two points has to be at least −1.25 or, in other words, z1 2−2≥−1.25z1 1+2.5, which implies z1 2+z1 1≥4.5 −0.25z1 1>3 where the last inequality is due to 3 ≥e1≥z1 1. Likewise, agent 2’s total demands are greater than 3, and such demands are infeasible. 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
2.9 The Give-and-Take Economy 59 the set of bundles (satisfying the standard division economy assumptions). 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.
60 Chapter 2. The Permissible and the Forbidden Claim: A Characterization of the Convex Y-equilibrium Consider a give-and-take economy with Σpeaki>0. There is a unique convex Y-equilibrium ‹Y,(yi)›. The set Ytakes the form [−1,m]for some m>0, and (yi)is Pareto optimal. Proof: Consider a permissible set of the form [−1,m]. If m<0, then all agents must give. If m≥0, then every agent who wants to give will select his peak, and every agent who wants to take is either at his peak or has a peak to the right of mand makes do with taking m. Let D(m)be the sum of all agents’ choices given the permissible set [−1,m]. The function Dis continuous, strictly increasing for any msmaller than max{peaki}, and is constant with value Σipeaki>0 for any larger m. In particular, D(0)≤0 and D(1)>0. Thus, there is a unique m∗≥0 for which D(m∗) = 0. The permissible set [−1,m∗], together with the agents’ optimal choices from that set, constitutes a convex para-equilibrium. It is also a convex Y-equilibrium because there is no convex para-equilibrium with a larger permissible set. If there were, it would have the form [−1,m] where m>m∗, but then agents would take too much (since D(m)>0). The profile (yi)is Pareto optimal: For each i,yiis at or to the left of his peak. Thus, if (zi)∈FPareto-dominates (yi), then yi≤zifor all iwith at least one strict inequality, thus 0 = Σyi<Σzi, violating feasibility. To prove uniqueness of the convex equilibrium, it remains to be shown that any closed convex para-equilibrium permissible set [x,y]is included in [−1,m∗]. In order for the social fund to be balanced, it must be that x≤0≤y. In equilibrium, agents who wish to give will do so at either their peak or at xif peaki<x. Therefore, the total giving in [x,y] is not more than that in [−1,m∗]. Since the social fund is balanced, the total taking in [x,y]must also be less than or equal to that in [−1,m∗], and therefore y≤m∗. Thus, [x,y]⊆[−1,m∗].
2.10 The Stay Close Economy 61 Comment: For this economy, while convex Y-equilibria are Pareto-optimal, a Y-equilibrium outcome need not be. A detailed example appears in Richter and Rubinstein (2020). The essence of the example is as follows: Let X={−2,−1,0,1,2}and n=2. The agents’ “convex” preference relations are 1101−11−212 and 1 22202−12−2. The “convex” permissible set Y={−2,−1,0}, together with the profile y1=y2=0, is a “convex” Y-equilibrium with a Pareto-optimal outcome. However, it is easy to verify that the non-convex permissible set Y={−2,2}with the profile y1=−2,y2=2 is a Y-equilibrium whose outcome is Pareto dominated by z1=−1, z2=1. 2.10 The Stay Close Economy The stay close economy is a convex Euclidean economy in which Xis a closed convex set of locations and Fis the set of profiles for which the distance between any two agents is at most d∗. That is, each member of the group chooses a position (for example, a political stance or a geographical location), and the group’s survival requires that the members “stay close” to each other. As always, each agent has strictly convex preferences for his own location without regard to the location of others. The potential source of conflict is that the group members have a diverse set of ideal locations which fails the closeness requirement. Note that the set Fsatisfies the imitation condition defined in Section 2.3. When d∗=0, this economy is called a consensus economy. In a centralized society, the authorities can coerce agents into occupying locations that guarantee survival. In a market, members would have to pay each other to stay close by. The Y-equilibrium idea is that there are norms that determine the borders of the permissible locations and strike a balance between societal harmony and individual liberty. Each agent chooses his most preferred location within the borders, and the outcome is that they all live close enough to one another. The borders are maximally liberal in the sense that if the borders are enlarged in any way, then the resulting individual choices would not be “close enough”.
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 Rmonotonic). 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 extends R, it follows that ajband therefore, ajcand so aDc. Therefore, the cycle can be shortened.
3.3 A Detour: Convex Preferences 71 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. 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.
72 Chapter 3. Status and Indoctrination 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 atb(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. (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”.
3.3 A Detour: Convex Preferences 73 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 an 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 generalizes standard convexity for Euclidean spaces, as will be shown later. (d) It does not require any algebraic structure. 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.
74 Chapter 3. Status and Indoctrination Suppose %is Λ-strictly convex. Since the preferences are continuous and Xis compact, there is a %-maximal element. It is unique since if there are two %-maximal elements y<z, and xis between yand z, then yDLxand zDRxand both yand zare weakly preferred to x. Therefore, there is no critical direction at x. 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. The next example shows that, for continuous preferences, the Λ-convexity notion used here generalizes the standard notion of convex preferences on Euclidean spaces. 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. Proof: Assume %is convex by the standard definition. That is, for every b∈X, the set U(b) = {z|zb}is convex. Since %is continuous, by the separating hyperplane theorem, there exists ≥v∈Ψsuch that for every x∈U(b)it holds that x>vb. That is, ≥vis a critical direction. Assume %is Ψ-convex. Let a,cbe elements in Xsuch that a,cb, and let zbe 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%c, and thus zb.
