Equilibrium in a civilized jungle

Rubinstein, Ariel; Yildiz, Kemal Article Equilibrium in a civilized jungle Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Rubinstein, Ariel; Yildiz, Kemal (2022) : Equilibrium in a civilized jungle, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 17, Iss. 3, pp. 943-953, https://doi.org/10.3982/TE4886 This Version is available at: https://hdl.handle.net/10419/296375 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. 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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/4.0/ Theoretical Economics 17 (2022), 943–953 1555-7561/20220943 Equilibrium in a civilized jungle Ariel Rubinstein School of Economics, Tel Aviv University and Department of Economics, New York University Kemal Yildiz Department of Economics, Bilkent University The jungle model with an equal number of agents and objects is enriched by adding a language, which is a set of orderings over the set of agents. An assignment of an agent to an object is justified within a group of agents if there is an ordering according to which that agent is the best suited in the group. A civilized equilibrium is an assignment such that every agent is the strongest in the group of agents consisting of himself and those who wish to be assigned to the object and can be justified within this group. We present (i) conditions under which the equilibrium in a civilized jungle is identical to the jungle equilibrium, (ii) a connection between the power relation and the language that is essentially necessary and sufficient for the existence of a Pareto efficient civilized equilibrium, and (iii) an analogue to the second welfare theorem. Keywords. Jungle equilibrium, justifiability, civilized equilibrium. JEL classification.C0,D0. 1. Introduction Consider a society consisting of an equal number of agents and objects. An agent has preferences over the objects and there are no externalities. Each agent is to be exclusively assigned to a single object. The agents are ranked by a power relation.Poweris not necessarily physical strength but can be, for example, social status or seniority. Up to this point, we have described the jungle model àlaPiccione and Rubinstein (2007) adapted to the object assignment model of Shapley and Scarf (1974). In a civilized jungle, the exercise of power requires some socially legitimate justification. We enrich the jungle model with a language, which specifies the legitimate criteria that can be used to justify the assignment of an agent to an object. For example, a legitimate criterion might rank the agents according to wealth, intelligence, or education level. The assignment of an agent to an object is justifiable if the agent is uniquely best suited—–according to one of the orderings in the language—from among the set consisting of himself and the agents who prefer the object to their own assigned object. Ariel Rubinstein: [email protected] Kemal Yıldız: [email protected] We wish to thank Michael Richter for valuable comments. Kemal Yıldız’s research is supported by the Scientific and Research Council of Turkey (TUBITAK) under Grant 1059B191601712 and the BAGEP Award of the Science Academy. ©2022 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4886 944 Rubinstein and Yıldız Theoretical Economics 17 (2022) Thus, when an agent claims that he should have been assigned to a particular object, he must justify his claim using a criterion according to which not only is he better suited than the agent who is assigned to the object, but also that he is better suited than any other agent who wishes to be assigned to the object. As is often the case in real life, an agent may self-servingly adopt a criterion that justifies assigning himself to his favorite object. He might justify his claim based on his wealth on one occasion and based on his intelligence on another. An agent wishing to be assigned to an object can use any criterion, an assumption that makes sense in situations where the objects differ but are nonetheless in the same category (such as office space, equally ranked positions in an organization, and time slots for lectures). This assumption would not make sense, for example, in modeling the assignment of hierarchical positions in an organization. Leadership skills might be a criterion for a high-ranking position, but are irrelevant in the case of a low-ranking position, while obedience may be a reasonable criterion for a low-ranking position, but is irrelevant in the case of a