Expressive mechanisms for equitable rent division on a budget
aa r X i v : . [ ec on . T H ] A p r Expressive mechanisms for equitable rent division on abudget ∗ Rodrigo A. Velez † Department of Economics, Texas A&M University, College Station, TX 77843
April 20, 2020
Abstract
We study the incentive properties of envy-free mechanisms for the allocationof rooms and payments of rent among financially constrained roommates. Eachagent reports her values for rooms, her housing earmark (soft budget), and anindex that reflects the difficulty the agent experiences from having to pay overthis amount. Then an envy-free allocation for these reports is recommended.The complete information non-cooperative outcomes of each of these mecha-nisms are exactly the envy-free allocations with respect to true preferences ifand only if the admissible budget violation indices have a bound.
JEL classification : C72, D63.
Keywords : budget constraints, equitable rent division, market design, algo-rithmic game theory, no-envy, quasi-linear preferences.
We study the problem of equitably allocating the rooms and payments of rent amongroommates who lease a house. An allocation is envy-free if no agent prefers theallotment of any other agent to her own (Foley, 1967). Envy-free allocations in ourenvironment coincide with the competitive allocations from equal endowments, anintuitively equitable institution, and are always Pareto efficient (Svensson, 1983). ∗ Thanks to Ariel Procaccia and seminar participants at the 2019 Economic Design and Algo-rithms in St. Petersburg Workshop and UT Austin for useful comments. An earlier version ofthis paper also studied computational complexity in equitable rent division with soft budgets. Wepresent the complexity results now in a companion paper (Velez, 2020). All errors are my own. † [email protected]; https://sites.google.com/site/rodrigoavelezswebpage/home Spliddit.org (Goldman and Procaccia, 2014), is based onthe elicitation of values for each room. Technically, these values determine a quasi-linear preference profile for which the system calculates an envy-free allocation (seeSec. 3 for details). This mechanism, and in general the mechanisms that elicit quasi-linear reports, have two appealing features. First, they can be deployed in practice.Agents’ reports are finite dimensional. Moreover, there are polynomial algorithmsto calculate an envy-free allocation for quasi-linear reports (c.f., Gal et al., 2017).Second, agents manipulation when reports are required to be quasi-linear has a clearlimit: If agents know each other well, the set of non-cooperative outcomes from theirmanipulation is exactly the set of envy-free allocations with respect to their true,potentially not quasi-linear, preferences (Velez, 2015, 2018). Requiring agents report quasi-linear preferences has a downside, however. Thisdomain of preferences requires that each agent be indifferent between increases ofrent on bundles that are indifferent for her. This may be at odds with agents’financial constraints. Suppose that an agent has a housing earmark of $500, and isindifferent between paying $500 for a large room and $300 for a small one. If theagent reports this preference, a quasi-linear based envy-free mechanism may assignthe agent the big room paying $700 while another agent receives the small roompaying $500. In this situation, the agent may prefer the allotment with the smallroom to her own allotment. Indeed, receiving the large room and paying $700 forit, requires the agent pays $200 above the amount she has earmarked for housing.This may require the agent accumulates debt, or repurposes other funds that shehad earmarked for entertainment, transportation, etc.The restrictive nature of quasi-linear preferences is not a simple theoretical possi- Incentives under incomplete info in our environment have been analyzed in the two-agent casein which the problem is equivalent to the allocation of a single object. With symmetric indepen-dent private values, there are efficient, individually rational, and incentive compatible mechanisms(Cramton et al., 1987). This efficiency result breaks down for more general information structures(see Moldovanu, 2002, for a survey). This example requires that there be at least another room that receives the residual rent rebate.
I’ve tried to use your fair calculator, but it did not work in my case.In our situation, I am the guy with the most tight [sic] budget. Unfor-tunately, your system does not take into account the maximum budgetrestrictions. I was assigned an option that was too expensive for me, soit did not help. Please advise if there is a way to use the system, takingthat kind of limitation into account.
There is a clear desire of this frustrated user to inform the mechanism designerabout his or her real preferences, and that the mechanism designer use this infor-mation in a meaningful way. A natural response to this request is to ask agents fortheir maximum budget restrictions and determine it they are compatible with no-envy. This is actually computationally feasible when the underlying preferences arequasi-linear (Procaccia et al., 2018). This does not resolve the issue when no-envyand maximum budget constraints are not compatible, however.One can think of different responses to this issue based on the principle of no-envy. For instance, one could proceed sequentially. Elicit valuations and maximalbudget constraints and determine whether no-envy is compatible with the budgetconstraints. If the answer is negative, no-envy requires that at least an agent needs toviolate her budget constraint. Thus, one needs to somehow elicit how difficult it is forthe agents to go over their budget and determine an allocation that is envy-free withrespect to these preferences. Alternatively, one could proceed with a simultaneous-move mechanism. Ask agents upfront to report their budget constraints and toquantify how difficult it is to go over these budgets. Then, compute an envy-freeallocation based on these preferences (one could give priority to allocations that donot violate the budget constraints).There is an obvious trade off between these two approaches. A sequential mech-anism introduces a layer of strategic complexity for the agents. Asking for richerreports upfront may also force more complex message spaces that are only necessaryin some cases in the sequential mechanism. We believe that both approaches are rea-sonable and its study is valuable. In this paper we concentrate on the simultaneous-move approach. There are two reasons to do so. First, this is the first step that onewould need to make the sequential mechanism operational. That is, one needs todetermine a reasonable preference domain that captures the difficulty that agents3ave to go over their budget. Second, as part of our analysis, we address the elic-itation of reports in our mechanisms. We believe that we succeed in keeping theircomplexity in check and thus our mechanisms have practical value.Our proposal is to extend the domain of admissible preferences in Spliddit’ssystem or any other direct revelation mechanism that restricts reports to be quasi-linear. In our domain, the budget-constrained quasi-linear preferences , an agent’spreference is determined by her valuation of rooms and two additional parameters:her housing earmark (soft budget) and an index that penalizes in a linear way theutility the agent gets from paying above this amount. Thus, the agent’s utility froma given allotment is the difference between the value of the room assigned to herand the real value of the rent paid including the penalty for going over the budgetif that is the case. Our scheme allows agents report more accurately their financialdifficulties.We proceed in two steps. In Sec. 3 we address the practical implementation ofour mechanisms. We propose explicit intuitive elicitation schemes for our domainof admissible preferences. Additionally, we discuss the computational complexity ofcalculating an envy-free allocation in our domain.Then, in Sec. 4 we analyze the incentive issues generated by our enlargement ofthe space of reports. Our benchmark is the complete information incentive propertyof the envy-free mechanisms with quasi-linear reports. That is, the set of non-cooperative outcomes of the complete information games induced by these mecha-nisms, are exactly the set of envy-free allocations with respect to true preferences.It turns out that if we do not impose a bound on the budget violation penalty,so agents can arbitrarily exaggerate their disutility of increases of rent over theirbudget, this property is lost. More precisely, if there are at least three agents whocan report arbitrarily high budget violation penalties, one can always constructan envy-free social choice function whose direct revelation game has Pareto ineffi-cient non-cooperative outcomes (Proposition 1). The good news is that if there isa bound on the value of the budget violation penalty that agents can report, the Budget constraints in rent division were first studied, in the related problem of partnershipdissolution, by Nicol´o and Velez (2017). Our domain of preferences was first proposed by Velez(2018) and independently studied from an algorithmic perspective by Arunachaleswaran et al.(2019). Procaccia et al. (2018) introduced the equitable rent division problem with hard budgetsand studied the complexity of determining whether there is an envy-free allocation that does notviolate these budgets. When there is no such allocation, their algorithm resorts to recommendan allocation with respect to the stated quasi-linear values without taking into consideration theagents’ preferences for paying over their budget. expressiveness of a mechanism. As we elaboratein detail in Sec. 3.1 and 5, a mechanism designer who constructs a direct revelationmechanism should look beyond the direct incentives of strategic agents who interactwith the system. The designer should take into account the extent to which agents’reports can encode their real preferences. At the same time, the designer needs tokeep incentives in check and computational complexity feasible.The remainder of the paper is organized as follows. Sec. 2 introduces the model.Sec. 3 introduces the mechanisms we propose and discusses their practical imple-mentation. Sec. 4 presents our results and discusses the technical challenges involvedin establishing them. Sec. 5 discusses the extension of our results to an environ-ment populated by both unconditionally truthful agents and strategic agents andconcludes.
