Equitable rent division∗ Rodrigo A. Velez† Department of Economics, Texas A&M University, College Station, TX 77843

March 3, 2018

Abstract How should a group of roommates allocate the rooms and contributions to rent in the house they lease? Economists have provided partial answers to this question in a literature that spans the last forty years. Unfortunately, these results were developed in a non-linear fashion, which obscures them to the non-specialist. Recently, computer scientists have developed an interest in this problem, advancing from an algorithmic complexity perspective. With this new interest gaining traction, there is an evident need for a coherent development of the results in economics literature. This paper does so. In particular, we build connections among results that were seemingly unrelated and considerably simplify their development, fill in non-trivial gaps, and identify open questions. Our focus is on incentives issues, the area in which we believe economists have more to contribute in this discussion. JEL classification: C91, D63, C72. Keywords: no-envy; efficiency; rent division; rental harmony; indivisible goods; equal-income competitive allocations.

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Introduction

A group of roommates need to allocate the rooms and contributions to rent in the house they lease. How should they do it? Think of a smartphone application, or in general an arbitrator, that intends to provide a protocol to solve these problems. What information should the system ask from the agents? How should the system process the information? What allocations result of the infinitely many alternative protocols that we can think of when agents pursue their self interest? Which of these protocols guarantee the allocations that result are equitable, efficient, etc.? The purpose of this paper is to survey the results in economics literature, developed since 1980 up to 2016, which provide partial answers to these questions. We fill in several non-trivial gaps, which ∗ All

errors are my own. https://sites.google.com/site/rodrigoavelezswebpage/home

† [email protected];

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became evident when writing this survey, and identify open questions for further research. Our presentation is essentially self-contained and our emphasis is on incentives issues. By presenting these results in a coherent linear progression, we make them easily accessible to researchers in computer science, a field in which there is a growing interest in them. For instance the not-for-profit fair allocation website Spliddit.org (Goldman and Procaccia, 2014) has attracted thousands of users to its fair rent calculator and has allowed researchers in computer science to identify open questions about the optimal way to make recommendations. Most prominently, the computational complexity of calculating an allocation satisfying certain normative criteria was recently addressed by Gal et al. (2017). The central form of equity that we consider is no-envy, i.e., the requirement that no agent prefer the allotment of any other agent to her own (Foley, 1967). In Sec. 3.2 we discuss the normative content of this property. Then, we present a unified development of seemingly disconnected results concerning existence and incentives properties of mechanisms that “target” envy-free allocations. The main issue that motivates this endeavor is that this literature is unnecessarily convoluted because it evolved non-linearly. For instance, a central result in the theory, the Strict Monotonicity Theorem (Theorem 4), was first proved “from scratch” by using linear approximations methods by Alkan et al. (1991). From here these authors prove general existence of envy-free allocations (Theorem 3). In our unified approach, Theorem 3 is an immediate consequence of the first basic result in the literature (Stromquist, 1980; Svensson, 1983). Then, we prove Theorem 4 from Theorem 3 and an elementary induction argument. As another example, the results concerning the so-called implementation of envy-free social choice correspondences were developed in an incremental way: first restricted to economies with only one good (Tadenuma and Thomson, 1995a), then in economies with n agents and n goods and quasi-linear preferences (Bevi´a, 2010; ¯ 2008), then for a particular subfamily of social choice functions in the Azacis, unrestricted domain of preferences (Velez, 2011), and finally arriving to a crisp result, the Maximal Manipulation Theorem (Theorem 5) of Andersson et al. (2014a,b) and Fujinaka and Wakayama (2015), which subsumes all previous literature and whose proof is strikingly simple given our unified approach to this problem.

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Road map and summary of results

To make our discussion concrete consider the following example, of Velez (2015), in which three agents, N ≡ {1, 2, 3}, lease a house with three rooms, A ≡ {1, 2, 3}, when rent is $1,200. Agent i’s utility from receiving room j and paying pi for it is vji − pi , where the different values of vji are given in Table 1. Quasi-linear preferences, as those in our example, will play a central role in our study. Since they are a finite dimensional space, it is reasonable to think that an arbitration protocol for this problem could asks agents to report a preference in this domain. In reality, agents’ true preferences may not be quasi-linear, however. For instance, an agent who is budget constrained feels 2

roommate 1 roommate 2 roommate 3

room 1 $400 $500 $400

room 2 $400 $700 $700

room 3 $400 $400 $600

Table 1: Valuations

differently about paying one dollar more when this dollar is still within her budget than when she has to borrow it (Sec. 8). Thus, in order to understand the effect of these possible income effects, we will study a general model in which admissible utility functions satisfy a form of continuity and that, other things equal, paying less money is better for each agent. The results we describe now apply to this general domain. Our objective is to construct and study arbitration protocols that result in an envy-free allocation. The first step to do so is to abstract from incentives issues and determine the feasibility of this endeavor when an arbitrator knows the true preferences of the agents. Indeed, in Sec. 4, we determine that for each possible admissible utility profile, there is an envy-free allocation. In our running example we can do so in a graphical intuitive way as follows. When preferences are quasi-linear, a necessary condition for no-envy is that the assignment of rooms maximizes aggregate value (Sec. 3.2). Thus, in our example, at each envy-free allocation, agent 1 receives room 1, agent 2 receives room 2, and agent 3 receives room 3. Consequently, we can simplify notation and describe each envy-free allocation by its rent assignment vector r ≡ (ri )i∈N . Then, r is envy-free, if and only if for each pair of different agents i 6= j the following linear inequality constraint is satisfied: vii − ri ≥ vji − rj . Since r1 + r2 + r3 = 1,200, we can easily solve this inequality system for r1 and r2 (Figure 1 (a)). After determining general existence of envy-free allocations, we consider incentive issues faced by an arbitrator who does not know the preferences of the agents. We approach this problem in three alternative forms of extreme-case scenario mechanism design. In each of these approaches, which we describe in more detail below, the essential element of analysis is to determine the extent to which the preferences of an agent affect the shape of the set of envy-free allocations for a given subprofile of preferences of the other agents. This, of course, has a clear non-cooperative flavor and could be embedded in the analysis of each incarnation of the mechanism design principles. This would create a lot of redundancy, however. For this reason, we collect the key results about the structure of the envy-free set in an independent section (Sec. 5). The key to understand how an agent’s preferences affect the shape of the envy-free set is to understand how the envy-free set changes when the rent to collect changes. This is surprising because these are seemingly unrelated problems. The intuition why this is so is the following. Imagine that you are trying to determine what allotments could be received by agent i in an envyfree allocation for a profile in which you are free to choose agent i’s preferences but the preferences of the other agents are fixed. Since there are finitely many

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Figure 1: (a) A ≡ {1, 2, 3}, N ≡ {1, 2, 3}, r1 + r2 + r3 = 1, 200, and valuations are given in Table 1. Let lij be the linear inequality constraint vii −ri ≥ vji −rj where vji is agent i’s value of room j. The figure displays each lij with its label inside the half space satisfying the inequality constraint. The shaded area contains all (r1 , r2 ) such that the allocation where agent i receives room i and rent payments are (r1 , r2 , 1, 200 − r1 − r2 ) is envy-free. Roommate 1’s indifference curves are vertical lines (she prefers lower r1 ); roommate 2’s indifference curves are horizontal lines (she prefers lower r2 ); roommate 3’s indifference curves are lines with constant r1 + r2 (she prefers lower r3 , i.e., higher r1 + r2 ). (b) Best envy-free allocations for each agent.

rooms, one can simplify this problem and only think about the allocations at which agent i receives a given room. Intuitively, starting from such an allocation, one can find another one by reshuffling rooms and changing the rent among the other agents. It turns out that these reshuffles and rent reassignments need to have the following structure (Lemma 1): the rooms that are assigned a higher rent are received (in both allocations) by the agents who are worse off with the change, the rooms that are assigned the same rent are received by the agents who are indifferent with the change, and the rooms that are assigned a lower rent are received by the agents who are better off with the change. Thus, one can partition any such a reallocation into sub-problems in which the individual rent payments are changing in the same direction. Thus our original task can be mapped into determining how the envy-free set changes when rent changes in any sub-problem (for this reason we need to consider a somehow more abstract formulation of our model that is necessary to handle variable population, rooms, and aggregate rent to collect). We advance this analysis in a series of results that culminate with a characterization, Theorem 5. By our assumption of continuity of preferences, the set of envy-free allocations for a given preference profile, which we interpret as the true preference profile, has well-defined welfare extremes for each agent. In our example, these extremes can be easily found graphically (see the agent’s indifference curves in 4

Fig. 1 (a) and the welfare extremes for each agent in Fig. 1 (b)). Theorem 5, states that for a given agent, say agent i, her corresponding true envy-free maximum welfare, say u∗i , has two key properties. First, agent i’s utility, measured with her true utility function, at an envy-free allocation in an economy in which, other things equal, one substitutes this agent’s utility function with some other admissible utility function is at most u∗i . Second, for each ε > 0 there is a quasi-linear utility function such that the value of agent i’s true utility at each envy-free allocation in the economy in which, other things equal, one substitutes agent i’s utility function with this utility function, is greater than or equal to u∗i − ε. Our example illustrates the second component of Theorem 5. For each agent, say agent 1, one can find alternative room valuations for this agent and substitute them in Table 1, guaranteeing that all the envy-free allocation for the new preferences are close to the best allocation for agent 1. Consider, for instance, the quasi-linear preference for agent 1 in which for some small ε > 0, v11 = 300 + ε, v21 = 500, and v31 = 400. At the unique assignment of rooms that maximizes aggregate value when agent 1 has this preference and agents 2 and 3’s preferences are given by Table 1, agent 1 receives room 1, agent 2 receives room 2, and agent 3 receives room 3. Thus, we can still represent envy-free allocations for these preferences in Fig. 1 (a). With these new preferences, the hyperplane determining the half space that satisfies the condition v11 − r1 ≥ v31 − r3 , which in Fig. 1 (a) is labeled l13 , moves towards the origin just above (300, 500). Thus, the set of envy-free allocations for these preferences is a tiny triangle that contains the corner (300, 500). With Theorem 5 at hand, we then follow a mechanism design approach (Sec. 6). First, we think of the arbitrator as selecting a mechanism, i.e., a protocol in which each agent will report some information, a message, and for a given profile of reports an outcome function determines an allocation. Note that the variables of design for the arbitrator are both the agents’ message spaces and the outcome function. Second, we specify the information structure of the problem, propose a prediction for the strategic situations that result when the mechanism is operated, and evaluate the performance of the mechanism with an extreme-case scenario analysis. First, we study dominant strategy implementation (Sec. 6.1). That is, we consider the possibility that the arbitrator designs a mechanism such that for each possible realization of preferences, the game that results when the mechanism is operated has a dominant strategy equilibrium whose outcome is envy-free with respect to the true preferences. This is the incarnation of the mechanism design paradigm in which one performs a best-case scenario analysis given the dominant strategy equilibrium prediction, which requires no information structure is determined. We first note that early results in the mechanism design literature imply dominant strategy implementation of envy-free allocations in our environment is impossible. That is, for each mechanism there is a realization of preferences for which the induced game has no dominant strategy equilibrium whose outcome is envy-free. Then we show that this impossibility is specially pronounced in our environment: For each envy-free direct revelation 5