high-ranking position. The proposed solution concept is civilized equilibrium (Cequilibrium), which is an assignment such that each agent (i) is justifiable within the group consisting of himself and the agents who envy him (ii) is stronger than any other agent who is justifiable within the same group. The Cequilibrium is related to the jungle equilibrium, which is an assignment such that if an agent envies another, then the latter agent is stronger than the former according to the power relation. In a civilized jungle, the language restricts the use of power by determining what can be viewed as a justifiable claim for or against an assignment. When an agent wishes to be assigned to an object that another agent is assigned to, it is not enough that he be stronger; his claim must be justifiable as well. In equilibrium, an agent who is assigned to an object must be the strongest agent in the group of agents consisting of himself and those who envy him and who can justify their claim within this group. Since justifiability is a prerequisite for the use of power, we refer to our equilibrium notion as “civilized equilibrium.” Before getting into the model, some words about the motivation of the paper. It is our view that the standard models in economic theory (whether they deal with markets, games, or choice problems) lack a critical real-life component, namely the language used by economic agents to communicate, to formulate their rules of behavior, or to maintain social norms.1Our first motivation is to incorporate this approach into the analysis of a jungle model by enriching it with a language that is used by the agents to justify their claims for being assigned to an object. The second motivation is related to the rhetoric of economic theory. The jungle model was intended to be—at least in part—a critique of the rhetoric used in standard economic theory that is used to extol the market system by way of the welfare theorems. Piccione and Rubinstein (2007) argued that one can employ similar rhetoric in order to extol the jungle system. 1See Rubinstein (1978) and Rubinstein (2000b, Chapter 4), who suggests the requirement that an agent’s preferences should be definable inagivenlanguage. Theoretical Economics 17 (2022) Equilibrium in a civilized jungle 945 Our results show that civilizing a jungle does not necessarily preserve the existence and efficiency of equilibrium. We observe that the first welfare theorem often fails in a civilized jungle. Therefore, one main focus of the analysis is to find when the equilibrium in a jungle is preserved as a Cequilibrium in a civilized jungle. The results indicate that in a civilized jungle, the power relation should respect the language in a specific way in order to achieve harmony in the form of a Pareto efficient equilibrium; otherwise, chaos or inefficiency might prevail. 2. The civilized jungle and the civilized equilibrium Acivilized jungle is a tuple N,X,(i)i∈N,,L.Thesetofagents is N={1, ,n}and the set Xconsists of nobjects.Eachagentihas a strict preference relation i,whichisa complete, transitive, and antisymmetric binary relation over X.Thepower relation is a strict ordering over N. The statement ijmeans that agent iis stronger than agent j. The language Lis a set of complete and transitive (but not necessarily antisymmetric) binary relations over the set of agents N.LetL={≥λ}λ∈,whereis the index set of L’s members. The set Lconsists of the criteria that can be used to justify the choice of an agent from within a group that is a nonempty subset of agents. We refer to a civilized jungle without a language as a jungle. An assignment (xi)i∈Nmaps each agent exclusively to an object. For brevity, we write xinstead of (xi)i∈N.Ajungle equilibrium is an assignment satisfying that there are no agents iand jsuch that xjixiand ij. Unlike in the model of the jungle, the use of the power relation in the civilized jungle is restricted such that a stronger agent can exercise his power in order to be assigned to an object only if he can justify being assigned to it by one of the criteria recognized as legitimate in the civilized jungle. An agent iis justifiable by ≥λfrom within the group Iif he is the unique maximizer of ≥λfrom within I.Anagentiis justifiable in group Iif he is justifiable by ≥λfrom within the group Ifor some ≥λ∈L.LetJL(I)be the set of agents justifiable in I. By definition, JL({i})={i}. A candidate for the solution concept of a civilized equilibrium is an assignment. For an assignment x,anagentjenvies agent iif xijxj. We denote the group consisting of agent iand all the agents who envy him by E(x,i). Definition 1. An assignment xis a civilized equilibrium (Cequilibrium) if each agent iis the -strongest agent in JL(E(x,i)). 