We briefly discussed a variation on the basic model of the jungle. The variation was called civilized jungle. The discussion was based on Ariel Rubinstein and Kemal Yildizs paper Equilibrium in a Civilized Jungle. Please read Sections 1 and 2 of their paper to familiarize yourself with the model. The questions in this problem are based on the examples in the paper.

1. Prove the claim in Example A that the unique C-equilibrium is obtained by executing the serial dictatorship as described, and prove that this equilibrium is Pareto efficient.

2. Does the conclusion of Example B hold if the dichotomous language described in the example is replaced with the language in which, for each agent, there exists a relation such that he is the unique maximal agent according to this relation?

3. Prove the claim in Example C. Is the identified outcome Pareto efficient?

4. Prove the claim in Example D. Is the outcome Pareto efficient?

Equilibrium in a Civilized Jungle Ariel Rubinstein School of Economics, Tel Aviv University and Department of Economics, New York University Kemal Yldz Department of Economics, Bilkent University February 13, 2021 ABSTRACT: The jungle model with an equal number of agents and objects is enriched by adding a set of orderings over the set of agents. The orderings provide potential criteria for determining the justifiability of an assignment of an agent to an object. A civilized equilibrium is an assignment such that every agent is the strongest within the set of agents who are justifiable in the group consisting of himself and agents who envy him. KEYWORDS: Jungle equilibrium, justifiability, civilized equilibrium. AEA classification: D0, C0. We wish to thank Michael Richter for valuable comments. This paper replaces "An tude in modeling the definability of equilibrium" which we distributed in the spring of 2020. 1 1. Introduction Consider a society consisting of an equal number of agents and objects. Each agent has preferences over the objects and there are no externalities. Each agent must be assigned to only one object. The agents are objectively ranked by a power relation. We think of power not necessarily as physical but instead it can reflect, for example, social status or seniority. Up to this point, what we have is the jungle model la Piccione and Rubinstein (2007) adapted to the object assignment model of Shapley and Scarf (1974). We enrich the model by including a language consisting of a set of orderings over the set of agents. The orderings can be thought of as potential criteria for arguing that an agent is the best-suited to be assigned to an object. For example, the criteria might rank the agents according to their economic status, intelligence, or level of education. The phenomenon that we aim to formulate is that the assignment of objects is not entirely based on who is stronger, but also requires some socially legitimate justification. The language specifies the legitimate criteria that can be used to justify a claim. As is often the case in real life, an agent may cynically adapt the criteria that justify assignment of himself to his favorite object. In one case an agent might justify his claim by being the wealthier and in another by being the most intelligent. A candidate for equilibrium in our model is an assignment that assigns each agent to a single object. Given an assignment, the claim of a particular agent to be assigned to an object is justifiable if the agent is the best-suited according to an ordering in the language from among the set of candidates who wish to be assigned to the object. Thus, when an agent claims that he should have been assigned to an object, he needs to justify his claim using a criterion according to which he is not only better-suited than the person who is assigned to the object but also better-suited than any other agent who wishes 2 to be assigned to the object. The solution concept we propose is civilized equilibrium (C-equilibrium). For an assignment to be a C-equilibrium, each agent must be justifiable within the group consisting of agents who envy him and must be stronger than any other agent who is justifiable within the same group. The C-equilibrium is related to the Jungle Equilibrium, in which the power relation determines the equilibrium, for given preferences of the agents. However, in a civilized jungle, power is restricted such that when an agent wishes to be assigned to an object that another agent is assigned to, even if he is stronger, he needs to come up with a justifiable claim. For an agent, a justifiable claim is a specific formula in the language of the form: "I am the best-suited agent from among those who wish to be assigned to the object." Thus, the language restricts the use of power by determining what can be viewed as a valid criterion for being assigned to an object. For that reason, we refer to our equilibrium notion as "civilized equilibrium". As mentioned, any justification can be used by an agent wishing to be assigned to an object. Therefore, the scope of the model is limited to situations where the objects are different but of the same type (such as office spaces, similar positions in an organization, time slots for lectures) and therefore it is reasonable to justify a claim using any relevant criterion. 