mechanism, i.e., each agent reports her preferences and for each profile of reports the prescribed outcome is an envy-free allocation for the reports, there is at most one preference profile for which truthful reports are a dominant strategy equilibrium (Proposition* 8).1 Then we consider non-cooperative implementation under complete information, i.e., under the assumption that the agents know each other well (Sec. 6.2 and 6.3). In this context we advance a worst-case scenario analysis based on Nash equilibrium and limit Nash equilibrium (Radner, 1980). First, we show that an arbitrator that insists on obtaining only envy-free allocations needs to accept a wide range of these allocations are Nash equilibrium outcomes of the allocation process (Lemma 6). The news are much better when we analyze noncooperative outcomes of envy-free direct revelation mechanisms. It turns out that these mechanisms produce in Nash equilibria, only envy-free allocations for true preferences (Propsitions 10). These Nash equilibria are not guaranteed to exists. However, they do exist if the arbitrator’s choices satisfy certain conditions that we characterize (Theorem 6 and Theorems* 7 and 8). In contrast to Nash equilibrium, limit Nash equilibria are always guaranteed to exist for each envy-free direct revelation mechanism. Remarkably, the analysis of these limit Nash equilibria alerts us about possible undesirable outcomes when the arbitrator allows agents to report preferences that are not quasi-linear. If this is so, there may be limit equilibria of the allocation process that are not envyfree (Example 4). This means that self-interest may lead to a profile of reports at which an agent’s gain by deviating is negligible and such that it produces an undesirable allocation. Indeed, these outcomes may violate the so-called Pareto efficiency. However, if reports are forced to be quasi-linear, each limit equilibrium outcome is envy-free (Theorem 9). Finally, we study the ranking of envy-free direct revelation mechanisms based on the extent to which agents are able to manipulate them (Sec. 7). Rankings based on ordinal criteria, e.g., set inclusion of profiles at which at least an agent can manipulate, point to the direct revelation mechanisms that for each profile of reports recommend the best envy-free allocation for an agent, as the least manipulable ones (Propositions 11-13). Remarkably, when preferences are quasilinear, there is essentially a unique way to break ties for the arbitrator among envy-free allocations that minimizes the maximal incentive, measured in money, to manipulate among agents (Theorem 10). In our example this tie-breaker would select the envy-free allocation that is equidistant to the indifference curves of the agents passing through their corresponding best envy-free allocation. The remainder of the paper is organized as follows. Sec. 3 presents the model and discusses the normative content of envy-free allocations. Sec. 4 presents results concerning existence of envy-free allocations. Sec. 5 presents results concerning the structure of the envy-free set. Sec. 6 presents results concerning the implementation of envy-free social choice correspondences. Sec. 7 presents results concerning the manipulability of envy-free direct revelation mechanisms. Sec. 8 discusses generalizations of the model and open questions. 1 We

label with a star the results that are first proved in this survey.

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3

Model

3.1

Environment

We study the allocation of rooms and rent among roommates. We are not interested in issues concerning changes in population of agents or their endowment of rooms. However, when proving results by means of an induction argument over the number of agents, it is convenient to consider problems with different set of agents and different endowment. In order to accommodate these needs we consider problems with agents from a set of potential agents N who are endowed with a subset of a set of potential rooms O. Generic agents are i, j, k, ... and generic rooms are a, b, c, .... More precisely, for a set of n agents N ⊆ N we study the allocation of n rooms A ⊆ O. Each agent receives exactly one room and pays, or receives, an amount of money. Thus, an allotment for agent i is a pair (ri , a) ∈ R × A. When ri ≥ 0 we interpret this allotment of money as a payment made by the agent in order to receive the room. When ri < 0 we interpret this allotment of money as a transfer that the agent receives together with the room. By leaving the sign of the agent’s allotment of money unrestricted, we can accommodate situations beyond rent division, as the allocation of tasks and salary. Each agent has a continuous preference on allotments that is represented by a continuous utility function. The generic utility function is ui . We assume throughout that preferences satisfy the following two properties:2 u1 money-monotonicity, i.e., for each consumption bundle (ri , a) and each δ > 0, ui (ri + δ, a) < ui (ri , a); and u2 no room is infinitely better than another (in terms of money), i.e., for each pair of rooms a and b, and each ri ∈ R, there is ti such that ui (ri , a) = ui (ti , b). Let U(A) be the space of continuous utility functions on R × A satisfying these two properties. Whenever it creates no confusion we simply write U for U(A). We denote by u ≡ (ui )i∈N ∈ U N the generic utility profile. For T ⊆ N , uT ≡ (ui )i∈T is the restriction of u to T and u−T ≡ (ui )i∈N \T is the restriction of u to N \ T . When T is a singleton, say T ≡ {i}, we write u−i instead of u−T . For two disjoint subsets T ⊆ N and S ⊆ N , and for each pair of utility profiles uS ∈ U S and uT ∈ U T , (uS , uT ) ∈ U S∪T is the corresponding joint utility profile. In applications in which preferences are private information, it is not realistic to require agents to provide reports that describe an arbitrary element in U, an infinitely dimensional set. In practice, agents are asked to report a preference out of a finite dimensional space. The most popular such restriction is the 2 We introduce utility representations only for convenience in the presentation. Properties u1 and u2 together imply continuity of preferences, i.e., weak upper and weak lower contour sets are closed. Thus, if one requires them, one can assume, without loss of generality, the preference is represented by a continuous utility function.

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following (see Sec. 8 for a discussion of alternatives). A preference is quasilinear whenever it is represented by a function of the form ui (ri , a) ≡ va − ri . Without loss of generality, we normalize mina∈A va = 0. In this way, for each a ∈ A, we can interpret va as the maximal amount of money that the agent would be willing to pay for receiving room a instead of receiving for free the worst room in A. Let Q be the domain of quasi-linear preferences. An economy is a list e ≡ (N, A, u, m) where N ⊆ N , A ⊆ O is such that |A| = |N |, u ∈ U(A)N , and m ∈ R is an amount we refer to as the budget. The set of economies with agent set N endowed with set of rooms A is E(N, A). Whenever it creates no confusion we write E instead of E(N, A). An allocation of A among N is a pair z ≡ (r, µ) where r ≡ (ri )i∈N ∈ RA and µ : N → A is a bijection. Agent i’s consumption at z is zi ≡ (ri , µ(i)), i.e., agent i receives consumption of money ri and room µ(i) at z. Whenever convenient we write z ≡ (zi )i∈N . We denote the consumption of money of the agent who receives room a ∈ A at z by ra . Let rA be the vector of consumptions of money associated with the different rooms at z, i.e., (ra )a∈A . We denote the set of allocations of A among N by Z(N, A). Whenever convenient we write u Pi (z) instead of ui (zi ). An allocation z ∈ Z(N, A) is feasible for budget m if i∈N ri = m. We denote the set of these allocations by Z(N, A, m). Given an economy e ≡ (N, A, u, m), whenever convenient, we write Z(e) instead of Z(N, A, m), and refer to these allocations as being feasible for e. Given two economies with different set of agents and rooms (N, A, uN , m) and (M, B, uM , l) and two allocations z ∈ Z(N, A, m) and s ∈ Z(N, A, l), we denote by (z, s) the allocation in Z(N ∪ M, A ∪ B, m + l) at which each i ∈ N receives zi and each i ∈ M receives si .

3.2

Equitable allocations

We concentrate our analysis on the following form of equity. Definition 1. Let e ∈ E. An allocation is envy-free for e if no agent prefers the consumption of any other agent to her own. We denote the set of envy-free allocations for e by F (e). A number of considerations lead to this principle as the most appealing and practical form of equity to consider. First, it is ordinal (Foley, 1967). Second, the test rules out all allocations that violate an obvious form of bias (Varian, 1974). Third, these allocations can be seen as the result of individual choices from a common budget set offered to all agents (Kolm, 1971). Indeed, in our environment, the set of envy-free allocations coincides with the set of allocations that an equal income market would produce (Svensson, 1983). More precisely, for each p ≡ (pa )a∈A and I ∈ R, let B(p, I) ≡ {(ra , a) : a ∈ A, ra = pa − I} be the set of consumption bundles that are affordable to an agent with income I at prices p.3 Let e ≡ (N, A, u, m) ∈ E. A pair (p, z) where p ∈ RA and z ∈ Z 3 An agent’s budget set for income I and prices p is the set of consumptions that are affordable to the agent given I and p. Since more money is better for the agent, it is without

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is an equal-income competitive equilibrium for e ≡ (N, A, u, m) if for P each i ∈ N , zi maximizes ui in B(p, ( a∈A pa − m)/n) and market clears, i.e., z is feasible. It is straightforward to prove the following equivalence. Remark 1 (Svensson, 1983). Let e ∈ E and z ∈ Z(e). Then, z ∈ F (e) if and only if z is an equal-income competitive allocation for e. Thus, the study of envy-free allocations in our environment can be simply seen as the study of the performance of an equal-income market, which is an intuitively equitable institution (Varian, 1976). Finally, the last reason to concentrate on the study of envy-free allocations in our environment is that they are Pareto efficient, i.e., there is no feasible way to make some agent better off without making another agent worse off (Svensson, 1983). This is easily proved as follows. Let e ≡ (N, A, u, m) ∈ E, z ≡ (r, µ) ∈ F (e), and s ≡ (t, σ) ∈ Z(e) such that each agent weakly prefers s to z. Since z ∈ F (e), for each i ∈ N and each a ∈ A, ui (ri , µ(i)) ≥ uiP (ra , a). By u1, for each i ∈ N , t ≤ r . Thus, for each a ∈ A, t ≤ r . Since i a a σ(i) a∈A ra = P t , r = t . Again, since z ∈ F (e), for each i ∈ N , u (z ) ≥ u (s ). Thus, a A A i i i i a∈A there is no feasible way to make an agent better off at z without making any agent worse off.

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Existence

Definition 2. Let l ∈ R and M < nl. Then, ui ∈ U satisfies Assumption A1 with respect to M and Pl if for each list of bundles {(ra , a)}a∈A such that for each a ∈ A, ra ≤ l and a∈A ra = M , the argmax of ui in this list contains a bundle (ra , a) such that ra < l. Theorem 1 (Stromquist, 1980; Svensson, 1983). Let e ≡ (N, A, u, M ) ∈ E with M < 0. Suppose that for each i ∈ N , ui satisfies Assumption A1 with respect to M and 0. Then there is z ≡ (r, µ) ∈ F (e) at which each agent gets a non-negative transfer of money, i.e., r ≤ 0. Theorem 1 can be proved by an argument based on Sperner’s Lemma, introduced by Su (1999). We refer the reader to Su (1999) for a didactic description of the method. One can easily verify that Su’s argument can be immediately adapted to prove the following result (c.f. Velez, 2016). Theorem 2. Let e ≡ (N, A, u, M ) ∈ E with M < nl. Suppose that for each i ∈ N , ui satisfies Assumption A1 with respect to M and l. Then, there is z ≡ (r, µ) ∈ F (e) such that for each i ∈ N , ri ≤ l. loss of generality that we define the budget set as the bundles that are exactly affordable to her. Note that a consumption bundle (xi , a), where xi ≥ 0 represents a transfer to the agent, is exactly affordable to the agent with income I at prices p whenever xi = I − pa . Thus, given I and p, the allotment, in our convention, that the agent can exactly afford for room a is ra = pa − I. This highlights that when one embeds our model in a general equilibrium environment, referring to the negative of the agent’s consumption of money as the rent of the room she receives is only appropriate if one can normalize each agent’s income to zero. This is always possible in our model.

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Interestingly, for a given utility function and a given budget, Assumption A1 is eventually satisfied for some upper bound on rent payments. Proposition 1 (Velez, 2016). Let ui ∈ U and M ∈ R. There is λ such that ui satisfies Assumption A1 with respect to M and each l ≥ λ. Proof. Let λ > 0 be such that M < λn and for each b ∈ A, ui (λ, b) ≤ mina∈A ui (M/(n − 1), a). Such a λ exists because ui satisfies u1 and u2. Let P l ≥ λ; b ∈ A; and (ra )a∈A such that rb = l, a∈A ra = M , and for each a ∈ A, ra ≤ l. Then, there is c ∈ A such that rc ≤ (M −l)/(n−1) < max{M/(n−1), l}. Thus, by u1, ui (rc , c) ≥ ui (M/(n − 1), c) ≥ ui (λ, b) ≥ ui (rb , b). An immediate consequence of Theorem 2 and Proposition 1 is the general existence of envy-free allocations. Theorem 3 (Alkan et al., 1991). For each e ∈ E, F (e) is non-empty.