2.1 Dichotomous languages A dichotomous language consists of properties (unary relations) that an agent may or may not have. Formally, a dichotomous language consists of orderings of Nwith two (nonempty) indifference sets, a top one and a bottom one, where every agent in the top set is superior to every agent in the bottom set. For each λ∈, we identify the ordering ≥λby means of a property λ, in the sense that agent ihas property λif he is in the top set of ≥λand fails to have property λif he is in the bottom set of ≥λ. A dichotomous language can be represented as a profile (φi)i∈N,whereφiis the set of properties in  946 Rubinstein and Yıldız Theoretical Economics 17 (2022) that agent ihas. Then the statement “agent iis justified by λin group I” means that i is the unique agent in Ifor whom λ∈φi.Thatis,thepropertyλmakes him “special” within the group I. 2.2 Examples Example A (Restrictive languages). Consider a civilized jungle with a restrictive language Lconsisting of a single strict ordering ≥over N. Then the unique Cequilibrium is obtained independently of the power relation by running serial dictatorship according to ≥and, therefore, it is Pareto efficient. ♦ Example B (The power relation is a member of the language). Consider a civilized jungle in which the power relation is a member of L. Then the assignment xobtained by running serial dictatorship according to is a Pareto efficient Cequilibrium. To see this, note that if an agent ienvies j, then ji. Therefore, since is a member of L,jis justified in E(x,j)by . It follows that jis the -strongest agent in JL(E(x,j)). Indeed, x is the unique Cequilibrium. Let ybe another assignment and let ibe the strongest agent who envies a weaker agent j.Theniis justified in E(y,j)by since iis the -strongest agent in E(y,j)and is a member of L. Therefore, yis not a Cequilibrium. ♦ Example C (Identical preferences). Assume that all agents share the same preferences a1a2···an. Suppose that the language Lcontains at least one strict ordering. This guarantees that for every group I,thesetJL(I)is not empty. Then inductively choose the sequence of agents such that ilis the -strongest agent in JL(N\{i1,,il−1}).The assignment of ilto alis the unique Cequilibrium. ♦ Example D (Justification by “I am who I am”). Consider a civilized jungle with the dichotomous language φi={mi}for every i∈N. The statement mistands for “my name is i.” Such a civilized jungle is extremely permissive in the sense that every agent ican justify being assigned to any object by arguing that he is the unique agent who deserves to be assigned to it by the criterion mi. Since every agent is justifiable in every group of agents, the unique Cequilibrium is obtained by running serial dictatorship according to the power relation. ♦ Example E (Nested dichotomous languages). Consider a dichotomous language where agents’ sets of properties are nested, in the sense that there is an ordering i1,,inof the agents such that φin⊂···⊂φi2⊂φi1. For each preference profile and independently of the power relation, the associated civilized jungle has a unique Cequilibrium obtained by running serial dictatorship according to the ordering i1,,in.♦ 3. The Cequilibrium and the jungle equilibrium As discussed in the Introduction,Cequilibrium is related to the jungle equilibrium. The key difference between them is that in a Cequilibrium, if an agent makes a claim on a different object, then not only must he be stronger than the current agent assigned to the Theoretical Economics 17 (2022) Equilibrium in a civilized jungle 947 object, he must also provide a justification. However, we will see below that if the power relation respects the language in a specific way, then the jungle equilibrium remains a C equilibrium and under additional conditions is even the unique Cequilibrium. Two properties of the power relation—given the language—turn out to be critical. Definition 2. A power relation is weakly L-concave if for every i,j∈N,wehaveij whenever for every ≥λ∈Lthere exists iλ∈N\{j}such that iλ≥λjand iiλ.Apower relation is strongly L-concave if for every i,j∈N,wehaveijwhenever for every ≥λ∈Lthere exists iλ∈N\{j}such that iλ≥λjand iiλ. ApowerrelationisL-concave if it respects the language in the following sense: Agent jcannot be stronger than agent iif