A set of n objects, A , that we refer to as rooms, is to be allocated among a set of n agents N ≡ { , ..., n } . Generic rooms are a, b, ... . Each agent is to receive a roomand pay an amount of money for it. The generic allotment is ( r a , a ) ∈ R × A . When r a ≥ a . Weallow for negative payments of rent, i.e., r a <
0. In this way we allow for alternativeinterpretations of our model, as the allocation of tasks and salary. Each agent hasa continuous preference on her outcome space, i.e., a complete and transitive binaryrelation on R × A that is represented by a continuous utility function. The genericutility function is u i . We assume throughout that preferences satisfy the followingtwo properties: (1) money-monotonicity , i.e., for each consumption bundle ( r a , a )and each δ > u i ( r a + δ, a ) < u i ( r a , a ); and (2) compensation assumption , i.e., for One can interpret budget constraints as self-reported subsistence wages. Our ordinal assumptions on preferences imply existence of continuous representations. a and b , and each r a ∈ R , there is t b ∈ R such that u i ( r a , a ) = u i ( t b , b ). We denote the domain of these preferences by U .Individual payments should add up to m ∈ R , the house rent. An allocation ofthe rooms and rent is a pair ( r, σ ) where σ : N → A is a bijection and r ≡ ( r a ) a ∈ A ∈ R A is such that P a ∈ A r a = m . The allotment assigned to agent i by allocation( r, σ ) is ( r σ ( i ) , σ ( i )). We denote the set of allocations by Z , with generic element z .With this notation, agent i receives z i at z . An allocation z is envy-free for u if noagent prefers the consumption of any other agent to her own at the allocation. Thatis, for each pair { i, j } ⊆ N , u i ( z i ) ≥ u i ( z j ). The set of these allocations is F ( u ).An allocation z ∈ Z is Pareto efficient for u if there is no z ′ ∈ Z that each agentweakly prefers to z and at least one agent strictly prefers. In our environment, eachenvy-free allocation for u is Pareto efficient for u (Svensson, 1983).Let D ⊆ U . A social choice function (scf) defined on D N associates an allocationwith each u ∈ D N . The generic scf is g : D N → Z . We say that g is envy-free if foreach u ∈ D N , g ( u ) ∈ F ( u ).An scf g : D N → Z induces a revelation mechanism in which each agent reports apreference in D and the outcome is the allocation recommended by g for the reports.We denote this mechanism by ( N, D N , g ) and the induced complete informationrevelation game for preference profile u ∈ U N by ( N, D N , g, u ). Our main results concern the non-cooperative outcomes of revelation mecha-nisms associated with envy-free scfs. It is well-known that these games may nothave pure-strategy Nash equilibria (see Velez, 2018). This happens because indi-vidual incentives in these games lead agents to profiles of reports in which multipleallocations achieve the same level of utility as the allocation chosen by the scf.This generates a discontinuity of payoffs because the scf necessarily acts as a tiebreaker on these allocations. An intuitive solution to this problem is to considerlimit Nash equilibrium outcomes, i.e., allocations that are in the proximity of rec-ommendations for reports that constitute almost Nash equilibria (see Velez, 2018,for an extensive discussion). Essentially, these are outcomes in which agents canalmost perfectly non-cooperatively coordinate by incurring a negligible sacrifice oftheir individual welfare. Formally, let
D ⊆ U , u ∈ U N , g : D N → Z , and ε >
0. Aprofile v ∈ D N is an ε -equilibrium of ( N, D N , g, u ) if no agent can gain more than ε in utility by choosing a different action, i.e., for each i ∈ N and each v ′ i ∈ D , u i ( g ( v − i , v ′ i )) ≤ u i ( g ( v )) + ε . An allocation z is a limit Nash equilibrium outcome of Note that we refer to ( N, D N , g, u ) as a revelation game, even though u may not be in D N . N, D N , g, u ) if there is a sequence of its ε -equilibrium outcomes that converges to z as ε vanishes. Agent i ’s preference on R × A is quasi-linear if it is represented by a function u i ( r a , a ) = v ia − r a , where ( v ia ) a ∈ A ∈ R A are the values that agent i assigns to each room. Let us denotethe domain of these preferences by Q .The current practice of equitable rent division is based on the elicitation ofquasi-linear reports and the computation of an envy-free allocation for these reports.These mechanisms have the following incentive property. Theorem 1 (Velez, 2018) . Let g be an envy-free scf defined on Q N . Then, for each u ∈ U N the set of limit Nash equilibrium outcomes of ( N, Q N , g, u ) is F ( u ) . As described in the introduction, direct feedback by Spliddit users points to theneed to allow agents report their financial constraints and to use this informationto select a recommendation. Doing this seems a natural reaction by a mechanismdesigner to these requests. As we argue below, this is somehow at odds with theparadigm of analysis in classical mechanism design, however. Thus, it is grantedthat we explore the normative foundation of this endeavor.The standard analysis of a mechanism consists on determining whether its in-dividual incentives are aligned with social objectives. It is irrelevant whether theusers of a system need to directly report their private information, or an abstractmessage that allows them to achieve the necessary coordination. Indeed, whenevera revelation principle applies to the environment of interest, the restriction to directrevelation mechanisms is considered only an analytical shortcut.In reality, mechanisms in which users directly provide, at least partially, theirprivate information, have special practical value. “Direct” reports have a clearmeaning for users. They are usually answers to questions for which there is a trueanswer that the user knows. These reports can also be tied to the normative goals themechanism designer targets and that may incentivize agents to use the system (see7rocaccia., 2019, for a related discussion). Spliddit, for instance, asks each agent toreport room values by distributing the rent of the house among the different rooms.Even if the agent’s preferences are not quasi-linear, this is equivalent to asking forthe agent’s true “indifference curve” composed of bundles whose rent add up tothe house rent (this is uniquely defined for very weak requirements on preferences).Spliddit also provides a plain words explanation of no-envy, and details in its demohow a user who truthfully answers the queries of the system should receive anallotment that is no worse than that of the other roommates.There is evidence that when agents are asked for their private information, somemay just report it truthfully, independently of the incentives not to do so. Thus,a mechanism designer who is taking advantage of the natural appeal of a directrevelation mechanism should not only align incentives with social objectives, butalso be mindful of the possibility that some users may be truthful. More precisely,we should not be content only with guaranteeing that agents can only coordinate onsocially optimal allocations. For instance, we know this is so for Spliddit. Indeed,if the agent raising the complaint that Procaccia et al. (2018) report (see Sec. 1) istruly frustrated because he or she finds that some other agent is receiving a betterdeal, there must be an alternative quasi-linear report that would allow the agentto obtain a comparable deal (Andersson et al., 2014a,b; Fujinaka and Wakayama,2015; Velez, 2018). The fact that this is so should not restrain us from taking theconcern of this agent at face value.One can think of different ways in which a mechanism designer can achieve this. For our equitable rent division problem, the mechanism designer aims to obtain anenvy-free allocation, i.e., is implicitly promising the agents that if they report theirprivate information, they will not prefer the allotment of the other agents to theirown. Thus, a natural minimal requirement is that this objective be achieved at theindividual level for truthful agents. That is, the mechanism designer should askagents report detailed enough private information, so when a given agent reports For instance in the “mind games” of Kajackaite and Gneezy (2017), a participant is asked tochoose a number in private, then roll a dice, and report if the choice and the dice coincided. Theparticipant receives a payoff $ x > For instance some authors have advocated for the use of strategy-proof direct revelation mech-anisms to protect the less sophisticated users who may be unconditionally truthful (Pathak andS¨onmez, 2008). Unfortunately, dominant strategy implementation is not available in our environ-ment (Alkan et al., 1991; Tadenuma and Thomson, 1995a; Velez, 2018).
Definition 1.