5

The structure of the envy-free set

Lemma 1 (Decomposition Lemma; Alkan et al., 1991). Let z ≡ (r, µ) ∈ F (N, A, u, m) and s ≡ (t, σ) ∈ F (N, A, u, l). Then, both µ and σ are bijections between: (i) {i ∈ N : ui (zi ) > ui (si )} and {a ∈ A : ra < ta }. (ii) {i ∈ N : ui (zi ) = ui (si )} and {a ∈ A : ra = ta }. (iii) {i ∈ N : ui (si ) > ui (zi )} and {a ∈ A : ta < ra }. Proof. We prove that µ is a bijection between the sets in statement (iii). The proof of the other statements is similar. Let N1 ≡ {i ∈ N : ui (si ) > ui (zi )} and A1 ≡ {a ∈ A : ta < ra }. Let i ∈ µ−1 (A1 ). Since s ∈ F (N, A, u, l), ui (si ) ≥ ui (tµ(i) , µ(i)) > ui (rµ(i) , µ(i)). Thus, i ∈ N1 and µ−1 (A1 ) ⊆ N1 . Let j ∈ N1 . Then, σ(j) ∈ A1 , for otherwise uj (sj ) ≤ uj (rσ(j) , σ(j)) ≤ uj (zj ). Then, σ(µ−1 (A1 )) ⊆ A1 . Since both µ and σ are bijections, σ(µ−1 (A1 )) = A1 . Let j ∈ N \ µ−1 (A1 ). Then, σ(j) 6∈ A1 . Thus, j 6∈ N1 . Thus, N1 ⊆ µ−1 (A1 ). Thus, N1 = µ−1 (A1 ). Proposition 2 (The envy-free set is a lattice; Alkan et al., 1991). Let z ≡ (r, µ) ∈ F (N, A, u, m) and s ≡ (t, σ) ∈ F (N, A, u, l). For each i ∈ N , let (z ∨ s)i ≡ zi if ui (zi ) ≥ ui (si ) and (z ∨ s)i ≡ si otherwise. For each i ∈ N , let (z ∧ s)i ≡ zi if ui (zi ) ≤ ui (si ) and (z ∧ s)i ≡ si otherwise. Then, z ∨ s and z ∧ s are envy-free allocations. Moreover, each agent weakly prefers z and s to z ∧ s, and weakly prefers z ∨ s to z and s. Proof. Let z ∨ s ≡ (x, η) and M ≡ {i ∈ N : ui (zi ) ≥ ui (si )}. Since for each pair {i, j} ⊆ M , η(i) = µ(i), then no two agents in M are assigned the same room by η. Since for each pair {i, j} ⊆ N \ M , η(i) = σ(i), then no two agents in N \ M are assigned the same room by η. By Lemma 1, σ is a bijection between N \ M and µ(N \ M ). Thus, for each i ∈ N \ M and each j ∈ M , σ(j) 6= µ(j). Thus, z ∨ s is well-defined. Now, let i ∈ M . For each j ∈ M , ui (z ∨ s) = ui (zj ) ≥ 10

ui (zj ). For each j ∈ N \ M , ui (z ∨ s) = ui (zj ) ≥ ui (rσ(j) , σ(j)) ≥ ui (sj ) (the last inequality holds by Lemma 1). Thus, for each i ∈ M and each j ∈ N , ui (z ∨ s) ≥ ui ((z ∨ s)j ). A symmetric argument shows that for each i ∈ N \ M and each j ∈ N , ui (z ∨ s) ≥ ui ((z ∨ s)j ). Thus, z ∨ s is envy-free. The last properties in the lemma directly follow from the definition of z ∨s. A symmetric argument applies to z ∧ s. Theorem 4 (Strict monotonicity; Alkan et al., 1991). Let e ≡ (N, A, u, m) ∈ E, z ∈ F (e), and ε > 0. Then, (i) there is an allocation in F (N, A, u, m − ε) that each agent prefers to z; and (ii) there is an allocation in F (N, A, u, m + ε) that is strictly worse than z for each agent. The following language is useful in the proof of Theorem 4 and subsequent results. Definition 3. Let e ≡ (N, A, u, m) ∈ E and z ≡ (r, µ) ∈ F (e). Let H ⊆ N and L ⊆ N be disjoint. Let s ≡ (t, σ) ∈ Z, ε > 0, and δ > 0. We say that: 1. s is obtained from z by rebating ε to the members of H while maintaining no-envy if s ∈ F (N, A, u, m − ε); for each i ∈ H, ui (si ) > ui (zi ); for each a ∈ µ(H), ta < ra ; and for each i ∈ N \ H, si = zi . 2. s is obtained from z by extracting δ from the members of L while maintaining no-envy if s ∈ F (N, A, u, m + δ); for each i ∈ L, ui (si ) < ui (zi ); for each a ∈ µ(L), ta > ra ; and for each i ∈ N \L, si = zi . 3. s is obtained from z by extracting δ from the members of L and rebating ε to the members of H while maintaining no-envy if s ∈ F (N, A, u, m − ε + δ); for each i ∈ H, ui (si ) > ui (zi ); for each a ∈ µ(H), ta < ra ; for each i ∈ L, ui (si ) < ui (zi ); for each a ∈ µ(L), ta > ra ; and for each i ∈ N \ (H ∪ L), si = zi . We prove Theorem 4 by induction on the cardinality of the set of rooms. The first step of the proof is the following lemma, which is of independent interest. Lemma 2. Let e = (N, A, u, m) ∈ E and z ≡ (r, µ) ∈ F (e). Suppose that H ⊆ N and L ⊆ N are (potentially empty) disjoint sets such that: • For each j ∈ N \ H, and each i ∈ H, uj (zj ) > uj (zi ). • For each i ∈ L, and each j ∈ N \ L, ui (zi ) > ui (zj ). Suppose also that Theorem 4 is valid for economies whose set of agents has cardinality less than or equal to |N | − 1. Then, there is ∆ > 0 such that for each ε < ∆ and each δ < ∆, 1. If 0 < |H| < |N |, then there is s ∈ Z that is obtained from z by rebating ε to the members of H while maintaining no-envy.

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2. If 0 < |L| < |N |, then there is s ∈ Z that is obtained from z by extracting δ from the members of L while maintaining no-envy. 3. If both H and L are non-empty, then there is s ∈ Z that is obtained from z by extracting δ from the members of L and rebating ε to the members of H while maintaining no-envy. Proof. Suppose that 0 < |H| < |N | is such that for each j ∈ N \ H, and each i ∈ H, uj (zj ) > uj (zi ). Let ∆ ≡ min{ri − ti : i ∈ H, j ∈ N \ H, uj (ti , µ(i)) = uj (rj , µ(j))}. Since for each j ∈ N \ P H and each i ∈ N , uj (zj ) > uj (zi ), ∆ > 0. Let 0 < ε < ∆ and mH ≡ i∈H ri . Since z ∈ F (e), then zH ∈ F (H, µ(H), uH , mH ). By our hypothesis, Theorem 4 is valid for economies whose set of agents has cardinality |H|. Thus, there is sH ≡ (tH , σH ) ∈ F (H, µ(H), uH , mH − ε) that each agent prefers to zH . By Lemma 1, for each a ∈ µ(H), ta < ra . We claim that (sH , zN \H ) ∈ F (N, A, u, m − ε). Let {i, j} ⊆ H. Since sH ∈ F (H, µ(H), uH , mH − ε), ui (si ) ≥ ui (sj ). Let i ∈ H and j ∈ N \ H. Then, ui (si ) > ui (zi ) ≥ ui (zj ), where the last inequality holds P because z ∈ F (e). Since for each a ∈ µ(H), ta < ra , and P t = ( a a∈µ(H) ra )−ε, 0 < rσ(i) −tσ(i) < ε < ∆. Thus, uj (zj ) > uj (si ). a∈µ(H) Let {i, j} ⊆ N \ H. Since z ∈ F (e), ui (zi ) ≥ ui (zj ). Thus, (sH , zN \H ) is obtained from z by rebating ε to the members of H while maintaining no-envy. The other statements in the lemma follow from similar arguments. Proof of Theorem 4. We prove statement (i). The proof of statement (ii) is symmetric. We prove (i) by induction over the cardinality of the set of agents n ≡ |N |. The statement is trivial when n = 1. Suppose then that the statement is true for economies with up to n − 1 ≥ 1 agents. Let e ≡ (N, A, u, m) ∈ E be such that |N | = n, z ≡ (r, µ) ∈ F (e), and ε > 0. Let s ≡ (t, σ) ∈ F (N, A, u, m − ε). Existence of this allocation is guaranteed by Theorem 3. Let B ≡ {a ∈ A : ta < ra }. Since ε > 0, B 6= ∅. Let H ≡ {i ∈ N : σ(i) ∈ B}. Suppose that N \ H 6= ∅, for otherwise the theorem would follow. By Lemma 1, for each j ∈ N \ H and each i ∈ H, uj (zi ) < uj (tµ(i) , µ(i)) ≤ uj (sj ) ≤ uj (zj ). By Lemma 2 there is 0 < δ < ε/2 and s′ ∈ F (N, A, u, m − δ) that is obtained from z by rebating δ to the members of H while maintaining no-envy. Thus, for each j ∈ N \ H and each i ∈ H, ui (s′i ) > ui (zi ) ≥ ui (zj ). By Lemma 2 there is 0 < η < ε/2 and s′′ ∈ F (N, A, u, m − δ − η) that is obtained from s′ by rebating η to the members of N \ H while maintaining no-envy. Thus, for each i ∈ N , ui (s′′i ) > ui (zi ). This proves that the set of δ > 0 such that δ ≤ ε and there is an element of F (N, A, u, m − δ) that each agent prefers to z, is non-empty. Since the set is bounded, it has a supremum. Let δ ∗ be this supremum. Since preferences are continuous, δ ∗ is the maximum of this set. Thus, δ ∗ = ε, for otherwise one can repeat the argument above rebating ε − δ ∗ from an allocation in F (N, A, u, m − δ ∗ ) that each agent prefers to z. The following graph, which is associated with an allocation at a particular economy, plays an important role in our understanding of budget monotonicity

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and incentives issues related with the implementation of envy-free allocations.4 Definition 4. The weak-envy graph for e ≡ (N, A, u, m) ∈ E and z ∈ Z(e) is Γ(u, z) ≡ (N, E) where (i, j) ∈ E if and only if ui (zi ) = ui (zj ). If there is a path from i to j in Γ(u, z) we write i →u,z j. Lemma 3. Let e ≡ (N, A, u, m) ∈ E, z ≡ (r, µ) ∈ F (e), {i, j} ⊆ N such that i →u,z j, l ∈ R, and s ≡ (t, σ) ∈ F (N, A, u, l). 1. If uj (sj ) > uj (zj ) or tµ(j) < rµ(j) , then ui (si ) > ui (zi ) and tµ(i) < rµ(i) . 2. If uj (sj ) = uj (zj ) or tµ(j) = rµ(j) , then ui (si ) ≥ ui (zi ) and tµ(i) ≤ rµ(i) . 3. If ui (si ) < ui (zi ) or tµ(i) > rµ(i) , then uj (sj ) < uj (zj ) and tµ(j) > rµ(j) . 4. If ui (si ) = ui (zi ) or tµ(i) = rµ(i) , then uj (sj ) ≤ uj (zj ) and tµ(j) ≥ rµ(j) . Proof. By Lemma 1, uj (sj ) > uj (zj ) if and only if tµ(j) < rµ(j) . Thus, suppose that tµ(j) < rµ(j) . Let {i0 , i1 , ..., im } ⊆ N be a path in Γ(u, z) from i to j, i.e., i0 = i; im = j; and for each l = 0, ..., m − 1, ul (zl ) = ul (zl+1 ). We can suppose without loss of generality that {i0 , i1 , ..., im } = {1, ..., j}. Since s ∈ F (N, A, u, l), uj−1 (sj−1 ) ≥ uj−1 (tµ(j) , µ(j)) > uj−1 (zj ) = uj−1 (zj−1 ). By Lemma 1, tµ(j−1) < rµ(j−1) . The recursive argument shows that ui (si ) = u1 (s1 ) > u1 (z1 ) = ui (zi ) and tµ(i) < rµ(i) . Definition 5. For each e ≡ (N, A, u, m) ∈ E and each continuous representation of the same preferences as u, v ∈ U(A)N , let F v (e) ≡ argmax min vi (si ). s∈F (e) i∈N