for each criterion, agent ican point to an agent who is weaker than himself and at least as suited as jaccording to the criterion. As explained by Richter and Rubinstein (2019), Lconcavity is not simply a technical condition and is closely related to standard notions of convexity and concavity. In our setting, Lconcavity represents an intuitive (though not necessarily realistic) relationship between the power relation and the language. The weak Lconcavity of the power relation—a weaker version of the strong Lconcavity defined by Richter and Rubinstein (2019)—is the restriction under which the jungle equilibrium is a Cequilibrium. Recall that this assignment always exists and is Pareto efficient. The strong Lconcavity of the power relation guarantees that the jungle equilibrium will be the unique Cequilibrium in a civilized jungle with a language of strict orderings. Proposition 1. Let N,X,(i),,Lbe a civilized jungle with a weakly L-concave power relation. (i) The jungle equilibrium of N,X,(i),is a Cequilibrium. (ii) If Lis a language of strict orderings and is a strongly L-concave power relation, then the Cequilibrium is unique. Proof. To prove (i), let xbe the assignment obtained by running the serial dictatorship according to . Then, for each j∈N,E(x,j)⊆{i|ji}. To show that xis a Cequilibrium, assume by contradiction that there is an agent jsuch that j/∈JL(E(x,j)). Then, for each λ∈,thereexistsjλ∈E(x,j)\{j}such that jλ≥λjand jjλ. Therefore, by the weak L concavity of ,wegetjj. To prove (ii), suppose that yis another Cequilibrium. Then there exists i,j∈Nsuch that ijand ienvies jin y. Since yis a Cequilibrium, i/∈JL(E(y,j)). Therefore, for each λ∈,thereexistsjλ∈JL(E(y,j)) such that jλ≥λi. Since yis a Cequilibrium, then jjλfor every λ∈. However, in that case, the strong Lconcavity of implies that ji, a contradiction. A weaker version of weak Lconcavity is Lreflectivity, which only requires that if an agent iis better suited than agent jaccording to all the criteria in L, then imust be 948 Rubinstein and Yıldız Theoretical Economics 17 (2022) stronger than j. Formally, is L-reflective if for every i,j∈N,wehaveijwhenever i> λjfor every ≥λ∈L. The following example demonstrates that if the power relation is not weakly L-concave, then the existence of a Cequilibrium is not guaranteed even if it is L-reflective. Example F. Let N={1, 2, 3}and X={a,b,c}. The preference profile (i), the language L={≥α,≥β}, and the power relation are specified as 123≥α≥β aba123 bac331 ccb212 Note that since 1 >α3and2>β3 while 3 1and32, weak Lconcavity implies that 3 3. Therefore, is not weakly L-concave. This civilized jungle does not have a Cequilibrium. Assume that xis a Cequilibrium. Agent 1 does not envy agent 2, since otherwise 1 ∈JL(E(x,2 )) (by ≥α)and12. Then 3 does not envy 2, since otherwise 3∈JL(E(x,2 )) (by ≥α)and32. This leaves the assignments [b,c,a]and [a,b,c].The former is not a Cequilibrium since 1 and 2 envy 3 who is not justifiable in N. The latter is not a Cequilibrium since only 3 envies 1, 3 is justifiable in {1, 3}(by ≥β)and31. ♦ 4. Existence of a Pareto efficient Cequilibrium Recall that a jungle equilibrium always exists and is Pareto efficient. It follows from Proposition 1that if the power relation in a civilized jungle with a language of strict orderings is strongly L-concave, then there is a unique Cequilibrium that is Pareto efficient. Thus, the first welfare theorem holds, as it does in a jungle. We will see that this is not the case in a civilized jungle. We start with an example showing that if the power relation is not weakly L-concave (although it is L-reflective), then a Cequilibrium may not be Pareto efficient even if it is unique. Example G. Let N={1, 2, 3, 4}and X={a,b,c,d}. The preference profile (i),the language {≥α,≥β}, and the power relation are specified as 1234≥α≥β abaa231 bacd412 ccbb143 dddc324 Note that since 4 >α1and3>β1 while 2 4and23, weak Lconcavity requires that 2 1. Therefore, is not weakly L-concave, since 1 2. It is easy to see that Theoretical Economics 17 (2022) Equilibrium in a civilized jungle 949 [b,a,c,d]is a Cequilibrium, which is Pareto dominated by [a,b,c,d]. To see that there is no other Cequilibrium, let ybe a Cequilibrium. Then it cannot be that y1=a, since otherwise both 3 and 4 would envy 1, and, therefore, he would not be justifiable. It cannot be that y4=a, since then 3 envies 4, is justifiable in E(y,4 ), and is stronger than 4. If y3=a,theny2=b(otherwise 2 envies 3, is justifiable in E(y,3 ), and is stronger than 3); however, then 1 envies 2, is justifiable in E(y,2 )(because 3 is absent) and is stronger than 2. Thus, y2=ain a Cequilibrium. It is then straightforward to show that y=[b,a,c,d].