3.3 A Detour: Convex Preferences 75 Construction of Convex Preferences: For a finite set X, if all orderings in Λ are strict, then the following procedure builds a Λ-convex preference relation: Take an alternative x1which is at the bottom of one of the primitive orderings, and place it at the bottom of %. Then, let x2be an alternative at the bottom of X−{x1}with respect to one of the primitive orderings, and place it (strongly or weakly) above x1. Continue this procedure until all alternatives are exhausted. The constructed preference is Λ-convex since the position of each b∈Xin % was determined when bwas at the bottom of some primitive ordering, which is then a critical direction at bsince any strictly preferred alternative is ranked strictly higher than bby that ordering. If Xis finite and all orderings in Λare strict, then every Λ-convex preference relation %can be constructed by the procedure described above (see Richter and Rubinstein (2019)): To apply the construction to obtain %, at every stage we must identify an alternative and a primitive ordering Dso that the alternative is both %-minimal and D-minimal from among the remaining alternatives. To start, pick x∈Xwhich is %-minimal, and let D∈Λbe a critical direction at x. If xis D-minimal, then set x1=x. If not, then pick ywhich is minimal according to the same D. The alternative yis also %-minimal since Dis a critical direction at xand xDy, and then set x1=y. Continue inductively with the remaining alternatives. Utility Representation: We say that a preference relation %over Xhas a Λmaxmin representation if there is a profile of functions (UD)D∈Λsuch that for every D∈Λthe function UDis a utility representation of Dand the function U(x) = minDUD(x)is well-defined and represents %. If Λis finite and %has a Λ-maxmin representation (UD), then %is Λ-convex: For any b∈X, take the ordering D∈Λfor whichUD(b)is minimal. The ordering Dis a critical direction at bbecause bDximplies UD(b)≥UD(x)and thus, U(b)≥U(x)and b%x. In Richter and Rubinstein (2019), it is shown that for finite X, the converse is also true: any Λ-strictly convex preference relation has aΛ-maxmin representation.
76 Chapter 3. Status and Indoctrination The existence of such a representation means that we can identify every alternative in the set Xby a vector of numbers in RΛsuch that: (i) for every primitive ordering, the values that are attached to the elements in Xat the corresponding coordinate are consistent with that primitive ordering’s ranking; and (ii) the preferences are represented by the minimum value attached to an alternative across the different dimensions. 3.4 Primitive Equilibrium In the canonical consumer model, the set of alternatives (bundles) is a subset of a Euclidean space and the following holds: (i) Agents have standard convex preferences. (ii) The “more expensive than” ordering is induced by a linear price system. In the language of this chapter: (i) Agents have Ψ-convex preferences where Ψis the set of all algebraic linear orderings (as shown in Section 3.3). (ii) The “more expensive than” ordering ≥pon the set of alternatives (defined by x≥pyif p∙x≥p∙y) is a member of Ψ. An important point is that the same set of primitive orderings appears in (i) and (ii) above. This suggests two new definitions. First, we enrich the notion of an economy with a set of primitives orderings Λand require that all agents’ preference relations are Λ-convex. We refer to such an economy by the term convex economy. Second, we refine the status equilibrium notion and require that the public ordering is one of the primitive orderings in Λ. We refer to such an equilibrium as a primitive equilibrium. Formally:
3.4 Primitive Equilibrium 77 Definition: Convex Economy Aconvex economy is a tuple ‹N,X,(%i)i∈N,F,Λ›where ‹N,X,(%i)i∈N,F› is an economy, Λis a set of primitive orderings over X, and all preferences are Λ-convex. Definition: Primitive Equilibrium Let ‹N,X,(%i)i∈N,F,Λ›be a convex economy. A primitive equilibrium is a status equilibrium ‹D,(xi)i∈N›where D∈Λ. Obviously, any primitive equilibrium is a status equilibrium, and when Λis the set of all orderings, any status equilibrium is a primitive equilibrium. Example: The Give-and-Take Convex Economy We return to the give-and-take economy. Recall that X= [−1,1]and F is the set of all profiles that sum up to 0. Let Λconsist of the two natural orderings: the rightist DR(which favours taking) and the leftist DL(which favours giving). Assume that every agent iholds continuous Λ-strictly convex preferences (with a single peak denoted by peaki). When Σipeaki=0, there is no conflict of interest in the economy and either primitive ordering, together with all agents choosing their peaks, is a primitive equilibrium. In fact, these are the only primitive equilibria (if ‹DL,(xi)i∈N›is an equilibrium, then peaki≤xifor all iand therefore, xi=peakifor all i). A more interesting case is Σipeaki>0 where agents wish to take more than they wish to give. Let F≤be the set of all feasible profiles with all agents at or to the left of their peaks. We now verify that F≤is equal to the set of all Pareto-optimal profiles. Any (xi)∈F≤is Pareto-optimal because any profile (yi)that Pareto-dominates it must rank xi≤yifor all i, with at least one strict inequality; however, such a profile is infeasible because 0 = Σxi<Σyi. On the other hand, if (xi)is feasible and not
84 Chapter 3. Status and Indoctrination utility function v(Θ) = Σz∈Θv(z). That is, v(Θ) is the sum of the vvalues attached to the individual items in the set Θ. Let Λbe the set of such strict orderings. It turns out (see Richter and Rubinstein (2019)) that 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 of goods 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 set allocation 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. 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).
