A New Approach to Fair Distribution of Welfare
aa r X i v : . [ ec on . T H ] S e p A New Approach to Fair Distribution of Welfare
Moshe Babaioff ∗ Uriel Feige † September 26, 2019
Abstract
We consider transferable-utility profit-sharing games that arise from settings in which agentsneed to jointly choose one of several alternatives, and may use transfers to redistribute the welfaregenerated by the chosen alternative. One such setting is the Shared–Rental problem, in whichstudents jointly rent an apartment and need to decide which bedroom to allocate to each student,depending on the student’s preferences. Many solution concepts have been proposed for suchsettings, ranging from mechanisms without transfers, such as Random Priority and the Eatingmechanism, to mechanisms with transfers, such as envy free solutions, the Shapley value, and theKalai-Smorodinsky bargaining solution. We seek a solution concept that satisfies three naturalproperties, concerning efficiency, fairness and decomposition. We observe that every solutionconcept known (to us) fails to satisfy at least one of the three properties. We present a newsolution concept, designed so as to satisfy the three properties. A certain submodularity condition(which holds in interesting special cases such as the Shared-Rental setting) implies both existenceand uniqueness of our solution concept.
We introduce a new solution concept for situations in which agents with cardinal preferences needto jointly choose one alternative from a set of alternatives, possibly compensating each other usingtransfers. This is a well studied setting in cooperative game theory, and we follow a normativeapproach that specifies properties that we wish our solution concept to have, and then design asolution concept that meets these specifications. To motivate our new solution concept and contrastit with well established previous solution concepts, we start with an example.Suppose that three students jointly rent a three bedroom apartment for a total rent of r unitsof money. The students need to do two things. One is to jointly pay the rent, and the other is tosolve the allocation problem, namely, decide which student gets which room, possibly compensatingeach other with money. We assume that the students are equals, in the sense that each studentbears equal responsibility in paying the rent, and equal eligibility in receiving a room. Being equals,each student first pays r/ Remark 1.1
In cases in which no student receives a transfer larger than r/ , transfers may beimplemented indirectly by having students pay unequal parts of the rent. However, in this paper we do ∗ Microsoft Research, [email protected]. † Weizmann Institute, Rehovot, Israel, [email protected]. Supported in part by the Israel Science Founda-tion (grant No. 1388/16). Part of this work was done at Microsoft Research, Herzeliya. ot constrain transfers to be smaller than r/ , and the question of whether transfers are implementedas direct transfers among the students or as modification to rent payments is not a concern of thecurrent paper. A common approach for allocating rooms (and other goods) is using the
Random Priority mecha-nism (a.k.a. random serial dictatorship ), that we abbreviate as RP. A total order among the studentsis chosen uniformly at random, and each student in her turn chooses a room among those that arestill available. RP has obvious advantages, being easy to implement in practice, agents (studentsin our case) have dominant strategies (given an agent’s turn to choose, she should simply choosethe available alternative that she most prefers), and being perceived as “fair” (all agents are treatedequally from the mechanism’s point of view). A significant drawback of RP is that it does not max-imize welfare – the resulting allocation may produce less welfare (sum of utilities) than alternativeallocations. Hence some economic efficiency is lost.Let us consider a concrete example. Suppose that the students can express their valuation forrooms in units of money, and that they are risk neutral (they wish to maximize the expected receivedvalue). Suppose further that for some small 0 < δ < , the value that each student derives by beinggiven each of the rooms is as in the following table:Example 1 Room 1 Room 2 Room 3Student 1 − δ δ Student 2 − δ δ Student 3 − δ + δ The maximum welfare allocation assigns room i to student i for every i , giving welfare of + 2 δ .However, RP will result with probability half with an assignment in which student 2 gets room 1,giving welfare , and hence the expected welfare of RP is δ lower than optimal.The RP mechanism does not involve transfer of money among agents. In our language, we refer toit as an ANT, which is an abbreviation for Allocation mechanism with No Transfers. To overcome itsweaknesses (shared by other ANTs as well), one often considers allocation mechanisms with transfers(abbreviated as AWT – Allocations With Transfers). AWTs allow for the following paradigm: firstchoose a maximum welfare allocation (thus creating the largest pie to divide: the maximum possiblewelfare to distribute among the agents), and then employ monetary transfers among the agents soas to distribute the high welfare to all agents, so as to satisfy some fairness criteria. In the exampleabove, this would mean assigning room i to student i for every i , and then figuring out what thetransfers should be so that the combination of allocation with transfers would be “fair”.To reason about transfers, we make the assumption that students have quasi-linear utilities: theutility of a student is simply the sum of her value for the room that she receives plus the transfer thatshe receives (the transfer may be negative if the student gives money rather than receives money).Moreover, we assume that the mechanism that computes the allocation and the transfers has accessto the true valuations of the students. (This full information assumption is standard in cooperativegame theory, and there are impossibility results showing that it cannot be avoided in our setting.See more details in Appendix R.) Within such a setting, there is a well studied class of AWTs thatis referred to as envy free solutions [8, 27, 9]). The basic principle is that one associates a transferwith each room (where the sum of transfers equals 0 – this is a budget balance condition) such thatgiven the transfers, each student (weakly) prefers a different room. Then each student gets the roomand associated transfers that she prefers, and no one prefers to switch with another agent. In theexample above we can associate the following transfers with the rooms:2 oom 1 Room 2 Room 3 − + 2 δ − δ − δ These transfers are indeed budget balanced and envy free, that is, each student i prefers herassigned room i (along with the associated payment) over any other room, leading to an allocationthat maximizes welfare and in which supposedly every student is happy (as she got her most preferredone out of the three available options).Let us consider a natural question. Suppose that the students initially intend to use the RPmechanism. Will the students be better off by using the envy free mechanism (that we abbreviateEF) instead of using RP? In some respects, the answer is no: RP is simpler to implement than EF,as it does not require students to disclose their valuation functions and to implement transfers. Inother respects the answer is yes: EF generates higher welfare. But let us consider this last aspectmore carefully. The social justification to maximize welfare is (in our opinion) the belief that theextra welfare will eventually get distributed to all members of the society that contributed to theincrease in welfare. Is it the case that the increase in welfare (generated by moving from RP to EF)is distributed over the three students in a reasonable way? The answer is negative in our opinion. • In RP, the sum of expected values derived by students 1 and 2 is 1. In EF, the sum of valuesincreases to 1 + δ , but the sum of what they lose due to transfers is − δ . If δ < , each ofthe two students gets higher expected utility from RP than from EF. It is not true that theincrease in welfare is distributed over all students in a way that every student (at least weakly)benefits. • Student 3 contributes nothing to the increase in welfare when changing from RP to EF (inboth cases her allocation is exactly the same – room 3). Nevertheless, under EF, student 3 notonly gets her most preferred room, but also gets paid. Moreover, this payment is even largerthan the total increase in welfare that EF offers compared to (the expected welfare of) RP.Another aspect that we find troublesome with the EF solution is the following. • In every Pareto efficient allocation, student 3 gets room 3 and the only question is which ofthe rooms 1 and 2 is allocated to which of the students 1 and 2. Hence the instance naturallydecomposes into two subinstances, I involving room 3 and student 3, and I , involving theother two students and two rooms. If one does this decomposition and then employs an EFmechanism on each component separately, student 3 does not receive any payments from theother students, and hence the resulting payments are completely different from those withoutthe decomposition. Likewise, suppose that we had started with two separate instances, I , and I as above, where every student prefers the rooms in her own instance over those inthe other instance (it may even be that each instance concerns a different apartment). If weuse EF mechanisms, then combining the two instances into one results in different paymentscompared to solving each of the instances separately. This sensitivity of the payments in EFmechanisms to composition and decomposition of instances (importantly, we are consideringhere cases in which composition and decomposition have no effect on the allocation itself) maylead to disagreements among the agents regarding what constitutes a single instance.An allocation instance may have several different envy free solutions, but the above shortcomingsare shared by all envy free solutions in the above example, provided that δ is sufficiently small . If δ is sufficiently small, then in every envy free solution the transfer associated with room 3 is positive: Let p i Shapley value , the
Nucleolus , the
Nashbargaining solution and the
Kalai-Smorodinsky (KS) bargaining solution. Every one of them suffersfrom at least one of the troubling aspects listed above, see sections 4.5, G.3 and Appendix I for moredetails. In fact, the same holds for every AWT approach that we could find in the literature. Hencedespite the many solution concepts that already exist, we find it appropriate to introduce a newAWT mechanism that does not suffer from any of the troubling aspects listed above.
We consider transferable-utility profit-sharing games , a setting that has been studied in previous work(e.g., by Moulin [20]). The room allocation problem of the previous section is a special case of thismore general setting.There is a set N of n agents (also referred to as players ) and a set A of alternatives . Everyagent i ∈ N has a valuation function v i : A → R . All valuation functions are expressed in thesame units (of money). We let v = ( v , . . . , v n ) denote the tuple of all the valuation functions. AnNT ( no transfers ) social choice function f receives as input the pair ( A , v ) that includes the set ofalternatives and the valuation functions, and outputs one of the alternatives from A . A randomizedNT social choice function may use randomization when choosing its output. Consequently, its outputis a probability distribution over alternatives.Given the tuple v of valuation functions, a set S ⊆ N of agents and an alternative A ∈ A , the welfare w S,v ( A ) that alternative A offers to S is defined as w S,v ( A ) = P i ∈ S v i ( A ). An NT socialchoice function f maximizes welfare (with respect to N ) if the alternative A ∗ ∈ A that f selectssatisfies w N ,v ( A ∗ ) ≥ w N ,v ( A ) for all A ∈ A .We allow transfer of money among agents. Such transfers are represented as a vector p =( p , . . . , p n ), where p i is the payment to agent i , measured in units of money. We refer to the caseof p i > in-payment (the amount of money of agent i increases), and to the case of p i < out-payment (the amount of money of agent i decreases). A transfer vector p is budget balanced if P ni =1 p i = 0. A transfer function g receives as input the triple ( A , v, A ∗ ) that includes the set ofalternatives, the valuation functions, and an alternative chosen by an NT social choice function, andoutputs a budget balanced transfer vector.We assume that the utility functions of the agents are quasi-linear . Namely, for agent i ∈ N with valuation function v i , her utility u i from the pair of alternative A and transfer vector p is u i ( A, p ) = v i ( A ) + p i . We further assume a setting of “full information upon request”: the socialplanner may request information about valuation functions of agents (this information might belimited to the ordinal preferences of an agent over a set of alternatives, or might be as general as denote the transfer associated with room i . For student 3 not to envy student 2 we must have p ≤ p + 2 δ . Forstudent 2 not to envy student 1 we must have p ≤ p − δ ≤ p − δ . Together with the budget balancerequirement we have that p = − p − p ≥ − p + 1 − δ − p − δ , implying that p ≥ − δ >
0, where the lastinequality holds when δ is sufficiently small. N corresponds to the set of three students, and A corresponds to the set of six possible permutations over rooms, matching one room to one student.The valuation functions v i are as in the example. An example of a randomized NT social choicefunction is the output of the Random Priority (RP) mechanism: once v is given (in fact, knowledge ofordinal preferences suffices here) RP induces a well defined probability distribution over alternatives.The envy-free allocation and transfers provided in the example are a solution (implicitly) involvingan NT social choice function f and a transfer function g . For the setting described above, we wish to design a solution concept that has two components: anNT social choice function, and an associated transfer function. We have three goals. One is economicefficiency . This goal is easily attainable in our full information framework – we simply select a welfaremaximizing alternative, which we denote by A ∗ . (If there are several welfare maximizing alternatives, A ∗ denotes one of them, selected arbitrarily.) Another goal is to achieve fairness , in the sense thatthe welfare will be shared “fairly” among all agents. Achieving this goal is made possible by the use oftransfers. Those agents for which alternative A ∗ is undesirable can be compensated by in-payments,and the budget balance requirement can be met by extracting an equal amount of out-paymentsfrom those agents who do desire alternative A ∗ . The assumption that agents have quasilinear utilityfunctions simplifies the accounting of the extent to which utility derived from payments can replaceutility derived from the selected alternative. The third goal is that of decomposability , which basicallymeans that if a large game involving multiple agents can be naturally decomposed into many smallergames over disjoint sets of agents, then the solution of the large game should also decompose intosolutions of the smaller games. Equivalently, one should be able to solve each smaller game separately,and obtain a solution to the large game as the concatenation of the solutions to the smaller games.Our contributions in this work are in setting the above three goals, proposing definitions for thefairness properties and decomposition properties that they refer to, proposing a solution concept thatattains the above three goals, and providing sufficient conditions for its existence and uniqueness.Here is an informal statement of our main result when specialized to the Shared–Rental problem. Theorem (informal) . The lex-max-WS solution (introduced in our work) for the Shared–Rentalproblem maximizes welfare and satifies the fairness and the decomposition properties alluded to above(and formally defined later in this paper). Moreover, in a well defined sense, it is the unique solutionthat satisfies these properties.
Here are more details regarding our contributions:1. We propose a new notion of fair solutions, the welfare-sharing core (abbreviated WS-core). SeeDefinition 2.1. It combines three principles that are briefly sketched below.(a) One principle is domination with respect to the utility agents can receive from a disagree-ment point , or reference point . This is mathematically similar to the familiar concept of individual rationality (IR), though conceptually there is a distinction between these twonotions. See Section A.1 for more details.(b) Another principle is that fairness entails not only lower bounds on the utilities that agentsderive from the solution, but also natural upper bounds. We introduce a set-function W max , where for a set S of agents, W max ( S ) is the welfare that S could derive from the5lternative that is best for S . The same notion appears in [21], where is is referred to as stand alone utility. We require that the utility that a solution (with transfers) offers toa set S of agents does not exceed W max ( S ). This leads to the notion that we (and [21])refer to as the anticore . See Section A.2 for more details.(c) Another principle is that of decomposability , as discussed above (see Section 3 for moredetails). A key property of the anticore is that it decomposes: the anticore of a decom-posable game is the concatenation of the anticores of each of the component games.2. We show that in our setting, if W max is submodular, then the WS-core is non-empty. SeeTheorem 2.2.3. We propose to use egalitarian considerations (specifically, the lexicographically-maximal welfare-sharing rule, denoted lex-max-WS ) for selecting a single solution from the WS-core, see Section4. When W max is submodular, we show (see Theorem 4.4, which relates to a previous resultof Dutta and Ray [6]) that different egalitarian considerations (e.g., also the min-square rule,defined in Section 4) all lead to the same unique solution.4. When W max is submodular, we show that computing the lex-max-WS solution can be done inpolynomial time. See Appendix O. Moreover, it is a continuous function (with a small Lipschitzconstant) of the valuation functions at points where the disagreement utility is a continuousfunction of the valuations. See Appendix P. The lex-max-WS solution may not be continuousat points in which the disagreement utility is not continuous.5. We explain the similarities and differences between our new solution concept and several relatednotions. These include coalitional games and imputations (Section A); cost-sharing games (Sec-tion A); notions related to our notion of decomposability, such as Separability (Section C) and consistency for reduced games (implicitly addressed in Section I.3); previous notions referredto as the anticore (Section A.2); egalitarian solution concepts and
Lorenz ordering (Section 4);the
Shapley value (Section 4.5); the
Nucleolus (Appendix I); envy free solutions (Section G.3.1);
Nash bargaining and
Kalai-Smorodinsky (KS) bargaining (Appendix I); population monotonicity and resource monotonicity (Appendix N).6. We show that for the Shared–Rental problem W max is submodular, and hence the lex-max-WS solution enjoys those properties shown above to be implied by submodularity. In addition, the lex-max-WS solution dominates Random Priority (by definition), and moreover, when instancesare “decomposable” it satisfies a strong notion of decomposability. See Section G.
