Rental harmony with roommates
aa r X i v : . [ c s . G T ] J un RENTAL HARMONY WITH ROOMMATES † YARON AZRIELI ∗ AND ERAN SHMAYA ∗∗ Abstract.
We prove existence of envy-free allocations in markets with heterogenousindivisible goods and money, when a given quantity is supplied from each of the goodsand agents have unit demands. We depart from most of the previous literature byallowing agents’ preferences over the goods to depend on the entire vector of prices.Our proof uses Shapley’s K-K-M-S theorem and Hall’s marriage lemma. We thenshow how our theorem may be applied in two related problems: Existence of envy-freeallocations in a version of the cake-cutting problem, and existence of equilibrium in anexchange economy with indivisible goods and money.Keywords: Envy-free; Assignment problem; Rental harmony; Cake cutting.JEL Classification: D63 Introduction
A central concept in the literature on economic fairness is envy-freeness [7, 21, 13] –an allocation is envy-free if no agent prefers the share allocated to another agent over hisown share. In this note we study existence of envy-free allocations when the goods to beallocated are indivisible and heterogenous, and when in addition there is one perfectlydivisible good (e.g., money). We assume that each agent has a demand for only one ofthe indivisible goods and that there is a given quantity supplied of each good.While there are many real-life examples that can fit into this framework, we will use forconcreteness the terminology of room-assignment and rent-division: Several rooms withdifferent characteristics and given capacities are available in a house, and the total rentfor the house needs to be divided between the rooms. In this context, envy-freeness boilsdown to a market clearing condition: A price is assigned to each room such that wheneach agent chooses his favorite room (given the prices) supply exactly equals demand andthe market clears. Following Su [18], we call such a situation rental harmony . Note that † We thank Herv´e Moulin for helpful discussions, and Chris Chambers and Rodrigo Velez for their com-ments on a previous draft of this paper. ∗ Department of Economics, The Ohio State University. email: [email protected]. ∗∗ Kellogg School of Management, Northwestern University, and School of Mathematics, Tel Aviv Uni-versity. email: [email protected]: June 11, 2018. even though we use the terms ‘rooms’ and ‘capacities’, we do not make the assumptionthat the agents to whom a given good is allocated, whom we call roommates , receive ajoint ownership of the same physical object. Rather, a room with capacity 7 stands foran indivisible good of which 7 units are supplied, and the roommates represent the 7agents who received these units. When we say that the price of the room is p we meanthat each unit costs p/ r, r ′ are two rooms with prices p and p ′ respectively, then each agent can say whether he prefers to get room r at price p or room r ′ at price p ′ . In our model an agent’s favorite room may be a function ofthe entire vector of prices . Thus, asking whether an agent prefers ( r, p ) to ( r ′ , p ′ ) is notmeaningful in our context, since the answer may depend on the prices of other rooms.There are several reasons why this is an important generalization. First, there may be‘rational’ reasons for agents’ preferences over rooms to be affected by the entire vector ofprices. This may be the case, for instance, if we view the choice of a room as only partof a larger ‘consumption plan’. For a concrete example, assume that a forward lookingagent needs to choose between three types of cars, say High ( H ), Intermediate ( I ) andLow ( L ), with corresponding prices p H > p I > p L . If p H is very high then an agent’spreferred option may be to buy type I and hold it for a long period of time. But if p H is reduced then the agent may prefer to buy L initially (saving a larger part of hisbudget) and upgrade to H later on when he has accumulated more wealth. Thus, hischoice shifted from I to L even though the prices of these cars did not change.Prices can also affect preferences if there is incomplete information about the qualityof the rooms, in which case prices may serve as a signaling device. For instance, real-estate prices in two neighboring suburbs may provide information about their relativequalities. An increase (or decrease) in the price of houses in one of them may thereforeaffect the desirability of the other. Another reason for a similar effect is when agents takeinto account the fact that prices affect choices of other agents. In such an interactivesituation there are plausible scenarios in which the entire vector of prices influencesagents’ optimal choices, for example if the price of a neighboring room indicates theidentity of its future inhabitants. Another reason to consider such general preferences is that framing effects and otherwell-documented ‘behavioral biases’ may be affecting