3.8 Initial Status Equilibrium 85 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 set allocation 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 the same Λ-convex preferences %∗satisfying that the sets ac and bd are %∗-superior to ab and cd . Then, ‹P=%∗,(ab,cd )›is a status equilibrium that is not Pareto-optimal. 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
86 Chapter 3. Status and Indoctrination 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. 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 of the two agents 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.
3.8 Initial Status Equilibrium 87 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. 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
88 Chapter 3. Status and Indoctrination 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. 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
3.8 Initial Status Equilibrium 89 •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. 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, zPy 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.
90 Chapter 3. Status and Indoctrination 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 ordering P−1, every agent ichooses an alternative xi≤eiand the sum of the chosen alternatives is non-positive since Σixi≤Σiei=0. Thus, by the continuity of the agents’ choices in z, there is a z∗∈[−1,1]for which the sum of the chosen elements is 0. The ordering Pz∗together with the profile of optimal choices from the corresponding budget sets is an initial status equilibrium.
4Biased Preferences Equilibrium In any economy, the core tension is between agents’ wants and societal feasibility, and an equilibrium notion finds a balance between them. In Chapters 2 and 3, we investigated equilibrium notions that invoke various social mechanisms to achieve that balance: norms emerge that affect agents’ opportunity sets such that if every agent optimizes his preference relation, then the profile of optimal choices is feasible. This chapter takes a different approach. We follow Rubinstein and Wolinksy (2022) who propose a solution concept which captures a different social mechanism that can resolve the fundamental conflict between wants and feasibility: agents’ preference relations are systematically biased. The bias does not affect the agents’ opportunity sets but, rather, their preferences, which are systematically biased in such a way that the profile of agents’ biased optimal choices is feasible. Recall Aesop’s classic fable (translation from Gibbs (2002)): Driven by hunger, a fox tried to reach some grapes hanging high on the vine but was unable to, although he leaped with all his strength. As he went away, the fox remarked “Oh, you aren’t even ripe yet! I don’t need any sour grapes.” In this fable, there is one agent, the Fox, and two alternatives, “picking the grapes” and “not picking the grapes”. The economic problem is that the Fox initially prefers the former alternative but only the latter alternative is feasible. The conflict in the fable is resolved not by restricting the Fox’s opportunities but, rather, by biasing his preferences so that he now prefers not to pick the grapes (which in his mind are turned to “sour grapes”). ©2024 Michael Richter and Ariel Rubinstein, CC BY-NC-ND 4.0 https://doi.org/10.11647/OBP.0404.04
92 Chapter 4. Biased Preferences Equilibrium Preference biases are not just a matter for fables. Introspection tells us that feasibility often influences our preferences in everyday life. We often assign greater value to what we can obtain (such as being an economist) and less to what we cannot (such as being a mathematician). However, we do not deny that there are also circumstances where the opposite is true, and the more unobtainable something is the more desirable it becomes. The Fox biased his preferences, and harmony was achieved. Similarly, we envision biases as a mechanism for bringing harmony to a multi-agent economy. These biases, like prices, will be systematic and apply uniformly to all agents. Every agent’s final preferences are determined by both the commonly shared bias and his initial preferences. Thus, in contrast to a competitive equilibrium where prices affect choice sets and preferences are fixed, in a biased preferences equilibrium biases affect preferences and choice sets are fixed. This illustrates the dual roles played by prices and preferences in standard economic settings. Note the difference between this chapter’s approach and the one taken by other economic models. In some of those models, the change in preferences is a side effect of an agent’s action (for example, smoking may influence the desire to smoke in the future, as modelled by Becker and Murphy (1988)). In others, the change in preferences is the outcome of a deliberate action by an interested party (for example, advertisers seek to influence customers’ preferences to their own advantage, as modelled by Bagwell (2007)). By contrast, this chapter models social situations in which preferences invisibly respond to feasibility pressures, just as price adjustments achieve harmony in a competitive market. 4.1 The Economy and the Equilibrium Concept In this chapter, the notion of an economy is modified to accommodate modelling systematic preference biases.