Our starting point is the (not necessarily new) premise that statements such as “this solution is fair”have no rigorous meaning on their own. Rather, the fairness of a solution needs to be judged inrelation to a reference context . In our definition of fairness, the reference context will be the set A ofalternatives together with a probability distribution π over A (which we will refer to as a referencepoint , or disagreement point ). We now present the definition of the WS-core, and then follow it upwith a discussion and comparison with related work.A solution ( A ∗ , p ) is composed of a welfare maximizing alternative A ∗ and a budget balancedtransfer vector p = ( p , . . . , p n ). The utility that agent i derives from solution ( A ∗ , p ) is u i ( A ∗ , p ) = v i ( A ∗ ) + p i . In our context, two solutions ( A ∗ , p ) and ( A ′∗ , p ′ ) are equivalent if u i ( A ∗ , p ) = u i ( A ′∗ , p ′ )6or every agent i . Consequently, we sometimes refer to the utility vector ( u ( A ∗ , p ) , . . . , u n ( A ∗ , p )) asthe solution.A solution will need to satisfy certain constraints, where these constraints are expressed as afunction of the utilities that agents derive from the solution. We shall use w S,v ( A ) = P i ∈ S v i ( A ) todenote the welfare derived by a set S of agents from an alternative A , and u S ( A ∗ , p ) = P i ∈ S u i ( A ∗ , p )to denote the utility derived by S from solution ( A ∗ , p ).We associate two classes of constraints with solutions ( A ∗ , p ):1. Domination:
We assume that a probability distribution π over A is given, where π ( A ) denotesthe probability associated with alternative A . This distribution represents the alternative thatwould be chosen in the absence of agreement to use a mechanism with transfers. As such,the distribution π may depend on the valuations v , and we shall sometimes use the notation π v to make this explicit. The value that agent i derives from π v is P A ∈A π v ( A ) v i ( A ), andwe refer to it as the agent’s disagreement utility. The domination constraints require that u i ( A ∗ , p ) ≥ P A ∈A π v ( A ) v i ( A ) holds for every agent i .2. The anticore:
We introduce a welfare function over sets of agents, which we denote by W max .For every S ⊆ N let W max ( S ) = max A ∈A [ P i ∈ S v i ( A )] indicate the maximum welfare achievableby S . The anticore constraints require that u S ( A ∗ , p ) ≤ W max ( S ) for every set S ⊆ N . Definition 2.1 (WS-core)
Suppose one is given a tuple v of valuation functions, a set A of al-ternatives, and a probability distribution π v over A . A solution ( A ∗ , p ) (composed of an alternative A ∗ ∈ A that maximizes welfare and a budget balanced vector p of transfers) is said to belong tothe welfare-sharing core (WS-core) if the solution ( A ∗ , p ) satisfies the above two sets of constraints(domination and anticore) with respect to the given v and π v . There are cases in which the WS-core is empty. Here is one such example. Suppose that thereare three agents and two alternatives. A is the disagreement alternative and all agents value it as 0,whereas A is the alternative that maximizes welfare, agent 1 values it as −
1, whereas each of theother two agents values it as 1. Hence there is welfare of − W max has value 0 both for the set { , } and for the set { , } , and hence there is no way of sharingthe welfare without violating at least one of the anticore constraints.Despite the above, in important special cases, the WS-core is nonempty. We first recall somestandard terminology. A set function f is monotone if f ( S ) ≥ f ( T ) for all T ⊂ S . A set function f is submodular if for every two sets S and T it holds that f ( S ) + f ( T ) ≥ f ( S ∩ T ) + f ( S ∪ T ).Equivalently, f is submodular if it has the decreasing marginal returns property: for every item i and two sets S ⊂ T it holds that f ( S ∪ { i } ) − f ( S ) ≥ f ( T ∪ { i } ) − f ( T ). A submodular functionneed not be monotone.Our main existence result is the following: Theorem 2.2
Given a tuple v of valuation functions, a set A of alternatives, and a probabilitydistribution π over A , either one of the following conditions suffices in order for the WS-core to benonempty.1. W max is submodular (though not necessarily monotone).2. W max − W π is monotone (though not necessarily submodular), where W π ( S ) = P i ∈ S P A ∈A π v ( A ) v i ( A ) is the expected value derived by set S from the disagreement distribution π . Note: if the dis-agreement utilities are 0, then a sufficient condition (though not necessary) for W max − W π tobe monotone is that the valuation functions are nonnegative. A central aspect in applying game theory, social choice and mechanism design in practice is that ofdecomposing large games into smaller games and reasoning about each small game separately. Onemay view all humanity (and other strategic living creatures) as participating in one huge game inwhich individuals pursue their own goals and have multiple interactions with other individuals. Thisgame is too heterogeneous and complicated to reason about as a whole. Moreover, the actions ofsome individuals have very low influence (if at all) on some other individuals, to the extent that theycan be ignored. Thus, to be able to reason about interactions between individuals, it is reasonableto decompose this huge game into smaller games, involving smaller numbers of individuals, andhaving a more homogeneous character. For example, a smaller game might be a particular auction,a particular room allocation problem, or elections for a particular position. In such a smaller gamewe specify who the players are, what actions are available to them, what the possible outcomes are,and assume that the value derived by the players from the game depends only on the outcome ofthat game. The decomposition of the huge “game of humanity” into smaller games is a modelingdecision that captures reality only in some approximate sense (the small games are not really isolatedfrom each other, there might be players affecting or affected by the game that we are not aware of,etc.), but seems to be an unavoidable modeling decision in areas such as social choice and mechanismdesign.Given the ubiquity of game decompositions, we think it is important that mechanisms (for profit-sharing games, in our context) will remain consistent throughout decompositions. Ideally, we wouldlike the solution to problems that have a natural partition to subproblems, to be the same whether ornot we consider the problem as a whole and find a solution, or consider each subproblem separatelyand find a solution to each.Motivated by the above view, in this section we introduce formal definitions for the notion ofan instance being decomposable, and for two notions of decomposability for mechanisms: weak andstrong.Let A be a set of alternatives, N be a set of agents, and let v = ( v , . . . , v n ) be a tuple specifyingthe valuation functions of the agents. We say that alternative A ∈ A is Pareto optimal with respectto a set S ⊂ N of agents if for every alternative B ∈ A , either there is some agent i ∈ S such that v i ( A ) > v i ( B ), or for all agents i ∈ S it holds that v i ( A ) = v i ( B ). Definition 3.1 (independent component, decomposable instance)
A set of players S ⊂ N is referred to as an independent component (or just component , for brevity) if for every alternative A ∈ A that is Pareto optimal with respect to S (given v ) and for every alternative B ∈ A that isPareto optimal with respect to ¯ S = N \ S , there is an alternative C ∈ A (possibly C = A or C = B )such that for every agent i ∈ S it holds that v i ( C ) = v i ( A ) , and for every agent j ∈ N \ S it holdsthat v j ( C ) = v j ( B ) . We say that an instance is decomposable if it has a component that is nontrivial(the component is neither empty, nor the whole instance).
8t is implicit in the above definition that if a decomposable instance has more than one Paretooptimal alternative, then there are agents that are indifferent among some choices of alternatives.Observe that if S ⊂ N is a component then so is N \ S . Definition 3.1 implies that if each of thetwo components S and N \ S selects a most preferred alternative on its own (such an alternative willbe Pareto optimal with respect to the component), then there will be no conflicts between the twochoices – we will be able to select a single alternative that is just as good, from the point of view ofevery player in every component.As an example to the decomposition concept introduced above, consider the Shared–Rental prob-lem example from Section 1.1, with valuation functions as in the table titled Example 1, and with δ < . In that example, there are two components, one containing Students 1 and 2, and the othercontaining Student 3. Every alternative A that is Pareto optimal for the first component assignsthe first two rooms to the first two students, and every alternative B that is Pareto optimal for thesecond component assigns the third room to the third student. The two alternatives A and B canbe replaced by one alternative C (in fact, in this simple example it will hold that C = A as thereis only one room in the second component), and every agent values C as being equally good as thealternative chosen by his own component.A solution involves two aspects: a choice of alternative, and transfers. In Proposition B.1 (Ap-pendix B) we show that any alternative that maximizes welfare also maximizes welfare for eachcomponent separately. Thus the proposition shows that every welfare maximizing solution respectsthe component structure of the given instance, as far as the choice of alternative is considered. Fora solution to qualify as “decomposable”, it makes sense to in addition require that there are notransfers between components. We refer to this forbidding of transfers between components as weakdecomposability . Definition 3.2 (weak decomposability)
Let N be the set of agents, let A be the set of alterna-tives, and let v be the tuple of valuation functions of the agents. A solution ( A, p ) , composed of analternative A ∈ A (in this definition we do not require A to be a welfare maximizing alternative,as decomposability is relevant also to mechanisms that do not maximize welfare) and a vector p oftransfers (summing up to 0), is weakly decomposable if for every component S ⊂ N it holds that P i ∈ S p i = 0 . Namely, the net transfer into the component is 0 (consequently, the same holds for thenet transfer out of the component). As a trivial example, every solution that involves no transfers is weakly decomposable.We also introduce a notion of strong decomposability that postulates that utilities of individualagents within a component are not influenced by decisions in other components. Unlike the notion ofweak decomposability which is the property of a single solution, the notion of strong decomposabilityis a property of a mechanism and not just of a single solution. In the context of our work in which weassume “full information upon request”, a mechanism M is a mapping from instances to solutions.The input to M is an instance I of arbitrary size, composed of a set N of agents, a set A ofalternatives, and a tuple v of valuation functions of the agents. The output M ( I ) is the proposedsolution for the instance I , where the solution is composed of a winning alternative (in general, it isnot required to be an alternative that maximizes welfare) and a vector of transfers. A mechanismcan be randomized, in which case, given an input instance, the mechanism generates a distributionover solutions, and the proposed solution is a random sample from this distribution. Definition 3.3 (strong decomposability)
We say that a mechanism M is strongly decomposable if for every decomposable instance I , the output of the mechanism is consistent with the outputs of themechanism on each of the components separately, in the following sense. Let N be the set of agentsin I , let A be the set of alternatives, and let v be the tuple of valuation functions of the agents. Given , A and v , let S ⊂ N be a component. Let I S be the instance that results from restricting the set ofagents of I to be just S (without changing the set of alternatives and the valuation functions of theagents in S ). Let M ( I ) ( M ( I S ) , respectively) denote the outcome (chosen alternative and vector oftransfers) when M is applied to instance I ( I S , respectively). Then for every agent i ∈ S , her utilityin both cases is the same. Namely, u i ( M ( I )) = u i ( M ( I S )) . (For randomized mechanisms, equalityneeds to hold for the expected utility.) Thus, strongly decomposable mechanisms essentially decide on the solution in each componentindependently of other components. Proposition B.2 in Appendix B shows that strong decompos-ability implies weak decomposability in the following sense: assume that M is a mechanism that forevery instance selects an alternative that maximizes welfare and a budget balanced vector of trans-fers. If M is strongly decomposable, then for every decomposable instance the solution produced by M is weakly decomposable.Observe that strong decomposability does not require that the same alternative be chosen in I and in I S , but rather only that agents in S receive the same utilities in both instances (and likewise,as ¯ S is also a component, that agents in ¯ S receive the same utilities in I and in I ¯ S ). In Section G.2we shall revisit decomposability in the context of the room allocation problem, and there we shalladditionally require (and achieve) that the choices made by M in different components give one singlealternative for all of I .We now present a proposition that describes the component structure of an instance.Given a set U , a collection C of subsets of U is a (distributive) lattice if for every two sets S, T ∈ C it holds that S ∩ T ∈ C and S ∪ T ∈ C . The minimal sets of a lattice (a set from the lattice is minimalif it is nonempty and does not contain any other nonempty set from the lattice) form a partition of U . The following proposition is proved in Appendix B. Proposition 3.4
Given a set A of alternatives, a set N of agents, and a tuple v of valuationfunctions, the components of N form a lattice. It follows from Proposition 3.4 that the minimal components form a partition of N . (It could bethat only N itself is a nonempty component.) It can be shown by induction that that if each of theminimal components S selects a most preferred alternative on its own, then there will be no conflictsbetween the choices – we will be able to select a single alternative that is just as good, from the pointof view of every player in every component. The anticore and decomposition:
A major benefit of the anticore is that it ensures decom-posability properties. We remark that even though our notion of the anticore is the same as thatof [21], the notion of decomposability was not defined in that or other previous work, and hence theconnection between anticore and decomposability is a new contribution of the current paper. Forweak decomposability (Definition 3.2) we have:
Proposition 3.5
Every solution in the anticore is weakly decomposable.
Proof:
Let N be the set of agents, let A be the set of alternatives, let v be the tuple of valuationfunctions of the agents, and let S ⊂ N be a component. Consider an arbitrary solution ( A ∗ , p )in the anticore, composed of a welfare maximizing alternative A ∗ ∈ A and a vector p of transfers.By Proposition B.1, A ∗ also maximizes the welfare of each of the components S and ¯ S separately.By the anticore constraints, the net transfer into S is at most 0, and so is the net transfer into ¯ S .Consequently, the net transfer into S is exactly 0. Hence the solution is weakly decomposable.It is premature at this stage to address strong decomposability (Definition 3.3). This will be donelater, in Proposition 4.3. 10 Selection from within the welfare-sharing core
The set of constraints corresponding to domination over a disagreement point are meant to achievethe property of having each agent (weakly) prefer (in terms of utility) every solution in the WS-coreover the disagreement point. Our guideline for selecting a unique solution from within the WS-core(when the WS-core is nonempty) is that we wish this property to hold not only in a qualitativemanner, but also in a quantitative manner, to the largest extent possible. Ideally, we would like it tobe that for every agent, switching to our mechanism offers a worthwhile increase in utility comparedto the disagreement point. This calls for an egalitarian distribution of the welfare gain among allagents, where the welfare gain is the difference in welfare between the maximum welfare alternative( W max ( N )) and the expected welfare generated by the disagreement point ( W π ( N )). However, equalsharing of the welfare gain might not be in the WS-core, because it might violate the constraintsof the anticore. Hence we aim to equalize the shares of the gain as much as possible, subject tosatisfying the anticore constraints. Before proceeding, let us establish some conventions and notation. We assume for convenience thatthe valuation function of each agent is such that at the disagreement point her expected value is 0.This can be enforced by applying an additive shift of u π ( i ) to each valuation function v i . Givena solution ( A ∗ , p ), we let u i denote the utility u i ( A ∗ , p ) = v i ( A ∗ ) + p i derived by agent i from thesolution (where the valuation function v i is such that the expected value offered by the disagreementpoint is 0). We shall sometimes refer to the vector u = ( u , . . . , u n ) (rather than to ( A ∗ , p )) as oursolution, as this vector summarizes what the agents care about in a solution. An egalitarian solutionwill give every agent utility u i = W max ( N ) n , but might not be in the WS-core. We present severaldifferent approaches for how to relax the egalitarian requirement so as to select a solution within theWS-core (when it is nonempty). • The min-square solution. Here we seek the unique solution within the WS-core minimizing P i ∈N ( u i ) . This solution minimizes the variance in the distribution of the welfare, subject tobeing in the WS-core. • The lexicographically-maximal ( lex-max-WS ) solution. Given a vector x ∈ R n , let ˆ x be thesame vector with coordinates rearranged such that in the new order ˆ x ≤ ˆ x ≤ . . . ≤ ˆ x n . Fortwo vectors x ∈ R n and y ∈ R n of equal sum of their entries, x ≥ Lex y denotes that for therearranged vectors ˆ x and ˆ y and for some 1 ≤ k < n it holds that ˆ x k > ˆ y k , with ˆ x i = ˆ y i forevery 1 ≤ i < k . A solution u in the WS-core is lexicographically maximal if u ≥ Lex u ′ for everyother solution u ′ in the WS-core. • A Lorenz-maximal solution. Given a vector x ∈ R n , let ˆ x be the same vector with coordinatesrearranged such that in the new order ˆ x ≤ ˆ x ≤ . . . ≤ ˆ x n . For two vectors x ∈ R n and y ∈ R n of equal sum, we say that x Lorenz dominates y (denoted by x ≥ Lor y ) if for therearranged vectors ˆ x and ˆ y it holds that P ki =1 ˆ x i ≥ P ki =1 ˆ y i , for every 1 ≤ k ≤ n . A Lorenzmaximal solution is a solution in the WS-core that Lorenz-dominates every other solution inthe WS-core. By definition, it also minimizes the so called Gini index of inequality [10].The next proposition presents some properties of the WS-core, for the proof see Appendix E.2.