choices in ways that the standardmodel cannot capture. For example, assume that rooms A and B have similar charac-teristics while room C is very different from the other two. Assume further that at agiven price vector p with p A = p B the agent’s preferred choice is room C . If the price of A increases then room B may become more attractive as it offers similar value as room A for a ‘bargain’ price. The agent may then choose B instead of C , even though theprices of these rooms have not changed.The two papers that are closest to ours are [17] and [18]. They allow for preferencesas general as in our model, but in both these papers the supply of each good (room)is one, i.e., the number of agents is equal to the number of goods. The proofs in boththese papers rely on this latter assumption. In another recently related paper Velez [23]studies envy-free allocations in a general model with externalities. The existence resultin that paper is based on the argument of [18]. This paper also makes the assumptionthat the supply of each good is one.Our contribution relative to these works is threefold. First, we allow for ‘roommates’,i.e., the supply of each of the indivisible goods in the market may be greater thanone, so that agents may be allocated different units of the same good. This extendsthe applicability of the result to many markets of interest. Notice that there is nostraightforward way to reduce the problem into one in which the number of agents androoms is the same. The reason is that, given the generality of preferences we allow overgoods, there is no way to lift a preference over goods to a preference over units.Second, our proof introduces a new tool to this literature. The proof relies on atopological result of Shapley [14] known as the K-K-M-S Theorem. Roughly speaking,our proof works as follows. For each subset of rooms T we consider the set of pricevectors at which the demand for rooms in T is sufficient to meet the capacity of theserooms. Our assumptions imply that for each T this is a closed set, and that everycollection of these sets corresponding to a balanced collection of subsets of rooms (seeAppendix A for the definition) covers the simplex of all possible price vectors. It thenfollows from the K-K-M-S theorem that there exists a price vector in which the demandfor every subset of rooms is sufficient. By Hall’s marriage lemma it is then possible toassign rooms to agents to exactly clear the market. Previous papers [1, 5] have usedHall’s marriage lemma for this purpose, but to our knowledge the use of the K-K-M-STheorem to establish the conditions required to apply the lemma is new, and as we showallows us to get a substantial generalization. Third, we show equivalence between the rental harmony environment and two otherproblems: The cake division or chore division problem and a model of a discrete exchangeeconomy with money introduced by Gale [8]. While these problem were studied inthe past using similar mathematical tools, there seems to be no direct argument forequivalence between them in the previous literature. Thus, we achieve a generalizationof the known results to these three problems and also establish the connection betweenthem.In the next section we state and prove our main result. In Section 3 we show how ourtheorem may be applied in the two related environments of cake/chore division and adiscrete exchange economy. Section 4 concludes with some final remarks.2.
Theorem and proof
Let N = { , , . . . , n } be the set of agents and let R be the finite set of available rooms .For each r ∈ R let c [ r ] > capacity of room r . Weassume that P r ∈ R c [ r ] = n (the case P r ∈ R c [ r ] > n trivially follows). Let F = (cid:8) f : N → R : (cid:12)(cid:12) f − ( r ) (cid:12)(cid:12) = c [ r ] ∀ r ∈ R (cid:9) be the set of all assignments of agents to rooms that respect the capacity constraints.The total rent for the house is normalized to 1, and we let∆( R ) = ( { p [ r ] } r ∈ R : X r ∈ R p [ r ] = 1 , p [ r ] ≥ ∀ r ∈ R ) be the set of possible ways to allocate the rent among the different rooms. We view∆( R ) as a subset of R R and work with the standard topology it inherits from that space.For p ∈ ∆( R ) the support of p is the set supp( p ) = { r ∈ R : p [ r ] > } . If T ⊆ R then∆( T ) = { p ∈ ∆( R ) : supp( p ) ⊆ T } is the face of ∆( R ) corresponding to T .Given a price vector p , each agent i has a set L i ( p ) ⊆ R of rooms she likes most atthese prices. We assume(A1) For each i and p , L i ( p ) = ∅ .(A2) For each i and p , supp( p ) c ⊆ L i ( p ).(A3) For each i , L i has a closed graph. That is, { p ∈ ∆( R ) : r ∈ L i ( p ) } is closed forevery r ∈ R and every i ∈ N .Assumption (A1) requires that every agent likes at least one of the rooms given eachprice vector. (A2) says that all agents like free rooms. We elaborate on this assumptionin Section 4.1. Finally, (A3) reflects continuity of preferences in prices. Theorem 1.