Let R ⊆ R + be a set containing . Then, u i ∈ U belongs to B ( R ) ifthere are ( v ia ) a ∈ A ∈ R A , b i ≥ , and ρ i ∈ R such that u i ( r a , a ) = v ia − r a − ρ i max { , r a − b i } . A preference in our domain has an intuitive simple structure. The agent has anunderlying quasi-linear preference that is modified by the existence of an amount b i that determines the marginal disutility of paying rent. If rent is below b i , theagent’s utility is simply its quasi-linear value. If rent is above b i , the agent’s utilityis its quasi-linear value minus a penalty determined as a linear function of the excesspayment. This domain admits two compatible interpretations. First, one can thinkof b i as a budget constraint and ρ i as the interest rate at which the agent hasaccess to credit, which is necessary to go over the budget constraint. Under thisinterpretation, agents’ utility is simply the present value of their consumption. Analternative interpretation is that b i is a housing earmark, i.e., the agent has a hardbudget B i > b i and has an expectation to pay at most b i for housing given all otherexpenses she has to cover. Going over her earmark for housing entails decreasingher expense for other needs, which could be done, but entails an additional sacrificethat is quantified by coefficient ρ i . We are specially concerned about the possible deployment of our mechanisms. Afirst practical issue that arises is the extent to which one can elicit the relevant Our requirement that 0 ∈ R guarantees that Q ⊆ B ( R ). It is common that students, who traditionally collectively lease houses an apartments, accumu-late debt to pay for their basic expenses including housing. B ( R ) with intuitive questions for whichthere is a true answer. One option is to proceed as if the agent had a preference inthis domain. As Spliddit, one can ask the agent to assign a rent to each room in away that the total rent is collected and the agent is indifferent between receiving eachroom with the corresponding rent. Then ask for their budget constraint. Finally,elicit the budget violation index whenever needed. There are three cases.1. The agent is budget constrained and the individual rents assigned to therooms are all weakly below the agent’s budget: In this case there is no need toask any further question. If one proceeds calculating an envy-free allocation for aquasi-linear preference with the reported values, the agent will always be assigneda room whose rent is below the budget constraint.2. The agent assigns different rents to at least two rooms and at least onerent is above the budget: Let a and b be rooms the agent assigns the highest andlowest rents, r a > r b . If r b < b i , one can ask the agent for the equivalent rebate in( r b , b ) of a rebate of one dollar in ( r a , a ). That is, we ask what τ makes the agentindifferent between ( r b − τ, b ) and ( r a − , a ). To enforce reported preferences are inour admissible domain, we can give the agent an option set τ ∈ { ρ i : ρ i ∈ R } . If r b ≥ b i , one can similarly ask the agent for the equivalent with room b of a rebateof r b − b i + 1 dollars for bundle ( r a , a ).3. The agent assigns equal rents to each room (and all above budget): In thiscase the agent has quasi-linear preferences represented by equal values. If we are tocalculate an ordinal selection from the envy-free set, as the one that maximizes theminimal payment of rent, we do not need to inquire for the budget violation index.If we are to calculate a cardinal selection, we need to ask the agent to asses howdifficult is for her to go over the budget compared to the roommates, or to a certainpopulation. Then, use the corresponding statistic calculated from the population oran available sample. Alternatively, one does not have to interpret this domain of preferences as rep-resenting the agent’s complete preference map. In the same way that quasi-linearutilities are only an approximation of an agent’s true preferences, one can think ofoffering a coarse ranking of an agent’s financial difficulties in a finite scale, say low, We could also use a lottery to elicit the index violation index assuming risk neutrality, or alsoelicit risk preferences and adjust accordingly. These are perhaps overcomplicated schemes thatprobably induce more noise in the reports than provide useful information. Then, map these reports to values of budget violation indicesthat are calibrated to increase performance of the system in either an experimentalenvironment or by means of surveys and empirical data (recall that no-envy can beevaluated ex-post with respect to true preferences).A second practical issue that one should be concerned with when consideringenlarging the space of reports from Q to B ( R ) is the complexity of calculating anenvy-free allocation. Fortunately, due to the central place of the equitable rent divi-sion problem in the literature on computational fair division, this issue has receivedattention in recent theoretical computer science literature. There is a polynomialalgorithm to calculate an envy-free allocation when preferences belong to B ( R ) withfinite R (Arunachaleswaran et al., 2019). This algorithm produces a “random” envy-free allocation, i.e. does not allow the designer to target a specific selection fromthe envy-free set. Not all envy-free allocations are intuitively equitable, however(Tadenuma and Thomson, 1995b; Gal et al., 2017). Thus, one would also like tobe able to calculate a particular selection from the envy-free set. As a responseto this, in a companion paper, we developed a polynomial algorithm to calculate amaxmin utility envy-free allocation (Spliddit’s current choice), and other prominentselections from the envy-free set, also for each B ( R ) with finite R (Velez, 2020). Incentive properties of allocation mechanisms are important to assess their potentialpractical value. In the case of the revelation mechanism of an envy-free scf definedin Q , Theorem 1 reveals that even though agents may not report their true rankings(e.g., their true indifference set in Spliddit’s questionnaire), if they are strategic, theirmisreports cancel out and in the end only envy-free allocations for true preferencescan be obtained.Strikingly, the assumption that the revelation game in Theorem 1 constrainsreports to be quasi-linear cannot be dropped altogether (when there are at leastthree agents). Velez (2015) proves this by constructing an scf g defined in U N and u ∈ U N for which there are limit Nash equilibrium outcomes of ( N, U N , g, u ) that One can also think of eliciting self-reported financial default rating indices, e.g., the agent’scredit score; or the interest rate at which the agent has access to credit. When there are only two agents, the direct revelation game of any envy-free scf produces onlylimit Nash equilibrium outcomes that are envy-free with respect to true preferences. The trivialcombinatorics in this case allows one to prove this result directly from Intermediate Value Theoremarguments. u . Intuitively, the example in Velez(2015) exploits that agents can arbitrarily exaggerate their sensitivity to changes inrent when preferences are unrestricted in U . This does not settle the issue whetherthe same phenomenon occurs when we only admit budget-constrained quasi-linearpreferences. The sequence of profiles in this example does not belong to B ( R ) forany R . However, it is possible to construct an example with the same features foreach admissible domain of preferences B ( R ) with unbounded R . Proposition 1.
Suppose that n ≥ . There is an envy-free g defined on B ( R + ) for which there is u ∈ Q N such that for each unbounded R , there is a limit Nashequilibrium of ( N, B ( R ) N , g, u ) that is not Pareto efficient for u . Proposition 1 reveals that enlarging the domain of preferences to account forsoft budget constraints in a direct revelation mechanism that forces reports to bequasi-linear, may not be innocuous in what concerns the incentive properties of thesystem. Indeed, if agents can arbitrarily exaggerate their sensitivity to increasesin rent above their budget, they can end up non-cooperatively coordinating on aPareto inefficient allocation (which is never envy-free).The following theorem, our main result, identifies boundedness of R as a generalcondition under which our proposal preserves the incentive property of envy-freequasi-linear mechanisms. Theorem 2.
Let g be an envy-free scf defined on B N ( R ) . If R is bounded, then foreach u ∈ U N the set of limit Nash equilibrium outcomes of ( N, B N ( R ) , g, u ) is F ( u ) . Proposition 1 and Theorem 2 imply that limiting the extent to which each agentcan exaggerate their sensitivity to increases of rent above their budget is necessaryand sufficient to guarantee that the non-cooperative outcomes of each direct reve-lation envy-free mechanism so defined are exactly the envy-free allocations for truepreferences.
Corollary 1.
Suppose that n ≥ and let ∈ R ⊆ R + . For each envy-free scf g defined on B ( R ) N and each u ∈ U N the set of limit Nash equilibrium outcomes of ( N, B ( R ) N , g, u ) is F ( u ) if and only if R is bounded. One can see Theorem 2 as the conjunction of two independent statements: (i)when reports are constrained to be in B ( R ) for bounded R , each limit Nash equilib-rium outcome of the direct revelation game of an envy-free scf is envy-free for the12rue preferences; (ii) each envy-free allocation for true preferences is a limit Nashequilibrium outcome of the direct revelation game of each envy-free scf defined in B ( R ) N . We will concentrate on providing intuition for (i), which requires a novelanalysis of incentives in the equitable rent division problem. The intuition why (ii)holds, heavily borrows from that behind the known proof of Theorem 1. The reason why extending the set of reports in an envy-free mechanism to includeall continuous preferences can create unwanted limit Nash equilibrium outcomes ina game, is the same reason why proving Theorem 2 is a somehow subtle problem:A limit Nash equilibrium outcome is not the precise recommendation for a givenreport; it is the limit of recommendations for a sequence of reports that can beincreasingly apart from each other as the outcome accumulates towards a limit.Thus, to understand the structure of the limit Nash equilibrium outcomes, it is notenough to determine what are the situations in which an agent can gain by changingher report considering the reports of the other agents are fixed. One needs to takeinto account the possibility that for each agent, the sequence of reports of the otheragents can make the gains that the agent can grab by changing her report vanish.Our characterization of limit Nash equilibrium outcomes can be divided in twosteps. If a given allocation is a limit Nash equilibrium outcome, a sequence of reportsthat sustains it as such necessarily has associated outcomes that are eventually inthe proximity of the allocation. Thus, our first step is to gain some control onthe structure of the sequences of reports that could possibly sustain an arbitraryoutcome. Our second step is to quantify the possibilities to manipulate an envy-freemechanism that arise because the profiles satisfy these properties.When reports are necessarily quasi-linear, Velez (2015) implicitly advances thesetwo tasks based on the limits of manipulation for an agent identified by Anderssonet al. (2014a,b) and Fujinaka and Wakayama (2015). The characteristic of problemsin this domain that makes this possible is that no-envy among some agents is pre-served when one increases their individual payments of rent by the same amount.Essentially, an agent, say agent i , can manipulate by forcing a reshuffle of roomsand a reassessment of rent by changing her report. In the crucial cases of analy- Essentially, since we assume that 0 ∈ R , we have that Q ⊆ B ( R ). It is known what the limitsof manipulations for an agent in an envy-free revelation mechanism are, and that these limits areachieved by quasi-linear reports (Andersson et al., 2014a,b; Fujinaka and Wakayama, 2015; Velez,2018). Thus, the construction that allows one to show that a given envy-free allocation for truepreferences is a limit Nash equilibrium outcome of the game of a mechanism that restricts reports tobe quasi-linear, also sustains the allocation as a limit Nash equilibrium outcome of any enlargementof the admissible domain of preferences (see Lemma 2). i forces a group of agents M ⊆ N \ { i } to pay more rent,say ∆ in aggregate, and distributes these gains among N \ M . If preferences arequasi-linear, this can be done simply by rebating ∆ in equal parts among the agentsin N \ M . Thus, agent i secures a utility gain of at least ∆ /n . Consequently, thestructure of preferences allows one to obtain a lower bound on the gain of agent i that is a linear function of the aggregate rent that agent i can extract from otheragents. It is intuitive that a result like this may be true when preferences are in B ( R ) for bounded R , because in this case the agents’ marginal utility of money hasa bound. This is not a consequence of an immediate computation, however. Indeed,one quickly realizes that a naive attempt to replicate the analysis with quasi-linearreports leads to challenging combinatorial problems. This forces us to advance asubtler approach that is based on better understanding the shape of the envy-freeset in our domain as the aggregate rent to pay changes. In particular, in Lemma 7in the Appendix, we state the following key result. The scf that selects an allocationwith maximal rent for a given room, say room a , within the envy-free set, has thefollowing property. There exists a universal constant θ , i.e., it depends only on n and the supremmum of R , such that if the rent decreases in aggregate by ∆, therent of room a decreases at least by θ ∆.Lemma 7 allows us to quantify the possible gain an agent has from manipulatingan envy-free revelation mechanism when the admissible domain of preferences is B ( R ) with bounded R . Intuitively, if an agent, say agent i , can extract an amountof money ∆ from a set of agents M ⊆ N \{ i } , then agent i can distribute these gainsamong N \ M guaranteeing that she receives an allocation that is no worse thana rebate of θ ∆ on her own room. We then determine that the possible gain of anagent at a particular profile of reports in an envy-free revelation mechanism whenthe admissible domain of preferences is B ( R ) for bounded R , is bounded below by alinear function of the difference in utility between the agent’s consumption and theconsumption of the other agents (Theorem 3). More precisely, consider an envy-freerevelation mechanism and a given profile of reports for which agent i , with her truepreference, envies another agent, say agent j . Let η > j so agent i is indifferentbetween this allotment and her consumption. There are universal proportions ω and ω for which the following holds. There is always a quasi-linear preference thatagent i can report and guarantees that as long as an envy-free allocation is chosenfor the preference report, her allotment will be at least as good as the worst of a14ent rebate in her room of at least ω η , or receiving the room of agent j but payingat most what j was paying plus ω η . Theorem 3 (Strong Manipulation) . Let ∈ R ⊆ R + be bounded. There are { ω , ω } ⊆ (0 , such that for each v ∈ B N ( R ) , each z ≡ ( r, µ ) ∈ F ( v ) , each i ∈ N , each u i ∈ U , and each j ∈ N such that u i ( z j ) − u i ( z i ) > , there is v ′ i ∈ Q such that, for each s ∈ F ( v − i , v ′ i ) , u i ( s i ) ≥ min (cid:8) u i (cid:0) r µ ( i ) − ω η, µ ( i ) (cid:1) , u i (cid:0) r µ ( j ) + ω η, µ ( j ) (cid:1)(cid:9) , where η > is such that u i ( r µ ( j ) + η, µ ( j )) = u i ( z i ) . Theorem 3 has the crucial consequence that the possibilities to manipulate for anagent never vanish when a sequence of reports lead to allocations that converge to anoutcome at which the agent envies another agent. Thus, if reports are constrained tobe in B ( R ) for bounded R , there is no envy in each limit Nash equilibrium outcomeof the game induced by an envy-free allocation mechanism. Lemma 1.