4.1 The Economy and the Equilibrium Concept 93 Definition: An Economy An economy is a tuple ‹N,(Xi)i∈N,K,((ui k)k∈K)i∈N,F›where: •For each agent i,Xiis his fixed personal choice set. •The set Kis a set of considerations common to all agents. •For each agent i,(ui k)k∈Kis a tuple of consideration functions over Xi such that i’s utility function over Xiis Σkui k(x). •The set of feasible profiles, F, is a subset of Πi∈NXi. This definition modifies our notion of an economy in two ways. First, and less importantly, different agents can have different choice sets. This allows for modelling a variety of settings. For example, an exchange economy with a set of goods K, a fixed price vector p, and initial endowment profile (ei) can be modelled by setting Xi={x∈RK +|p∙x=p∙ei}and F={(xi)|Σxi= Σei}. Another example is a two-sided matching market with two equally-sized populations Aand B. This can be modelled by setting Xi=Bfor any i∈A and Xj=Afor any j∈B, while Fis the set of all profiles (xi)for which for every i,j,xi=jimplies xj=i. The second and more important modification of the original definition of an economy is the use of a different notion of preferences. Rather than specifying an ordinal preference relation over the set of alternatives, we use the following type of utility function that enables us to model systematic biases. All agents share the same set of considerations K. Each agent iis characterized not by an ordinal preference relation, but by a vector of consideration functions ui= (ui k)k∈Kwhere ui k(x)represents the impact of consideration kon his overall evaluation of the alternative x. The consideration functions are not constant and, where applicable, are differentiable. Agent i’s overall utility from an alternative xis the sum of the utilities obtained from those considerations, i.e. Σk∈Kui k(x). A preference bias is modelled as a systematic and uniform change in the weights placed on the considerations. Let Λ = RK ++ be the set of biases. A bias
100 Chapter 4. Biased Preferences Equilibrium 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. 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
4.3 The Fixed-Prices Exchange Economy 101 consideration function ui kat 0 is infinity. Let (x1,x2)be a Pareto-optimal feasible 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, ‹λ= (1,1),(xi)›is a biased preferences equilibrium. If μ6=p1/p2, then the bias λ= (p1,p2μ)modifies both preferences so that the MRS1,2 of the biased preferences of each iat xi is μλ1/λ2=p1/p2. Thus, ‹λ,(xi)›is a biased preferences equilibrium. (x1,x2) p1 p2 u1u1 u2 u2 (e1,e2) λ(x1,x2) p1 p2 T(u1,λ) T(u2,λ)(e1,e2) Figure 4.1 Equilibrium in an Edgeworth Box For the case of more than two goods and any number of agents, 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 lthat he consumes and any other good k, it holds that MRSk,lat xiis bounded from above by qk/ql. Consequently, by setting λ= (pk/qk)k∈K, it holds that for agent i’s biased preferences and any good lthat he consumes, the MRSk,lat xi is bounded from above by pk/pl=λkqk/λlql(the bound is an equality if xi k>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, ‹λ,(xi)›is a biased preferences equilibrium. (ii) Let ‹λ,(xi)›be a biased preferences equilibrium. Then, for each agent iand any good lthat he consumes, the MRSk,lof the biased preferences T(λ,ui)at xiis bounded from above by pk/pl. Therefore, the MRSk,lof
102 Chapter 4. Biased Preferences Equilibrium his initial preferences at xiis bounded from above by pk/λk pl/λl. Thus, (xi) is a Walrasian equilibrium outcome in the unbiased economy with price vector pk/λkand initial endowment (xi). Therefore, by the standard First Welfare Theorem, (xi)is pre-Pareto optimal. 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(λ,u1)T(λ,u2)u1u2u1u2T(λ,u1)T(λ,u2) p1 p2 u1 u1 u2 u2 T(u1,λ) T(λ,u1) T(u2,λ) T(λ,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.
4.3 The Fixed-Prices Exchange Economy 103 (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. 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=λ1/(αiλ2). 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=λ1/(α2λ2). 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.
104 Chapter 4. Biased Preferences Equilibrium 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 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 λ= (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 λ, 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 actually 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 (λh), an agent iderives utility λhvi(h)from house h. Example: The following table presents the consideration function values in a housing economy with two agents.
4.4 Housing-Type Economies 105 h1h2 v1(h)4 3 v2(h)3 1 Table 4.2 House utilities 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
106 Chapter 4. Biased Preferences Equilibrium ln(vi(xi)) −pxi, and therefore, it is also a maximizer of vi(xi)/epxi. Thus, ‹(λh=1/eph)h∈H,(hi)i∈N›constitutes a biased preferences equilibrium. In the other direction, let ‹λ,(hi)›be a biased preferences equilibrium and (xi)be any other assignment. For each i,λhivi(hi)≥λxivi(xi)and therefore, Πiλhivi(hi)≥Πiλxivi(xi). Since Πiλhi= Πiλxi, it follows that Πivi(hi)≥Πivi(xi), that is, (hi)is Nash maximal. We proceed with two modifications to the housing economy: Example: The Partnership Economy As in Shapley and Shubik (1971), the agents are composed of two equallysized populations, Aand B. Each agent chooses a unique partner from the other population, that is, Xi=Bfor any i∈Aand Xj=Afor any j∈B. A profile is feasible if for every i,j, if ichooses j, then jchooses i. An agent i’s valuation of a partnership with jis vi(j)>0. Importantly, the ex-ante valuations are assumed to be symmetric, that is, vi(j) = vj(i), but the biased valuations might not be. Example: Let A={1,2}and B={3,4}. Table 4.3 presents the original valuations (left bi-matrix) and the equilibrium biased valuations (right bi-matrix). Each cell gives the values of iand jof being matched. The Nash-maximal matching is 1↔4 and 2↔3 (depicted), which is a biased preferences equilibrium with the bias λ= (2,1,2,1). 3 4 1 1,1 3,3 2 3,3 4,4 T(∙,λ) → 3 4 1 2,2 3,6 2 6,3 4,4 Table 4.3 A Biased Preferences Equilibrium Claim: In the partnership economy, the set of biased preferences equilibrium outcomes is the set of all Nash-maximal profiles.