Proposition 4.1
When the WS-core is nonempty:1. The min-square solution exists and is unique (in terms of the utility that it offers each agent). . The lexicographically-maximal solution exists and is unique.3. The min-square solution and the lexicographically-maximal solution need not coincide.4. A Lorenz dominating solution need not exist.5. If a Lorenz dominating solution exists, it is unique, and moreover, it coincides both with thelexicographically-maximal solution and with the min-square solution. Thus, the min-square and the lexicographically-maximal solutions exist whenever the WS-core isnonempty, whereas a Lorenz dominating solution need not exist. Out of the two solutions that doexist, we suggest picking the lexicographically-maximal-Welfare-Sharing solution, which we denote by lex-max-WS . (This choice is not of major significance to our work. Proposition 4.2 (with a differentalgorithm) and Proposition 4.3 also hold with respect to the min-square solution, and Corollary 4.5shows that the two solutions coincide in many cases of interest.)
When W max is submodular, the utilities in the lex-max-WS solution can be computed using analgorithm that we refer to as water filling (this is a generic name, used also elsewhere, for algorithmsthat increment variables at a uniform rate, subject to constraints). It proceeds in iterations. Initially(at iteration 0), all agents are free and every agent i starts with her disagreement utility u π ( i ). Ifany of the constraints of the anticore are tight (satisfied with equality) by this initial solution, thenthe set S of agents involved in the tight constraints become locked . Thereafter, in every iteration j ≥ x j , where x j > x j is the largest increase that does not violate any of the anticore constraints).At this point, the set S j of agents involved in a newly tight constraint become locked (some of theseagents may have been locked already earlier), and iteration j ends. Proposition 4.2
When W max is submodular, the water filling algorithm computes the lex-max-WS solution. Proof:
By design, the water filling algorithm satisfies all domination constraints and all con-straints of the anticore. Likewise, it produces a lexicographically maximal solution subject tothese constraints. It remains to show that by the end of the algorithm, the anticore constraint P i ∈N u ( i ) ≤ W max ( N ) is tight (meaning that all welfare has been shared by the agents), where here u ( i ) = P jt =1 x j is the utility of agent i that was locked right after iteration j (and u ( S ) will serve asshorthand notation for P i ∈ S u i ).When the water filling algorithm ends, every agent is involved in a tight anticore constraint.Suppose for the sake of contradiction that the constraint u ( N ) ≤ W max ( N ) is not tight. Let S j (for j = 1 , , . . . ) be the sets of agents whose constraints became tight in iteration j of the executionof the water filling algorithm. Consider a set function W ′ max that is identical to W max , except that W ′ max ( N ) = u ( N ) < W max ( N ). Then W ′ max is submodular, all the sets S j referred to above are tightwith respect to it, and so is N . By Lemma P.2, all unions of these sets S j are also tight with respectto W ′ max . By repeatedly taking unions, we arrive at two sets S and T such that S ∪ T = N , both u ( S ) = W ′ max ( S ) = W max ( S ) and u ( T ) = W ′ max ( T ) = W max ( T ) hold, and u ( S ∩ T ) = W ′ max ( S ∩ T ) Agent 2 Agent 3 lex-max-WS solution. However, the water filling algorithm will start by giving evert agent a utilityof . At this point every agent is part of a tight anticore constraint), and the algorithm ends beforedistributing all the welfare to the agents. lex-max-WS The lex-max-WS solution lies in the anticore, and hence by Proposition 3.5 it is weakly decomposable.We now consider strong decomposability (see Definition 3.3). This property involves comparing thesolutions generated for different instances. As the lex-max-WS solution for an instance depends onthe disagreement point for the instance, we need to also relate between the disagreement points ofdifferent instances. For this purpose, we assume that there is a mechanism that given an instanceoutputs the disagreement point for that instance. For example, RP served as such a mechanism inSection 1.1. Proposition 4.3 When W max is submodular, every mechanism M that satisfies both following prop-erties is strongly decomposable.1. For every instance M selects the respective lex-max-WS solution.2. The disagreement utilities (that define the domination constraints for the WS-core) are theoutput of a disagreement mechanism that is strongly decomposable. Proof: The outcome of the water filling algorithm on the whole instance is identical to theconcatenation of its outcomes on each component separately, because of Proposition D.1. We next show that in the important case that W max is submodular (e.g., in the Shared–Rentalproblem), a Lorenz dominating solution necessarily exists. Theorem 4.4 below is an adaptation of a13heorem of Dutta and Ray [6], which considers Lorenz minimal solutions and supermodular charac-teristic functions (we consider Lorenz maximal solutions and submodular characteristic functions).We provide a detailed proof of Theorem 4.4 rather than attempt to use the results of [6] as a blackbox,because in our setting we need to ensure that the solution dominates a given disagreement point, andthis issue does not seem to have an analog in the setting of [6]. The proof of the following theoremappears in Appendix F. Theorem 4.4 If W max is submodular, then the lex-max-WS solution (which is in the WS-core)Lorenz-dominates all other solutions in the WS-core. From Proposition 4.1 and Theorem 4.4 we derive the following corollary, implying that our so-lution lex-max-WS satisfies all three properties: it is a Lorenz dominating solution, the min-squaresolution and the lexicographically-maximal solution. Corollary 4.5 If the function W max is submodular then the WS-core is non-empty, a Lorenz dom-inating solution exists, it is unique, and it coincides with both the min-square solution and thelexicographically-maximal solution. Summarizing, when selecting a solution from the WS-core, we employ the egalitarian paradigm.As we shall see, in several natural settings (such as the Shared–Rental problem, see Section G) W max is submodular. In these cases, Theorem 4.4 offers a natural choice of a unique solutionwithin the WS-core, because (essentially) all natural relaxations of the notion of being egalitarian(min-square, lexicographically-maximal, Lorenz-maximal) coincide. Moreover, in these cases thesolution is computable in polynomial time, and also is continuous with a Lipshitz constant of 1 (seeAppendices O and P for exact statements. We find it particularly instructive to compare our lex-max-WS solution concept with that of theShapley value [25]. In our context, it is natural to let W max play the role of a characteristic functionin the definition of the Shapley value. Given a permutation σ over the agents, let S denote the set ofagents that precede agent i in σ . The marginal contribution of agent i in σ is W max ( S ∪{ i } ) − W max ( S ).The Shapley value of agent i is her expected marginal contribution in a random permutation overthe players. The sum of Shapley values of all players is exactly the maximum welfare W max ( N ). TheShapley value mechanism selects an allocation that maximizes welfare, and arranges the transfers sothat the utility of every player equals her Shapley value.When W max is submodular, the Shapley value resides in the anticore (this follows from theknown fact that the Shapley value is in the cost-sharing core whenever the characteristic function issubmodular). However, being oblivious to the reference point, the Shapley value is sometimes not inthe WS-core, even when W max is submodular. We shall see such examples in Appendix H.The use of the Shapley value as a solution concept for problems such as Shared–Rental problemwas advocated in [21]. There is was shown that the Shapley value solution satisfies four propertiesthat are referred to as individual rationality , resource monotonicity , population monotonicity , and stand alone test . Our lex-max-WS solution satisfies the stand alone test (which is equivalent tobeing in the anticore) and satisfies the domination property that is considerably stronger and moreversatile than the notion of individual rationality used in [21]. However, it does not satisfy thatresource monotonicity and the population monotonicity properties, see Appendix N for details.In Appendix M we compare between the solutions offered by the Shapley value and the solutionsoffered by lex-max-WS , covering all instances of two agents and two alternatives. In these instances14he Shapley value solution does lie in the WS-core, and for some range of values it coincides with lex-max-WS . For those ranges of values for which the two solutions differ, the lex-max-WS solutionoffers a more egalitarian sharing of welfare compared to that offered by the Shapley value. Other solution concepts: Previous work proposes many other solution concepts for profit-sharing and cost-sharing games. See for example [20], [19], [23] and references therein, as well asAppendix I in which we discuss the nucleolus and some known bargaining solution concepts. Itwould be too space consuming to discuss all these solution concepts in an informative manner, solet us just note that despite the many existing solution concepts, our solution concept appears tobe new, and the first to satisfy the combination of properties that we seek. It is our opinion thatin the setting studied in the current paper (which in particular includes a disagreement point), oursolution concept is preferable over any of the previous solution concepts in terms of the balancethat it achieves between fairness properties and the incentives that it provides to switch (from amechanism with no transfers) to our mechanism. Acknowledgments We thank Herve Moulin and Eyal Winter for helpful discussions. References [1] S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardnessof approximation problems. J. ACM , 45(3):501–555, 1998.[2] A. Bogomolnaia and H. Moulin. A new solution to the random assignment problem. Journal ofEconomic Theory , 100(2):295–328, October 2001.[3] O. N. Bondareva. Some applications of linear programming methods to the theory of cooperativegames (in russian). Problemy Kybernetiki , (10):119–139, 1963.[4] X. Deng and C. H. Papadimitriou. On the complexity of cooperative solution concepts. Mathe-matics of Operations Research , 19(2):257–266, 1994.[5] J. Derks, H. Peters, and P. Sudh¨olter. On extensions of the core and the anticore of transferableutility games. International Journal of Game Theory , 43(1):37–63, Feb 2014.[6] B. Dutta and D. Ray. A concept of egalitarianism under participation constraints. Econometrica ,57(3):615–635, 1989.[7] U. Feige. On maximizing welfare when utility functions are subadditive. SIAM J. Comput. , 39(1):122–142, 2009.[8] D. K. Foley. Resource allocation and the public sector. Yale Econ Essays , 7(1):45–98, 1967.[9] Y. K. Gal, M. Mash, A. D. Procaccia, and Y. Zick. Which is the fairest (rent division) of themall? J. ACM , 64(6):39:1–39:22, 2017.[10] C. Gini. Concentration and dependency ratios. Rivista di Politica Economica , 87:769–789, 1909.English translation, 1997.[11] M. X. Goemans, S. Gupta, and P. Jaillet. Discrete newton’s algorithm for parametric submodularfunction minimization. In IPCO , volume 10328 of Lecture Notes in Computer Science , pages212–227. Springer, 2017. 1512] J. Goldman and A. D. Procaccia. Spliddit: Unleashing fair division algorithms. ACM SIGecomExchanges , 13(2):41–46, 2014.[13] E. Kalai and M. Smorodinsky. Other solutions to nash’s bargaining problem. Econometrica , 43(3):513–518, 1975.[14] S. Khot, D. Minzer, and M. Safra. Pseudorandom sets in grassmann graph have near-perfectexpansion. Electronic Colloquium on Computational Complexity (ECCC) , 25:6, 2018.[15] B. Lehmann, D. J. Lehmann, and N. Nisan. Combinatorial auctions with decreasing marginalutilities. Games and Economic Behavior , 55(2):270–296, 2006.[16] M. Maschler, J. A. M. Potters, and S. H. Tijs. The general nucleolus and the reduced gameproperty. International Journal of Game Theory , 21(1):85–106, Mar 1992.[17] J. Milnor. Reasonable Outcomes for N-person Games . Research memorandum. Rand Corpora-tion, 1953.[18] D. Monderer, D. Samet, and L. S Shapley. Weighted values and the core. International Journalof Game Theory , 21:27–39, 1992.[19] H. Moulin. Axioms of Cooperative Decision Making . Econometric Society Monographs.[20] H. Moulin. Egalitarianism and utilitarianism in quasi-linear bargaining. Econometrica , 53(1):49–67, 1985.[21] H. Moulin. An application of the shapley value to fair division with money. Econometrica , 60(6):1331–49, 1992.[22] K. Nagano. A strongly polynomial algorithm for line search in submodular polyhedra. DiscreteOptimization , 4(3-4):349–359, 2007.[23] B. Peleg and P. Sudholter. Introduction to the theory of cooperative games (second edition) .Springer, 2007.[24] D. Schmeidler. The nucleolus of a characteristic function game. SIAM Journal of AppliedMathematics , 17:1163–1170, 1969.[25] L. S. Shapley. A value for n-person games. In H. W. Kuhn and A. W. Tucker, editors, Con-tributions to the Theory of Games II , pages 307–317. Princeton University Press, Princeton,1953.[26] L. S. Shapley. On balanced sets and cores. Naval Research Logistics Quarterly , (14):453–460,1967.[27] F. E. Su. Rental harmony: Sperner’s lemma in fair division. The American MathematicalMonthly , 106(10):930–942, 1999.[28] M. E. Yaari and M. Bar-Hillel. On dividing justly. Social Choice and Welfare , 1(1):1–24, 1984.16 Discussion of domination and of the anticore In our setting, one alternative among A must be selected. Our solution concept postulates that analternative A ∗ of maximum welfare is selected, and budget-balanced transfers are used in order toshare the welfare fairly among the agents. The reason for choosing a maximum welfare A ∗ is tocreate as large as possible pool of welfare to be distributed among the agents. The assumption thatutility functions are quasi-linear allows the transfers to redistribute the welfare in an arbitrary wayamong the agents, regardless of the value of A ∗ to each of the agents. The outcome of this welfaredistribution is summarized by the utilities of the agents, which combine the values derived from A ∗ and from the transfer. Hence our setting can be described in the following equivalent way. Thereis a set N of agents, and a given amount of welfare which equals w N ,v ( A ∗ ). This welfare needs tobe shared among the agents, where the share of each agent i is her utility u i , under the conditionthat P i ∈N u i = w N ,v ( A ∗ ). The reference context that is available to us is the tuple v of valuationfunctions, the set A of alternatives, and a probability distribution π v over A . Our welfare-sharing(WS) core (Definition 2.1) is based on two sets of constraints. Below we justify each set of constraints,and also discuss its relation to other notions in cooperative game theory. A.1 Domination We assume that a probability distribution π v over alternatives A is given. This π v serves as a referencepoint (also referred to as a disagreement point ). Namely, π v represents what the agents intend todo (select one alternative A ∈ A according to probability distribution π v ) if they are restricted touse a (randomized) social choice function without transfers. For example, students allocating roomsin a shared apartment may intend to use the Random Priority mechanism, which is quite simple toimplement. We propose to them that they use a mechanism with transfers instead, a mechanism thatgenerates more welfare, but may be more complicated to implement. However, we cannot enforcethat the agents switch from the reference point to our mechanism. We can only try to convince themto do so, and moreover, we might need to convince each and every one of them before the switchto our mechanism actually happens. In order for an individual agent to be convinced to switch, itdoes not suffice that the utility of other agents will increase by the switch – we need to guaranteethat she herself will get higher utility from the new mechanism (or at least not lose utility). Here wemake the assumption that agents are risk neutral, and hence the distribution over utilities that anagent derives from the randomized reference point can be summarized by one number – the expectedutility. Hence we impose the domination constraint u i ( A ∗ , p ) ≥ P A ∈A π v ( A ) v i ( A ) for every agent i .We remark that if agents are risk averse (rather than risk neutral) then the utility that theyderive from the randomized reference point becomes smaller, making our deterministic mechanismseven more attractive.We assume that a reference point is given for the original game, but make no assumption regardinghow this reference point is chosen. In particular, the reference point, which may be a function of thevaluations, need not be a continuous function of the valuation functions. Indeed, in many naturalcases (reference points derived from Random Priority , or from the Eating mechanism of [2]) thereference point is not continuous in the valuation functions.There are other solution concepts that also require domination over a disagreement point. Forexample, this is the case for Nash bargaining solution, and for the Kalai-Smorodinsky bargainingsolution (see Appendix I). Often, the disagreement point represents the bargaining power of agents– the utility that they can obtain by not participating in the mechanism. Hence it is natural thatthe mechanism needs to offer them at least their disagreement utility, as otherwise they would notparticipate. Considerations of this sort are referred to as individual rationality (IR). However, in17ur context, agents do not have an outside option of not participating. Rather, the disagreementpoint itself is the outcome of a mechanism that involves all agents, referred to as the disagreementmechanism. Hence it is not clear what utility, if any, an agent will derive by refusing to participate inour mechanism, because this may depend on what the other agents do. Hence we view