Proposition 3 (Velez, 2011). Let e ≡ (N, A, u, m) ∈ E and v ∈ U(A)N be a continuous representation of the same preferences as u. Then, z ∈ F v (e) if and only if z ∈ F (e) and for each i ∈ N there is j ∈ argmini∈N vi (zi ) such that i →u,z j. Proof. Let z ∈ F v (e). Then, z ∈ F (e). Let T ≡ argmini∈N vi (zi ). Let M ≡ {i ∈ N : ∃j ∈ T, i →u,z j}. We prove that M = N . Suppose by contradiction that N \ M 6= ∅. Let i ∈ N \ M . We claim that for each j ∈ T , ui (zi ) > ui (zj ). Suppose by contradiction that ui (zi ) ≤ ui (zj ) for some j ∈ T . Since z ∈ F (e), ui (zi ) = ui (zj ). Thus, i →u,z j. This is a contradiction. By Lemma 2 there is ∆ > 0 such that for each 0 < ε < ∆ there is s that is obtained by extracting ε from the members of N \ M and rebating ε to the members of M while maintaining no-envy. Since preferences are continuous, ε can be selected such that argmini∈N vi (si ) ⊆ M . Thus, mini∈N vi (zi ) < mini∈N vi (si ). 4 Alkan (1994) defined a binary relation on the set of rooms, which is equivalent to our graph formulation here. Velez (2011) pioneered the use of this binary relation in the study of incentives issues related with envy-free allocations. Andersson et al. (2014a) and Fujinaka and Wakayama (2015) also use this binary relation, in its equivalent form for the set of agents, in the study of incentives issues related with envy-free allocations.

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This contradicts z ∈ F v (e). Now, suppose that z ≡ (r, µ) ∈ F (e) and for each i ∈ N there is j ∈ argmini∈N vi (zi ) such that i →u,z j. We claim that z ∈ F v (e). Suppose by contradiction that there is s ≡ (t, σ) ∈ F (e) such that mini∈N vi (si ) > mini∈N vi (zi ). Let a ∈ A and i ∈ N be such that µ(i) = a. Let j ∈ argmini∈N vi (zi ) be such that i →u,z j. Thus, P vj (sj )P> vj (zj ). Thus, uj (sj ) > uj (zj ). By Lemma 3, ta < ra . Thus, m = a ta < a ra = m. This is a contradiction. Proposition 4 (Maxmin envy-free selections are essentially single valued; Alkan et al., 1991). Let e ≡ (N, A, u, m) ∈ E and v ∈ U(A)N be a continuous representation of the same preferences as u. Let z ≡ (r, µ) and s ≡ (t, σ) be two elements in F v (e). Then, rA = tA and for each i ∈ N , ui (zi ) = ui (si ). Proof. Let i ∈ N . We claim that rµ(i) ≤ tµ(i) . Suppose by contradiction that rµ(i) > tµ(i) . By Proposition 3, there is j ∈ argmink∈N vk (zk ) such that i →u,z j. By Lemma 3, rµ(j) > tµ(i) . By Lemma 1, uj (zj ) < uj (sj ). Thus, vj (zj ) < vj (sj ) and z 6∈ F v (e). This is a contradiction. Thus, rA = tA . By Lemma 1, for each i ∈ N , ui (zi ) = ui (si ). Proposition 5 (Maxmin envy-free selections are budget monotone; Alkan et al., 1991). Let e ≡ (N, A, u, l) ∈ E, v ∈ U(A)N be a continuous representation of the same preferences as u, and z ≡ (r, µ) ∈ F v (e). For each h < l, each s ≡ (t, σ) ∈ F v (N, A, u, h), and each i ∈ N , ui (si ) > ui (zi ). Proof. Let j ∈ argmini∈N vi (zi ). By Theorem 4, uj (sj ) > uj (zj ). By Lemma 1, tµ(j) < rµ(j) . By Proposition 3 for each i ∈ N there is j ∈ argmini∈N vi (zi ) such that i →u,z j. By Lemma 3, for each a ∈ A, ta < ra . By Lemma 1, for each i ∈ N , ui (si ) > ui (zi ). Definition 6. Let e ≡ (N, A, u, m) ∈ E and i ∈ N . The agent-i-optimal envy-free set for e is F i (e) ≡ argmax ui (si ). s∈F (e)

Lemma 4. e ≡ (N, A, u, m) ∈ E and i ∈ N . Then z ∈ F i (e) if and only if z ∈ F (e) and for each j ∈ N , j →u,z i. Proof. Let v ∈ U(A)N be continuous, represent the same preferences as u, and such that for each j 6= i, vi < vj . Then, F i (e) = F v (e). By Proposition 3, z ∈ F i (e) if and only if z ∈ F (e) and for each j ∈ N , j →u,z i. Proposition 6 (Fujinaka and Wakayama, 2015). Let e ≡ (N, A, u, m) ∈ E, i ∈ N , and z ≡ (r, µ) ∈ F i (e). For each vi ∈ U(A) and each s ≡ (t, σ) ∈ F i (N, A, (u−i , vi ), m), rA = tA . Proof. By Lemma 4 for each j ∈ N , j →u,z i. Let v ≡ (u−i , vi ). Then, for each j ∈ N , j →v,z i. Let a ∈ argmaxb∈A vi (rb , b) and i0 be the agent who receives room a at z. Let s be the allocation at which agent i receives (ra , a) and is 14

obtained by reshuffling consumptions along a shortest path defining i0 →u,z i. More precisely, suppose without loss of generality that M ≡ {1, 2, ..., i} are different agents such that µ(1) = a and for each l = 1, ..., i − 1, ul (zl ) = ul (zl+1 ). Let s ≡ (t, σ) ∈ Z(N, A, u, m) be such that for each j ∈ N \ M , sj = zj ; for each l ∈ {1, ..., i − 1}, sl = zl+1 ; and si = z1 . Since a ∈ argmaxb∈A vi (rb , b) and for each agent j 6= i, uj (sj ) = uj (zj ), s ∈ F (N, A, v, m). By construction, for each j ∈ M , j →v,s i. Let j ∈ N \ M . Recall that j →u,z i. Thus, there is {j0 , ..., jk }, different agents, such that j0 = j, jk = i, and for each l = 0, ..., k − 1, ul (zl ) = ul (zl+1 ). Let h be the first index in 0, ..., k such that jh ∈ M . Since jk ∈ M , jh is well defined. Then, j →v,s jh . Since jh →v,s i, j →v,s i. By Lemma 4, s ∈ F i (N, A, v, m). Let s′ ≡ (t′ , γ) ∈ F i (N, A, v, m). By Proposition 4, t′A = tA = rA . The following theorem is the central result concerning incentives in this literature. The first two statements, in equivalent formulations, are due to Andersson et al. (2014a,b) and Fujinaka and Wakayama (2015). The third statement is a slightly stronger form of the second. Theorem 5 (Maximal Manipulation Theorem). Let e ≡ (N, A, u, m) ∈ E, i ∈ N , and z ≡ (r, µ) ∈ F i (e). Then, 1. For each vi ∈ U(A) and each s ∈ F (N, A, (u−i , vi ), m), ui (si ) ≤ ui (zi ). 2. For each ε > 0 there is vi ∈ Q(A) such that for each s ∈ F (N, A, (u−i , vi ), m), ui (si ) ≥ ui (zi ) − ε. 3. For each δ > 0 there is vi ∈ Q(A) such that for each s ≡ (t, σ) ∈ F (N, A, (u−i , vi ), m), σ(i) = µ(i) and ti ≤ ri + δ. Proof. Let z ≡ (r, µ) be as in the statement of the theorem. We prove 1 first. t = r. Let vi ∈ U(A), and sb ≡ (b t, σ b) ∈ F i (N, A, (u−i , vi ), m). By Proposition 6, b Let s ≡ (t, σ) ∈ F (N, A, (u−i , vi ), m). Then, vi (si ) ≤ vi (b si ). By Lemma 1, tσ(i) ≥ b tσ(i) . Thus, ui (si ) ≤ ui (b tσ(i) , σ(i)) = ui (rσ(i) , σ(i)) ≤ ui (zi ). Now we prove 2. Let δ > 0. Let vi ∈ Q(A) be the preference such that for each j ∈ N \ {i}, vi (ri + δ, σ(i)) = ui (rj − δ/(n − 1), σ(j)). By Proposition 6, z ∈ F i (N, A, (u−i , vi ), m). Let s ≡ (t, σ) ∈ F (N, A, (u−i P, vi ), m). By Lemma 1, ti ≥ rσ(i) . Thus, σ(i) = µ(i), for otherwise since ri +δ+ j6=i (rj −δ/(n−1)) = m, there is j ∈ N \ {i} such that vi (sj ) > vi (si ). For the same reason, ti ≤ ri + δ. Thus, ui (si ) ≥ ui (ri + δ, µ(i)). Since ui is continuous, for a given ε, one can select δ such that ui (si ) ≥ ui (ri , µ(i)) − ε.

6

Implementation of envy-free sccs

We consider now the problem of an arbitrator who is in charge of recommending an allocation of the resources available. The arbitrator knows with certainty the set of agents who are involved, the set of rooms with which these agents are endowed, and the aggregate rent that has to be collected. The arbitrator 15

knows the set of preferences that are admissible for each agent. For instance, the arbitrator may know that agents have quasi-linear preferences. We will assume that the set of admissible preferences is the same for each agent and denote it by D ⊆ U. The arbitrator does not know the actual preferences of the agents, because this is their private information. We assume complete information, i.e., each agent knows the preferences of the other agents. Since the only unknown in an economy for the arbitrator is the agents’ preferences, we will identify the set of all possible economies that the arbitrator needs to consider with the corresponding set of preference profiles. More precisely, we fix N ⊆ N , A ⊆ O such that |N | = |A|, and m ∈ R, and identify the set of economies {(N, A, u, m) : u ∈ DN } with the corresponding set of preferences DN . We also write Z for Z(N, A, m). A social choice correspondence (scc) is a set valued function that assigns to each preference profile a set of feasible allocations for it. It is informative to think of an scc as collecting the allocations that the arbitrator identifies as “optimal” for each economy. The generic scc is u ∈ DN 7→ G(u) ⊆ Z. We refer to the scc that assigns to each u ∈ U N the set F (u) as the envy-free scc. We denote this correspondence simply by F . Consistently, we denote by F i the scc that assigns to each u ∈ U the best allocations for agent i in F (u). A social choice function (scf) is a single-valued scc. The generic scf is u ∈ DN 7→ g(u) ∈ Z. Let G, H, and g be two sccs and an scf defined on a domain DN . We write G ⊆ H if for each u ∈ DN , G(u) ⊆ H(u). We write g ∈ G, if for each u ∈ D, g(u) ∈ G(u). A (simultaneous-form) mechanism is a list (M, f ) where M ≡ (Mi )i∈N is a list of message spaces (unrestricted sets) for the agents and f : M → Z is an outcome function. For each u ∈ DN , mechanism (M, f ) induces a simultaneous complete information game in which (1) the set of agents N have action spaces (Mi )i∈N ; (2) conditional on action profile M ∈ M being selected by the agents, the outcome of the game is f (M ); and (3) agents’ preferences on outcomes, which are common knowledge, are u. Let (N, M, f, u) be this game. A solution is a function that assigns to each game (N, M, f, u) a subset of M. The generic solution is S. The set of outcomes predicted by solution S in game (N, M, f, u) is f (S(N, M, f, u)). A mechanism (M, f ) implements in solution S an scc G in domain D if for each u ∈ DN the set of outcomes predicted by solution S in game (N, M, f, u) is G(u).5

6.1

Dominant strategies

Definition 7. Let (M, f ) be a mechanism, u ∈ DN , and i ∈ N . A message Mi ∈ Mi is a dominant strategy for ui if for each M−i ∈ M−i and each Mi′ ∈ Mi , ui (f (M )) ≥ ui (f (M−i , Mi′ )). A message profile M ∈ M is a dominant strategy equilibrium for (N, M, f, u) if for each i ∈ N , Mi is a dominant 5 Our notion of implementation is some times called full implementation. It corresponds to a worst-case scenario approach to mechanism desing.