4.4 Housing-Type Economies 107 Proof: Let (ai)be a Nash-maximal profile, that is, one that maximizes Σi∈Nln(vi(xi)) over F. Since the utility functions are symmetric, i.e. (vi(j) = vj(i)), the profile (ai)also maximizes both Σi∈Aln(vi(xi)) and Σi∈Bln(vi(xi)) over F. Taking the agents to be Aand the houses to be B,Shapley and Shubik (1971) showed that a price vector (pj)j∈Bexists, such that for every agent i∈A, the choice of aimaximizes ln(vi(j)) −pj over all j∈Xi=Band therefore, maximizes vi(j)/epjas well. Reversing roles, there is a price vector (pj)j∈Awith analogous optimality properties. Therefore, ‹(λj=1/epj)j∈N,(ai)i∈N›constitutes a biased preferences equilibrium. In the other direction, let ‹(λj)j∈N,(ai)i∈N›be a biased preferences equilibrium, and let (xi)∈F. For every i, it holds that λaivi(ai)≥ λxivi(xi). Therefore, Πi[λaivi(ai)] ≥Πi[λxivi(xi)]. Since Πiλai= Πiλxi, it follows that Πivi(ai)≥Πivi(xi). That is, (ai)is Nash maximal. The condition that the value of a match between any two agents is the same for both of them is sufficient for the A-Nash-maximal matching to be B-Nash-maximal as well. Without this condition, a biased preferences equilibrium may not exist: Example: Consider an assignment economy where A={1,2},B={3,4}, and 3 14, 1 42, 4 23, 2 31. No utility presentation of these preferences is consistent with the assumption that the value of a match is identical for both partners (since it requires that v1(3)>v1(4) = v4(1)> v4(2) = v2(4)>v2(3) = v3(2)>v3(1) = v1(3)). Suppose that 1↔3 is a match in a biased preferences equilibrium. Then, λ1> λ2(so that agent 3 chooses agent 1 over agent 2). But then, agent 4 will also choose agent 1, violating feasibility. Likewise, 1↔4 cannot be a match in a biased preferences equilibrium.
108 Chapter 4. Biased Preferences Equilibrium Example: A Production Economy In the production economy (related to Atakan et al. (2023)), there is a set of indivisible goods Kand two equally-sized groups of agents: consumers (C) and producers (P). Every i∈Cconsumes exactly one unit of a single good, that is Xi=K, and ui k>0 is consumer i’s utility from consuming good k(which he wishes to maximize). Every producer i∈P must produce exactly one unit of a single good, that is, Xi=K, and ci k>0 is producer i’s utility-cost from producing good k(which he wishes to minimize). The set Fconsists of all profiles satisfying that, for every good k, the number of its consumers is equal to the number of its producers. Note that this economy differs from the partnership economy in that consumers and producers choose a good rather than a partner and the biases are applied to the goods rather than to the agents. A bias vector λ= (λk)k∈Kalters consumer i’s utility vector from (ui k)k∈Kto (λkui k)k∈Kand producer i’s utility-cost vector from (ci k)k∈Kto (λkci k)k∈K. Thus, a bias λsimultaneously rescales both the consumers’ utility and the producers’ utility-costs of good kby the same factor λk. Thus, an increase in λkis analogous to that of a decrease in the price of good kin a regular exchange economy: it makes the good more desirable to buyers and less desirable to sellers. Underlying a bias could be some trait such as quality: a high bias, like a high quality level, makes the good more desirable to consumers and increases the utility-cost to produce it. The following claim again uses a Shapley and Shubik (1971)-style argument to characterize the biased preferences equilibrium profiles. It implies that they exist and are pre-Pareto optimal. Claim: In the production economy, the biased preferences equilibrium profiles are precisely the solutions of: max (xi)∈F Π i∈Cui xi Π i∈Pci xi (*)
4.4 Housing-Type Economies 109 Proof: Let ‹λ,(xi)›be a biased preferences equilibrium, and let (yi)∈F. It follows that λxiui xi≥λyiui yifor every i∈Cand λxici xi≤λyici yifor every i∈P. Combined with the equality Πi∈Cλzi= Πi∈Pλzi, which holds for all (zi)∈F, we conclude that: Π i∈Cui xi Π i∈Pci xi = Π i∈Cλxiui xi Π i∈Pλxici xi≥ Π i∈Cλyiui yi Π i∈Pλyici yi = Π i∈Cui yi Π i∈Pci yi and therefore (xi)is a solution of (*). In the other direction, let (xi)be a solution of (*). Let (xi k)be the allocation matrix with a row for each agent and a column for each good, where xi k=1 if ichooses kand xi k=0 otherwise. The matrix solves the following linear maximization problem: max (mi k) ΣkΣi∈C[ln(ui k)mi k]+ΣkΣi∈P[−ln(ci k)mi k] such that Σi∈Cmi k−Σi∈Pmi k=0∀k(μk) mi k≥0∀i,k(γi k) Σkmi k=1∀i(ψi) The above problem always has a solution which is a binary matrix (that is, xi k=0 or 1, for every i,k). To understand why, Birkhoff (1946) (and his extensions in Budish et al. (2013)) shows that any matrix of real numbers that satisfies the above constraints is a convex combination of binary matrices that also satisfy them. Since the target function is linear, any such binary matrix is also a solution to the linear programming problem. The constraints in the above optimization are labelled by their shadow values, which appear in the parenthesis to the right. Let λ= (eμk)k∈K. We will now verify that ‹λ,(xi)›is a biased preferences equilibrium.