our dominationproperty as implementing a fairness principle rather than reflecting bargaining power: switching fromthe disagreement mechanism to a new mechanism with transfers generates extra welfare, and it is“fair” that this extra welfare be shared by all participants, or at the very least, that no agent suffersa loss in utility.There are solution concepts that fix a particular reference point (e.g., two possible reference pointsare considered in [21], leading to notions that the paper refers to as weak and strong IR). Our workdiffers from these works in the sense that our mechanisms can receive an arbitrary reference pointas an input parameter, and dominate the given reference point. A.2 The anticore When agent i receives an in-payment of p i , we wish it to be the case that agent i could justify toothers why she deserves such a payment. Such a justification is needed because against every in-payment to one agent there is an equal amount of out-payment from other agents, and these otheragents need to be convinced that their out-payments are extracted for a good reason. A justificationthat agent i can provide is that A ∗ is not her preferred alternative, and so she should be compensatedfor not contesting the choice of A ∗ . This justification has a limit. Without transfers, the highestutility that agent i can hope to achieve is W max ( i ) = max A ∈A v i ( A ). Hence there is no justificationfor p i to exceed W max ( i ) − v i ( A ∗ ).More generally, we require that also every set S of agents (e.g., all agents of a certain gender) willbe able to justify receiving net in-payment into S . Against every in-payment to S there is an equalamount of out-payment from the set ¯ S = N \ S of remaining agents (e.g., all agents of the othergender), and this other set needs to be convinced that their out-payments are extracted for a goodreason. Without transfers, there is no alternative that can offer S total utility higher than W max ( S ).Hence it is difficult to justify extracting out-payments from ¯ S if their use is to increase the utility of S beyond W max ( S ). This gives the constraints of the anticore. Related notions: The constraint u i ≤ W max ( i ) is referred to as reasonable from above (REAB)by Milnor [17] (see page 21 in [23]). The anticore extends the REAB constraint to sets, requir-ing u S ( A ∗ , p ) ≤ W max ( S ) for every set S . The anticore was defined in [21], where the constraint u S ( A ∗ , p ) ≤ W max ( S ) was referred to as the stand alone test for set S . The term anticore also ap-peared in some other work, though not necessarily with the same interpretation. Let us elaborateon this.We first recall the notion of core in transferable utility cooperative games . Suppose that there is aset N of players and a characteristic function f : N → R , specifying for each coalition of players thepayoff that the coalition can achieve on its own. An imputation I = ( I , . . . , I n ) is a vector of payoffs,distributing the payoff f ( N ) of the grand coalition among the players. Namely, P i ∈N I i = f ( N ). Animputation is said to be in the core (which we shall call here the imputation core , so as to distinguishit from others notions of core used in this paper) if for every set S ⊆ N of players, P i ∈ S I i ≥ f ( S ).In our context, the utilities u i play the same mathematical role as the imputations I i . Thefunction W max indirectly plays the role of the characteristic function, by defining a characteristicfunction D ( S ) = W max ( N ) − W max ( N \ S ) (technically referred to as the dual of W max ) and requiring u S ( A ∗ , v ) ≥ D ( S ). This dual definition is more in line with our setting being that of a profit game (theinequalities ensure some minimum utility for each set of agents). However, we find it inconvenient towork with the dual definition (in particular when it is combined with the domination constraints),18nd hence we use the equivalent definition of u S ( A ∗ , v ) ≤ W max ( S ).The anticore resembles the notion of core in cost-sharing games , and indeed such a core is some-times referred to as an anticore (see [18]). Suppose that there is a set N of players and a characteristicfunction f : N → R , specifying for each coalition of players the cost of receiving service on its own.A cost-sharing vector c = ( c , . . . , c n ) is a vector of costs, distributing the cost f ( N ) of the grandcoalition among the players. Namely, P i ∈N c i = f ( N ). A cost-sharing vector is said to be in the anticore (which we shall call here the cost-sharing anticore , so as to distinguish it from our notion ofanticore) if for every set S ⊆ N of players, P i ∈ S c i ≤ f ( S ). Our anticore has the same mathemat-ical structure as a cost-sharing anticore, equating W max with the characteristic function f , and theutilities u i ( A ∗ , p ) with the cost shares c i . A major difference is that we impose this mathematicalstructure in a profit-sharing game in which agents wish to maximize their utility, rather than in a cost-sharing game in which agents wish to minimize their cost.The term anticore has been used in [5] for profit-sharing games. There, the characteristic function f : N → R specifies for each coalition of players the profit that it can generate on its own. A profit-sharing vector u = ( u , . . . , u n ) is a vector of profits, distributing the profit f ( N ) of the grandcoalition among the players. Namely, P i ∈N u i = f ( N ). In [5], a profit-sharing vector is said to be inthe anticore if for every set S ⊆ N of players, P i ∈ S u i ≤ f ( S ). Derks et al [5] provide the followingjustification for the anticore: “if one coalition obtains less than its worth, then it is only fair thatall coalitions obtain at most their worths”. Our anticore has the same mathematical structure asthat of [5], equating W max with the characteristic function f , and the utilities u i ( A ∗ , p ) with theprofit shares u i . However, our function W max does not carry the same interpretation that one usuallyassociates with a characteristic function of a profit-sharing game. In our setting, a set S of players cannot generate for itself a utility W max ( S ) if it breaks from the grand coalition. Rather, W max ( S )is the maximum utility that S can obtain (without transfers) if it stays in the grand coalition, andall players, including the players not in S , agree to choose the alternative that maximizes the welfareof S .Depending on the nature of the characteristic function f , the anticore in [5] might be either emptyor nonempty. (In fact, the main content of that work concerns how to handle cases in which boththe core and the anticore are empty.) In contrast, in our setting, the anticore is always nonempty.In particular, not using any transfers (equivalently, using the all 0 transfer vector) is always in theanticore. By nature of its construction, our function W max , when viewed as a characteristic functionin a profit-sharing game, guarantees non-emptiness of the anticore. B Decomposability The following proposition shows that the goals of maximizing welfare and of satisfying decompositionproperties are compatible with each other. Proposition B.1 Any alternative that maximizes welfare also maximizes welfare for each componentseparately. Proof: Let A be an alternative that maximizes welfare, namely, for which P i ∈N v i ( A ) is largestpossible. For a component S , let B be an alternative that maximizes the welfare of S , namely, forwhich P i ∈ S v i ( B ) is largest possible. We need to show that P i ∈ S v i ( A ) = P i ∈ S v i ( B ).Suppose for the sake of contradiction that P i ∈ S v i ( A ) < P i ∈ S v i ( B ). Let C be an alterna-tive that maximizes the welfare of ¯ S . Then necessarily P i ∈ ¯ S v i ( C ) ≥ P i ∈ ¯ S v i ( A ). By the factthat S is a component, there must be an alternative D for which P i ∈ S v i ( D ) = P i ∈ S v i ( A ) and19 i ∈ ¯ S v i ( D ) = P i ∈ ¯ S v i ( C ). It follows that P i ∈N v i ( D ) > P i ∈N v i ( A ), contradicting the assumptionthat A maximizes welfare.We next prove that strong decomposability implies weak decomposability. Proposition B.2 Let M be a mechanism that for every instance selects an alternative that maxi-mizes welfare and a budget balanced vector of transfers. If M is strongly decomposable, then for everydecomposable instance the solution produced by M is weakly decomposable. Proof: Let S be a component. Let A be the alternative (maximizing welfare for N ) chosen by M ininstance I , and let A S be the alternative (maximizing welfare for S ) chosen by M in instance I S . ByProposition B.1 we have that P i ∈ S v i ( A ) = P i ∈ S v i ( A S ). By strong decomposability of M we havethat u i ( M ( I )) = u i ( M ( I S )) for every i ∈ S , and consequently P i ∈ S u i ( M ( I )) = P i ∈ S u i ( M ( I S )).As a utility of an agent is the sum of value for the chosen alternative and the transfer, we have thatthe sum of transfers of the agents in S must be the same in I and in I S . But in I S the sum oftransfers is 0, because of budget balance. Hence in I the net transfer into S is 0 as well, as requiredby weak decomposability.We now prove Proposition 3.4, which is restated here for convenience. Proposition B.3 Given a set A of alternatives, a set N of agents, and a tuple v of valuationfunctions, the components of N form a lattice. Proof: For every component S , also its complement ¯ S = N \ S is a component as well. Consequently,if we prove that for every pair of components their intersection is also a component, this will implythat their union is a component as well.Let S and T be components. We need to show that S ∩ T is a component. Let A be an alternativethat is Pareto optimal for S ∩ T . Let the vectors of values that A gives to the sets S ∪ T , S \ T , T \ S and N \ ( S ∪ T ) be a , a , a and a , respectively. Let B be an alternative that is Pareto optimal for N \ ( S ∩ T ). Let the vectors of values that B gives to the sets S ∪ T , S \ T , T \ S and N \ ( S ∪ T )be b , b , b and b , respectively. We need to show the existence of an alternative C for which therespective vectors of values are a , b , b , b .Let A S be an alternative that is Pareto optimal for S and for every agent in S offers at least asmuch value as A does. Such an alternative must exist for the following reason. If A is Pareto optimalfor S , then take A S = A . If A is not Pareto optimal for S , then there must be an alternative A ′ thatdominates A for all agents in S . Continue the argument with A ′ .Likewise, let B ¯ S be an alternative that is Pareto optimal for ¯ S and for every agent in ¯ S offersat least as much value as B does. Such an alternative must exist as well. By the fact that S is acomponent, there must exist an alternative C S whose vector of values dominates a , a , b , b .Repeating the above argument with T instead of S , there must exist an alternative C T whosevector of values dominates a , b , a , b .Now take an alternative A ′ T that is Pareto optimal for T and for every agent in T offers at leastas much value as C S does, and an alternative B ′ ¯ T that is Pareto optimal for ¯ T and for every agentin ¯ T offers at least as much value as C T does. By the fact that T is a component, there must bean alternative C whose vector of values dominates a , b , b , b . In fact, it must equal a , b , b , b ,because otherwise either A was not Pareto optimal for S ∩ T , or B was not Pareto optimal for N \ ( S ∩ T ). The existence of C shows that S ∩ T is a component, as desired.20 Decomposability versus separability Recall the notion of decomposable solutions and mechanisms defined in Section 3. This notionappears to be new, but there are other known concepts that bear superficial similarity to decompos-ability. The purpose of this section is to present one such concept, that of separability, and clarifythe differences between separability and decomposability.Separability (see for example [19] and references therein) of profit-sharing games is an internalconsistency property for profit-sharing mechanisms. Suppose that that there is a profit-sharingmechanism M that given a set N of players, disagreement utilities u i for each i ∈ N , and a totalwelfare W ≥ P i ∈N u i to be divided among the players, assigns shares x i ≥ u i of the welfare to therespective agents, satisfying P i ∈N x i = W . Then M is separable if it has the property that for every S ⊂ N , had M been applied to an instance in which S is the set of players, in which the same u i asabove (for i ∈ S ) are their disagreement utilities, and in which the total welfare to share is P i ∈ S x i ,then M would propose the same shares x i as above (for i ∈ S ). A simple example of a separablemechanisms is the egalitarian mechanism that equalizes the profit (above disagreement point) of allagents.A major difference between decomposability and separability is that decomposability (of a so-lution, or a mechanism) is a property that needs to hold only for decomposable instances, whereasseparability needs to hold for all instances. Consequently, decomposable mechanisms need not beseparable, and in fact our lex-max-WS mechanism is not separable. If S ⊂ N is not a component,then the way the part of the welfare allocated to a subset S of players is distributed among the playersof S is affected by the way the remaining welfare is distributed among members of N \ S , because theanticore constraints involve constraints on sets T of players that intersect both S and N \ S , and theseconstraints affect which share allocations within S are feasible. Likewise, separable mechanisms neednot be decomposable (and in general are not). For example, the egalitarian profit-sharing mechanism(equalizing x i − u i ) is separable, but fails to satisfy even the weak decomposability property.A property that is related to separability and can take into account the anticore constraintsis consistency for reduced games . This property is satisfied by a solution concept referred to as theNucleolus. This property can coexist with decomposability and can be incorporated into our solutionconcept by changing the rule that selects a solution from the WS-core, but we prefer not to enforcethis property for reasons explained in Section I.3. D More on the anticore The following property of the anticore plays a central role in decomposability aspects. Proposition D.1 For a given decomposable instance, let T , . . . T t be a partition of the set N ofagents into components (in the sense of Definition 3.1). Then a solution ( A, p ) , composed of a chosenalternative A and vector p of transfers, satisfies the anticore constraints ( u S ( A, p ) ≤ W max ( S ) forevery set S ⊆ N ) if and only if it satisfies them for every set S that is fully contained in a component(namely, S ⊆ T j for some ≤ j ≤ t ). Proof: The “only if’ direction is a triviality. For the “if” direction, assume for the sake ofcontradiction that there is a set S such that u S ( A, p ) > W max ( S ), but for every 1 ≤ j ≤ t itholds that u S ∩ T j ( A, p ) ≤ W max ( S ∩ T j ). Then for every j there is an alternative A j such that u S ∩ T j ( A, p ) ≤ u S ∩ T j ( A j ), and then Definition 3.1 implies that there is some alternative A ∗ such that u S ∩ T j ( A, p ) ≤ u S ∩ T j ( A ∗ ) simultaneously for all j . Consequently, u S ( A, p ) ≤ u S ( A ∗ ) ≤ W max ( S ),which contradicts our assumption. 21 Properties of the WS-core E.1 Proof of Theorem 2.2 Recall that the welfare-sharing (WS) core contains those solutions that are in the anticore anddominate the reference point. We now turn to prove Theorem 2.2, that the WS-core is nonemptywhen either W max is sobmodular, or W max − W π is monotone. Our proof is based on the wellknown Bondareva-Shapley theorem [3, 26], though our setting involves some subtleties that mightbe overlooked in attempts to directly apply the Bondareva-Shapley theorem.We first recall some standard terminology. A set function f is additive if for every set S , f ( S ) = P i ∈ S f ( i ). A set function f is submodular if for every two sets S and T it holds that f ( S ) + f ( T ) ≥ f ( S ∩ T ) + f ( S ∪ T ). Equivalently, f is submodular if it has the decreasing marginal returns property:for every item i and two sets S ⊂ T it holds that f ( S ∪ { i } ) − f ( S ) ≥ f ( T ∪ { i } ) − f ( T ).A collection of sets T , . . . , T k and nonnegative coefficients λ , . . . , λ k is said to be a fractionalcover for set S if for every item i ∈ S it holds that P j | i ∈ T j λ j ≥ 1. This fractional cover for S will be referred to as proper if furthermore T i ⊂ S for every 1 ≤ i ≤ k . A set function f is(proper) fractionally subadditive if for every S , whenever T , . . . , T k and λ , . . . , λ k form a (proper,respectively) fractional cover for S , the inequality P kj =1 λ j f ( T j ) ≥ f ( S ) holds.A set function f is XOS [15] if for some t there are additive set functions g , . . . , g t such thatfor every set S , f ( S ) = max tj =1 g j ( S ). Observe that by definition, the function W max belongs to theclass XOS.Given π = π v (the previously defined reference point), the set function W π ( S ) = P A ∈A π ( A ) P i ∈ S v i ( A )describes the expected total utility that a set of agents receives from the reference point. Observe that W π is an additive set function. We now define a characteristic function f as f ( S ) = W max ( S ) − W π ( S ).This function belongs to the class XOS (because W max is in XOS and W π is additive) and is non-negative.It was shown in [7] that for nonnegative monotone set functions, the class of XOS functions isthe same as the class of fractionally subadditive functions. However, this correspondence need notapply to the XOS function f defined above, because f need not be monotone. The example providedafter Definition 2.1 can serve to illustrate this difficulty. For this reason Theorem 2.2 considers twospecial cases. In the second of these special cases f ( S ) is indeed monotone, and hence as explainedabove, it is fractionally subadditive (which implies that it is also proper fractionally subadditive). Inthe first special case, W max is submodular, and for this case we can use the following proposition. Proposition E.1 If f is a nonnegative submodular function then it is proper fractionally subadditive. Proof: Suppose for the sake of contradiction that the proposition is false. Then there is a counterexample: a set S , a fractional cover for S composed of sets { T i } with nonnegative coefficients { λ i } ,such that P λ i f ( T i ) < f ( S ), and T i ⊂ S for all i (this is the property of being a proper fractionalcover). Consider the smallest such counter example, with smallest | S | , and conditioned on | S | , withsmallest P λ i . For each item j ∈ S , let c j = P i | j ∈ T i λ i . By virtue of being a fractional cover we havethat c j ≥ j ∈ S .We claim that there must be at least one item j in S for which c j = 1. This holds because otherwisewe could divide all λ i by min j c j and obtain a smaller counter example ( P λ i would decrease, we willstill have a fractional cover, and P λ i f ( T i ) will decrease because all terms are nonnegative).Given that there is an item j ∈ S with c j = 1, we may remove j both from S and from all sets T i , and we remain with a proper fractional cover. The condition P λ i f ( T i ) < f ( S ) together withthe decreasing marginal returns property imply that P λ i f ( T i − { j } ) < f ( S − { j } ). Hence we havea smaller counter example, which is a contradiction.22 orollary E.2 If the function W max is submodular, then the function f ( S ) = W max ( S ) − W π ( S ) asdefined above is nonnegative and proper fractionally subadditive. Proof: As W max ( S ) is submodular and W π ( S ) is additive, their difference f ( S ) = W max ( S ) − W π ( S )is submodular. The function f ( S ) is nonnegative because W π ( S ) is the expected welfare that S derives from a distribution over the alternatives, whereas W max ( S ) is the welfare derived from thebest alternative. The proper fractional subadditivity property follows from Proposition E.1.We are now ready to prove Theorem 2.2. Proof: (of Theorem 2.2) Observe that the theorem is equivalent to the statement that the cost-sharing core with respect to the characteristic function f ( S ) = W max ( S ) − W π ( S ) contains a non-negative cost-sharing vector c = ( c , . . . , c n ), with P i c i = f ( N ). Being in the cost-sharing corewith respect to f implies being in the anticore with respect to W max , and being nonnegative impliesdomination over W π .Consider the primal linear program (LP) with variables x , . . . , x n (which in a feasible solutionwill give the desired cost shares): Maximize P i ∈N x i subject to: • P i ∈ S x i ≤ f ( S ) for every set S • x i ≥ i The dual to the above LP has variables y S for every set S : Minimize P S ⊆N f ( S ) y S subject to: • P { S | i ∈ S } y S ≥ i • y S ≥ f is proper fractionally subadditive implies that the value of the dual is at least f ( N ). Being a minimization problem, the value of the dual is in fact exactly f ( N ), by taking y N = 1,and y S = 0 for S = N . Hence the primal LP is feasible and has value f ( N ), as desired. E.2 Proof of Proposition 4.1 We next restate and prove Proposition 4.1. Proposition E.3 When the WS-core is nonempty:1. The min-square solution exists and is unique (in terms of the utility that it offers each agent).2. The lexicographically-maximal solution exists and is unique.3. The min-square solution and the lexicographically-maximal solution need not coincide.4. A Lorenz dominating solution need not exist.5. If a Lorenz dominating solution exists, it is unique, and moreover, it coincides both with thelexicographically-maximal solution and with the min-square solution. Proof: 23. The WS-core is a bounded polytope (with constraints u ≥ P i u i = W max ( N ), and the anti-core constraints). The function P i ∈N ( u i ) is strictly convex. Hence it has a unique minimumin the WS-core.2. Assume for the sake of contradiction that there are two lexicographically maximal solutions u and u ′ with u = u ′ . As the WS-core is a convex set, it holds that also u ” = u + u ′ is in theWS-core. It is not difficult to see that either u ” > Lex u or u ” > Lex u ′ (with a strict inequality).This contradicts the assumptions that both u and u ′ are lexicographically maximal.3. To see that the min-square solution and the lexicographically-maximal solution need not coin-cide, consider the following example with four alternatives and six players:Example2 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6Alternative A 1 1 1 1 3 3Alternative B 0 2 2 2 2 2Alternative C -1 -1 -1 -1 4 -9Alternative D 0 -2 -2 -2 -9 4The disagreement distribution π is uniform over the four alternatives. Under this distribution,the expected utility of each agent is 0. (In the above example π is supported on Pareto-optimal alternatives. If this is not required, we may replace Alternatives C and D by a singledisagreement alternative that gives utility 0 for every agent, thus simplifying the example.)Any of the first two alternatives maximizes welfare, giving a welfare of 10. An egalitariandistribution of the welfare will give each agent utility 5/3, but this is not in the anticorebecause it violates the constraint for Agent 1 (whose utility is upper bounded by 1). Thelexicographically maximal solution is to choose Alternative A and have no transfers. Thisgives Agent 1 his maximum possible value of 1, and then conditioned on that, it gives each ofAgents 2, 3 and 4 their maximum utility of 1 (because agent 1 and agent j ∈ { , , } combinedare not allowed to have utility above 2). However, this is not a min-square solution: choosingAlternative B with no transfers gives a lower value for the sum of squares (20 instead of 22).4. In the above example, there is no Lorenz dominating solution, because the lexicographicallymaximal solution (utilities as in Alternative A) does not Lorenz dominate the solution corre-sponding to the utilities under Alternative B.5. If follows essentially by definition that x ≥ Lor y implies that also x ≥ Lex y . Hence Lorenzdomination implies being lexicographically maximal. To obtain the Lorenz dominating vectorof utilities from any other vector of utilities (that maximizes welfare and is in the WS-core),one needs to shift utility from coordinates of higher utility to coordinates of lower utility, bythis lowering the sum of the squares of the utilities. This shows that a Lorenz dominatingsolution is also a min-square solution. F Proof of Theorem 4.4 – Lorenz domination In this section we prove Theorem 4.4: If W max is submodular, then the lex-max-WS solution (whichis in the WS-core) Lorenz-dominates all other solutions in the WS-core.24 roof: Let f be the characteristic function associated with the WS-core. Namely, f = W max , whenthe valuation function of each agent is such that the expected value of the disagreement point is 0(recall that this can be enforced by applying an additive shift to the valuation functions). Recall that f is nonnegative, though it need not be monotone. By our assumption, f is submodular. We need toshow that the core of the corresponding cost-sharing game contains a solution that Lorenz-dominatesall other solutions in the core.Let x = ( x , x , . . . , x n ) denote the lex-max-WS solution, as found by the water filling algorithm.We now show that x ≥ Lor y for every y in the core. We need to sort the coordinates of x by order ofnondecreasing values (equivalently, sort the agents by nondecreasing utilities, above the disagreementpoint, under x ). In this order the sets S i appear in the order S , S , . . . S m . More precisely, denoting S j \ S i 1, the sets T i appear in order T , T , . . . , T m . For y , we may pick anorder of our choice (not necessarily nondecreasing) on its coordinates. If in this arbitrary order wehave P j ≤ i x j ≥ P j ≤ i y j for every i , then x ≥ Lor y . The order we choose for y is also T , T , . . . T m ,and within each T i we sort the coordinates by order of nondecreasing values. Observe that for every N i = ∪ j ≤ i T i it holds that x ( N i ) ≥ y ( N i ), because by the proof of Proposition 4.2 (replacing N by N i in the proof of the proposition), x gives the set N i its maximum possible value given theconstraints of f (and y has to be in the core). Consider now any intermediate coordinate k withinan interval S i (starting at N i − and ending at N i ). At the left endpoint of the interval S i we have x ( N i − ) ≥ y ( N i − ). The x values are constant within the set S i whereas the y values are increasing.Hence if it would happen that P j ≤ k x j ≥ P j ≤ k y j fails to hold, this would lead to x ( N i ) < y ( N i ),which would be a contradiction. Hence it must be that x ≥ Lor y . G Shared Rental (Unit-Demand Matching) A motivating example for our new solution concept is the Shared–Rental problem . In the Shared–Rental problem n ≥ n bedrooms and jointly need to pay the rent r and decide on the matching of rooms to the agents. We assume that they are already committedto rent the apartment and that each will first pay an equal share of the rent, that is, will pay r/n ,and then they will need to decide on the matching of the agents to the rooms (each getting exactlyone room), and the transfers between the agents to make the outcome ”fair”. The Shared–Rentalproblem can be restated more abstractly as a matching problem for unit-demand agents which wedescribe next.In the problem of matching with unit-demand agents , the goal is to allocate a set M of n ≥ n unit-demand agents, giving each of them one item. Each unit-demand agent i assignsa value v i ( j ) to each item j ∈ M . We would like to match agents with items - each agent willreceive exactly one item. Mapping this to the general framework, the set of alternatives is the setof permutations over the n items, and the value that an agent assigns to an alternative is his valuefor the item he receives in that permutation. That is, permutation σ assigns item σ ( i ) to agent i ,and his value for σ is v i ( σ ( i )).Note that we can indeed frame the Shared–Rental problem as a matching with unit-demandagents problem, although in the Shared–Rental problem there is the additional component of payingthe rent. The rent payment from each agent can be encoded by some shifted unit-demand valuationsobtained from the original valuations by decreasing the valuation of each item by the rent amount We will not handle the question of how they have entered this commitment and if it was rational for them to doso. In our framework no items can be left unallocated. We make no assumption that the value of an item is necessarily non-negative. Note that even if that was the case /n . Thus, the two problems are essentially equivalent and we will go back and forth between thetwo. G.1 The WS-core and the lex-max-WS solution To define the welfare-sharing core (WS-core), we first discuss reasonable distributions that the agentsmight consider as their reference point and expect the solution to dominate. Such distributionsnaturally arise from some standard mechanisms (without transfers) that may be considered natural,for example, for the Shared–Rental problem.1. Uniform ( U ). The allocation π is a permutation chosen uniformly at random, independent of v .2. Random priority ( RP ). One selects a random permutation over agents, and then the agents inturn each select one item from those remaining.3. Eating ( Eat ), also known as probabilistic serial [2]. Each item has unit volume. Each agent“eats” items at the same rate, starting at his most preferred item. Whenever an item is totallyconsumed, each agent eating it switches to his highest priority item that still has some volumeleft. When all items are consumed, we have a fractional allocation. It is decomposed into aweighted sum of integral allocations, and one of them is chosen (with probability proportionalto its weight).It is easy to see that Uniform might actually have Pareto dominated allocations in its support, so itseems less attractive to use. In fact, it is shown in [2] that Eating Pareto dominated RP which inturn Pareto dominates Uniform . As Uniform might select an allocation that is Pareto dominated,we take RP as a natural reference mechanism (yet all our claims below will also hold for the Eating mechanism). The disagreement point we take will be the expected utility of the output of the RPmechanism on the specific valuations. We note that this utility is not continuous in valuationswhen the preference order of some agent changes, but it is easy to see it is continuous (and evenLipschitz continuous with a small constant of 1) as long as the ordinal preferences remain the same.Observe that we can again shift the unit-demand valuations by decreasing from each agent valuationhis expected utility from the reference distribution, normalizing his utility in the new disagreementpoint to zero. We make this normalization assumption in the rest of the section.To complete the definition of the welfare-sharing core, we consider the anticore constraints W max ( · ) for these unit-demand valuations. We observe that in the matching with unit-demandsetting, W max ( S ), the maximal achievable welfare that a set S ⊆ N of agents can obtain, is themaximal weighted matching of the set S to any subset T ⊆ M with | T | = | S | . The set function W max ( · ) (when the sets are sets of agents) satisfies the OXS property defined by [15]. They haveproved that any such set function is submodular. We immediately get the following implications byTheorem 2.2, Theorem 4.4 and Proposition 4.1: Proposition G.1 For any unit-demand setting, the set function W max : N → ℜ is submodular.Thus, within the WS-core there is a Lorenz dominating solution. This solution is unique, and more-over, it coincides both with the lexicographically-maximal solution ( lex-max-WS ) and with the min-square solution. before the shift, it might fail to hold after the shift. 26o we see that all three above solutions exist and coincide for the unit-demand case, and recallthat we call it the lex-max-WS solution. We next observe that in the Shared–Rental problem thissolution satisfies some desirable decomposability properties. Remark G.2 We note that unit-demand valuations are submodular, and W max is submodular forunit-demand valuations. Another case where W max is submodular is when the valuation functionsof the agents are additive over the items (which is also a class of submodular valuation functions).However, there are submodular valuations for which W max is not submodular. See Appendix J formore details. G.2 Decomposability Recall the notion of decomposability from Definition 3.1. This notion, when specialized to of unit-demand matching instances (such as the Shared Rental problem) is equivalent to the definitionbelow. Definition G.3 We say that a unit-demand matching instance I = ( N , M , v ) is decomposable ifthere is some partition of N into P , P , . . . , P t (with t > ) and of M into M , . . . , M t , with theproperties that | P ℓ | = | M ℓ | for all ≤ ℓ ≤ t , and that in every Pareto optimal allocation, in everypart P ℓ (referred to also as a component ), each agent of P ℓ receives an item from M ℓ . A sufficient condition for P , P , . . . , P t and M , . . . , M t to serve as a decomposition of an instanceis that for every 1 ≤ ℓ ≤ t , every agent in P ℓ prefers every item in M ℓ over every item not in M ℓ .Consider a large house with five rooms, three on the east wing and two on the west wing, and fiverenters. Hence for the room assignment problem, which is a unit demand matching instance , thereare 5! = 120 alternatives, one for each permutation. Assume that the five renters can be partitionedto two groups, the “east group” with three renters and the “west group” with two renters. Theagents have the following preferences: each agent in the east group prefers any room on the eastwing over any on the west wing, and each agent on the west group prefers any room on the west overany on the east. In such a case, given a mechanism that provides solutions to the room assignmentproblem, there are two natural options regarding how to use it. One is to apply the mechanism onthe whole input instance. The other is to first assign each group of agents to its preferred wing, andthereafter apply the mechanism to each wing independently (one such subproblem has three agentsand 3! = 6 alternatives, the other has two agents and 2! = 2 alternatives), without further exchangeof information or transfer of money between the two groups. If the mechanism enjoys the strongdecomposability property (Definition 3.3) then every agent is indifferent regarding which of the twooptions is used, in the sense that both options give her the same utility. So the agents may as welldecompose the instance.Conversely, if the strong decomposability property fails, then for some agents the first optiongives higher utility than the second, and for some other agents the second option gives higher utilitythan the first (this will necessarily happen for a budget balanced mechanism that maximizes welfare,because the sum of utilities in both options is the same). Hence it might be difficult to reachagreement among the agents regarding which option to choose. Likewise, had we started with twoseparate instances, one for the east group and one for the west group, with no group interested inrooms in the other wing, we would be faced with the question of whether or not to compose thesetwo instances into one larger instance for the whole house, as this affects the distribution of welfareamong the agents. The use of a strongly decomposable mechanisms eliminates the source of suchconflicts. 