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strategy for ui .6 Proposition 7 (Alkan et al., 1991; Tadenuma and Thomson, 1995a). Let g ∈ F and D ⊇ Q. There is no mechanism (M, f ) such that for each u ∈ DN , (N, M, f, u) has a dominant strategy equilibrium whose outcome is envy-free for u. Proposition 7 is a corollary of a general impossibility result developed in the mechanism design literature (Green and Laffont, 1979). Since Proposition 7 only refers to envy-free scfs, one can provide a direct proof of it as follows (c.f. Alkan et al., 1991; Tadenuma and Thomson, 1995a).7 The well-known revelation principle states that if an scf g admits a mechanism with the property in the proposition, then for each u ∈ DN and each agent, the true report is a dominant strategy in (N, DN , g, u). Definition 8. Let D ⊆ U and g defined on DN . Then, g is strategy-proof in D if for each u ∈ DN , each i ∈ N , and each vi ∈ D, ui (g(u)) ≥ ui (g(u−i , vi )). Thus, one can prove Proposition 7 by showing that no g ∈ F is strategyproof in D ⊇ Q. In order to do so one can exhibit a profile u ∈ DN such that for each g ∈ F , u is not a dominant strategy equilibrium of (N, DN , g, u). With our knowledge about the structure of the set of envy-free allocations we can prove a stronger result: Each g ∈ F fails to induce truthful dominant strategy equilibria for at most one utility profile. Proposition* 8. Let g ∈ F and D ⊇ Q. There is at most one u ∈ QN such that u is a truthful dominant strategy equilibrium of (N, DN , g, u). If such a profile u ∈ QN exists, then for each pair {i, j}, ui = uj . Proof. Suppose that u ∈ QN is a truthful dominant strategy equilibrium of (N, DN , g, u). Let (r, µ) ≡ g(u). Then, for each i ∈ N , each vi ∈ DN , and each v−i ∈ DN \{i} , ui (g(v−i , vi )i ) ≤ ui (g(v−i , ui )i ). Since Q ⊆ D and g ∈ F , by Theorem 5, for each i ∈ N and each v−i ∈ DN \{i} , g(v−i , ui ) ∈ F i (N, A, (v−i , ui ), m).

(1)

We claim that for each pair {i, j} ⊆ N , ui = uj . Since g(u) ∈ F (u), for each pair {i, j} ⊆ N , ui (g(u)i ) ≥ ui (g(u)j ). We prove that for each pair {i, j} ⊆ N , ui (g(u)i ) = ui (g(u)j ). Since u ∈ QN , this implies our claim. Suppose by means of contradiction that there are {i, j} ⊆ N such that ui (g(u)i ) > ui (g(u)j ). If N = {i, j}, by Lemma 4, g(u) 6∈ F j (N, A, u, m). This contradicts (1). Suppose then that N \ {i, j} 6= ∅. Let s ≡ (t, µ) be the allocation such that ti ≡ rµ(i) − ε, tj ≡ rµ(j) − ε, and for each l ∈ N \ {i, j}, tl ≡ rµ(l) + 2ε/(n − 2). For each k ∈ N \ {i, j} let vk ∈ Q be the preference that is indifferent between the 6 Note that the dominant strategy equilibrium prediction does not make use of our complete information structure. 7 Tadenuma and Thomson (1995a) prove a version of Proposition 7 for the allocation of a single good. This corresponds to a domain restriction in our model in which agents are indifferent among n − 1 rooms.

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bundles in s. Let v ≡ (ui , uj , vN \{i,j} ). Since (ri , µ(i)) is the best bundle for i among the bundles in (r, µ), the best bundle for i among the bundles in s is (ti , µ(i)). For the same reason, the best bundle for j among the bundles in s is (tj , µ(j)). Since each agent in N \ {i, j} is indifferent among all bundles in s, s ∈ F (v). Since ui (g(u)i ) > ui (g(u)j ), then ui (si ) > ui (sj ). Now, since for each l ∈ N \ {i, j}, ui (g(u)i ) ≥ ui (g(u)l ), ui (ti , µ(i)) > ui (tl , µ(l)). Thus, it is not the case that i →v,s j. Thus, s 6∈ F j (N, A, v, m). Let z ≡ (t′ , σ) ∈ F j (N, A, v, m). Then, uj (zj ) > uj (sj ). By Lemma 1, t′µ(j) < tµ(j) . Thus, for each l ∈ N \ {i, j}, vl (zl ) > vl (sl ). By Lemma 1, t′µ(l) < tµ(l) . Thus, σ(i) = µ(i) and t′i > ti . Thus, uj (zj ) > vj (zi ) and for each l ∈ N \ {i, j}, vl (zl ) > vl (zi ). Thus, there is no l ∈ N \ {i} such that l →v,s i. Thus, z 6∈ F i (N, A, v, m). By Lemma 1, no allocation obtained by reshuffling consumptions at z is in F i (N, A, v, m). By Proposition 4, F i (N, A, v, m) ∩ F j (N, A, v, m) = ∅. However, by (1), g(v) ∈ F i (N, A, v, m) ∩ F j (N, A, v, m). This is a contradiction. Now, suppose that v ∈ QN is a truthful dominant strategy equilibrium of (N, DN , g, v) and v 6= u. Then, there is i ∈ N such that vi 6= ui . Since u ∈ QN is a truthful dominant strategy equilibrium of (N, DN , g, u) and v ∈ QN is a truthful dominant strategy equilibrium of (N, DN , g, v), then (u−i , vi ) is a truthful dominant strategy equilibrium of (N, DN , g, (u−i , vi )). By our claim above, for each j ∈ N \ {i}, vi = uj . This is a contradiction. Remark 2. Let D ⊆ U and u ∈ DN such that for each pair {i, j} ⊆ N , ui = uj . There is g ∈ F such that u is a dominant strategy equilibrium of (N, DN , g, u). Proof. By Lemma and each v ∈ DN such that for each T 4, fori each M ⊆ N S i ∈ M , vj = uj , i∈M F (N, A, v, m) = i∈M F i (N, A, v, m) 6= ∅. Thus, there is g ∈ F such that for each v ∈ DN such that there is i ∈ N with vi = ui , g(v) ∈ F i (N, A, v, m). By Theorem 5, ui is a dominant strategy for ui in (N, DN , g, u).

6.2

Nash implementation

Definition 9. Let (M, f ) be a mechanism and u ∈ DN . A message profile M ∈ M is a Nash equilibrium of (N, M, f, u) if for each i ∈ N and each Mi′ ∈ Mi , ui (f (M )) ≥ ui (f (M−i , Mi′ )). The key to understand what the limits of Nash implementation are for envyfree sccs is the following definition and subsequent well-known result. Definition 10. Let u ∈ U N and z ∈ Z. Then, u′ ∈ U N is obtained by a monotonic transformation of u at z if for each i ∈ N , each xi ∈ R, and each a ∈ A, u′i (zi ) ≥ u′i (xi , a) implies ui (zi ) ≥ ui (xi , a). An scc G is invariant under monotonic transformations in domain D ⊆ U if for each u ∈ DN , each z ∈ G(u), and each u′ ∈ DN obtained by a monotonic transformation of u at z, z ∈ G(u′ ).

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Lemma 5 (Maskin, 1999). If there is a mechanism that implements an scc G in Nash equilibria in domain D, then G is invariant under monotonic transformations in domain D. It is impossible to implement an envy-free scf in Nash equilibria in a domain of preferences that contains the quasi-linear domain. Lemma 6 (Velez, 2017). Let Q ⊆ D ⊆ U. No g ∈ F can be implemented in Nash equilibria in D. Proof. We show that no g ∈ F is invariant under monotonic transformations in domain D. By Lemma 5, this implies the result. Suppose by means of contradiction that g ∈ F is invariant under monotonic transformations in domain D. For convenience denote A ≡ {a1 , ..., an }. Assume without loss of generality that m = 0 (the argument can be reproduced with the appropriate translation for any m ∈ R). Let u ∈ QN be the profile such that for each r1 ∈ R, u1 (r1 , a1 ) ≡ (n + n/(n − 1)) − r1 and for each t ∈ {2, ..., n}, u1 (r1 , at ) ≡ −r1 ; and for each i ∈ N \ {1}, each ri ∈ R, and each t ∈ {1, ..., n}, ui (ri , at ) ≡ −ri . Let z ≡ (r, µ) ∈ F (u). Recall that z is Pareto efficient (Svensson, 1983). Thus, µ(1) = a1 . Since for each i ∈ N \ {1}, u1 (z1 ) ≥ u1 (zi ) and ui (zi ) ≥ ui (z1 ), r1 ∈ [0, n]. Let z ≡ (r, µ) ≡ g(u). Let α ∈ [0, n] \ {r1 }. Let uα ∈ QN be the profile such that for each i ∈ N , each ti ∈ R, ui (ti , a1 ) ≡ α + α/(n − 1) − ri and for each i ∈ {2, ..., n}, ui (ti , at ) ≡ −ti . Let z α ≡ (rα , µα ) ≡ g(uα ) (since all agents have the same utility function at uα , z α is such that an agent receives room a1 and pays α for it, and all other agents receive the other rooms and are compensated with α consumption of money). We claim that µα (1) 6= a1 . Suppose by contradiction that µα (1) = a1 . Then, u is a monotonic transformation of uα at z α . Since g is monotonic, z α ∈ g(uα ). This contradicts g being an scf, i.e., being single-valued. Thus, µα (1) 6= a1 . Suppose without loss of generality that for infinitely many α ∈ [0, n] \ {r1 } agent 2 receives room a1 in g(uα ). Let X ⊆ [0, n] \ {r1 } be this set. Let v ∈ QN be such that v2 ≡ u1 and for each i ∈ N \ {2}, vi ≡ u2 . Let s ≡ (t, σ) ≡ g(v). Then, for each α ∈ X \ {t2 }, v is a monotonic transformation of uα at z α . Since g is monotonic, g(v) ⊇ {z α : α ∈ X \ {t2 }}. This contradicts that g is single-valued. Definition 11 (Tadenuma and Thomson, 1995a). Let D ⊆ U. An scc G ⊆ F defined on D satisfies complete indifference if for each u ∈ DN for which there is z ∈ Z such that for each pair {i, j} ⊆ N , ui (zi ) = ui (zj ), we have that z ∈ G(u). Sccs satisfying complete indifference can be essentially single valued, i.e., for each profile of preferences recommend a unique level of welfare. However, among envy-free sccs satisfying this property, F is the only one that can be implemented in Nash equilibria in a domain containing quasi-linear utilities. We present mechanisms that do so in the next section. Proposition 9 (Tadenuma and Thomson, 1995a). Let Q ⊆ D ⊆ U and G ⊆ F satisfy complete indifference. Suppose that there is a mechanism (M, g) that 19

implements G in Nash equilibria in domain D. Then, for each u ∈ D, G(u) = F (u). Proof. By Lemma 5, if G can be implemented in Nash equilibria in domain D, then G is monotonic in domain D. Let u ∈ DN and z ≡ (x, σ) ∈ F (u). Let v ∈ Qn be the profile such that for each pair {i, j} ⊆ N , vi (zi ) = vi (zj ). Thus, for each pair {i, j} ⊆ N , vi = vj . Since G satisfies complete indifferent, G(v) = F (v). Since z ∈ F (v), z ∈ G(v). Since u is a monotonic transformation of v at z, z ∈ G(u).