♦ The observation in Example Gis not just a coincidence. The following proposition shows that in a civilized jungle with a language of strict orderings, it is “essentially” true that if the power relation is not weakly L-concave, then we can find a preference profile for which there is no Pareto efficient Cequilibrium. There are exceptions, as shown in Example A,inwhichthesetofCequilibria is determined by the language independently of the power relation. More precisely, we show that our claim holds unless there is an agent iwho is ranked by right above another agent j(i.e., there is no k∈Nsuch that ikj)whoL-dominates iin the sense that jis better suited than iaccording to every criterion in the language L(i.e., j> λifor every ≥λ∈L, denoted by jD Li).2 Proposition 2. Let N,X,(i),Lbe a civilized jungle with a language Lof strict orderings such that there are no i,j∈Nsuch that iis ranked right above jby and jD Li. If the power relation is not weakly L-concave, then there is a preference profile (i)such that there is no Pareto efficient Cequilibrium. Proof.Step 1. Suppose that is not L-reflective. Then there exist i,j∈Nsuch that jD Libut ij. Assume that among all such pairs, the number of agents who are ranked between iand jaccording to is minimal. Suppose that iis ranked right above agent k by . We know that k= jand kDLi. It follows from our choice of iand jthat jDLk. Next we construct a preference profile such that there is no Pareto efficient Cequilibrium in the associated civilized jungle. Let {i,j,k}be specified as ijk aa c . . . cbai bc bk ···j . . .. . .. . .. . . 2It is easy to see that if iis ranked right above jby the power relation and jL-dominates i, then the set of Cequilibria of N,X,(i),,Lis identical to that of N,X,(i),,L,whereis the power relation obtained from by swapping the positions of iand j. 950 Rubinstein and Yıldız Theoretical Economics 17 (2022) Assume that all other agents’ most preferred objects are distinct and different from a,b,c. By contradiction, assume that xis a Pareto efficient Cequilibrium. Since xis Pareto efficient, {i,j,k}must be assigned to {a,b,c}while all other agents are assigned to their most preferred objects. If xk=a, then either xi=cor xj=c, contradicting that x is Pareto efficient. If xi=a,theniis not justifiable in E(x,i), since jenvies iand jD Li. This contradicts that xis a Cequilibrium. Therefore, [xi,xj,xk]must be either [c,a,b] or [b,a,c]. Suppose that [xi,xj,xk]=[c,a,b].ThenE(x,j)={i,j,k}. Next we argue that kis justifiable in E(x,j). Since jDLk,thereexists≥λ∈Lsuch that k> λj. Then, since jD Li,wealsohavek> λi. Therefore, kis justifiable in E(x,j)by ≥λand is stronger than j. This contradicts that xis a Cequilibrium. Suppose that [xi,xj,xk]=[b,a,c].ThenE(x,k)={i,k}. Since kDLi,wehaveiis justifiable in E(x,k). Since iis stronger than k, this contradicts that xis a Cequilibrium. Step 2. Suppose that is L-reflective but not weakly L-concave. Then there exist i,j∈Nsuch that for every λ∈,thereexistsjλ∈N\{i}such that jλ>λiand jjλ,but ij.LetI={jλ}λ∈∪{i}. Recall that the set JL(I)consists of agents in Iwho are the maximizers of ≥λfor some λ∈. Since ≥λis a strict ordering for every λ∈,wehavefor each j∈I\{i},ifj/∈JL(I),thenJL(I)=JL(I\{j}).LetI∗={i,j1,,jm}be a subset of Isuch that JL(I∗)=I∗\{i}. We assume without loss of generality that j1j2···jm. Let Z={z0,z1,,zm}be a set of distinct alternatives. Define a preference profile (i)such that the following statements hold: i. Every j∈I∗prefers every alternative in Zto every alternative in X\Zand every j∈N\I∗prefers every alternative in X\Zto every alternative in Z. ii. The preferences of the agents in I∗restricted to Zare ij1j2j3···  jm z0z0z1z2zm−1 zmz1z0z0z0 ··z2z3zm . . .. . .. . .. . .. . . ·· ·zm· ·zmzmz1· Let xbe a Pareto efficient Cequilibrium. Then, by Pareto efficiency, xj∈Zfor every j∈I∗. For each jk,jl∈I∗,ifk<l,thenjkdoes not envy jl. Otherwise, since jk∈JL(I∗), we have jk∈JL(E(x,jl)) and jkjl, contradicting that xis a Cequilibrium. Therefore, either ior j1must be assigned to z0and the agents in I∗\{i}must be assigned to objects by running serial dictatorship according to their indices and in ascending order. Thus, we are left with two cases, both of which lead to a contradiction of xbeing a C equilibrium. Case 1: [xi,xj1,,xjm]=[z0,z1,z2,,zm].ThenE(x,i)=I∗,buti/∈JL(I∗).