27or unit demand matching instances, decomposition goes beyond the weak and strong decom-position properties of Section 3. The added feature is that when an instance decomposes, then alsoeach (Pareto optimal) alternative by itself can also be decomposed. For example, an allocation ofrooms to agents in the above example can be decomposed into the allocation of the east wing roomsand the allocation of the west wing rooms. In these settings, the fact that a solution decomposesis not only a statement about the end result of the solution, but also about the physical procedureby which the solution can be obtained. We can indeed partition the players into disjoint groups,let each group solve its own subproblem with no communication with the other groups, and theconcatenation of the separate solutions derived independently by each group gives back one welfaremaximizing alternative (and a vector of budget balanced transfers).Considering allocation mechanisms without transfers for unit-demand matching, we observe that Uniform does not respect the component structure (agents need not get an item from their own partin the partition), whereas both RP and Eating are strongly decomposable.When addressing strong decomposability of our lex-max-WS mechanism, we postulate that thereference point used by the lex-max-WS comes from a strongly decomposable reference mechanism(like RP ). The reference point for each part of the decomposition is the outcome of the execution ofthe reference mechanism on the corresponding sub-problem separately. The following Corollary is aspecial case of Proposition 4.3. Corollary G.4 The lex-max-WS mechanism for unit-demand matching problems is strongly decom-posable whenever the reference mechanism is strongly decomposable. In particular, this holds whenthe reference mechanism is either RP or Eating. G.3 Comparison to other solutions It will be insightful to compare our solution to two standard solutions from the literature: the envy-free solution, and the Shapley solution (defined in Section 4.5). In this section we show that neitherone of them satisfies all the properties we are after, even for the Shared–Rental problem. We alsopresent a complete comparison of the solutions for the case that n = 2 in Appendix M. G.3.1 Envy-free solutions Recall that an allocation is envy free if no agent prefers some other agent’s allocation and paymentover her own. We have already seen in Section 1.1 examples for unit demand matching in which everyenvy free solution is not in the anticore, does not dominate RP , and does not satisfy decomposability.Additional related examples are provided by Proposition L.1 in Appendix L. In that appendixwe also prove the following: Proposition G.5 There are instances of the Shared–Rental problem in which the valuation functionof each player is nonnegative and sums up to the total rent (in particular, this is the setting studiedin [9]), but nevertheless there is a player that in every envy-free solution both gets his most desiredroom and receives more money than his rent share. G.3.2 The Shapley value solution The Shapley value solution is unique. The next proposition lists some of its properties, see AppendixH for the proof. Proposition G.6 The Shapley-value solution is in the anticore of the Shared–Rental problem andsatisfies strong decomposability, yet it does not dominate RP (and hence also does not dominateEating). Properties of the Shapley-value solution A theoretical justification given for using the Shapley value is that it is the unique solution thatsatisfies three properties referred to as symmetry (agents with the same valuation function receivethe same utility), zero player (an agent whose valuation function is 0 for all alternatives does notreceive nor pay any transfer) and linearity (linear changes to W max lead to linear changes in thedistribution of welfare).When the WS-core in nonempty, the solution offered by the Shapley value will in general bedifferent than our lex-max-WS solution, and the reason for this is that lex-max-WS does not satisfythe linearity property with respect to W max . Not satisfying properties associated with W max alone(linearity or other) is a natural consequences of the fact that our WS-core is not defined only byconstraints derived from W max , but also by constraints derived from the disagreement point. Henceeven if W max remains unchanged and only the disagreement point changes, our solution will change,whereas the Shapley value solution would not change.It is known that in general cost-sharing games, the Shapley value solution might be outside thecost-sharing core. Likewise, in our more specialized setting (in which W max is not arbitrary, butrather derived from valuations over alternatives) the Shapley value solution might be outside theanticore. This is shown in [21]. For completeness, we also provide such an example. Consider threeagents and four alternatives, with valuation functions as in the following table:Example 3 Alternative 1 Alternative 2 Alternative 3 Alternative 4Agent 1 Agent 2 Agent 3 W max is not submodular: the marginal contribution of agent 1 to the set { , } is 1, whereas her marginal contribution to the set { } is 0. H.1 Proof of Proposition G.6 We restate and prove Proposition G.6. Proposition H.1 The Shapley-value solution is in the anticore of the Shared–Rental problem andsatisfies strong decomposability, yet it does not dominate RP . Proof: As noted earlier, the function W max is submodular for the Shared–Rental problem, and theShapley value for submodular cost-sharing games lies in the cost-sharing core. As mathematicallythe anticore has the same definition as the cost-sharing core, it follows that the Shapley value is inthe anticore.To see that the Shapley value is strongly decomposable, recall that the Shapley value of an agent i is computed by considering a uniform distribution over all permutations, and taking the expectedmarginal value (with respect to W max ) of the agent over a random choice of permutation. As theinstance is decomposable, these marginals only depend on those agents from the part P j that contains i that arrived before agent i . The uniform distribution for permutations over all agents indices auniform distribution over the permutations for the agents in part P j . Hence the vector of Shapley29alues for all agents is simply the concatenation of the vectors of Shapley values for each of the parts,implying strong decomposability.The fact that it does not dominate RP follows from the fact that the Shapley value mechanismis continuous, whereas the RP mechanism is not, not even when its output maximizes welfare (seeProposition K.1). Consequently, computing the utility of the agents under the Shapley value mech-anism for the example used in the proof of Proposition K.1 will provide an example proving thecurrent proposition. I Comparison to some prior solution concepts Many bargaining solutions have been suggested in the past. Below we discuss some of the most knownones, and show that none of them satisfy all the properties we consider desirable. The solutions wediscuss include the Nash bargaining solution and the Kalai-Smorodinsky bargaining solution [13]. Astransfers are allowed, each of them picks a division of the utility generated by the social maximizingoutcome among the players. These solutions can be defined to dominate RP, but need not lie in theanticore. This last fact has the following undesirable consequences. Proposition I.1 The Nash bargaining solution is not reasonable from above ( u i ≤ W max ( i ) mightfail for some agent i ), and it fails to satisfy even the weak decomposition property. Proposition I.2 The Kalai-Smorodinsky bargaining solution is reasonable from above, but it failsto satisfy even the weak decomposition property. Below we present examples proving the above propositions. We also discuss another solutionconcept, the nucleolus . We explain how a version of this notion can be defined so that it will be inour WS-core. As such, it can potentially serve as an alternative to the solution that we propose toselect from the WS-core, namely, the lex-max-WS solution. Faced with these two alternatives, weexplain why we prefer lex-max-WS over the version of the nucleolus that resides in the WS-core. I.1 The Nash bargaining solution The Nash bargaining solution, when applied in our setting, becomes identical to the egalitariansolution, equalizing the gains of all agents above their respective disagreement utilities. As such, itdominates RP, but need not lie it the anticore. Moreover, the Nash bargaining solution need notbe reasonable from above. Namely, it may hold that u i > W max ( i ) for some agent i . The followingexample illustrates this point.Consider a setting with two agents and a two alternatives. The first agent has values of 24 and0 for the two alternatives, while the second agents has value of 0 and 4. Random priority will givethe agents expected utilities of 12 and 2, respectively. These are the disagreement utilities, and theirtotal is only 14, which is 10 less than the maximum welfare of 24. The Nash bargaining solution willgive each of them an additional utility of 5 so the final utilities will be 17 and 7. In particular, theutility of the second agent is higher than the utility offered to her by the best alternative.Likewise, the Nash bargaining solution fails to satisfy the even the weak decomposition property.The example in Appendix I.2 can serve to illustrate this last point (details omitted). I.2 The Kalai-Smorodinsky bargaining solution The Kalai-Smorodinsky (KS) solution has the following geometric interpretation. Each alternativecorresponds to a point in ℜ n + , specifying the utility of each of the n agents. The point of best tilities is the point with coordinate for each agent equal to the best utility she gets in any feasiblealternative. The Kalai-Smorodinsky solution is the intersection of the Pareto frontier with the linefrom the disagreement point to the point of best utilities.We note that while our solution aims to equalize utility gains (compared to disagreement utilities)as much as possible (subject to anticore constraints), the KS solution aims to equalize the fractionof the utility improvements (from the disagreement to the best point).Like in our solution, the KS solution never gives any agent more than her utility in the bestalternative for her. Yet, the KS solution might fail to satisfy the anticore constraints for groups ofmore than one player. Consequently, it fails to satisfy even the weak decomposition property. Thisis illustrated by the following example:Consider an item allocation setting in which there are four agents and four items, and in eachalternative each agent receives one item. Hence there are 4! = 24 possible alternatives. The valuationof each agent for each item is presented in the following table:Example Item 1 Item 2 Item 3 Item 4Agent 1 Agent 2 Agent 3 Agent 4 , , , , , , , , I.3 The Nucleolus The nucleolus [24] is a solution concept that gives a unique solution to a coalitional game. If thecore of the game is nonempty, the nucleolus resides in the core. Part of its attractiveness is due topossessing a property referred to as consistency for reduced games (see [19] or [16] for more details).To apply this solution concept, one needs to describe our setting as a coalitional game, wherethere is a characteristic function associated with sets of agents. The question becomes which charac-teristic function to use. In general, readers may propose whatever characteristic function they findappropriate, and at this level of generality we have nothing to say about the nucleolus. However, partof our work concerns exactly the issue of selecting characteristic functions that we find appropriatefor our setting, and they can serve as the basis for the definition of the nucleolus.Our notion of anticore and the fact that we have a profit game (in which agents share thewelfare increase that results from selecting a maximum welfare alternative) suggests the use of D ( S ) = W max ( N ) − W max ( N \ S ) as a characteristic function (see the end of Section A.2). Applyingthe nucleolus framework to this characteristic function is then equivalent to selecting a solution that31atisfies the anticore constraints with as large a margin as possible (measured in terms of a lexico-graphic vector of the margins). This solution satisfies the weak decomposability property (becauseit is in the anticore, see Proposition 3.5), but need not satisfy the domination constraints (and hencemight not be in the WS-core). Even in the room (item) allocation setting (which is submodular),there are examples in which the nucleolus fails to dominate the Random Priority mechanism. Theinstance described in Appendix K serves as one such example (for the same reason that it showsthat the Shapley value solution does not dominate RP – see the last paragraph of the proof ofProposition H.1 for an explanation).To force the nucleolus solution to lie within the WS-core, one may modify the characteristicfunction so that its value on each set S is the maximum between D ( S ) and the sum of the disagreementutilities of agents in S . We refer to this function as the WS characteristic function, and to theresulting nucleolus concept as nucleolus-WS . It selects a solution that satisfies all constraints of theWS characteristic function with as large a margin as possible (measured in terms of a lexicographicvector of the margins). In contrast, our lex-max-WS solution satisfies only the domination constraintswith as large a margin as possible, subject to not violating the anticore constraints. Having a largemargin for the domination constraints serves the purpose of giving agents as strong as possibleincentives to give up the reference point and adapt our mechanism. Keeping a large margin from allconstraints (including the anticore constraints) serves a different purpose: it attempts to keep theselected solution away from all boundaries of the feasible region. We prefer the lex-max-WS solutionover nucleolus-WS, though using nucleolus-WS is also a reasonable option for selecting a solutionfrom the WS-core. In particular, like the lex-max-WS , the nucleolus-WS mechanism satisfies thestrong decomposability property. This is a consequence of being in the anticore (that ensures weakdecomposability), together with the fact that the nucleolus satisfies consistency for reduced games(details omitted).Section M presents (among other things) a comparison between lex-max-WS and nucleolus-WSwhen there are two agents. J Is W max submodular for submodular Combinatorial Valuations? Proposition G.1 is the result of the fact that for unit-demand valuations the set function W max issubmodular. Thus, a similar claim is true for other classes of combinatorial valuation functionsover the items that ensure that the set function W max is submodular. One would be tempted tothink that the fact that unit-demand valuations are submodular was sufficient to prove that W max is submodular. Yet care should be taken. The submodularity of unit-demand valuations is for eachvaluation function, with respect to sets of items , while the submodularity of W max is for the totalwelfare, and with respect to sets of agents . Indeed, we next show that the fact that the class ofsubmodular valuation functions does not ensure that the set function W max is submodular. Proposition J.1 There are instances of allocation problems in which all agents have valuation func-tions that are submodular (and in fact, also belonging to the classes of budget additive valuationsand coverage valuations) but nevertheless, the associated set function W max is not submodular. Proof: Consider an example with four agents and three items. The values that agents associatewith individual items are presented in the following table.32xample 4 Item 1 Item 2 Item 3Agent A Agent B Agent C Agent D A , B and C are additive , and the valuation function of agent D is budget additive with a budget of 2 (it also happensto be a coverage function ). Hence any set of either two or three items has value as 2 for agent D , evenif the sum of item values is larger than 2. All four valuation functions are submodular. However, theset function W max is not submodular. Consider the sets S = { A, B, D } and T = { A, C, D } . Then4 + 4 = W max ( S ) + W max ( T ) < W max ( S ∩ T ) + W max ( S ∪ T ) = 3 + 6, violating the submodularityinequality.We conclude by remarking that if all agents have additive valuation functions then is is true thatthe set function W max is submodular, and Proposition G.1 applies to this setting as well. K Both Random Priority and Eating are not continuous Proposition K.1 Neither the Random Priority (RP) mechanism nor the Eating mechanism arecontinuous, not even on instances on which they nearly maximize welfare. Proof: Consider an instance with three players and three items. For small ǫ > 0, the valuationvectors of the players are (1 , − ǫ, ǫ ), (1 , − ǫ, ǫ ), (1 , , ǫ ). The maximum welfare allocation haswelfare 2, and so does each allocation that might result from RP and Eat (because it is never thecase that player 3 gets item 2). The expected utilities under RP are roughly ( , , ) (up to O ( ǫ )),and likewise for Eat (in both cases player 3 has probability of getting item 1). Consider now aslightly modified set of valuation vectors: (1 − ǫ, , ǫ ), (1 − ǫ, , ǫ ), (1 − ǫ, , ǫ ). Again, the maximumwelfare allocation has welfare 2, and so does each allocation that might result from RP and Eat (because it is never the case that player 3 gets item 2). The expected utilities under RP are roughly( , , ) (up to O ( ǫ )), and likewise for Eat (in both cases player 3 has probability of getting item 1).As ǫ tends to 0, every player suffers discontinuity in its utility. L Properties of envy-free mechanisms Proposition L.1 There are instances of the unit-demand matching problem in which in each in-stance there is unique envy-free solution, and this solution fails to satisfy some desirable properties,as follows.1. It does not satisfy decomposability.2. It does not dominate the Random Priority allocation, and moreover, some agent i exceeds hisown upper bound on utility W max ( i ) = max j v i ( j ) .3. It does dominate the Random Priority allocation and even the Eating mechanism, but never-theless, some agent i exceeds his own upper bound on utility W max ( i ) .4. It is in the anticore but still fails to dominate Random Priority . . It dominated Random Priority (and even Eating ) and no agent i exceeds his own upper boundon utility W max ( i ) , but is not in the anticore. Proof: We write valuation functions as vectors of item values. In all cases an allocation thatmaximizes welfare is the identity permutation. (It can be made unique by adding a small ǫ > v i ( i ), but this is omitted from the examples so as to keep the notation simple.)1. See example in the introduction.2. Valuations (6,0,0), (6,0,0), (0,6,6). The only envy free transfer is ( − , , Random Priority (2 instead of 3), and agent 3 gets more utility than hishighest value (8 instead of 6).3. Valuations (6,0,0), (6,0,0), (1,0,0). The unique envy free transfer is ( − , , RandomPriority and Eating each agent get the first item with probability 1 / 3. Agent 3 gets moreutility than his highest value (2 instead