116 Chapter 5. A Comparison to Game Theory In a J1-equilibrium pairing, it can be that an agent prefers another to his current partner. But, the agent is prohibited from making such an approach because the agent to be approached is stronger than him. This contrasts with pairwise stability where what prevents him from acting is that the agent to be approached will reject him. The following proposition compares the J1-equilibrium concept with that of pairwise stability. It is found that the J1-equilibrium concept is stricter: any J1-equilibrium outcome is pairwise stable (and therefore also Pareto-optimal). Since pairwise-stable pairings do not always exist, neither will J1-equilibria. Proposition 5.1: J1-equilibrium Properties (i) Every J1-equilibrium pairing is pairwise stable. (ii) A pairwise-stable pairing might not be a J1-equilibrium pairing. Proof: (i) Let ‹B,(xi)›be a J1-equilibrium. 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 pairing 1↔2 and 3↔4 is pairwise stable, but there is no agent who 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 (highlighted) that is not a J1-equilibrium outcome.
5.2 The Jungle Equilibrium 117 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 N2topranks 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 the jungle model (Chapter 1), the ability of one agent to take the house of another and, likewise, in a J1-equilibrium the ability of one agent to approach another, depends solely on the power relationship between the two agents. 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 two solution concepts take this into account. In a J2-equilibrium, an agent can approach another in a different pair only if he is stronger than both the desired agent and that agent’s partner. Definition: J2-Equilibrium AJ2-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 is more powerful than both j and j’s partner (that is, iBjand iBxj). Like the J1-equilibrium, the J2-equilibrium does not allow for pairings in which no agent gets his first best: in any J2-equilibrium, the strongest agent is matched with his most-preferred partner. Obviously, every J1-equilibrium is also a J2-equilibrium. We will now see that a J2-equilibrium always exists, unlike a J1-equilibrium.
118 Chapter 5. A Comparison to Game Theory Proposition 5.2: J2-equilibrium Properties (i) A J2-equilibrium always exists. (ii) Every J2-equilibrium pairing is Pareto-optimal. (iii) A Pareto-optimal pairing (even if it is pairwise stable) need not be a J2-equilibrium pairing. Proof: (i) Choose an arbitrary agent i1, and make him the strongest agent. Call his first-best partner j1, pair them together, and make j1the weakest agent. Continue in this manner to obtain a pairing in which every agent ikis paired with jk, who is ik’s favourite partner from N−{i1,j1,...,ik−1,jk−1}, and set the power relation to be i1Bi2B∙∙∙B j2Bj1. Thus, every ikis stronger than every jl. This procedure generates 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 any other agent because every other couple, ik↔jk, has at least one member who is stronger than him, namely ik. (ii) Let ‹B,(xi)›be a J2-equilibrium. Assume that (yi)Pareto-dominates (xi). Let jbe the strongest agent in D≡ {i|xi6=yi}. By Paretodominance, yjjxj(recall that preferences are strict). Agent yjand yj’s original partner xyjare both in D. Therefore, jyj,xyj, which violates ‹B,(xi)›being a J2-equilibrium. (iii) For the economy depicted in Table 5.2, the highlighted pairing {1↔2,3↔4}is pairwise stable but is not a J2-equilibrium pairing since no agent gets his first-best.
5.2 The Jungle Equilibrium 119 Example: The Common-ranking Two-sided Matching Economy Every mixed pairing (xi)is a J2-equilibrium pairing supported by assigning the power relations of agents in each side by the rank of their matches (that is, for every two members iand jfrom the same side assign iBjif xiis higher-ranked than xj). While every mixed pairing is part of a J2-equilibrium, not every power relation is. For example, for the case of four agents, there is no J2equilibrium 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. Finally, in a J3-equilibrium, an agent can force a partnership with jonly if he is stronger than j,j’s partner, and his own abandoned partner. Definition: J3-Equilibrium AJ3-equilibrium is a tuple ‹B,(xi)›for which there are no iand jsuch that jixiand iBj,xi,xj. Obviously, every J2-equilibrium is also a J3-equilibrium. The J3-equilibrium requires stronger conditions for an agent to be able to disturb society’s harmony, and we will see that every power relation is part of some J3equilibrium (unlike the J2-equilibrium case). Nonetheless, in terms of equilibrium pairings, the J2and J3-equilibrium notions are equivalent. Proposition 5.3: J3-equilibrium Properties (i) For every power relation B, there is a J3-equilibrium ‹B,(xi)›. (ii) The set of J3-equilibrium pairings is equal to the set of J2-equilibrium pairings (thus, every J3-equilibrium pairing is Pareto-optimal, though not every Pareto-optimal pairing is a J3-equilibrium pairing).