2. The civilized jungle and the civilized equilibrium A civilized jungle is a tuple N,X,(i)iN ,,L . The set of agents is N = {1, . . . ,n} and the set X consists of n objects. Each agent i has a strict preference relation i , which is a complete, transitive, and anti-symmetric binary relation over X. The power relation is 3 a strict ordering over N. The statement i j means that agent i is stronger than agent j . The language L is a set of complete and transitive (but not necessarily antisymmetric) binary relations over the set of agents N. We represent L = {} where is the index set of L s members. The set L is the stock of 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. Unlike 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 i is justifiable by from within the group I, if i is the unique maximizer of from within I. An agent i is justifiable in group I if i is justifiable by from within the group I for some L . Let JL (I) denote the set of agents who are justifiable in group I. That is, JL (I) = {i I | i is the unique -maximal agent in I for some L }. By definition, JL ({i }) = {i }. A candidate for our solution concept of a civilized equilibrium is an assignment (xi)iN that maps each agent to an exclusive object. For brevity, we write (xi) instead of (xi)iN . For an assignment (xi), an agent j envies agent i if xi j x j . For an assignment (xi) and an agent j , we denote the group consisting of agent j and all the agents who envy him by E((xi), j ). An assignment (xi) is a civilized equilibrium if each agent j is the -strongest agent in the set of agents that are justifiable in the group E((xi), j ). Definition 1 An assignment (xi) is a civilized equilibrium (C-equilibrium) if each agent j is the -strongest agent in JL (E((xi), j )). 4 2.1 Dichotomous languages A dichotomous language consists of properties (unary relations) that an agent may or may not satisfy. Formally, a dichotomous language consists of orderings with two 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 , we identify the ordering by means of a proposition in the sense that agent i satisfies if i is in the top set of and fails to satisfy if he is in the bottom set of . We treat total indifference in two ways: If all agents are in the top set of , then they all satisfy , and if all agents are in the bottom set of , then none do. A dichotomous language can be represented as a profile (i)iN where i is a nonempty subset of propositions in that are valid for agent i. Note that in the case of a dichotomous language (i)iN , "an agent i is justified by in the group I " means that i is the unique agent in I for whom i . Therefore, in a civilized jungle with a dichotomous language N,X,(i),,(i)iN , an assignment (xi) is a C-equilibrium if and only if for every j N either no one envies j or j is the strongest from among the justifiable agents who envy him, that is the set of agents who can find a proposition that he uniquely satisfies (that "makes him special") within the group E((xi), j ). 2.2 Examples Example A (Restrictive languages) Consider a civilized jungle with a restrictive language L consisting of a single strict ordering over N. Then, the unique C-equilibrium is obtained by running the serial dictatorship according to , independently of . Example B (Justification by "I am who I am") Consider a civilized jungle with the dichotomous language i = {mi } for every i N. The statement mi stands for "my name 5 is i ". Such a civilized jungle is extremely permissive in the sense that every agent can justify being assigned to any object by arguing that he is the unique agent who deserves to be assigned to it. Since every agent is justifiable in every group of agents, the unique C-equilibrium is obtained by running serial dictatorship according to the power relation , and therefore is Pareto efficient. Example C (Identical preferences) Assume that all agents share the same preferences a1 a2 a n . Suppose that L is a language that contains at least one strict ordering, which guarantees that there is always a justifiable agent in a group. Then, inductively choose the sequence of agents such that il is the -strongest agent in JL (N \ {i 1, . . . ,il 1}). Then, the assignment of il to al is the unique C-equilibrium. Example D (Nested dichotomous languages) Consider a dichotomous language where the sets of orderings in which agents in the top sets are nested, that is i n i 2 i 1 . For each preference profile and independently of , the associated civilized jungle has a unique C-equilibrium obtained by running the serial dictatorship according to the ordering i 1, . . . ,i n .