of 1).4. Valuations (2,1,0), (2,1,0), (0,1,0). The unique envy free transfer is (-1,0,1). This satisfiesthe anticore constraints, but does not dominate Random Priority (each of agents 1 and 2 getsutility 1, whereas in Random Priority they get expected utility ).5. Valuations (7,0,0,0), (7,0,0,0), (2, 0, 1, 0), (2, 0, 1, 0). The unique envy free transfer is( − , , , Eating mechanism and no agent i exceeds his own upperbound on utility W max ( i ), but it violates the core upper bound constraints on some group (thegroup { , } has utility 4, higher than their upper bound of 3). L.1 Envy-free solutions might pay a renter that got his top room We restate and prove Proposition G.5. Proposition L.2 There are instances of the Shared–Rental problem in which the valuation functionof each player is nonnegative and sums up to the total rent (this is the setting studied in [9]), butnevertheless there is a player that in every envy-free solution both gets his most desired room andreceives more money than his rent share. Proof: Consider an instance with five players (and five rooms) in which they need to pay rent of1. For 0 < ǫ < , the valuations functions are (in vector notation): v = v = (1 − ǫ, , ǫ, , v = v = (0 , − ǫ, , ǫ, v = (0 , , − ǫ , − ǫ , ǫ − ǫ above room 3. Likewise, room 2 needs to be pricedexactly 1 − ǫ above room 4. To avoid envy for agent 5, room 5 needs to be priced at most ǫ aboveany of rooms 3 and 4. Using the above information, the maximum that player 5 can be charged is atmost − ǫ (with players 1 and 2 getting rooms 1 and 2 and charged − ǫ each, and players 3 and 4getting rooms 3 and 4 and charged − ǫ each). For 0 < ǫ < players 1 and 2 more than coverthe rent, and each of players 3, 4 and 5 receives money rather than pays rent. In a sense, players 1and 2 are paying the other players so that the other players agree to give them rooms 1 and 2. It isperhaps justified that players 3 and 4 get paid – they really wanted rooms 1 and 2, and instead gotrooms of almost no value. Hence they sacrificed something to enable the solution, and may deservesome compensation. In contrast, player 5 got his most desirable room and sacrificed nothing, butnevertheless, is also getting paid. M A complete analysis for unit-demand matching with two agents In this section we present a complete analysis of unit demand matching when n = 2 (two agents andtwo items). This is fact captures all settings in with two agents and two alternatives (here the twoalternatives are the two possible permutations over the items). We compare between the followingfour mechanisms:1. Envy-free. To select a unique solution among all envy-free solutions, we use max-min as theselection rule, as was recommended in [9].2. The Shapley value solution.3. Kalai-Smorodinsky bargaining. Here we use RP (or equivalently for the case n = 2, Eating ) asthe disagreement mechanism.4. Our lex-max-WS , again with RP as the disagreement mechanism.Let us denote the agents by { A, B } and the items by { a, b } . After some normalization, we mayassume that the valuation function for A is v A ( a ) = 1 and v A ( b ) = − 1, and for B is v B ( a ) = δ and v B ( b ) = − δ , with − ≤ δ ≤ 1. The maximum welfare allocation (with a ties in case that δ = 1)allocates item a to agent A and item b to agent B . The following table presents the pair of utilities(for agent A and B ) under the mechanisms max-min-EF (maxmin envy free), Shapley value , KS (Kalai-Smorodinsky bargaining) and our lex-max-WS . max-min-EF Shapley value KS lex-max-WS δ ≤ | δ | , | δ | ) (1 , | δ | ) (1 , | δ | ) (1 , | δ | )0 ≤ δ ≤ ( − δ , − δ ) (1 − δ, 0) ( − δ δ , δ (1 − δ )1+ δ ) (1 − δ, δ ) ≤ δ ≤ − δ , − δ ) (1 − δ, 0) ( − δ δ , δ (1 − δ )1+ δ ) ( − δ , − δ )We wish to draw the attention of the reader to the following facts.When δ < 0, implying that each agent desires a different item, the max-min-ES solution dictatesa transfer from agent A to agent B . Thus agent B not only gets his most desired item, but also getspaid for taking it. Agent A is better of in the RP disagreement mechanism, where he gets his mostdesired item without having to pay agent B .The utility that agent B gets from the Shapley value solution is equal to his expected utility atthe disagreement point (the expected output of RP). This goes against our perception of fairness, inwhich increase in the general welfare should be shared by all those who contributed to the increase.35oth the Kalai-Smorodinsky bargaining solution and the lex-max-WS solution share the increasein welfare among the two agents, though to different extents. We leave it to the reader to decidewhich of the two does it better. The case n = 2 is too small to illustrate our main reason for preferring lex-max-WS over KS, which is the fact that KS does not satisfy decomposition properties. This isillustrated in the proof of Proposition I.2), by an example where n = 4.In Section I.3 we explained how the notion of the nucleolus can be adapted to our WS-core, givingthe nucleolus-WS mechanism. For the two agents case, nucleolus-WS gives the same solution as lex-max-WS , except in the range 0 < δ < , where nucleolus-WS gives the pair of utilities (1 − δ , δ ).Interestingly, there is a unique value ( δ = ) in the range 0 < δ < N Population and resource monotonicity We consider here the unit-demand matching setting. Population monotonicity means that by in-troducing an additional agent, it cannot be that a different agent gains utility. Resource mono-tonicity means that by introducing an additional item, it cannot be that an agent looses utility.Moulin [Econometrica 1992] showed that the Shapley value satisfies both population and resourcemonotonicity. Here is an example showing the the lexmax-WS solution does not satisfy populationmonotonicity and does not satisfy resource monotonicity. In the example both random priority andthe Eating mechanism give the same disagreement utility, and hence the example applies to both.Example 5 Item A Item B Item C Item DAgent 1 12 0 6 0 Agent 2 12 6 0 0 Agent 3 24 12 0 25There are three agents { , , } and four items { A, B, C, D } . The valuation function of agents isas described in the table.If only agents { , } participate and only items { A, B, C } , then for each agent, both the dis-agreement utility and the lexmax-WS utility are 9. If the set of agents is changed to { , , } , thedisagreement utilities become 8 for agent 1, only 7 for agent 2, and 14 for agent 3. The lexmax-WSsolution has welfare 36, giving agent 1 utility of 9.5 and agent 2 utility of 8.5 (at this point theanticore constraint for the set { , } is tight). Hence population monotonicity for agent 1 does nothold upon introducing agent 3.Changing now the set of items to { A, B, C, D } changes the disagreement utilities of each ofagents 1 and 2 to be 9. This is also their utility in the respective lexmax-WS solution, hence itemmonotonicity does not hold – introducing item D caused the utility of agent 1 to drop from 9.5 to 9. O Algorithms and computational complexity The water filling algorithm of Section 4 can be seen to imply the following proposition. Proposition O.1 Suppose that the valuation functions of the players are expressed as rational num-bers (namely, every v i ( j ) is expressed as pq for some integers p and q .) If W max is submodular, thetransfers of the lex-max-WS solution are also rational. In this section we make the convention that valuation functions take only integer values in therange [ − M, . . . , M ], where M is taken to be sufficiently large. (If valuation functions take arbitrary36ational values, they can be scaled to give integer values, by multiplying by the lowest commondenominator.) As to the output of the algorithm, we shall not insist on getting the exact lex-max-WS solution, but rather a solution that gives every agent a utility that differs by at most ǫ from herutility in the lex-max-WS solution. Here ǫ > M is very large, it may wellbe that ǫ > 1, in which case we assume that ǫ is an integer). Consequently, the numerical valuesmanipulated by algorithms can be restricted to numbers expressible by O (log M + log(1 + ǫ )) bits,even though the true lex-max-WS solution might require much higher precision. We view the useof ǫ as justified in essentially all practical situations, as payments can practically be made only at aprecision determined by the smallest denomination accepted in the relevant currency.In general, running times of algorithms can be expressed as functions of the number of players n , number of alternatives |A| , range of valuations M , the precision parameter ǫ , and the descriptionlength of the disagreement distribution π v . To reduce the number of parameters in Definition O.2,we do not state explicitly the dependency on |A| and π v . Later, in contexts in which it matters, wewill also consider the effect on |A| and π v . Definition O.2 Using the above notation, we consider the following classes of running times foralgorithms: • Weakly polynomial. The number of operations performed is polynomial in ( n, M, ǫ ) . • Polynomial. The number of operations performed is polynomial in ( n, log M, log ǫ ) . • Strongly polynomial. The number of operations performed is polynomial in n and independentof M and ǫ , though each operation may involve numbers with O (log M + log(1 + ǫ )) bits,and hence the time per operation (for example, adding two numbers) might be polynomial in log M + log(1 + ǫ ) . To extract an algorithm out of the water filling algorithm, one needs subroutines for the followingthree tasks:1. Compute the disagreement utilities u π v ( i ) for every agent i .2. Compute the increments x j for every iteration j .3. Determine at each iteration which agents are involved in constraints that become tight, so asto lock these agents.Suppose first that the disagreement utilities are easy to compute. A specific case when thishappens is when there is one designated disagreement alternative , and π is supported only on thisalternative. In fact, whenever the disagreement utilities are easy to compute, we may add a “dummy”alternative (which will serve as the disagreement alternative) whose value to each agent exactly equalsthe computed disagreement utility of the agent, and shift π to be supported only on the dummyalternative. (Note that adding this dummy alternative does not change any of the constraints of theanticore.) Hence Theorem O.3 extends to all cases in which the disagreement utilities are easy tocompute. Theorem O.3 Consider instances in which the number of alternatives is bounded by some poly-nomial in n , and one is given a disagreement alternative that forms the support of disagreementdistribution π . Then: . If there is even a weakly polynomial time algorithm for computing the lex-max-WS solution onsuch instances, then P = N P .2. Nevertheless, if W max is submodular, then lex-max-WS can be computed in strongly polynomialtime. Moreover, this result extends also to the case where the number of alternatives is notbounded by a polynomial in n , provided that there is a value oracle for computing W max (namely,given S , the value of W max ( S ) can be computed in time polynomial in n ). Proof: We first prove the NP-hardness result. It is by reduction from the maximum independentset problem MIS. Let α ( G ) denote the maximum size of an independent set in a graph G . Weshall consider the standard gap version M IS c,s , where c is the completeness parameter, s is the soundness parameter, and 0 < s < c < 1. The input to M IS c,s is a graph G on n vertices, andthe computational task is to output yes if α ( G ) ≥ cn , to output no if α ( G ) ≤ sn , and any outputis allowed if sn < α ( G ) < cn . We may assume without loss of generality that G has no isolatedvertices. It is known that M IS c,s is NP-hard for some values of 0 < s < c < s = for some c > . (For example, startingwith M IS c,s with s < / 2, add to the graph (1 − s ) n isolated vertices, and connected them all tothe same vertex in the graph.)We reduce an instance of M IS c, with < c < lex-max-WS as follows. Givenan input graph G ( V, E ) with n vertices (an instance of M IS c, ), every vertex v ∈ V corresponds toan agent and every edge ( u, v ) ∈ E corresponds to an alternative. Every agent v derives value n fromeach of the alternatives that correspond to the edges incident with v , and value 0 from every otheralterative. Hence every alternative has welfare exactly 2 n . In addition, there is the disagreementalternative D that gives each agent value c (where c is the completeness parameter of the M IS c, instance).This completes the description of the lex-max-WS instance. Observe that the number of agents is n and that M = n . Fix ǫ = − c = c − c . Hence a weakly polynomial time algorithm for lex-max-WS simply needs to run in time polynomial in n .The amount of welfare offered by the maximum welfare alternative (any of the edges) is 2 n . If thewelfare could be distributed evenly over all agents, it would give each agent a utility of 2. However,such a solution might not be in the anticore. So let us consider the value of W max ( S ) for variousnonempty subsets S of agents. If S forms an independent set in G , then W max ( S ) = max[ n, | S | c ].Else, W max ( S ) = 2 n .It follows that if G is a no instance of M IS c, , giving each agent a utility of 2 is feasible (itis in the anticore). However, if G is a yes instance, there is a set S of cn > n/ W max ( S ) = n . As the disagreement utility is c , all members of this set S get utility exactly c < lex-max-WS solution is 2 when G is a no instance and c < G is a yes instance. Computing lex-max-WS with error smaller than c − c will allow us to distinguishbetween these cases. This completes the proof of the NP-hardness result.We now show that lex-max-WS can be computed in strongly polynomial time when W max issubmodular. For this, we explain how each of the three tasks of the water filling algorithm can becomputed in strongly polynomial time.1. Compute the disagreement utilities u π ( i ) for every agent i . This can be done in stronglypolynomial time because the disagreement alternative is given.2. Compute the increments x j for every iteration j . This task is a special case of a problem known38s line search in submodular polyhedra , and can be solved in strongly polynomial time, givenvalue oracle access for the underlying submodular function [22, 11].3. Determine at each iteration which agents are involved in constraints that become tight, soas to lock these agents. Once x j for iteration j has been computed, the agents involved intight constraints are those agents whose utility cannot be increased, unless either an anticoreconstraint is violated, or the utility of some other agent is decreased. Computing the maximumincrease of utility of an agent, subject to keeping the utility of other agents unchanged and notviolating an anticore constraint, is again a special case of line search in submodular polyhedra .Going over all agents not locked in previous iterations and determining for which of them themaximum increase is 0 gives us the newly locked agents. P Continuity of the lex-max-WS solution In this section we consider continuity properties of the lex-max-WS solution as a function of thecardinal valuations of the agents. First, it is important to note that when the disagreement utilitiesare a function of the valuations, this function might have discontinuity points. For example, thismight happen if the disagreement utilities are computed as the outcome of the Random Prioritymechanism, and cardinal valuations change to the extent that ordinal preferences over alternativesalso change. At discontinuity points for disagreement utilities we shall not require (and do not expect)that the lex-max-WS solution will be continuous. Hence we shall assume in this section that thedisagreement utility is the outcome of some distribution π v over the set A alternatives, and that thisdistribution does not change when cardinal valuations of agents change (though the disagreementutility itself might change, due to the change in valuations of the alternatives). Hence for agent i thedisagreement utility can be expressed as E A ← πv A [ v i ( A )]. We remark that if the disagreement utilitiesare computed as the outcome of the Random Priority mechanism, our results hold with respect tochanges of cardinal valuations that do not alter the ordinal preferences over the alternatives.As a convention, we shall apply additive shifts to the valuations so that the disagreement utilitiesare 0. This is done without loss of generality, because the allocation and the transfers of the lex-max-WS solution remain unchanged when an additive shift is applied to the valuation function of anagent. After these additive shifts, E A ← πv A [ v i ( A )] = 0 for every agent i . We let u i denote the utilityof agent i in the lex-max-WS solution.We now introduce additional notation for the purpose of discussing continuity, and the associatedLipshitz constant. We shall consider the effects of the change of the valuation function of a singleagent i (while keeping the valuation functions of all other agents fixed) on the utilities of each ofthe agents. Let I be an instance, and let v i be the valuation function of agent i (viewed as a vectorin R m where m = |A| ). Let e ∈ R m be a modification vector, leading to a new valuation function v ′ i = v i + e , and correspondingly a new instance I ′ . To satisfy our convention that the disagreementutility of agent i is 0, we require the modification vector to satisfy E A ← πv A [ e ( A )] = 0, which thenimplies that also E A ← πv A [ v ′ i ( A )] = 0.Introduce a parameter t (for time ) that changes gradually from − − ≤ t ≤ 1, letinstance I t be the instance in which the valuation function of i is v i + te . Hence I = I and I = I ′ .Let | e | denote the maximum difference between entries in vector e (the maximum value minus theminimum value). For every set S and every − ≤ t ≤ 1, the change in W max ( S ) from instance I toinstance I t is at most | t | · | e | if i ∈ S , and there is no change if i S .39or an arbitrary player j (it can be that j = i ), let u j ( t ) denote the utility of player j under the lex-max-WS solution on instant I t . Proposition P.1 For instance I and modification vector e as above, for every agent j the associatedutility