Under assumptions (A1), (A2) and (A3) there exists p ∗ ∈ ∆( R ) and anassignment f ∗ ∈ F such that f ∗ ( i ) ∈ L i ( p ∗ ) for every i ∈ N . Proof.
For every p ∈ ∆( R ) and every T ⊆ R let A T ( p ) = { i ∈ N : L i ( p ) ∩ T = ∅} be the set of agents who like one of the rooms in T at prices p . Also, for T ⊆ R define K T = ( p ∈ ∆( R ) : | A T ( p ) | ≥ X r ∈ T c [ r ] ) to be the set of price vectors at which the demand for rooms in T is sufficient to meetthe capacity of these rooms. Claim 1.
Each K T is closed in ∆( R ). Proof.
Note that K T = [ B,g \ i ∈ B { p ∈ ∆( R ) : g ( i ) ∈ L i ( p ) } , where the union ranges over all pairs ( B, g ) such that B is a set of agents with | B | ≥ P r ∈ T c [ r ] and g : B → T is an assignment of rooms in T to the agents in B .The sets { p ∈ ∆( R ) : g ( i ) ∈ L i ( p ) } are closed for every i and g by (A3). Therefore, K T is closed as a finite union of intersections of closed sets. (cid:3) Claim 2. If T is a balanced collection of subsets of R (see Appendix A for the definition)then S T ∈T K T = ∆( R ) Proof.
Let T be a balanced collection and let { λ T } T ∈T be non-negative coefficients sat-isfying P T ∈T λ T T = R . Taking scalar product with arbitrary u ∈ R R we have that(1) X T ∈T λ T X r ∈ T u [ r ] = X r ∈ R u [ r ] . Fix some p ∈ ∆( R ) and let g : N → R be a choice of optimal rooms for the playersat prices p (here we use (A1)), so that in particular g − ( T ) ⊆ A T ( p ) for every T ∈ T .Then X T ∈T λ T | A T ( p ) | ≥ X T ∈T λ T | g − ( T ) | = X T ∈T λ T X r ∈ T | g − ( r ) | = X r ∈ R | g − ( r ) | = n = X r ∈ R c [ r ] = X T ∈T λ T X r ∈ T c [ r ] , where the second and last equalities follow from (1) with u [ r ] = | g − ( r ) | and u [ r ] = c [ r ],respectively. It follows that there is T ∈ T such that | A T ( p ) | ≥ P r ∈ T c [ r ], so that p ∈ K T . (cid:3) It follows from Claims 1 and 2 that the collection of sets { K T } T ⊆ R satisfies theconditions of Corollary 1 in Appendix A. Thus, there exists p ∗ ∈ ∆( R ) such that p ∗ ∈ T T ⊆ supp( p ∗ ) K T .Now, consider a bipartite graph with sets of vertices N and R , where a node i ∈ N isconnected to a node r ∈ R if r ∈ L i ( p ∗ ). If T ⊆ supp( p ∗ ) then | A T ( p ∗ ) | ≥ P r ∈ T c [ r ] since p ∗ ∈ K T , and if T * supp( p ∗ ) then | A T ( p ∗ ) | = n ≥ P r ∈ T c [ r ] since all the players likefree rooms. It follows that the graph satisfies the condition of Hall’s Marriage Theorem(See Theorem 3), so there is a subgraph in which each agent is connected to at most oneof the rooms in R and each room in R is exactly full. Since P r ∈ R c [ r ] = n each agent isconnected to exactly one room. This defines the required assignment f ∗ . (cid:3) Variations of the problem
Cake cutting and chore division.