Let g be an envy-free scf defined on B N ( R ) for bounded R . Then foreach u ∈ U N , each limit Nash equilibrium outcome of ( N, B ( N ) N , g, u ) belongs to F ( u ) . A converse of this result, which holds for any admissible domain containingquasi-linear preferences, completes our proof of Theorem 2. Lemma 2.
Let
Q ⊆ D ⊆ U and g an envy-free scf defined on D N . Then for each u ∈ U N , each z ∈ F ( u ) is a limit Nash equilibrium outcome of ( N, D N , g, u ) . The careful reader can notice an inconsistency in our motivation and analysis. Onthe one hand, we are preoccupied by the possibility that some agents may truthfullyanswer the queries of our mechanisms. This motivates us to make the space ofreports more informative, so the agent can tell us about her financial constraints.On the other hand, we analyze the incentives of our mechanisms assuming that all agents engage in their manipulation. Note that Lemmas 1 and 2 together also imply Theorem 1. This theorem was stated withoutproof by Velez (2018).
15 more consistent formulation of our manipulation model should account for theexistence of the so-called behavioral type agents who are unconditionally truthful.If we assume complete information about the identity of truthful agents amongstrategic agents (the mechanism designer has no knowledge of this), one can advancethe incentive analysis with the tools that we developed for the case when all agentsare strategic. Again, our benchmark here is the incentive property of quasi-linearenvy-free mechanisms. Suppose that a set of agents T are uncondionally truthfuland S ≡ N \ T are strategic. If agents’ true preferences are quasi-linear, then the setof non-cooperative outcomes of a quasi-linear envy-free mechanism for the inducedgame for agents S given the truthful reports of agents in T , is welfare equivalentto the set of envy-free allocations for true preferences that are Pareto undominatedfor S within the envy-free set (Velez, 2015). It turns out that this property isagain retained by our budget-constrained quasi-linear preferences mechanisms. That is, if agents’ true preferences can be encoded in our message space, the setof non-cooperative outcomes from the manipulation of our mechanisms by S is, inwelfare terms, the space of envy-free allocation for true preferences that are Paretoundominanted within this set for S . This means that our mechanisms provide ameaningful welfare lower bound to truthful agents. They receive an allotment thatis no worse than the worst they could obtain in some non-cooperative outcome whenthey strategically engage in the manipulation of the mechanism (see Velez, 2015, fora detailed analysis of this welfare lower bound).In conclusion, we introduced a family of mechanisms for the allocation of roomsand rent among roommates. The mechanisms that are currently in use are practicaland have desirable incentive properties. Unfortunately, these mechanisms do notallow agents inform the mechanism designer about their financial difficulties. Themechanisms we propose alleviate these issues while they also retain the main prop-erties of the mechanisms that are currently in use. They retain their practicality. Inparticular, reports are finite dimensional and can be elicited with intuitive questions.Moreover, some of these mechanisms’ computational complexity is polynomial. Most Our proof of Lemma 1 generalizes to the environment with truthful agents. Thus, only envy-freeallocations with respect to true preferences can be obtained as a limit equilibrium of the manipula-tion game for agents S . The proof that each envy-free allocation that is Pareto undominated for S within the envy-free set can be obtained in welfare terms as a limit equilibrium of the manipulationgame for agents S can be completed along the lines of Lemma 3 in Velez (2015). Step 3 of this proofrequires that one account for the preferences of T not being quasi-linear. This can be done basedon our Lemma 6. The proof that an allocation that is Pareto dominated for S within the envy-freeset cannot be a limit equilibrium outcome of the manipulation game for S can be completed basedon our Lemma 7 with a similar argument to that we use to prove Theorem 2. expressiveness of a direct revelation mech-anism. In the spirit of the market design paradigm, we are concerned about thedetails of practical implementation. We are not content with considering an ab-stract elicitation scheme in which agents magically report their preferences in anadmissible domain. We acknowledge the subtlety of elicitation. We envision themechanism designer will uniquely identify a preference in the domain of admissiblepreferences for each agent by asking questions that have a true answer and partiallyreveal the private information of the agents. We then realize that the mechanismdesigner is exploiting the nature of direct revelation mechanism to have intuitive andsimple message spaces. We believe that a mechanism designers who takes advantageof this feature of direct revelation mechanisms, is also invested with a fiduciary dutywith the users of the systems he or she designs. This is so because there is evidencethat when agents are asked for their private information they actually truthfullyreveal it quite often independently of the incentives not to do so. Thus, we needto add to our desiderata of design that if an agent tells us the truth, i.e., takes atface value the queries of our systems, we should reciprocate. That is, we should beable to deliver, to a reasonable degree, on the promises we make. In the equitablerent division problem, this gives us the simple but concrete objective of design thatthe domains of admissible preferences we use should be expressive enough to encodethe most common preference phenomena in the environments of interest. At thesame time we need to keep elicitation feasible, and computational complexity andincentives in check.It would be interesting to further study the complexity and incentive propertiesof alternative approaches to account for more general preferences in our environmentand other problems of interest. It would be also interesting to implement our mech-17nisms in laboratory experiments and field applications and measure the extent towhich they can improve on the performance of the mechanisms currently in use. Appendix
It is convenient for our proofs to consider a variable set of agents, rooms, and rent.This allows us to describe redistribution of resources among subgroups. For C ⊆ A ,let U ( C ) be the space of preferences on R × C satisfying money-monotonicity andthe compensation assumption . Consistently, let Q ( C ) be the sub-domains of U ( C )of quasi-linear preferences; and for R ⊆ R + containing 0, let B ( C, R ) the subdomainof preferences as in Definition 1 for the consumption space R × C . An economy is atuple e ≡ ( M, C, u, l ) where M ⊆ N , C ⊆ A is such that | C | = | M | , u ∈ U ( C ) M , and l ∈ R . Consistently with our notation in the body of the paper, we simply describean economy ( N, A, u, m ) by the profile u ∈ U N . An allocation for ( M, C, u, l ) is apair ( r, σ ) where σ : M → C is a bijection and r ≡ ( r a ) a ∈ C is such that P a ∈ C r a = l .The allotment assigned to agent i by allocation ( r, σ ) is ( r σ ( i ) , σ ( i )). We denote theset of allocations for ( M, C, u, l ) by Z ( M, C, l ). Recall that we simplify and write Z for Z ( N, A, m ). For z ∈ Z ( M, C, l ) and s ∈ Z ( L, D, m ) where M and L are disjointand C and D are disjoint, ( z, s ) is the allocation in Z ( M ∪ L, C ∪ D, l + m ) whereeach i ∈ M receives z i and each j ∈ L receives s j . An allocation z ∈ Z ( M, C, l ) is envy-free for e ≡ ( M, C, u, l ) if for each pair { i, j } ⊆ M , u i ( z i ) ≥ u i ( z j ). The set ofthese allocations is F ( e ). Definition 2.
For each u ∈ U N and i ∈ N let F i ( u ) ≡ argmax s ∈ F ( u ) u i ( s i ) . The following results play a central role in our analysis.