6.3

Direct revelation mechanisms

The direct revelation mechanism associated with an scf g and domain R ⊆ U is (RN , g). These direct revelation mechanisms are intuitive and natural alternatives for an arbitrator. For instance, Spliddit.org, as of 2017, asks agents for values of rooms (see Sec. 8) and then recommends an allocation that maximizes the minimum utility among agents in the set of envy-free allocations for the reports (Gal et al., 2017). We have concluded that envy-free direct revelation mechanisms are all manipulable (Sec. 6.1). However, an arbitrator may not be concerned that people may lie about their private information. What matters to her is that one of the allocations, or the allocation, that she considers optimal is achieved. In the remainder of this section we show that envy-free direct revelation mechanisms actually perform satisfactorily, in this sense, when agents know each other well. 6.3.1

Direct Nash implementation

If an arbitrator allows agents to report quasi-linear preferences, the direct revelation games associated with each scc G ⊆ F can sustain only envy-free allocations. Proposition 10 (Fujinaka and Wakayama, 2015). Let Q ⊆ R ⊆ U and g ∈ F . Let u ∈ U N T and v ∈ RN a Nash equilibrium of (N, RN , g, u). Then, g(v) ∈ F (u) and g(v) ∈ i∈N F i (v−i , ui ). Proof. Let v ∈ RN be a Nash equilibirum of (N, RN , g, u) and z ≡ (r, µ) = g(v). Let i ∈ N and s ≡ (t, σ) ∈ F i (v−i , ui ). For each vi′ ∈ R, s′ ≡ g(v−i , vi′ ) ∈ F (v−i , vi′ ). By statement 1 in Theorem 5, ui (s′i ) ≤ ui (si ). Thus, ui (zi ) ≤ ui (si ). Since Q ⊆ R and vi is a best response for ui , by statement 2 in Theorem 5, for each ε > 0, ui (zi ) ≥ ui (si ) − ε. Thus, ui (zi ) = ui (si ). Let vi′′ ∈ R be such that zi is the unique maximizer of vi′′ among the bundles in z. Let s′′ ≡ (t′ , σ ′ ) ∈ F i (v−i , vi′′ ). We claim that t′µ(i) = ri . Since z ∈ F (v−i , vi′′ ), vi′′ (s′′i ) ≥ vi (zi ). By Lemma 1, t′µ(i) ≤ ri . By Proposition 6, tA = t′A . If t′µ(i) < ri , then tµ(i) < ri . Thus, ui (si ) = ui (zi ) < ui (tµ(i) , µ(i)). Thus, σ(i) 6= µ(i) and s 6∈ F (v−i , ui ). This is a contradiction. Since t′µ(i) = ri , by Lemma 1, vi′′ (zi ) = vi′′ (s′′i ). Thus, z ∈ F i (v−i , vi′′ ). By Proposition 6,

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rA = t′A = tA . Since s ∈ F (v−i , ui ) and ui (zi ) = ui (si ), z ∈ F (v−i , ui ). Thus, z ∈ F i (v−i , ui ). Thus, for each j ∈ N , ui (zi ) ≥ ui (zj ). Thus, z ∈ F (u). The Nash equilibrium prediction may not be well-defined for the games induced by some scc G ⊆ F . The following lemma allows us to identify the source of this problem. Lemma 7. Let Q ⊆ R ⊆ U and g ∈ F . Let u ∈ U N and v ∈ RN a Nash equilibrium of (N, RN , g, u). Then, for each pair {i, j} ⊆ N , i →v,g(v) j. Proof. By Proposition 10, for each j ∈ N , g(v) ∈ F j (v−j , uj ). Let i ∈ N \ {j}. By lemma 4, i →v−j ,uj ,g(v) j. Thus, i →v,g(v) j. We can conclude that if g ∈ F and z is a Nash equilibrium outcome of (N, RN , g, u), there has to be a profile v ∈ RN such that g(v) = z and for each pair {i, j} ⊆ N , i →v,z j. This imposes some challenges to the existence of equilibria of manipulation games of sccs and scfs. The two-agent case when an arbitrator requires preferences be quasi-linear illustrates it. Example* 1. Suppose that N ≡ {i, j} and A ≡ {a, b}. Let Q ⊆ D ⊆ U and g ∈ F be an scf that assigns room a to agent i whenever possible. Let u ∈ DN be such that agent j receives room a in each allocation in F (u). We claim that the set of Nash equilibria of (N, QN , g, u) is empty. Suppose by contradiction that v ∈ QN is a Nash equilibrium of (N, QN , g, u) and z ≡ g(v). By Lemma 7, i →v,z j. Thus, vi (zi ) = vi (zj ). Symmetrically, vj (zj ) = vj (zi ). Thus, there is an allocation in F (v) in which agent i receives room a. Thus, agent i receives room a at z. This is a contradiction. The following inverse form of Lemma 7 allows us to identify sufficient conditions for the implementation of envy-free correspondences. Lemma 8. Let R ⊆ U, g ∈ F defined on RN , u ∈ U N , and z ∈ F (u). Suppose that there is v ∈ RN such that g(v) = z and for each pair {i, j} ⊆ N , i →v,z j. Then, z is a Nash equilibrium outcome of (N, RN , g, u). Proof. Let i ∈ N . Since z ∈ F (u) ∩ F (v), z ∈ F (v−i , ui ). Since for each j ∈ N , j →v,z i, j →v−i ,ui ,z i. Thus, for each i ∈ N , z ∈ F i (v−i , ui ). By statement 1 in Theorem 5, vi is a best response to v−i for ui in (N, RN , g, u). Thus, v is a Nash equilibrium outcome of (N, RN , g, u) whose outcome is z. This suggests the following definition. Definition* 12. Let R ⊆ U and G ⊆ F defined on R. Then G’s connectedprofile range is the set of allocations z ∈ Z, denoted by C(G, R), for which there is v ∈ RN such that z ∈ G(v) and for each pair {i, j} ⊆ N , i →v,z j. It is always possible to construct envy-free scfs with full connected-profile range when there are at least three agents and quasi-linear reports are possible.8 8 See Azacis ¯ (2008) for another example of an envy-free scf that has full connected-profile range when n ≥ 3.

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Example 2 (Fujinaka and Wakayama, 2015). Suppose that N = {1, ..., n} with n ≥ 3. Let Q ⊆ R ⊆ U. We construct g ∈ F defined on RN whose connectedprofile range is Z. For each z ≡ (x, µ) ∈ Z and for each i ∈ N , let vi ∈ Q be such that vi (zi ) = vi (zi+1 ) and for each l 6= i + 1, vi (zi ) > v(zl ) (in this circular arrangement n + 1 = 1). Then, z ∈ F (v) and for each pair {i, j} ⊆ N , i →v,z j. Moreover, since n ≥ 3, the function that assigns to each z its corresponding profile v is one-to-one. Let g(v) ≡ z. For each other profile u ∈ RN for which g(u) has not been defined yet, let g(u) ∈ F (u). Theorem 6 (Fujinaka and Wakayama, 2015). Let D ⊆ U, R ⊆ U, and g ∈ F defined on RN whose connected-profile range contains F (DN ). Then, (RN , g) implements F in Nash equilibria in D. Proof. Let u ∈ DN . By Proposition 10, each Nash equilibrium outcome of (N, RN , g, u) is in F (u). Let z ∈ F (u). Since g’s connected-profile range contains F (DN ), there is v ∈ RN such that g(v) = z and for each pair {i, j} ⊆ N , i →v,z j. By Lemma 8, z is a Nash equilibrium outcome of (N, RN , g, u). ¯ Theorem 6 generalizes the main result in Azacis (2008), which states that when the admissible domain of preferences is Q, the Nash equilibrium outcome correspondence of the mechanism associated with a particular envy-free scf, which has full connected-profile range, is F .9 Theorem* 7. Let R ⊆ U and g ∈ F defined on RN . Then, (RN , g) implements F in Nash equilibria in U if and only if g’s connected-profile range is Z. Proof. Suppose that (RN , g) implements F in Nash equilibria in U. Let z ∈ Z. Let u ∈ QN ⊆ U N be such that each agent is indifferent among all bundles in z. Thus, z ∈ F (u). Let v ∈ U N be a Nash equilibrium of (N, RN , g, u) such that g(v) = z. By Lemma 7, for each pair {i, j} ⊆ N , i →u,z j. Thus, z ∈ C(g, R). Now suppose that F (U N ) = Z = C(g, R). By Theorem 6, (RN , g) implements F in Nash equilibria in U. An scf’s connected-profile range may not span the whole feasible set. Indeed, when there are only two agents, and reports are required to be quasi-linear, there is no envy-free scf with full connected-profile range. This is so because, in this case, for each z ≡ (x, σ) there is only one v ∈ QN such that z ∈ F (v) and for each pair {i, j} ⊆ N , i →v,z j. Moreover, this v is also the only such profile for the allocation obtained by swapping consumptions at z. An scf’s connected-profile range needs not to be the whole feasible set in order to induce a direct revelation mechanism with non-empty Nash equilibrium correspondence, however. Theorem* 8. Let D ⊆ U, R ⊆ U, G ⊆ F , and f ∈ F . 1. For each u ∈ DN , the set of Nash equilibrium outcomes of (N, RN , f, u) is C(f, R) ∩ F (u). 9 Azacis ¯ (2008) result also states that the “strong” Nash equilibrium outcome correspondence of the mechanism associated with an envy-free scf with full connected-profile range is F.