A closely related problem to the one we con-sider is the problem of allocating pieces of a cake to a group of agents in a way that everyagent is happy with the piece he got. There are several formulations of this problem,starting with the classic works [6] and [16]. In the version closest to our model (see, e.g.,[18, Section 3]) the cake has a rectangular shape and one can only use n − n pieces ( n is the number of agents). Each possible cake-cutcorresponds then to a point in the n − n = | R | and c [ r ] = 1 for each r ∈ R .A similar problem is that of chore division, in which a set of undesirable entities(‘chores’) is to be allocated to a group of agents. Each chore comes with a monetarycompensation attached to it as well as the number of agents that should be performingit. One is interested in finding compensations for the various chores such that wheneach agent chooses a favorite chore there are enough agents performing each chore.One example of this situation would be the allocation of administrative tasks to facultymembers in an academic department.What is common to both these problems, and different from the rental harmony prob-lem we considered, is that higher amounts of the divisible good are desired by the agents. In the rental harmony problem we interpreted the transfers as rent that an agent paysfor his room, and we assumed in (A2) that agents like free rooms. On the other hand,in the cake cutting problem p [ r ] = 0 means that the r th piece is empty, and so a hungryagent would not want to get it. Similarly, in the chore division problem a chore withoutcompensation is unlikely to be the favorite of any agent.Consider the following alternative to (A2), which requires that agents never like anempty piece of cake (or a chore with no compensation):(A2 ∗ ) For each i and p , L i ( p ) ⊆ supp( p ). Proposition 1.
Under assumptions (A1), (A2 ∗ ) and (A3) there exists p ∗ ∈ ∆( R ) andan assignment f ∗ ∈ F such that f ∗ ( i ) ∈ L i ( p ∗ ) for every i ∈ N . Proof.
Consider preferences L ∗ i over ∆( R ) that satisfy (A1), (A2 ∗ ) and (A3). We trans-form these preferences to preferences L i over ∆( R ) that satisfy (A1), (A2) and (A3) inthe following way: For each room r ∈ R let v r be the vertex of ∆( R ) corresponding tothat room, and let F r = ∆( R \ { r } ) be the face of ∆( R ) opposite to v r . Denote by w r thebarycenter of F r , that is w r = | R |− P s ∈ R \{ r } v s . Let ϕ : ∆( R ) → ∆( R ) be the uniqueaffine embedding such that ϕ ( v r ) = w r . Then ϕ maps ∆( R ) onto a smaller copy of thissimplex, which lies inside ∆( R ). In particular, ϕ maps the boundary of ∆( R ) onto theboundary of ϕ (∆( R )), and the interior of ∆( R ) onto the interior of ϕ (∆( R )).Define L i by L i ( p ) = L ∗ i ( ϕ − ( p )) , if p ∈ interior( ϕ (∆( R ))) ,L ∗ i ( ϕ − ( p )) ∪ { r ∈ R : p [ r ] ≤ / | R |} , if p ∈ boundary( ϕ (∆( R ))) , { r ∈ R : p [ r ] ≤ / | R |} , otherwise . In words: If p is in the interior of the image of ϕ (the interior of the small copy of thesimplex) then the favorite rooms under L i are the same as those under L ∗ i when pricesare ϕ − ( p ); if p is not in the image of ϕ then only the relatively cheap rooms are thefavorites; if p is on the boundary of the image of ϕ then both rooms that are favoriteunder L ∗ i when prices are ϕ − ( p ) and the relatively cheap rooms are preferred.It is straightforward to verify that L i satisfies assumptions (A1), (A2) and (A3). It fol-lows from Theorem 1 that there exists an envy-free allocation for these preferences. Let ¯ p be the price vector associated with this allocation. We claim that ¯ p ∈ interior( ϕ (∆( R ))).To see why notice first that ¯ p must be in the image of ϕ , since outside the image onlyrelatively cheap rooms are liked, and there is always at least one room with price greaterthan 1 / | R | that no agent would choose. Second, assume by contradiction that ¯ p is on the boundary of the image of ϕ . Then ¯ p ∈ ϕ ( F r ) for some room r . But then no agent likesroom r at prices ¯ p since by (A2 ∗ ) r / ∈ L ∗ i ( ϕ − (¯ p )) and since ¯ p [ r ] = 1 / ( | R | − > / | R | .To conclude, ¯ p ∈ interior( ϕ (∆( R ))) and therefore L i (¯ p ) = L ∗ i ( ϕ − (¯ p )) for each agent i .It follows that there is an envy-free allocation for preferences L ∗ i with prices ϕ − (¯ p ). (cid:3) Equilibrium in a discrete exchange economy.