120 Chapter 5. A Comparison to Game Theory Proof: (i) Let be a strict ordering. We inductively construct a pairing (xi)for which ‹,(xi)›is a J3-equilibrium using a “generalized serial dictatorship” procedure: First, the B-strongest agent picks his mostpreferred partner and they are matched. In each subsequent step, the B-strongest remaining agent is matched with his most-preferred partner from among those remaining. By this procedure, half of the agents “make a choice” while the other half “are chosen”. Any agent who “makes a choice” can only prefer agents who match before him, i.e. those who are stronger than him or are paired with a stronger partner. Any agent who “is chosen” is neutralized because he is matched with a stronger partner. (ii) As mentioned, any J2-equilibrium is also a J3-equilibrium. We now show that any J3-equilibrium pairing is a J2-equilibrium pairing (perhaps with a different power relation). Consider a J3-equilibrium ‹B,(xi)›. In every couple, there is a stronger agent and a weaker one. Let Sbe the set of n/2 stronger agents and Wbe the set of n/2 weaker ones. Define a new power relation B0by preserving BonSand on Wand pushing all members of Wbelow all members of S. The tuple ‹B0,(xi)› is a J2-equilibrium: If not, then there would be iand jsuch that jixi and iB0xj,j. It must be that i∈Ssince iis B0-stronger than a pair of agents, j↔xj. Thus, iBxi. Since the power relation is preserved on S, iis -stronger than the -stronger agent in {j,xj}who is in S. Thus, ixi,j,xj, contradicting ‹B,(xi)›being a J3-equilibrium. 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, the social norm determines which pairs are permitted and which are forbidden.
5.3 Restricting Partnerships: Pairwise Y-equilibrium 121 Definition: Y-Equilibrium Let Mbe the set of all doubletons (sets of size 2). A para-Y-equilibrium is a tuple ‹Y,(xi)›where Y⊆Mand (xi)is a pairing such that, for every agent i,xiis i-maximal in {j|{i,j}∈Y}. A Y-equilibrium is a para-Yequilibrium 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 the Y-equilibrium notion of 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 also 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. This implies that, in term of outcomes, the Y-equilibrium notion is more permissive than pairwise stability or the J-equilibrium notions. In particular, it always exists. Proposition 5.4: Y-equilibrium and Pareto Optimality The set of Y-equilibrium pairings =The set of Pareto-optimal pairings. Proof: Given any pairing (xi), define L((xi)) to be the set of all doubletons {i,j} such that xi%ijand xj%ji. Notice that for every i, the doubleton {i,xi} is in L((xi)). Let ‹Y,(xi)›be a Y-equilibrium and (yi)be a pairing that Paretodominates (xi). Obviously, L((xi)) ⊇Y. Clearly, L((yi)) ⊇L((xi)). In fact, the inclusion is strict since at least one agent, say j, is strictly better off in (yi)and therefore the pair {j,yj}is in L((yi)) −L((xi)). The
122 Chapter 5. A Comparison to Game Theory tuple ‹L((yi)),(yi)›is a para-Y-equilibrium with a larger set of permissible pairs, which contradicts ‹Y,(xi)›being a Y-equilibrium. On the other hand, let (xi)be a Pareto-optimal pairing. We now show that the tuple ‹L((xi)),(xi)›is a Y-equilibrium. If not, then there is a paraY-equilibrium ‹Z,(yi)›with Z⊃L((xi)). All agents are weakly better off in (yi)than in (xi). The set Zcontains at least one pair {i,j}which is not in L((xi)). Without loss of generality, suppose that jixi. In that case, yi%ijixiand therefore, (yi)Pareto-dominates (xi). Example: The Common-ranking Two-sided Matching Economy Every mixed pairing is Pareto-optimal and therefore, is a Y-equilibrium pairing. The Y-equilibrium pairing {i1↔j1,i2↔j2,...}is supported by a maximally restricted permissible set which contains only the equilibrium matches (and all the matches between any two agents from the same side). 5.4 Prestige by Partner: Status Equilibrium We now turn to the status equilibrium concept discussed in Chapter 3 (referred to as an S-equilibrium in Richter and Rubinstein (2024)). Harmony is established by a status ordering of the agents that blocks an agent from approaching certain other agents. In a status equilibrium, agents are paired up, and no agent can approach any other agent who has a higher status than his current partner. The only agents he has the courage to approach are those with a (weakly) lower status than his own partner. An equilibrium is harmonious in that no agent can find a different partner who is both approachable and more desirable.