Theorem 4 (Maximal Manipulation Theorem; Andersson et al., 2014a,b; Fujinakaand Wakayama, 2015; see Theorem 5.15, Velez, 2018) . Let u ∈ U N , i ∈ N , and z ≡ ( r, µ ) ∈ F i ( u ) . Then,1. For each v i ∈ U and each s ∈ F ( u − i , v i ) , u i ( s i ) ≤ u i ( z i ) .2. For each δ > there is v i ∈ Q such that for each s ≡ ( t, σ ) ∈ F ( u − i , v i ) , σ ( i ) = µ ( i ) and t i ≤ r i + δ . efinition 3. The envy-free graph for u ∈ U and z ∈ Z is Γ( u, z ) ≡ ( N, E )where ( i, j ) ∈ E if and only if u i ( z i ) = u i ( z j ). If there is a path from i to j in Γ( u, z )we write i → u,z j . Lemma 3 (Lemma 5.13, Velez, 2018) . Let u ∈ U N and i ∈ N . Then z ∈ F i ( u ) ifand only if z ∈ F ( u ) and for each j ∈ N , j → u,z i .Proof of Proposition 1. We prove the proposition by means of an example. Let N := { , ..., n } with n ≥ A := { a, b, c , ..., c n − } . In the example we willconstruct, agents 3 , , ..., n and objects C := { c , ..., c n − } are replicas. That is,each such an agent has the same preferences and for each i ∈ N and each pair { c j , c ′ j } ⊆ C and each x ∈ R , u i ( x, c j ) = u i ( x, c ′ j ). In what follows wheneverconvenient we use, without loss of generality, representative agent 3 and object c .Let g be an scf that for each u ∈ B ( R + ) N selects g ( u ) ∈ F i ( u ). Furthermore,assume that whenever possible agent 1 receives room a , agent 2 receives room b ,agent 3 receives room c , agent 4 receives room c , and so on. Let R be unbounded.Let u ∈ Q N be the preference profile and x := ( x d ) d ∈ A the set of consumptionsof money in the bundles shown with square dots in Fig. 1 (a). Let z (cid:3) := ( x, σ )be the allocation such that σ (1) = a , σ (2) = b , σ (3) = c , σ (4) = c , and soon. The essential features of z (cid:3) and u are the following. First, x a > x c > x b and there is a parameter δ > x b + ( n − δ < x c − δ . Thisparameter is fixed in the construction. Second, agent 1 envies agent 2. Indeed, u ( x b + δ/ , b ) = u ( x a , a ). Agents 2 and 3 receive their best bundle at the allocation.Moreover, u ( x b , b ) = u ( x a − δ/ , a ) and agent 1 prefers her bundle to ( x c − δ, c ).Then, z (cid:3) is not Pareto efficient for u , because if agents 1 and 2 exchange rooms soagent 1 is indifferent with the change and agent 2 pays x a − δ/ a , thisagent is better off.We claim that z (cid:3) is a limit Nash equilibrium outcome of ( N, B ( R ) N , g, u ). Toprove so we will construct a sequence of profiles u ε ∈ B ( N ) N , for some sequence ofvanishing ε s such that the possibilities to manipulate for each agent vanish in thesequence and as ε → g ( u ε ) → z (cid:3) .Let 0 < ε < δ be such that x b + ( n − δ + ε < x c − δ and ( n − ε < ( n − δ . Let z △ be the allocation in which z △ = ( x a − ε/ , a ), z △ = ( x b + ( n − δ + 2 ε/ , b ),and z △ = ( x c − δ, c ). The bundles in z △ are shown in triangle dots in Fig. 1 (c)-(e).Denote the room prices at this allocation by x △ . Let z • be the allocation in which z • = ( x a − ε, a ), z • = ( x b + ( n − ε, b ), and z • = ( x c − ε, c ). The bundles in z • x • .Preferences u ε , u ε , and u ε are shown in Fig. 1 (b)-(d), respectively. Preference u ε is quasi-linear and indifferent between all bundles ( z i ) i ∈ N . Preference u ε is suchthat u ε ( z △ ) = u ε ( z △ ) = u ε ( z △ ) and u ε ( z • ) = u ε ( z • ) . (1)We claim that there is indeed u ε ∈ B ( R ) satisfying these two conditions. Intuitively,since x a − ε > x c − δ > x b + ( n − δ + ε , we can choose ρ i high enough so the gain ofutility from a rebate at ( x a , a ) for the agent is arbitrarily high. Moreover, since wecan choose the budget report for the agent we can calibrate the required indifference.More precisely, consider a preference in B ( R ) associated with parameters v i , b i ∈ [ x • a , x △ a ], and ρ i ∈ R . Let τ := x △ b − x • b . Then, τ = x b + ( n − δ + 2 ε/ − ( x b + ( n − ε ) ≥ ε/
3. Note that x △ a − x • a = ε/
3. We will show that these parameters can bechosen such that (1) holds. Since x a − ε > x b + ( n − δ + ε , we have that x △ b < x • a .Thus, (1) holds whenever v ia − x △ a − ρ i ( x △ a − b i ) = v ib − x △ b and v ia − x • a = v ib − x • b . Wecan choose then v i such that v ia − v ib = x • a − x • b = x △ a − ε/ − x △ b + τ . Thus, we need tobe able to choose b i and ρ i such that x △ a − ε/ − x △ b + τ − x △ a − ρ i ( x △ a − b i ) + x △ b = 0.That is, (for ρ i > b i = x △ a − ρ i ( τ − ε/ . Since R is unbounded, one can choose ρ i ∈ R such that b i determined by the equationabove belongs to [ x • a , x △ a ]. Finally, preference u ε is such that u ε ( z (cid:3) ) = u ε ( z (cid:3) ) = u ε ( z (cid:3) ) and u ε ( z △ ) = u ε ( z △ ) . (2)Since x a > x c , we can choose ε small enough so x △ a = x a − ε/ > x c and ε < δ .Thus, the construction for u ε can be exactly replicated to prove that there is actually u ε ∈ B N ( R ) satisfying (2).We will show now that as ε → g ( u ε ) → z (cid:3) . Indeed, we prove that for each ε , g ( u ε ) = z . Consider the graph Γ( u ε , u , u ε , z (cid:3) ). We show this graph in Fig. 1 (e)where link ( i, j ) is shown as an arrow from z (cid:3) i to z (cid:3) j (note that for simplicity we donot show the arrows ( i, i ) in the graph). Since the preferred bundle in z (cid:3) for both u and u ε is z (cid:3) , we have that Γ( u ε , u , u ε , z (cid:3) ) = Γ( u ε , z (cid:3) ). Moreover, note thatboth u ε and u ε are indifferent among each bundle in z (cid:3) . Thus, z (cid:3) ∈ F ( u ε ). Sincefor each j ∈ N , there is a path in Γ( u ε , z (cid:3) ) that flows to agent 2, by Lemma 3,20 (cid:3) ∈ F i ( u ε ). Since z (cid:3) ’s assignment of rooms coincides with the one chosen by g whenever possible, we have that g ( u ε ) = z (cid:3) .We finally show that for each i ∈ N and each u ′ i ∈ B ( R ), u i ( g ( u ε − i , u ′ i )) ≤ u ( g ( u ε )) + ε = u ( z (cid:3) i ) + ε . This proves that z (cid:3) is a limit equilibrium outcome of( N, A, g, u ). Consider first agent 1. Observe that z △ ∈ F ( u ε − , u ). Moreover, thereis a path from each j ∈ N to agent 1 in Γ( u ε − , u , z △ ). Thus, z △ ∈ F ( u ε − , u ). ByTheorem 4, for each u ′ ∈ B ( R ), u ( g ( u ε − , u ′ )) ≤ u ( z △ ) = u ( z (cid:3) ) + 2 ε/
3. Considernow agent 2. Observe that z (cid:3) ∈ F ( u ε − , u ). Moreover, there is a path from each j ∈ N to agent 2 in Γ( u ε − , u , z (cid:3) ). Thus, z (cid:3) ∈ F ( u ε − , u ). By Theorem 4, for each u ′ ∈ B ( R ), u ( g ( u ε − , u ′ )) ≤ u ( z (cid:3) ). Finally consider agent k ∈ { , ..., n } . Observethat z • ∈ F ( u ε − k , u k ). Moreover, there is a path from each j ∈ N to agent k in Γ( u ε − k , u k , z • ). Thus, z • ∈ F k ( u ε − k , u k ). By Theorem 4, for each u ′ k ∈ B ( R ), u k ( g ( u ε − k , u ′ k )) ≤ u k ( z • k ) ≤ u k ( z (cid:3) k ) + ε .The following lemma is used in the proof of the subsequent results. Lemma 4.
Let u ∈ U N , i ∈ N , v i ∈ U , and z ≡ ( r, µ ) ∈ F ( u − i , v i ) . Then, for each δ > , there is v ′ i ∈ Q such that for each s ≡ ( t, σ ) ∈ F ( u − i , v ′ i ) , σ ( i ) = µ ( i ) and t µ ( i ) ≤ r µ ( i ) + δ .Proof. Let a ≡ µ ( i ) and ( r ′ , γ ) ∈ F i ( u − i , v i ). By Alkan et al. (Lemma 3, 1991), r ′ a ≤ r a . Let u ′ i ∈ U be a preference that prefers ( r ′ a , a ) to all other bundles at ( r ′ , γ ),i.e., such that for each b ∈ A \{ a } , u ′ i ( r ′ a , a ) > u ′ i ( r ′ b , b ). Let ( r ′′ , γ ′ ) ∈ F i ( u − i , u ′ i ). ByFujinaka and Wakayama (Lemma 3, 2015), r ′′ = r ′ . Thus, γ ′ ( i ) = a . Let δ >
0. ByTheorem 4 (Statement 3), there is v ′ i ∈ Q such that for each s ≡ ( t, σ ) ∈ F ( u − i , v ′ i ), σ ( i ) = a and t a ≤ r ′′ a + δ = r ′ a + δ ≤ r a + δ . Lemma 5.