Our result can be used to proveexistence of equilibrium in an exchange economy with indivisible goods and money, as inthe model studied by Gale [8]. While Gale assumes that supply of each of the indivisiblegoods is 1, we allow for arbitrary quantities. The essential difference between the rentalharmony problem we consider and Gale’s exchange economy is that the prices of therooms need not sum up to 1. Instead, it is only assumed that the price of each room isnon-negative and bounded above (by 1, without loss). Thus, C ( R ) = [0 , R is the set of possible ways to price the different rooms. To stay consistent with ourprevious terminology we keep calling the indivisible goods ‘rooms’, even though theinterpretation of the model is somewhat different now. Agents’ preferences are stillrepresented by the sets L i ( p ) ⊆ R , so that r ∈ L i ( p ) means that agent i likes room r at prices p . We keep assumptions (A1) and (A3) unchanged, but we replace themonotonicity assumption (A2) with the following arguably more compelling assumption:(A2’) For each i and p , L i ( p ) ⊆ { r ∈ R : p [ r ] < } .Thus, instead of assuming that one of the rooms with price 0 will be chosen we assumethat a room with price 1 (the maximal possible price) will not be chosen. We also weaken(A1) to allow for the possibility that no room is desirable when the prices of all roomsis 1. As should be clear from the proof we could weaken (A1) further.(A1’) For each i and p such that p [ r ] = 1 for some r ∈ R , L i ( p ) = ∅ . Proposition 2.
Under assumptions (A1’), (A2’) and (A3) there exists p ∗ ∈ C ( R ) andan assignment f ∗ ∈ F such that f ∗ ( i ) ∈ L i ( p ∗ ) for every i ∈ N . Proof.
Let B ( R ) = { p ∈ C ( R ) : p [ r ] = 0 for some r } and let ϕ : B ( R ) ↔ ∆( R )be a homeomorphism with the property that ϕ ( p )[ r ] = 0 whenever p [ r ] = 1. Sucha homeomorphism is constructed in Gale’s proof of his theorem. Then ϕ transformspreferences over B ( R ) that satisfy (A1’), (A2’) and (A3) into preferences over ∆( R ) that We thank Rodrigo Velez for pointing us to this Gale paper. satisfy (A1), (A2 ∗ ) and (A3). By Proposition 1 these preferences admits an envy-freeallocation. (cid:3) Final comments
On assumption (A2).