function u j ( t ) is continuous in the interval − ≤ t ≤ (as long as the WS-core remainsnonempty). Proof: This is a consequence of the water filling algorithm. Details omitted.Having established continuity, we now analyse the associated Lipshitz constant. Let B denote thetotal welfare of the maximum welfare alternative, and let b S stand for W max ( S ). The lex-max-WS solution satisfies the following set of linear constraints that we refer to as the WS-constraints:1. P i ∈ S u i ≤ b S for every S ⊂ [ n ].2. P ni =1 u i = B (where B = b [ n ] ).3. u i ≥ ≤ i ≤ n .Given a feasible solution u to the WS-constraints (the lex-max-WS solution is one such solution),we say that a set S ∈ [ n ] is tight if its corresponding constraint is satisfied with equality (namely, P i ∈ S u i = b S ). Observe that the set [ n ] is always tight, by constraint 2, and we may treat the emptyset as tight as well.A collection S of sets is a (distributive) lattice if for every S ∈ S and T ∈ S it holds that S ∩ T ∈ S and S ∪ T ∈ S . Lemma P.2 If W max is submodular, then for every feasible solution u , the collection of tight sets(w.r.t. the WS-constraints) forms a lattice. Proof: Recall that b S = W max ( S ). Let S and T be two sets that are tight under the feasiblesolution u , and for Y ⊂ [ n ] let U ( Y ) denote the sum of utilities derived in u by the players in set Y . The tightness implies that U ( S ) = W max ( S ) and U ( T ) = W max ( T ). Additivity of U implies that U ( S ) + U ( T ) = U ( S ∩ T ) + U ( S ∪ T ). Submodularity of W max implies that W max ( S ) + W max ( T ) ≥ W max ( S ∩ T ) + W max ( S ∪ T ). Consequently, it must hold that U ( S ∩ T ) = W max ( S ∩ T ) and U ( S ∪ T ) = W max ( S ∪ T ), since for every set Y ⊆ [ n ] it holds that U ( Y ) ≤ W max ( Y ). Namely, both S ∩ T and S ∪ T are tight. Remark P.3 If W max is submodular, then given only the collection of sets that are tight for lex-max-WS solution (but not lex-max-WS itself ), we can compute lex-max-WS as follows. Process thetight sets in an order consistent with the natural partial order over these sets (a tight set is processedonly after all tight sets that it contains are processed). Given a set S in the collection, let S ′ ⊂ S denote those variables u i ∈ S whose value was already determined by sets previously visited in thepartial order. Then every variable in S \ S ′ gets value b S − P j ∈ S ′ x j | S |−| S ′ | . Theorem P.4 When W max is submodular and the disagreement utilities are the outcome of somedistribution π v over the alternatives, the lex-max-WS solution (which is continuous in t , see Propo-sition P.1) has Lipshitz constant at most 1. roof: We shall refer to a value − ≤ t ≤ breakpoint if at that point (either approachingit from the left of from the right or both) some set that was not tight (with respect to the WS-constraints for lex-max-WS ) becomes tight. There are only finitely many breakpoints. Removingthese breakpoints, the interval − ≤ t ≤ t = ± T of tight sets forms alattice. Given two instances I t and I t + ǫ in the subinterval, by how much could the solutions change?Recalling Remark P.3 (and the notation S and S ′ from that remark), this entails checking by howmuch b S − P j ∈ S ′ x j might change (due to the change in v i between the two instances). If i S then there is no change. If i ∈ S but i S ′ then b S changes by at most ǫ | e | and P j ∈ S ′ x j does notchange, and hence the change is at most ǫ | e | . If i ∈ S ′ then consider the collection of sets { T k } suchthat for every k it holds that T k ∈ T , T k ( S and there is no set T ∈ T such that T k ( T ( S (the T k are maximal). By the lattice structure, the sets T k are disjoint. Without loss of generality, i ∈ T . Then the change of value for each variable in S \ S ′ is exactly b S [ I t + ǫ ] − b S [ I t ] − P k ( b Tk [ I t + ǫ ] − b Tk [ I t ]) | S |−| S ′ | .Observe that b S [ I t + ǫ ] − b S [ I t ] is nonzero only as a result of an alternative changing value for i , andlikewise for b T [ I t + ǫ ] − b T [ I t ] (and for the rest of the T k we have that b T k [ I t + ǫ ] − b T k [ I t ] = 0). Bythe definition of | e | , the difference in these changes cannot exceed ǫ | e | . (The change in b S [ I t + ǫ ] isupper bounded by the change in value of the alternative allocated to i when computing b S [ I t + ǫ ], andsimilarly for the change in b T [ I t + ǫ ]. Hence the difference between these two changes cannot exceed ǫ | e | .) Consequently, we get in all cases a Lipshitz constant of at most 1. Remark P.5 If the WS-core is nonempty and W max is not submodular, then the Lipshitz constantof lex-max-WS might depend on n , the number of agents. For example, suppose that there are n agents and four alternatives, A , A , A , A . Let v = (1 , , , , v = . . . = v n − = (1 , , , , and v n = (1 , , , n ) . Alternative A serves as the disagreement alternative, and alternative A maxi-mizes welfare (which is n ). The function W max is not submodular. In particular, W max ( { } ) = 2 , W max ( { , } ) = 6 , W max ( { , } ) = 6 , W max ( { , , } ) = 12 , showing that W max ( { , } )+ W max ( { , } ) The water filling algorithm for computing the lex-max-WS solution requires the computation of thedisagreement utilities u π v ( i ). Let us discuss briefly the computational complexity of this task in thespecial case of the room (item) allocation problem. Suppose that there are n agents and n items,that the valuation functions of the agents for the items are given (where v i ( j ) is the value that agent i associates with item j , and is an integer with absolute value at most M ), and one needs to allocateone item to each agent. In our setting, the disagreement utility for agent i in lex-max-WS is theexpected utility that agent i derives from some default allocation mechanism with no transfers. Weconsider here the three candidate default mechanisms that were presented in Section G.1, sketch howthey can be implemented algorithmically, and briefly discuss the modifications employed in thesemechanisms to handle situations in which the valuation function of an agent might have ties. • Uniform (U) . The disagreement utility of agent i is n P nj =1 v i ( j ). It can be computed exactlyin time polynomial in n and log M . 41 Random priority (RP) . Suppose first that for every agent, her valuation function has no ties(there is no agent i and items j = j ′ such that v i ( j ) = v i ( j ′ )). The naive approach for computingthe disagreement utilities (exactly) involves considering all n ! permutations over the agents,and for each permutation determining which item is received by which agent, based on theordinal preferences of the agents. This procedure can be implemented in polynomial space (in n ), but it is not polynomial time, and we (the authors) have no reason to believe that thereis an alternative algorithm that does compute the disagreement utilities in polynomial time.(Computation of the Shapley value, which is also defined in terms of all possible permutations,is known to be P complete in some settings [4].)The RP mechanism is adapted as follows to allow for valuation functions that have ties. Recallthat an alternative A is a matching of items to agents. Given a permutation over the agents(as before, all n ! permutations are considered), each agent in her turn is faced with a list ofalternatives that are still available, and items that are still available to her (matched to herin an least one of the remaining alternatives). Of the items available to her, the agent selectsone or more items as most desirable, and discards those alternatives that match to her an itemthat is not one of the most desirable available items. Implementing this mechanism naivelyseems to require n ! space to store all alternatives. However, it can also be implemented inpolynomial space (though still not polynomial time) as follows. When it is the turn of an agent i to select an item, and several of the remaining items are tied in being most desirable (all havethe highest value under v i ), the agent is temporarily put on hold, allowing subsequent agentsto select items. At every step, if the set of agents on hold contains a subset S of agents whoseunion of desirable items is also of size | S | (we call such a set tight ), the members of S each getone of their desired items. Finding tight sets can be done in polynomial time (using standardalgorithms for bipartite matching).We remark that one can also compute the disagreement utilities up to precision ǫ using arandomized weakly polynomial time algorithm that succeeds with high probability. This is doneby randomly sampling O ( M √ log nǫ ) permutations, for each of them computing the allocationobtained when agents serially select their most preferred item, and for every agent averagingover the utilities that she derives from all the allocations. • Eating mechanism (EAT) . The Eating mechanism can naturally be adapted to the case thatthe valuation function of an agent may have ties: instead of “eating” one item at rate 1, theagent can “eat” all k tied items, each at rate 1 /k . EAT has at most n phases, where a phaseends when some item becomes fully consumed. The length of each phase can be computed ina number of operations that is polynomial in n . The precision required in order to express theexact length of a phase may grow significantly as phases progress. Hence practically one wouldcompute the output up to some desirable precision ǫ . This can be done in strongly polynomialtime (details omitted).Combining the above discussion on the Eating mechanism and Theorem O.3, we have the followingcorollary. Corollary Q.1 In the room allocation problem with n agents and with the Eating mechanism servingas a disagreement point, the lex-max-WS solution can be computed in strongly polynomial time. We remark that for small values of n , the polynomial time algorithms of Theorem O.3 andCorollary Q.1 might be slower than other simpler to implement algorithms that are not polynomialtime. Consider for example the room allocation problem. If the disagreement point is the random42riority mechanism, then the disagreement utility of all agents can be determined by considering all n ! orders over players. The anticore has 2 n − j of the water fillingalgorithm, one can check what upper bound each of these constraints places on x j , and take for x j the smallest of these upper bounds. Shared apartments rarely have more than n = 4 rooms, and theabove simple algorithm will run very quickly on such instances. R Full information upon request In this work we assume (as is often assumed in literature on cooperative games and in systemslike Spliddit [12]) that knowledge of the true valuation functions of the agents is available to ourmechanisms. This should not be interpreted as if we assume that all agents know the valuationfunctions of all other agents. Rather, the interpretation is that a mechanism can request informationfrom the agents about their valuation functions (e.g., the most preferred alternative from a set ofalternatives in the random priority mechanism, full ordinal preferences for the eating mechanism,cardinal valuations for lex-max-WS ) and obtain truthful replies. In practice, presumably agents willbe asked to report their valuation functions to the mechanism in private – in our mechanisms thereis no need for an agent to know the valuation functions of other agents.Below we explain why we view it as both necessary and reasonable to assume that agents willsupply truthful information to our mechanisms. Dominant strategies. Informally, a (deterministic) mechanism has dominant strategies , if forevery agent, whenever the agent needs to provide information to the mechanism, then the combinationof the information previously provided to the agent by the mechanism and the valuation function ofthe agent itself suffice in order for the agent to figure out a “best response”. Here the response isthe information that the agent provides, and being “best” means that regardless of the informationnot available to the agent (such as valuation functions of other agents), no other response will leadto higher end utility for the agent. (The definition for randomized mechanisms is somewhat morecomplicated, but not needed for the discussion here.) It is well known (and easy to prove) that inour setting, even in the special case of Shared–Rental problem with only two agents, there is nomechanism that satisfies the following three properties simultaneously: maximizing welfare, beingbudget balanced, and having dominant strategies. Hence it is unavoidable to give up at least one ofthree properties, and we choose to give up having dominant strategies. Why would agents report their true valuation function to the mechanism? As remarkedearlier, our mechanisms do not have the property that being truthful is a dominant strategy (as thisis theoretically impossible). So why is it reasonable to assume that agents will be truthful? Wepropose here several practical reasons why this may be (approximately) the case in some real worldsettings. One reason is that games are not played in isolation, but in a larger social context thatinvolves various educational processes and social norms, and this context may encourage truthfulbehavior in the game. For example, a common social norm is that cheating by someone who hasbeen treated unfairly is more socially acceptable than cheating by someone who was treated fairly.Our solution concept incorporates fairness features (an agent is guaranteed utility at least as high asthe disagreement utility, and is furthermore guaranteed that her monetary payments are used onlyso as to compensate those agents for which the chosen alternative is less desirable, and not so asto provide agents with profits beyond what they could obtain from their best alternative), and thismay help reduce the drive to cheat. Another reason why agents may report their true valuationfunctions is because in our mechanisms, being truthful is a strategy that is not dominated by anyother strategy. Unless an agent knows the valuation function of other agents, being untruthful mightcause the agent to lose utility. 43 he burden of reporting valuations. Two features of our lex-max-WS mechanism alleviatesome of the cognitive/computational burden that an agent might suffer when computing what toreport to the mechanism. One aspect is the continuity property, and moreover, the small Lipshitzconstant. For example, in situations where Theorem P.4 applies, an agent may provide the requestedinformation up to an additive error of ǫ of her choice, and be guaranteed that the effect of this errorof her final utility will be a difference of at most ǫ . So an agent that is not sensitive to a differenceof ǫ in her utility can afford to compute only ǫ -approximations to her valuation function. The otheraspect that sometimes alleviates the burden of reporting valuations is the issue of decomposability,especially in settings like Shared–Rental problem. The mechanism can be broken into two phases,where in the first phase it suffices to report only ordinal valuations, and in the second phase, cardinalvaluations need to be reported only for the component to which the agent ends up belonging, andnot for the whole input instance. S Discussion of some other modeling assumptions Quasi-linear utilities. We assume that the utility functions of the agents are quasi-linear . It isdesirable to limit quasi-linearity assumptions so that they need to hold only in a limited range ofvalues. All solutions in the WS-core satisfy the property that for every agent i her utility lies betweenmin A [ v i ( A )] = W min ( i ) and max A [ v i ( A )] = W max ( i ). Hence it suffices for our purposes that for everyagent i , her utility function is quasi-linear in the range [ W min ( i ) , W max ( i )]. The role of monetary transfers. Monetary transfers are used in our solution concept in orderto make it beneficial for all agents to move from the disagreement point to a solution that maximizeswelfare. Monetary transfers can be used for other purposes as well, such as taxing those agents thathappen to be rich and subsidizing those agents that happen to be poor, but these uses of monetarytransfers are beyond the scope of this work, and can be applied (if desired) independently of oursolution concept. Outcome versus process. We assume that what the agents care about is the final outcome– the alternative chosen and the transfers. However, sometimes agents care also about the processby which the outcome was reached. For example, players may derive satisfaction not only from“winning”, but also from a sense “playing well” (e.g., making clever moves in challenging situation,regardless of the outcome). In our setting, by changing the mechanism (e.g., from RP to lex-max-WS ) the nature of the “game” changes, and the amount of “pleasure” (or displeasure) derived fromplaying the game changes. Aspects of this nature are not captured by our work. Valuation functions and fairness. The mechanisms discussed in this paper are based oneither ordinal preferences or cardinal valuations of the agents. We remark that there are studies thatsuggest that even full knowledge of cardinal valuations is insufficient information if the goal is toachieve a solution that is deemed fair by humans. It turns out (see [28], for example) that dependingon additional annotation that is provided for the same cardinal valuations, such as whether thevaluation is based on needs , on preferences or on beliefs , humans tend to choose different solutionsas being fair. Disagreement point for Shared–Rental problem. We assumed a situation in which thestudents who are faced with the room allocation problem already rented the apartment, and for thissetting we used RP as a disagreement mechanism. One may consider also a situation in which thestudents are contemplating the possibility of renting the apartment, but have not yet committed torenting it. In this case the problem changes because another alternative is introduced, that of notrenting the apartment. The value of this alternative for each student is rn (where r is the total rentand n is the number of students), because this is the amount of money saved by the student by not44enting. It is natural to treat this new alternative as the disagreement point. As our lex-max-WSlex-max-WS