Let u ∈ U N , i ∈ N , v i ∈ U , z ≡ ( r, µ ) ∈ F ( u − i , v i ) , and j ∈ N such that j → u,z i . Then, for each δ > , there is v ′ i ∈ Q such that for each ( t, σ ) ∈ F ( u − i , v ′ i ) , σ ( i ) = µ ( j ) and t µ ( j ) ≤ r µ ( j ) + δ .Proof. Let u ′ i ∈ U be a preference that prefers ( r µ ( j ) , µ ( j )) to all other bundlesat ( r, µ ), i.e., such that for each a ∈ A \ { µ ( j ) } , u ′ i ( r µ ( j ) , µ ( j )) > u ′ i ( r a , a ). Since j → u,z i , j → u − i ,u ′ i ,z i . Thus, there is an allocation in s ∈ F ( u − i , u ′ i ) that is obtainedby reshuffling z along the path that defines j → u − i ,u ′ i ,z i and assigning ( r µ ( j ) , µ ( j ))to agent i . Thus, s i = ( r µ ( j ) , µ ( j )). By Lemma 4, for each δ >
0, there is v ′ i ∈ Q such that for each ( t, σ ) ∈ F ( u − i , v ′ i ), σ ( i ) = µ ( j ) and t µ ( j ) ≤ r µ ( j ) + δ .21 bC δ u u u δ δ rsrs rs (a) abC u ε rsrs rs (b) abC δ u ε rsrs rs utut ut bb b ε ε ε ( n − δ ( n − ε ε (c) abC δ u ε rsrs rs utut ut bb b ε ε ε ( n − δ (d) abC Γ( u , u ε , u ε , z △ )Γ( u ε , u , u ε , z (cid:3) )Γ( u ε , u ε , u , z • ) rsrs rs utut ut bb b (e) Figure 1:
An inefficient limit Nash equilibrium outcome of (
N, A, u, g ). A point in each axis, say x a in the axis corresponding to object a , represents the bundle ( x, a ). Axis C is a representativeaxis for objects C = { c , ..., c n − } . The consumptions at allocation z (cid:3) are represented by squaredots. Similarly for z △ and z • . In panel (e), arrows represent the links in the corresponding graphfor the agents who receive the respective bundles in the corresponding allocation. emma 6. There is a constant η > such that for each M ⊆ N , C ⊆ A such that | C | = | N | , u ∈ B ( R, C ) M , s ≡ ( t, σ ) ∈ F ( M, C, u, l ) , and z ≡ ( r, µ ) ∈ F ( M, C, u, l ′ ) ,if j → u,s i and t σ ( i ) < r σ ( i ) , then r σ ( j ) − t σ ( j ) ≤ η ( r σ ( i ) − t σ ( i ) ) . Proof.
Fix z ≡ ( r, µ ) and s ≡ ( t, σ ) as in the statement of the lemma. Then, j → u,s i and t σ ( i ) < r σ ( i ) . We first prove the result when ( j, i ) ∈ Γ( u, s ). We do this in Steps1-3 below. Step 1 : For each j ∈ N , r µ ( k ) − t µ ( k ) ≤ max { , (1 + ρ )( r σ ( k ) − t σ ( k ) ) } . Sup-pose first that r σ ( k ) ≤ t σ ( k ) . That is, the room of agent k at s receives a non-negative rent rebate at z . By Alkan et al. (Lemma 3, 1991), the room of agent k at z receives also a non-negative rent rebate at z . Thus, r µ ( k ) ≤ t µ ( k ) and r µ ( k ) − t µ ( k ) ≤ max { , (1 + ρ )( r σ ( k ) − t σ ( k ) ) } . We can suppose then that r σ ( k ) > t σ ( k ) and r µ ( k ) > t µ ( k ) . Since s ∈ F ( u ), u j ( t µ ( k ) , µ ( k )) ≤ u j ( t σ ( k ) , σ ( k )). Since z ∈ F ( u ), u j ( r σ ( k ) , σ ( k )) ≤ u j ( r µ ( k ) , µ ( k )). Thus, u j ( t µ ( k ) , µ ( k )) − u j ( r µ ( k ) , µ ( k )) ≤ u j ( t σ ( k ) , σ ( k )) − u j ( r σ ( k ) , σ ( k )) . (3)Now, u j ( t µ ( k ) , µ ( k )) − u j ( r µ ( k ) , µ ( k )) = r µ ( k ) − t µ ( k ) − ρ j (cid:0) max { , t µ ( k ) − b j } − max { , r µ ( k ) − b j } (cid:1) ≥ r µ ( k ) − t µ ( k ) , (4)where the last inequality follows from r µ ( k ) ≥ t µ ( k ) . Let ¯ ρ = sup R . Similarly, u j ( t σ ( k ) , σ ( k )) − u j ( r σ ( k ) , σ ( k )) = r σ ( k ) − t σ ( k ) + ρ i (cid:0) max { , r σ ( k ) − b j } − max { , t σ ( k ) − b j } (cid:1) ≤ (1 + ρ j )( r σ ( k ) − t σ ( k ) ) ≤ (1 + ¯ ρ )( r σ ( k ) − t σ ( k ) ) , (5)where the first inequality follows because since t σ ( k ) < r σ ( k ) , max { , r σ ( k ) − b j } − max { , t σ ( k ) − b j } is equal to r σ ( k ) − t σ ( k ) when b j ≤ t σ ( k ) , equal to r µ ( k ) − b j when t σ ( k ) < b j ≤ r σ ( k ) , and equal to zero when r σ ( k ) < b j . Putting together equations233), (4), and (5), we have that r µ ( k ) − t µ ( k ) ≤ (1 + ρ )( r σ ( k ) − t σ ( k ) ) . (6) Step 2 : For each ( j, i ) ∈ Γ( u, s ) , r µ ( j ) − t µ ( j ) ≤ max { , (1+ ρ )( r σ ( i ) − t σ ( i ) ) } . Supposefirst that r σ ( i ) ≤ t σ ( i ) . By Velez (Lemma 5.7, 2018), r µ ( i ) ≤ t µ ( i ) . Thus, r µ ( j ) − t µ ( j ) ≤ max { , (1 + ρ )( r σ ( i ) − t σ ( i ) ) } . We can suppose then that r σ ( i ) > t σ ( i ) and r µ ( i ) > t µ ( i ) .Let x ∈ R be such that u j ( x, σ ( i )) = u j ( z j ). Since u j ( z j ) < u j ( s j ) = u j ( t σ ( i ) , σ ( i ))and preferences are money-monotone, x > t σ ( i ) . Since z ∈ F ( u ), u j ( x, σ ( i )) = u j ( z j ) ≥ u j ( r σ ( i ) , σ ( i )). Thus, r σ ( i ) ≥ x > t σ ( i ) . Thus, x − t σ ( i ) ≤ r σ ( i ) − t σ ( i ) .Since s ∈ F ( u ), u i ( t µ ( j ) , µ ( j )) ≤ u j ( t σ ( i ) , σ ( i )). Since u j ( r µ ( j ) , µ ( j )) = u j ( z j ) = u j ( x, σ ( i )), r µ ( j ) − t µ ( j ) ≤ u i ( t µ ( j ) , µ ( j )) − u j ( r µ ( j ) , µ ( j )) ≤ u j ( t σ ( i ) , σ ( i )) − u j ( x, σ ( i )) ≤ (1 + ρ )( r σ ( i ) − t σ ( i ) ) , (7)where the first and last inequalities follow from the argument leading to equations(4) and (5). Step 3 : For each ( j, i ) ∈ Γ( u, s ) , such that r σ ( i ) > t σ ( i ) , r σ ( j ) − t σ ( j ) ≤ (1 + ρ ) n ( r σ ( i ) − t σ ( i ) ) . By Step 2, r µ ( j ) − t µ ( j ) ≤ (1 + ρ )( r σ ( i ) − t σ ( i ) ). Let j ∈ N be such that σ ( j ) = µ ( j ). Thus, r σ ( j ) − t σ ( j ) ≤ (1 + ρ )( r σ ( i ) − t σ ( i ) ). Thus, if j = j , our step is proved. Suppose that j = j . By Step 1, r µ ( j ) − t µ ( j ) ≤ max { , (1 + ¯ ρ )( r σ ( j ) − t σ ( j ) ) } . Thus, r µ ( j ) − t µ ( j ) ≤ (1 + ρ ) ( r σ ( i ) − t σ ( i ) ) . Suppose that we have found { j , ..., j m } ⊆ N \{ j } , different agents such that σ ( j ) = µ ( j ), σ ( j ) = µ ( j ),..., σ ( j m ) = µ ( j m − ); and r µ ( j m ) − t µ ( j m ) ≤ (1 + ρ ) m +1 ( r σ ( i ) − t σ ( i ) ). Let k ∈ N such that σ ( k ) = µ ( j m ). Then, r σ ( k ) − t σ ( k ) ≤ (1 + ρ ) m +1 ( r σ ( i ) − t σ ( i ) ). We claim that k
6∈ { j , ..., j m } . Suppose by contradiction that k = j l forsome l ∈ { , ..., m } . If l = 1, σ ( k ) = µ ( j ). Thus, j m = j . A contradiction.If l > σ ( k ) = µ ( j l − ). Thus, j m = j l − . This contradicts { j , ..., j m } are alldifferent agents. Thus, either k = j or k ∈ N \ { j, j , ..., j m } . In the former case, r σ ( j ) − t σ ( j ) ≤ (1 + ρ ) m +1 ( r σ ( i ) − t σ ( i ) ), which proves our step. In the later caselet j m +1 = k . Recall that σ ( j m +1 ) = µ ( j m ). Thus, { j , ..., j m +1 } ⊆ N \ { j } are alldifferent agents. By Step 1, r µ ( j m +1 ) − t µ ( j m +1 ) ≤ max { , (1+ ¯ ρ )( r σ ( j m +1 ) − t σ ( j m +1 ) ) } .24hus, r µ ( j m +1 ) − t µ ( j m +1 ) ≤ max { , (1 + ¯ ρ )( r µ ( j m ) − t µ ( j m ) ) } . Thus, r µ ( j m +1 ) − t µ ( j m +1 ) ≤ (1 + ρ ) m +2 ( r σ ( i ) − t σ ( i ) ) . Since N is finite, by repeating the preceding argument we eventually find that r σ ( j ) − t σ ( j ) ≤ (1 + ρ ) n ( r σ ( i ) − t σ ( i ) ). Step 4 : Concludes . Suppose that j → u,s i . Let ( j, k ) , ( k , k ) , ..., ( k m , i ) bethe path in Γ( u, s ) that defines j → u,s i . If r σ ( k l ) ≤ t σ ( k l ) for some l = 1 , .., m ,then ( k, j ) ∈ Γ( u, s ) implies, r σ ( j ) ≤ t σ ( j ) (Lemma 5.7, Velez, 2018). Thus, we cansuppose without loss of generality that for each l = 1 , ..., m , r σ ( k l ) > t σ ( k l ) . By Step3, we have that r σ ( j ) − t σ ( j ) ≤ (1 + ¯ ρ ) n × n ( r σ ( i ) − t σ ( i ) ) . Definition 4.