5.4 Prestige by Partner: Status Equilibrium 123 Definition: Status Equilibrium A status equilibrium is a tuple ‹P,(xi)›where (xi)is a pairing and Pis a weak ordering of the agents such that, for every agent i, there is no j such that jixiand xiPj . Status equilibrium pairings have the properties that at least one agent (the agent with the highest-ranked partner) gets a partner whom he most prefers, and if preferences are strict, then at most one agent gets his least-preferred partner (it can only be the agent with the lowest-ranked partner). We now establish some relationships between the status equilibrium concept and the J2-equilibrium, Pareto optimality, and pairwise stability. Proposition 5.5: Status Equilibrium Properties (i) Every status equilibrium pairing is a J2-equilibrium pairing and therefore, is Pareto-optimal. (ii) The notions of status equilibrium and pairwise-stability are distinct; it is possible for either notion to exist when the other one does not. Proof: (i) Let ‹P,(xi)›be a status equilibrium and break ties so that Pis strict. Define a power ranking by ranking agents according to the status of their partners: ijif xiPx j. The tuple ‹,(xi)›is a J2-equilibrium: If jixifor some i,j, then it must be that j Pxisince ‹,(xi)›is a status equilibrium. But then xji, and iis prevented from approaching jby the power of j’s partner. By Proposition 5.2, the pairing is also Paretooptimal. (ii) In the economy depicted in Table 5.1, there is no pairwise-stable pairing, but the ordering 1P2P3P4 supports the pairings {1↔3,2↔4} and {1↔4,2↔3}as status equilibria.
124 Chapter 5. A Comparison to Game Theory In the economy depicted in Table 5.2, the highlighted pairing {1↔2,3↔4}is pairwise stable, but there is no status equilibrium: The highlighted pairing is not a status equilibrium pairing because no agent gets his first-best. Neither are the other two pairings because in each of them, two agents get their last choice. Example: The Common-ranking Two-sided Matching Economy By Proposition 5.5, only mixed pairings can be status equilibrium pairings. In fact, any mixed pairing is a status equilibrium pairing with any ranking Pthat satisfies i1Pi2P...Pin/2and j1Pj2P...Pjn/2(for any agent i, every agent that idesires more than his partner has a higher status than i’s partner). 5.5 Prestige by Self: Initial Status Equilibrium We adapt the initial status equilibrium concept to the matching economy by taking an agent’s initial status to be himself. Recall that this concept belongs to the choice group of solution concepts (see Section 0.4). Every agent chooses his partner, but the status ranking only allows an agent to approach agents with the same status or lower. In equilibrium, the status ranking is such that the individual choices form a pairing (that is, if ichooses j, then jchooses i). Formally, a candidate for an initial status equilibrium is a tuple ‹P,(xi)›where iPj is interpreted as “i’s status is at least as high as j’s” and (xi)is a pairing. This adapted initial status equilibrium concept is referred to as a C-equilibrium in Richter and Rubinstein (2024). Definition: Initial Status Equilibrium An initial status equilibrium is a tuple ‹P,(xi)›such that, for every agent i, his partner xiis i’s most-preferred partner in {j∈N|iPj }.
5.5 Prestige by Self: Initial Status Equilibrium 125 Obviously, any two matched agents in an initial status equilibrium must have the same status. Therefore, every initial status equilibrium is also a status equilibrium and, by Proposition 5.5, its pairing is Pareto-optimal. Of particular interest is the relationship between the initial status equilibrium and the J1-equilibrium. If ‹P,(xi)›is an initial status equilibrium, then ‹,(xi)›is a J1-equilibrium where is any strict tie-breaking of P(that is, ij only if iPj ). However, unlike the initial status equilibrium concept, a J1-equilibrium does not require that an agent be weakly stronger than his partner. Many pairings can be a J1-equilibrium pairing but not an initial status equilibrium pairing. For example, consider the two-sided matching economy with N1={1,3},N2={2,4}, and a tragedy: 1 loves 2, 2 loves 3, 3 loves 4, and 4 loves 1. Both mixed pairings are J1-equilibrium pairings. One is the first-best for N1’s members. It is supported by any power relation that ranks N1’s members above N2’s members. The other is the opposite. Neither is an initial status equilibrium pairing (shortly, we will see why). We need an additional concept. We say that the matching economy is pairrankable if the set of agents Ncan be partitioned into doubletons I1,...,In/2, such that each agent in Iqprefers his partner in the doubleton to any member of Iq+1∪∙∙∙∪In/2. In other words, the agents can be partitioned into a sequence of doubletons where every agent’s partner is his best choice from those who are not ahead of him. Pair-rankability is a strong property of a matching economy which emerges in some natural settings. Two classical families of pair-rankable matching economies are: (i) Agents live in a metric space and rank partners by their closeness. The first doubleton can consist of the two closest agents, and each subsequent doubleton consists of the two closest among those remaining. (ii) Agents are positioned on a line, and each has single-peaked preferences over the other agents with a peak at one of his neighbours. This implies that an extreme agent top-ranks his only neighbour and, for any set of agents, there are
132 Chapter 5. A Comparison to Game Theory 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. 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.
5.11 Comparing our Approaches with Nash Equilibrium 133 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 Yequilibrium, 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 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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About the Team Alessandra Tosi was the managing editor for this book. Jennifer Moriarty copyedited the book. The cover image was created by Ariel Rubinstein from a concept by Michael Richter and Ariel Rubinstein. The cover of this book was produced by Jeevanjot Kaur Nagpal in InDesign using Fontin and Calibri fonts. Michael Richter and Ariel Rubinstein typeset the book in LaTeX and produced the paperback and hardback editions. Cameron Craig produced the PDF and HTML editions. Conversion was performed with open source software freely available on our GitHub page at https://github.com/ OpenBookPublishers.
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