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2. (RN , f ) implements G in Nash equilibria in D, if and only if, for each u ∈ DN , G(u) = C(f, R) ∩ F (u). Proof. We prove statement 1. Let u ∈ U N . Suppose that v is a Nash-equilibrium of (N, RN , g, u) and z ≡ g(v). By Proposition 10, z ∈ F (u). By Lemma 7, for each pair {i, j} ⊆ N , i →v,z j. Thus, z ∈ C(f, R). Suppose that z ∈ C(f, R) ∩ F (u) is non-empty. Then, there is v ∈ RN such that z = g(v) and for each pair {i, j} ⊆ N , i →v,z j. By Lemma 8, z is a Nash equilibrium outcome of (N, RN , g, u). Statement 2 follows from statement 1. There are always scfs whose direct revelation mechanism has a non-empty Nash equilibrium outcome correspondence when reports are restricted to be quasi-linear. Example 2 and Theorem 6 guarantee it for more than three agents. The following example and Theorem* 8 guarantee it for the two-agent case.10 Example* 3. Suppose that N ≡ {1, 2} and A = {a, b}. Let Q ⊆ R ⊆ U. There are two of bijections between N and A, which we label {µ, σ}. Let {Dµ , Dσ } be a partition of R into two subsets such that each is dense in R.11 Let g ∈ F be such that for each v ∈ QN such that for each pair {i, j} ⊆ N , vi = vj , g(v) ≡ (t, η) is the unique allocation in F (v) such that ta ∈ Dη . For each other profile v ∈ RN , let g(v) ∈ F (v). We claim that for each u ∈ U n , C(g, Q) ∩ F (u) 6= ∅. Let u ∈ R and z ≡ (r, η) ∈ F (u). Suppose first that for each i ∈ N , ui (zi ) = ui (zj ) where j 6= i. For each i ∈ N , let vi ∈ Q be the quasi-linear preference that shares the zi indifference curve with ui . Thus, z ∈ F (v). By Lemma 3, for each s ≡ (t, η ′ ) ∈ F (v), tA = rA . Thus, g(v) ∈ F (u). Now, suppose that for some i ∈ N , ui (zi ) > ui (zj ) where j 6= i. Then, there is ε > 0 such that for each δ ∈ [0, ε], denoting rδ ≡ (ri + δ, rj − δ), sδ ≡ (rδ , η) ∈ F (u). Since Dη is dense in R, there is δ ∈ [0, ε] such that raδ ∈ Dη . Let v be the profile of queasi-linear preferences for which each agent is indifferent between the allotments at sδ . Then, g(v) = sδ and g(v) ∈ F (u). 6.3.2

Limit Nash equilibrium implementation

It is unsatisfactory that Nash quilibrium is not a well-defined prediction for the mechanisms associated with all the envy-free scfs and sccs. Lemma 7 shows that individual incentives lead to equilibrium reports for which there is always one additional allocation that is welfare equivalent to the equilibrium outcome. This means that in order for an allocation to be an equilibrium outcome in a game induced by an envy-free scf, the scf needs to allow agents to signal the allocation they would like to choose among those that are welfare equivalent for their reported preferences. For instance, when there are only two agents, the simple and intuitive scf in Example 2 does not induce a mechanism with well-defined Nash equilibrium prediction. Indeed, when there are two agents, the arbitrator needs to allow agents to report utilities that are not quasi-linear 10 Example* 11 Take

3 can be generalized to n ≥ 2. We omit the lengthy proof. this partition to be, for instance, the sets of rational and irrational numbers.

23

or condition the allocation that she recommends, for instance, on whether the values reported by the agents are rational numbers or not, as in Example 3. An alternative prediction that has been successful in other environments in which intuitive games fail to have Nash equilibria due to the effect of tiebreakers, as in imperfect competition pricing games, is the so-called limit equilibria (Radner, 1980). Intuitively, this solution concept predicts the allocations that have arbitrarily close outcomes of profiles of reports at which agents play “almost” best responses to the actions of the other players. It turns out that this prediction is well-defined for the mechanism induced by each envy-free scf. Surprisingly, there are significant differences between Nash equilibrium and limit equilibrium prediction for these games. Definition 13. Let R ⊆ U, u ∈ U N , g an scf, and ε > 0. A profile v ∈ RN is an ε-equilibrium of (N, RN , g, u) if no agent can gain more than ε in utility by choosing a different action in R.12 An allocation z is a limit Nash equilibrium outcome of (N, RN , g, u) if there is a sequence of its ε-equilibrium outcomes that converges to z as ε vanishes. Contrary to the Nash equilibrium, limit equilibria of the mechanism associated with an envy-free scf may not be envy-free when reports are unrestricted. Indeed, they may even violate Pareto efficiency. δ 2

2ε 3

2δ

c

0

2δ

c rs

2δ

rs

bc

u1

ut

rs

b

u2 a

uε1 b

ut

bc

rs

uε2

u3

uε3

a rs

0

rs

2ε (a)

ut

bc

2ε ε 3 3

(b)

Figure 2: Point 0 on the axis corresponding to a room, say a, represents the bundle (0, a). Each point y < 0 on the axis corresponding to a room, say a, represents the bundle in which the agent pays |y| to receive the room, i.e., the bundle (−y, a). Each point y > 0 on the axis corresponding to an room, say a, represents the bundle in which the agent receives a transfer |y| with the room, i.e., the bundle (−y, a). Segments connect bundles that are indifferent for ui (“indifference curve”); (a) illustrates indifference sets for utility profile u; (b) illustrates indifference sets for utility profile uε , which is a 2ε/3-equilibrium of (N, U N , g, u).

Example 4 (Velez, 2015). Let N ≡ {1, 2, 3}, A ≡ {a, b, c}, and m ≡ 0. Let g ∈ F 2 be defined as in Example 2. Furthermore, let g be such that it assigns room a to agent 1 for each profile for which it is possible and has not been determined in the example. Fix δ > 0. Let u ∈ U N be a utility profile with indifference sets as in Fig. 2 (a). Let z ∈ Z be the allocation at which z1 ≡ (0, a), z2 ≡ (0, b), and z3 ≡ (0, c). Since u1 (z2 ) > u1 (z1 ), we have that z 6∈ F (u). 12 That

is, for each i ∈ N and each vi′ ∈ R, ui (g(v−i , vi′ )) ≤ ui (g(v)) + ε.

24

Moreover, since u1 (δ/2, b) = u1 (z1 ) and u2 (−δ/2, a) > u2 (z2 ), z is not Pareto efficient for u. Let uε be a utility profile with indifference sets shown in Fig. 2 (b); uε1 is quasi-linear. Notice that both agents 1 and 3, under preferences uε , are indifferent among all three bundles z1 , z2 and z3 . Agent 2 strictly prefers z2 to both z1 and z3 under preferences uε2 . Thus, z ∈ F (uε ), 1 →uε ,z 2, and 3 →uε ,z 2. By Proposition 4, z ∈ F 2 (uε ). Thus, g(uε ) = z. Now, since agent 2 also strictly prefers z2 to both z1 and z3 under the true preferences u2 , then z ∈ F (uε−2 , u2 ), 1 →uε−2 ,u2 ,z 2, and 3 →uε−2 ,u2 ,z 2. Thus, z ∈ F 2 (uε−2 , u2 ). By Theorem 5, uε2 is a best response for agent 2 to uε−2 . Let z ′ be the allocation such that z1′ ≡ (−2ε/3, a), z2′ ≡ (2ε/3 + 2δ, b), and z3′ ≡ (−2δ, c). Observe that u1 (z1′ ) > max{u1 (z2′ ), u1 (z3′ )}; uε2 (z2′ ) = uε2 (z1′ ) = uε2 (z3′ ); and uε3 (z3′ ) = uε3 (z1′ ) > uε3 (z2′ ). Thus, z ′ ∈ F (uε−1 , u1 ), 2 →uε−1 ,u1 ,z′ 1, and 3 →uε−1 ,u1 ,z′ 1. Thus, {z ′ } = F 1 (uε−1 , u1 ). By Theorem 5, for each u′1 ∈ U, u1 (z1′ ) ≥ u1 (g(uε−1 , u′1 )). Thus, by changing her report at uε , agent 1 cannot achieve an allocation that is preferred to (−2ε/3, a), i.e., the bundle obtained by rebating her 2ε/3 at z. Finally, let zb be the allocation such that zb1 ≡ (ε, a), zb2 ≡ (2ε, b), and zb3 ≡ (−ε, c). Observe that uε1 (b z1 ) = uε1 (b z3 ) > uε1 (b z2 ); uε2 (b z2 ) = uε2 (b z1 ) > uε2 (b z3 ); ε and u3 (b z3 ) > max{u3 (b z1 ), u3 (b z2 )}. Thus, zb ∈ F (u−3 , u3 ) and 2 →uε−3 ,u3 ,bz z} = F 3 (uε−3 , u3 ). By Theorem 5, for each u′3 ∈ U, 1 →uε−3 ,u3 ,bz 3. Thus, {b u3 (b z3 ) ≥ u3 (g(uε−3 , u′3 )). Thus, by changing her report at uε , agent 3 cannot achieve an allocation that is preferred to (ε, c), i.e., the bundle obtained by rebating her ε at z. Thus, uε is a 2ε/3-Nash equilibrium of (N, U N , g, u). Recall that z = g(uε ). Thus, z is the limit of a sequence of ε-equilibrium outcomes of (N, U N , g, u) as ε vanishes. When an arbitrator restricts agents’ reports to quasi-linear utility functions, the mechanism associated with each envy-free scf implements the envy-free scc in limit equilibria. Theorem* 9. Let g ∈ F be defined on QN . Then (QN , g) implements F in limit equilibria in U. This theorem generalizes a result of Velez (2015) that states only implementation in Q. We note that Velez (2015)’s argument applies unmodified to implementation in U. We omit its proof.

7

Manipulability of envy-free direct revelation mechanisms

A straightforward consequence of Proposition* 8 is that no g ∈ F is strategyproof in a domain D ⊇ Q. In particular, this proposition shows that with a possible single exception, for each profile u ∈ QN , at least one agent may benefit by lying about her preferences in (N, QN , g, u). This does not imply that all envy-free scfs are “equally manipulable,” however. First, suppose that 25

{f, g} ⊆ F and u ∈ QN . It is still possible that lying is in the best interest of a different set of agents at u when one uses, in a direct revelation form, f than when one uses g. Second, even if at u lying is in the best interest for the same number of agents, or the same set of agents, when one uses either f or g, it is still possible that the intensity of preference for the manipulation by these agents is different for these scfs. Definition 14 (Andersson et al., 2014a). Let D ⊆ U, f a scf defined on D, u ∈ DN , and i ∈ N . Agent i can manipulate f at u if there is vi ∈ D such that ui (f (u−i , vi )) > ui (f (u)). We would like to rank scfs in terms of their manipulability. There are several ordinal natural alternatives to do so. Definition 15 (Andersson et al., 2014a). Let D ⊆ U and f and g defined on DN . For each u ∈ U N , let Mf (u) be the set of agents who can manipulate f at u. 1. f is profiles-inclusion less manipulable than g if for each u ∈ DN for which Mf (u) 6= ∅, Mg (u) 6= ∅. 2. f is agents-inclusion less manipulable than g if for each u ∈ DN , Mf (u) ⊆ Mg (u); f is agents-inclusion strictly less manipulable than g, if additionally the inclusion is proper for some u ∈ U N . 3. f is agents-counting less manipulable than g if for each u ∈ DN , |Mf (u)| ≤ |Mg (u)|; f is agents-counting strictly less manipulable than g, if additionally the inequality is strict for some u ∈ U N . The following is a straightforward consequence of Theorem 5. Lemma 9 (Andersson et al., 2014a). Let Q ⊆ D ⊆ U, f ∈ F defined on D, u ∈ DN , and i ∈ N . Then, agent i cannot manipulate f at u if and only if f (u) ∈ F i (u). The profiles-inclusion manipulation order does not discriminate among envyfree scfs. Proposition 11 (Andersson et al., 2014a). Let Q ⊆ D ⊆ U and f ∈ F defined on D. For each g ∈ F defined on D, f is profiles-inclusion less manipulable than g. Proof. Let u ∈ DN be such that Mf (u) 6= ∅. Let i ∈ Mf (u). By Lemma 9, f (u) 6∈ F i (u). Denote (r, µ) = f (u) and let s ≡ (t, σ) ∈ F i (u). Thus, ui (si ) > ui (f (u)). By Lemma 1, rA 6= tA . By Lemma 1, there is j ∈ N such that uj (sj ) > uj (f (u)). Thus, F i (u) ∩ F j (u) = ∅. Let g ∈ F . By Lemma 9, if i cannot manipulate g at u, j can manipulate g at u. Among all envy-free scfs, the agents-inclusion minimally manipulable scfs select, for each profile, an allocation that is optimal for an agent among the envy-free allocations for the profile. 26