Assumption (A2) is probably the most restrictive of ourconditions. Because we did not assume that agents’ preference are monotonic in theprices, (A2) is the only assumption that captures the intuition that agents are tightfisted:Free rooms are always at least as good as non-free roomsIt is possible to relax (A2) somewhat without affecting the result. Consider the fol-lowing assumption:(A2 ◦ ) If supp( p ) = R then supp( p ) c ∩ L i ( p ) = ∅ for every agent i .This weaker version requires that if there are free rooms then every agent likes at leastone of them. Our result holds unchanged if (A2) is replaced by (A2 ◦ ). The reason isthat, given (A3), (A2 ◦ ) implies (A2). To see why, fix some p with | supp( p ) c | ≥ | supp( p ) c | ≤ r ∈ supp( p ) c . For every α ∈ (0 , p α defined by p α [¯ r ] = p [¯ r ] = 0 and p α [ r ] = αp [ r ] + (1 − α ) | R |− for each r = ¯ r . Then ¯ r is the only free room at every p α , so by (A2 ◦ ) every agent likes¯ r . But p α → p as α →
1, so by (A3) every agent likes ¯ r at prices p as well.Still, even this weaker version rules out many standard preferences. In particular,any quasi-linear preferences in which two rooms have different values do not satisfythis assumption, and indeed envy-free allocations need not exist with such preferences(see, for example, [1, Section 6]). However, starting from any preferences L i satisfying(A1) and (A3) (in particular, quasi-linear preferences), it is possible to obtain modifiedpreferences that satisfy all three assumptions by altering L i only on the boundary of thesimplex. Specifically, the correspondence ˜ L i ( p ) = L i ( p ) ∪ supp( p ) c satisfies (A1), (A2)and (A3) whenever L i satisfies (A1) and (A3).4.2. Efficiency.
Some of the previous papers on fair allocations have studied the rela-tionship between envy-freeness and efficiency. In our model, however, it is not clear whatefficiency means. The reason is that preferences of agents are defined over the indivisiblegoods (rooms) conditional on the vector of prices. Thus, one cannot compare allocationsacross different price vectors.An alternative approach, which allows to consider efficiency, would be to start frompreferences over pairs ( p, r ) where p is the vector of prices and r is the room assigned to the agent. From such preferences one can derive the preferences over rooms conditionalon prices. However, such across-prices comparisons are not relevant for the question ofexistence of envy-free allocations, which is the focus of this note. We therefore preferredto simplify the exposition and notation by using the conditional preferences as primitive.4.3.
Manipulation.
While we proved existence of envy-free allocations, we did notstudy whether such allocations can be implemented when agents’ preferences are theirprivate information. This aspect of the problem has been analyzed under the type ofpreferences allowed in the previous literature – see for example [22] and the referencestherein. It would be interesting to see which of the results obtained in that literatureapply in our set-up as well.4.4.
Constructing a solution.
Our proof is not constructive, as the K-K-M-S theo-rem guarantees existence of the desired price vector p ∗ without showing how to find it.However, one could construct algorithms that approximate p ∗ up to an arbitrary levelof precision. For instance, Shapley’s original proof of the K-K-M-S theorem relies on acombinatorial result in the spirit of Sperner’s lemma, which can be used approximate p ∗ .Once p ∗ is found it is easy to construct the envy-free assignment f ∗ . Appendix A. K-K-M-S Theorem
A collection T of subsets of R is called balanced if there are non-negative coefficients { λ T } T ∈T such that X T ∈T λ T T = R . The following result by Zhou [25] is a variant of Shapley’s [14] ‘K-K-M-S theorem’,the only difference being that the covering sets are open rather than closed. See also[3, 9, 10, 11, 15, 24, 25] for alternative proofs of the K-K-M-S theorem and related results,as well as for applications of this result in the theory of cooperative games.
Theorem 2. [25] Let { L T } T ⊆ R be a collection of open subsets of ∆( R ) with the propertythat ∆( S ) ⊆ S T ⊆ S L T for every S ⊆ R . Then there exists a balanced collection T suchthat T T ∈T L T = ∅ . Corollary 1.