For each M ⊆ N , C ⊆ A , u ∈ U ( C ) N , l ∈ R , and a ∈ C let F a ( M, C, u, l ) ≡ argmin ( r,µ ) ∈ F ( M,C,u,l ) r a . Lemma 7.
There is a constant θ > such that for each M ⊆ N , C ⊆ A suchthat | C | = | N | , u ∈ B ( R, C ) N , a ∈ A , ( t, σ ) ∈ F a ( M, C, u, l ) , ε > , and ( r, µ ) ∈ F a ( M, C, u, l + ε ) , r a − t a ≥ θε .Proof. Let i ∈ N be such that σ ( i ) = a . By Velez (Proposition 1, 2011), for each j ∈ N , j → u,s i . By Lemma 6, there exists η > n and¯ ρ ≡ sup R , such that r σ ( j ) − t σ ( j ) ≤ η ( r σ ( i ) − t σ ( i ) ). Thus, ε = X j ∈ N r σ ( j ) − t σ ( j ) ≤ nη ( r a − t a ) , and for θ = nη , r a − t a ≥ θε . Proof of Theorem 3.
Let v ∈ B ( R ) N , z ≡ ( r, µ ) ∈ F ( v ), i ∈ N , and u i ∈ U . Let j ∈ N . Suppose that u i ( z j ) > u i ( z i ). Let y be such that u i ( r µ ( j ) − y, µ ( j )) = u i ( z i ).Since preferences are continuous and satisfy money-monotonicity there is a uniquesuch a y and y >
0. There are two cases.
Case 1 : j → v,z i . Then, j → v − i ,u i ,z i . Let u ′ i ∈ U be indifferent among allbundles in z . Thus, there is an allocation in F ( v − i , u ′ i ), obtained by reshuffling25long the chain that defines j → v − i ,u i ,z i , at which agent i receives z j . By Lemma 4,for each ω ∈ (0 ,
1) there is v ′ i ∈ Q such that for each s ≡ ( t, σ ) ∈ F ( v − i , v ′ i ), σ ( i ) = µ ( j ) and u i ( s i ) ≥ u i (cid:0) r µ ( j ) + ω y, µ ( j ) (cid:1) . Case 2 : It is not the case that j → v,z i . Let x > u ′ i ∈ U a utilityfunction that is indifferent between z i and for each k ∈ N \ { i } , ( r µ ( k ) − x, µ ( k )).By choosing x sufficiently large we can guarantee that for each k ∈ N \ { i } , if u ′ i ( t µ ( k ) , µ ( k )) ≥ u ′ i ( z i ), then u i ( t µ ( k ) , µ ( k )) ≥ u i ( z j ) . (8)Let u ′ ≡ ( v − i , u ′ i ). Since z i is the preferred bundle of u ′ i among all bundles at z and z ∈ F ( v ), z ∈ F ( u ′ ). Let M ≡ { k ∈ N : k → u ′ ,z i } and L = N \ M . Then i ∈ M and for each k ∈ M and each h ∈ N \ M , u ′ h ( z h ) > u ′ h ( z k ). Since it is notthe case that j → v,z i , it is not the case that j → u ′ ,z i . Thus, j ∈ L = ∅ . Let ϕ be a function that assigns to each l ∈ R an allocation ϕ ( l ) ∈ F µ ( i ) ( M, µ ( M ) , u ′ M , l ).For each b ∈ µ ( M ), let r b ( l ) be the rent payment of the agent who receives room b at ϕ ( l ). Then, by Velez (Proposition 2, 2017), r b ( · ) is a continuous strictly increasingfunction and l u ′ i ( ϕ ( l ) i ) is a continuous strictly decreasing function. By Velez(Corollary 1 and Proposition 2, 2017), there is a function ψ that assigns to each l ∈ R an allocation ψ ( l ) ∈ F ( L, µ ( L ) , u L , l ) such that (i) ψ ( P a ∈ µ ( L ) r a ) = z L , (ii)if for each b ∈ µ ( L ) and each x ∈ R , r b ( x ) is the rent payment of the agent whoreceives room b at ψ ( x ), r b is a continuous strictly increasing function; and (iii)for each i ∈ L , x u i ( ψ ( x ) i ) is a continuous strictly decreasing function. Let l M ≡ P a ∈ µ ( M ) r a and l L ≡ P a ∈ µ ( L ) r a . Consider the set D ≡ n δ ∈ R : ∀ k ∈ M, ∀ j ∈ L, u ′ j (cid:16) ψ ( L L + δ ) j (cid:17) ≥ u ′ j (cid:16) ϕ ( l M − δ ) j (cid:17)o . Recall that for each k ∈ M and each h ∈ L , u ′ h ( z h ) > u ′ h ( z k ). Thus, 0 ∈ D and allthe inequalities that define D hold strictly for δ = 0. Since for each b ∈ µ ( M ), thefunction r b ( · ) is continuous and for each k ∈ L , the function l u ′ k ( ψ ( l ) k ) is alsocontinuous, there is δ > D . Since for each k ∈ M , l u ′ k ( ϕ ( l ) k ) is increasing, ϕ ( l M − δ ) ∈ F ( M, µ ( M ) , u ′ M , l M − δ ), and ψ ( l L + δ ) ∈ F ( L, µ ( L ) , u L , l L + δ ), foreach δ ∈ D , ( ϕ ( l M − δ ) , ψ ( l L + δ )) ∈ F ( u ′ ). Since F ( u ′ ) is a compact set, D isbounded above. Let δ be the supremum of D . By the same argument above, ifall inequalities that define D are strict at δ , there is δ > δ in δ . Thus, δ isthe maximum of D . Let z ≡ ( r , µ ) ≡ ( ϕ ( l M − δ ) , ψ ( l L + δ )) ∈ F ( u ′ ). Thus,26here is h ∈ L and k ∈ M such that u ′ h ( z h ) = u ′ h ( z k ). Let i be the agent whoreceives µ ( i ) at z . Since z M ∈ F µ ( i ) ( M, µ ( M ) , u ′ M , l M ), by Velez (Proposition 1,2011), for each k ∈ M , k → u ′ ,z i . Thus, there is h ∈ L such that l → u ′ ,z i . Let M ≡ { k ∈ N : k → u ′ ,z i } and L = N \ M . Since z = ( ϕ ( l M − δ ) , ψ ( l L + δ )), M ) M and µ ( M ) ) µ ( M ). By repeating this process 1 ≤ m ≤ n times, oneconstructs { z ≡ ( r , µ ) , z ≡ ( r , µ ) , ..., z m ≡ ( r m , µ m ) } ⊆ F ( u ′ ) and { δ , ..., δ m } all positive amounts, such that: (i) z = z ; (ii) for each k ∈ { , ..., m } , denoteby i k the agent who receives object µ ( i ) at z k and by j k the agent who receivesobject µ ( j ) at z k , M k ≡ { h ∈ N : h → u ′ ,z k i k } , and L k ≡ N \ M k ; then for each k = 1 , ...m , M k − ( M k , µ ( i ) ∈ µ k − ( M k − ) ( µ k ( M k ), L k ( L k − ; for each k = 0 , ..., m − µ ( j ) ∈ µ k ( L k ); and µ ( j ) ∈ µ m ( M m ); and (iii) for each k = 1 , ..., m , z k ∈ F ( u ′ ); for each h ∈ M k − , r k − µ ( h ) > r kµ ( h ) ; for each h ∈ L k − , r k − µ ( h ) < r kµ ( h ) ;and z kM k − = ϕ ( l M k − − δ k ) where l M k − ≡ P a ∈ µ k − ( M k − ) r k − a and ϕ is a functionthat assigns to each l an allocation ϕ ( l ) ∈ F µ ( i ) ( M k − , µ k − ( M k − ) , u M k − , l ). Let∆ ≡ δ + · · · + δ m . By Lemma 7, there exists θ > n and¯ ρ ≡ sup R , such that for each k = 1 , ..., m , r k − µ ( i ) − r kµ ( i ) ≥ θδ k . Thus, r µ ( i ) − r mµ ( i ) ≥ θ ∆.Since for each k = 0 , , ..., m −