Proposition 12 (Andersson et al., 2014a). Let Q ⊆ D ⊆ U and f ∈ F defined on D. Then, there is no g ∈ F that is agents-inclusion strictly less manipulable than f if and only if for each u ∈ DN , there is i ∈ N such that f (u) ∈ F i (u). Proof. Let u ∈ DN . Suppose that there is i ∈ N such that f (u) ∈ F i (u). By Theorem 5, i ∈ N \ Mf (u). Let g ∈ F . Suppose that Mg (u) ⊆ Mf (u). Thus, i ∈ N \Mg (u). By Theorem 5, g(u) ∈ F i (u). Let j ∈ N \Mg (u). By Theorem 5, g(u) ∈ F j (u). Since {f (u), g(u)} ⊆ F i (u) and F i is a maxmin envy-free scf, by Proposition 4, uj (f (u)) = uj (g(u)). Thus, f (u) ∈ F j (u). Theorem 5, j ∈ N \ Mf (u). Thus, N \ Mg (u) ⊆ N \ Mf (u). Thus, Mg (u) = Mf (u). Thus, f is agents-inclusion less manipulable than g. Suppose that there is no g ∈ F that is agents-inclusion strictly less manipulable than f . Let u ∈ U N . We claim that there is i ∈ N such that f (u) ∈ F i (u). Suppose first that N \ Mf (u) 6= ∅, and let i ∈ N \ Mf (u). By Theorem 5, f (u) ∈ F i (u). Suppose then that N \ Mf (u) = ∅. Let g be the scf defined as follows. For each v ∈ U N \ {u}, let g(v) = f (v), and g(u) ∈ F i (u) for some i ∈ N . Let v ∈ U N \ {u} and j ∈ Mg (v). By Theorem 5, g(v) = f (v) 6∈ F j (v). By Theorem 5, j ∈ Mf (v). Now, let i ∈ N be such that g(u) ∈ F i (u). By Theorem 5, i ∈ N \ Mg (u). Thus, Mg (u) ( Mf (u) = N . Thus, g is agents-inclussion striclty less manipulable than f . This is a contradiction. Among all envy-free scfs, the agents-counting minimally manipulable scfs select, for each profile, an envy-free allocation that is optimal for a set of agents with maximal cardinality. Proposition 13 (Andersson et al., 2014a). Let Q ⊆ D ⊆ U and f ∈ F defined on D. Then, there is no g ∈ F that is agents-counting strictly less manipulable than f if and only if for each u ∈ DN , there is i ∈ N such that f (u) ∈ F i (u) and i ∈ arg max |{j ∈ N : F j (u) = F i (u)}|. Proof. Let u ∈ U N and g ∈ F . By Theorem 5, N \ Mg (u) = {j : g(u) ∈ F j (u)}. By Proposition 4, if F i (u) ∩ F j (u) 6= ∅, then F i (u) = F j (u). Thus, if there is i ∈ N \ Mg (u), g(u) ∈ F i (u) and N \ Mg (u) = {j : F j (u) = F i (u)}

(2)

Suppose that for each u ∈ DN there is i ∈ N such that f (u) ∈ F i (u) and i ∈ arg max |{j ∈ N : F j (u) = F i (u)}|. Thus, for each u ∈ DN , |N \ Mg (u)| ≤ |N \ Mf (u)|. Thus, |Mf (u)| ≤ |Mg (u)|. Thus, f is agents-counting less manipulable than g. Suppose that there is no g ∈ F that is agents-counting strictly less manipulable than f . Thus, there is no g ∈ F that is agents-inclusion strictly less manipulable than f . By Proposition 12, for each u ∈ U N , there is i ∈ N such that f (u) ∈ F i (u). Let u ∈ U N . We claim that i ∈ arg max |{j ∈ N : F j (u) = F i (u)}|. Suppose by contradiction that there is k ∈ N such that |{j ∈ N : F j (u) = F k (u)}| > |{j ∈ N : F j (u) = F i (u)}|. Let g be the scf defined as follows. For each v 6= U N \ {u}, let g(v) = f (v), and g(u) ∈ F k (u). By (2), Mg (u) = |N | − |{j ∈ N : F j (u) = F k (u)}| < |N | − |{j ∈ N : F j (u) = F i (u)}| 27

and for each v 6= u, Mf (v) = Mg (v). Thus, g ∈ F is agents-counting strictly less manipulable than f . This is a contradiction. In order to measure intensity of preference to manipulate a mechanism let us restrict to agents whose preferences are quasi-linear. Definition 16. Let f be defined on QN , u ∈ QN , and i ∈ N . Agent i’s maximal gain from manipulation of f at u is χ(f, u, i) ≡ sup ui (f (u−i , vi )) − ui (f (u)). vi ∈Q

By Theorem 5 an agent’s maximal gain from manipulation of an envy-free scf is finite for each utility profile. Definition 17. Let f and g be defined on QN . Then, f is less manipulable than g with respect to the maximal gain from manipulation if for each u ∈ QN , max χ(f, u, i) ≤ max χ(g, u, i); i∈N

i∈N

f is strictly less manipulable than g with respect to the maximal gain from manipulation if additionally the inequality is strict for some utility profile. Surprisingly, welfare-wise there is a unique minimally manipulable envy-free scf with respect to the maximal gain from manipulation. Definition 18. Let f ∗ be the scf defined on Q as follows. Let u ∈ QN . For each i ∈ N , let z i ∈ F i (u). Let (r, µ) ≡ z 1 ∨ (z 2 ... ∨ (z n−1 ∨ z n )...). Let fP∗ (u) be the feasible horizontal translation of (r, µ), i.e, f ∗ (u) ≡ ((ri + (m − 13 i∈N ri )/n), µ). Theorem 10 (Andersson et al., 2014b). Let g ∈ F be defined on QN . If there is u ∈ QN and i ∈ N such that ui (g(u)) 6= ui (f ∗ (u)), then f ∗ is strictly less manipulable than g with respect to the maximal gain from manipulation. Proof. Let u ∈ QN and z ≡ (x, µ) = g(u). Since each envy-free allocation is Pareto efficient, there is i ∈ N such that ui (g(u)) ≤ ui (f ∗ (u)). By hypothesis, this inequality holds strictly for at least one profile. Let z i ∈ F i (u). By Theorem 5, χ(g, u, i) = ui (z i ) − ui (g(u)) ≥ ui (z i ) − ui (f ∗ (u)) = χ(f ∗ , u, i). Thus, max χ(g, u, j) ≥ max χ(f ∗ , u, j). j∈N

j∈N

The inequality holds strictly when ui (g(u)) < ui (f ∗ (u)). Thus, f ∗ is strictly less manipulable than g with respect to the maximal gain from manipulation. 13 The computational complexity of calculating an element of F i (u) when u ∈ QN is polynomial (Andersson and Ehlers, 2013; Gal et al., 2017).

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8

Discussion

Given our emphasis on the study of individual incentives, we have presented only the results about the structure of the set of envy-free allocations that are essential for this purpose. Existence of envy-free allocations has been extended to environments with consumption externalities (Velez, 2016) and with price externalities (Gale, 1984).14 These allocations may not be Pareto efficient when there are externalities. However, for some restrictions on preferences, one can guarantee no-envy implies Pareto efficiency (Velez, 2016). The complexity of calculating an envy-free allocation when preferences are quasi-linear has been determined to be polynomial (Aragones, 1995; Klijn, 2000; Gal et al., 2017). Several papers have studied the existence and structure of envy-free sccs and scfs satisfying the principle of solidarity, i.e., a change in parameters of a problem leads to a change in welfare of the agents who are not responsible for the new conditions in the same direction (Thomson, 1983). Most notably, Alkan et al. (1991) and Alkan (1994) study solidarity notions under the addition of rooms; Alkan (1994) and Tadenuma and Thomson (1995b) study solidarity notions under the arrival of new agents; Tadenuma and Thomson (1991) study the consistency axiom, which can be interpreted as a form of solidarity when allocations have to be reassessed for a subset of agents with their corresponding initial allotments; and Velez (2017) studies solidarity when budget changes, i.e., when rent increases, all agents should contribute in welfare terms. We have restricted our analysis to situations in which agents may report quasi-linear preferences and to situations in which agents are required to provide quasi-linear reports. The market design practitioner may find significant reasons to ask for reports in different domains. The following are two concrete examples for which it would be interesting to study incentive issues. Example 5. In order to elicit preferences, Spliddit.org asks agents to distribute the rent to collect among the different rooms in the house and interprets these amounts as values. In our formal language, for a given house rent m, this system asksPagents to report a vector of numbers p ≡ (pa )a∈A satisfying that p ≥ 0 and a∈A pa = m. Spliddit.org then interprets this message as the report that the agent has preferences represented by the utility function (ri , a) ∈ R × A 7→ pa − ri .15 This preference elicitation is arguably intuitive and also guarantees that at each envy-free allocation for each profile of reports no agent is paid to receive a room, which is not guaranteed if one elicits preferences in the quasi-linear domain. Example 6. A relevant practical issue is that agents may be budget constrained and thus a quasi-linear report, which neglects any income effect, may not allow an agent to report her true preferences. The results we surveyed imply that quasi-linear reports are expressive enough if agents know each other well and 14 No-envy in these results with externalities means that no agent prefers the allocation obtained by swapping her consumption with another agent. 15 P Spliddit.org preferences are those for which our normalized representation satisfies that a∈A va ≤ m.

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are strategic. However, there is a value in allowing agents report their true preferences. For instance, an agent who is sincere will never prefer the allotment of any other agent to her own when an envy-free mechanism is operated. It is, of course, not practical to ask agents for reports in an infinite dimensional space as U. However, there are variations of the quasi-linear domain that may be rich enough and allow agents make reports closer to their true preferences. For instance, Procaccia et al. (2018) propose to also ask for budget constraints. Then they construct polynomial algorithms that produce an allocation that maximizes some criteria of justice, prominently the minmax utility, among the envy-free allocations, with respect to the reported quasi-linear preferences, that minimize the maximal budget violation. Another practical variation would be to elicit from each agent an underlying quasi-linear preference, a budget constraint, and an additional number ρ that reflects the agent’s degree of difficulty to violate the budget constraint (as the interest rate that the agent needs to pay over the amount of money borrowed above her budget constraint). Our formulation of the model also requires that the number of agents be equal to the number of rooms. The situations in which there are more agents than rooms can be seen as a preference restriction in our model. All results generalize to these environments. If there are more rooms than agents, but agents retain unit demand of rooms, envy-free allocations may not be Pareto efficient (Alkan et al., 1991). There are always envy-free and Pareto efficient allocations for each economy, however (Alkan et al., 1991; Velez, 2017). It is an open question to determine the extent to which the results concerning incentives that we present generalize to these environments. We have concentrated on the implementation of envy-free scfs and sccs by means of simultaneous-move mechanisms. Two of our general conclusions are that dominant strategy implementation is impossible and that no finite subsolution of F can be implemented. Two alternative approaches that qualify these results are worth mentioning. First, an extended notion of strategy-proofness to sccs is compatible with no-envy in a special case of our model in which all rooms are equal (Bochet and Sakai, 2007). Second, if one considers sequential mechanisms, it is possible to implement envy-free scfs, in sub-game perfect equilibrium, in the two-agent two-room case (Nicol´ o and Velez, 2017). It is an open question to study the extent to which these results extend to our general environment. Finally, in our study of direct revelation mechanisms we have restricted our attention to protocols induced by social choice functions, i.e., the judgement of an arbitrator who is able to select as optimal one particular allocation for each preference profile. One can also think of an arbitrator who has a coarse judgement about the optimality of allocations and wants to use this judgement as the basis for an allocation mechanism. There are multiple ways to model and analyze this situation. One alternative is to consider generalized games that allow for multiple outcomes and to define solutions for these generalized games (c.f Tadenuma and Thomson, 1995a; Bevi´a, 2010). A second approach is to enlarge the strategy space of the agents and consider mechanisms based on 30

an scc in which agents are allowed to report preferences, an additional message that allows them to coordinate on an allocation that is optimal for the reported preferences, and a tie breaker that is used in case agents do not coordinate on an allocation (c.f. Velez, 2011; Velez and Thomson, 2012). Our analysis can be easily adapted to these approaches, obtaining similar conclusions. That is, Nash equilibria of the manipulation of envy-free sccs are envy-free with respect to true preferences; and if an scc has a “rich” connected profile range, its Nash equilibrium outcome correspondence is non-empty.

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