Let { K T } T ⊆ R be a collection of closed subsets of ∆( R ) such that S T ∈T K T =∆( R ) whenever T is a balanced collection. Then there is p ∗ ∈ ∆( R ) such that p ∗ ∈ T T ⊆ supp( p ∗ ) K T . This is essentially the approach taken in [23]. In [23] preferences are defined over the entire allocation(including how rooms are assigned to other agents), but the impersonality axiom implies that agentsonly care about the vector of prices and the room assigned to them. Proof.
For each T define L T = K cT . Then each L T is open and T T ∈T L T = ∅ for everybalanced collection T . By Theorem 2 there is S ⊆ R and p ∗ ∈ ∆( S ) such that p ∗ L T for every T ⊆ S . Thus, p ∗ ∈ T T ⊆ S K T ⊆ T T ⊆ supp( p ∗ ) K T . (cid:3) Appendix B. Marriage Theorem with polygamy
Theorem 3. [4, Corollary 3.11] Let G be a bipartite graph with vertex sets X and Y ,and let c : Y → N . Then G contains a subgraph H such that d H ( y ) = c [ y ] for every y ∈ Y and d H ( x ) ∈ { , } for every x ∈ X if and only if for every S ⊆ Y d G ( S ) ≥ X y ∈ S c [ y ] . References [1] Abdulkadiroglu, A., T. S¨onmez and M. U. ¨Unver, “Room assignment-rent division: A marketapproach,”
Social Choice and Welfare
22 (2004), 515-538.[2] Alkan, A., G. Demange and D. Gale, “Fair allocation of indivisible goods and criteria for justice,”
Econometrica
59 (1991), 1023-1039.[3] Billera, L. J., “Some theorems on the core of an n -person game without side payments,” SiamJournal of Applied Mathematics
18 (1970), 567-579.[4] Bollob´as, B.,
Modern graph theory (New York: Springer-Verlag, 1998).[5] Demange, G., D. Gale and M. Sotomayor, “Multi-item auctions,”
The Journal of Political Economy
94 (1986), 863-872.[6] Dubins, L. E. and E. H. Spanier, “How to cut a cake fairly,”
The American Mathematical Monthly
68 (1961), 1-17.[7] Foley, D., “Resource allocation and the public sector,”
Yale Economic Essays
International Journal of GameTheory
13, 61-64.[9] Herings, P. J. J., “An extremely simple proof of the K-K-M-S theorem,”
Economic Theory
Proceedingsof the American Mathematical Society
104 (1988), 759-763.[11] Komiya, H., “A simple proof of the K-K-M-S theorem,”
Economic Theory
Arrow and the founda-tions of the theory of economic policy (London: Macmillan, 1987), 341-349.[13] Moulin, H.,
Fair division and collective welfare (Cambridge: MIT press, 2003).[14] Shapley, L. S., “On balanced games without side payments,” in Hu, T. C. and S. M. Robinson,Eds.
Mathematical Programming (New York: Academic Press, 1973), 261-290.[15] Shapley, L. S. and R. Vohra, “On Kakutani’s fixed point theorem, the K-K-M-S theorem and thecore of a balanced game,”
Economic Theory
Econometrica
17 (1949), 315-319. [17] Stromquist, W., “How to cut a cake fairly,” The American Mathematical Monthly
87 (1980), 640-644.[18] Su, F. E., “Rental harmony: Sperner’s lemma in fair division,”
The American MathematicalMonthly
106 (1999), 930-942.[19] Svensson, L., “Large indivisibilities: An analysis with respect to price equilibrium and fairness,”
Econometrica
51 (1983), 939-954.[20] Tadenuma, K. and W. Thomson, “No-envy and consistency in economies with indivisible goods,”
Econometrica
59 (1991), 1755-1767.[21] Varian, H., “Equity, envy, and efficiency,”
Journal of economic Theory
Journal of Economic Theory
146 (2011),326-345.[23] Velez, R. A., “Fairness and externalities,” mimeo, Texas A&M University, 2013.[24] Zhou, L., “A new bargaining set of an N-person game and endogenous coalition formation,”
Gamesand Economic Behavior