1, and each h ∈ L k , r kµ ( h ) < r k +1 µ ( h ) , and µ ( j ) ∈ µ k ( L k ),we have that r mµ ( j ) − r µ ( j ) < ∆. There are two cases. Case 2.1. µ m ( i ) = µ ( i ). Since r mµ ( i ) < r µ ( i ) , and { z , z m } ⊆ F ( v − i , u ′ i ), byAlkan et al. (Lemma 3, 1991), u ′ i ( z mi ) > u ′ i ( z i ) = u ′ i ( z i ). By (8), u i ( z mi ) > u i ( z j ).By Lemma 4, for each η >
0, there is v ′ i ∈ Q such that for each s ≡ ( t, σ ) ∈ F ( v − i , v ′ i ), σ ( i ) = µ m ( i ) and u i ( s i ) ≥ u i (cid:0) r µ m ( j ) + η, µ m ( j ) (cid:1) . Thus, for each ω ∈ (0 , v ′ i ∈ Q such that for each s ≡ ( t, σ ) ∈ F ( v − i , v ′ i ), u i ( s i ) ≥ u i (cid:0) r µ ( j ) , µ ( j ) (cid:1) >u i (cid:0) r µ ( j ) + ω y, µ ( j ) (cid:1) . Case 2.2. µ m ( i ) = µ ( i ). Since r µ ( i ) − r mµ ( i ) ≥ θ ∆, u i ( z mi ) ≥ u i ( r µ ( i ) − θ ∆ , µ ( i )).Since r mµ ( j ) − r µ ( j ) < ∆, u i ( r mµ ( j ) , µ ( j )) ≥ u i ( r µ ( j ) + ∆ , µ ( j )). Since µ ( j ) ∈ µ m ( M m ), j m → v − i ,u ′ i ,z m i where µ m ( j m ) = µ ( j ). Suppose that ∆ ≤ y/
2. Since z m ∈ F ( v − i , u ′ i ), by Lemma 5, there is v ′ i ∈ Q such that for each s ≡ ( t, σ ) ∈ F ( v − i , v ′ i ), σ ( i ) = µ ( j ) and u i ( s i ) ≥ u i (cid:0) r µ ( j ) + ∆ + y/ , µ ( j ) (cid:1) ≥ u i ( r j + 3 y/ , µ ( j )). Fi-nally, suppose that ∆ > y/
2. Since z m ∈ F ( v − i , u ′ i ), by Lemma 4, there is v ′ i ∈ Q such that for each s ≡ ( t, σ ) ∈ F ( v − i , v ′ i ), σ ( i ) = µ ( i ) and u i ( s i ) ≥ u i (cid:0) r µ ( i ) − θ ∆ + θ ∆ / , µ ( i ) (cid:1) = u i (cid:0) r µ ( i ) − θ ∆ / , µ ( i ) (cid:1) Thus, for each s ≡ ( t, σ ) ∈ F ( v − i , v ′ i ), u i ( s i ) ≥ u i (cid:0) r µ ( i ) − θy/ , µ ( i ) (cid:1) .Summarizing, in all cases, there is v ′ i ∈ Q such that for each s ≡ ( t, σ ) ∈ ( v − i , v ′ i ), u i ( s i ) ≥ min (cid:8) u i (cid:0) r µ ( i ) − θy/ , µ ( i ) (cid:1) , u i (cid:0) r µ ( j ) + 3 y/ , µ ( j ) (cid:1)(cid:9) . Proof of Lemma 1.
Let z ≡ ( r, µ ) be a limit Nash equilibrium of ( N, D N , g, u ). Weclaim that z ∈ F ( u ). Suppose by contradiction that there are { i, j } ⊆ N , suchthat u i ( z j ) > u i ( z i ). Let δ > u i ( r j + δ, µ ( j )) = u i ( z i ). Let ( v ε , x ε )be a sequence of ε -equilibria of ( N, D N , g, u ), whose respective outcomes are z ε ≡ ( r ε , µ ε ) ∈ g ( v ε ) ∈ F ( v ε ), such that as ε vanishes, z ε → z . We can suppose withoutloss of generality that for each ε , µ ε = µ . Let δ ε ∈ R be such that u i ( r εµ ( j ) − δ ε , µ ( j )) = u i ( z εi ). Let { ω , ω } ⊆ (0 ,
1) be the coefficients in Theorem 3. Sincepreferences are continuous and as ε vanishes, z ε → z , there is η > ε < η , δ + (1 /ω − δ/ > δ ε ≥ δ/ > r εµ ( i ) ≤ r µ ( i ) + ω δ , and r εµ ( j ) < r µ ( j ) + − ω . Since D ⊆ B , by Theorem 3 there is v ′ i ∈ Q ⊆ D such that, foreach s ∈ F ( v ε − i , v ′ i ), u i ( s i ) ≥ min n u i (cid:16) r εµ ( i ) − ω δ ε , µ ( i ) (cid:17) , u i (cid:16) r εµ ( j ) + ω δ ε , µ ( j ) (cid:17)o . Since δ + (1 /ω − δ/ > δ ε ≥ δ/ > u i ( s i ) ≥ min (cid:26) u i (cid:16) r εµ ( i ) − ω δ, µ ( i ) (cid:17) , u i (cid:18) r εµ ( j ) + 1 + ω δ, µ ( j ) (cid:19)(cid:27) . Since r εµ ( i ) ≤ r µ ( i ) + ω δ and r εµ ( j ) < r µ ( j ) + − ω , u i ( s i ) ≥ min (cid:26) u i (cid:16) r µ ( i ) − ω δ, µ ( i ) (cid:17) , u i (cid:18) r εµ ( j ) + 3 + ω δ, µ ( j ) (cid:19)(cid:27) . Let ¯ ε ≡ min (cid:26) u i (cid:16) r µ ( i ) − ω δ, µ ( i ) (cid:17) , u i (cid:18) r εµ ( j ) + 3 + ω δ, µ ( j ) (cid:19)(cid:27) − u i ( z i ) . Since ω > ω <
1, ¯ ε >
0. Thus, for each ε < η , u i ( g ( v ε − i , v ′ i )) ≥ u i ( z i ) + ¯ ε .Since as ε vanishes, z ε → z , we can further select ε < ¯ ε/ u i ( z εi ) − u i ( z i ) < ¯ ε/
2. Thus, u i ( g ( v ε − i , v ′ i )) ≥ u i ( z i ) + ¯ ε > u i ( z εi ) + ¯ ε/ > u i ( z εi ) + ε . Thus, v ε is notan ε -equilibrium of ( N, D N , g, u ). This is a contradiction.28 roof of Lemma 2. Let u ∈ U N and z ≡ ( r, µ ) ∈ F ( u ). Let v ε ∈ Q N be theprofile such that for each i ∈ N and each j ∈ N \ { i } , v εi ( r µ ( i ) + ε/ ( n − , µ ( i )) = v εi ( r µ ( j ) − ε/ ( n − , µ ( j )). Thus, each agent i assigns a value to the object theyconsume at z that is ε (1 / ( n −
1) + 1 / ( n − ) greater than the value of each otherobject. We claim that v ε is an ε -equilibrium of ( N, D N , g, u ). Let s ∈ F ( v ε ), then s is Pareto efficient. Thus, each agent receives µ ( i ) at s . Thus, s = ( t, µ ) for some t ∈ R A . We claim that for each i ∈ N , t µ ( i ) ≤ r µ ( i ) + ε/ ( n − i ∈ N such that t µ ( i ) > r µ ( i ) + ε/ ( n − P j ∈ N t µ ( j ) = P j ∈ N r µ ( j ) , thereis j ∈ N such that t µ ( j ) < r µ ( j ) − ε/ ( n − . Thus, u i ( s j ) > u i ( s i ) and s F ( v ε ).This is a contradiction. Since for each i ∈ N , t µ ( i ) ≤ r µ ( i ) + ε/ ( n − i ∈ N , t µ ( i ) ≥ r µ ( i ) − ε . Let i ∈ N . By Fujinaka and Wakayama (Lemma3, 2015), for each s ∈ F i ( v ε − i , u i ), t µ ( i ) ≥ r µ ( i ) − ε . Thus, for each s ∈ F i ( v ε − i , u i ), u i ( s i ) ≤ u i ( z i ) + ε . By Fujinaka and Wakayama (Statement 1, Theorem 1, 2015),for each v ′ i ∈ U and each s ∈ F ( v ε − i , v ′ i ), u i ( s i ) ≤ u i ( z i ) + ε . Thus, for each v ′ i ∈ U s ≡ g ( v ε − i , v ′ i ) ∈ F ( v ε − i , v ′ i ) is such that u i ( s i ) ≤ u i ( z i ) + ε . Let ( t ε , µ ) = g ( v ε ). Then,for each i ∈ N , r i − ε ≤ t εi ≤ r i + ε/ ( n − ε vanishes, ( t ε , µ ) → z . Thus, z is a limit Nash equilibrium of ( N, D N , g, u ). References
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