Fractional Top Trading Cycle on the Full Preference Domain
aa r X i v : . [ ec on . T H ] M a y Fractional Top Trading Cycle on the Full Preference Domain ∗ Jingsheng Yu † Jun Zhang ‡ May 18, 2020
Abstract
Efficiency and fairness are two desiderata in market design. Fairness requires ran-domization in many environments. Observing the inadequacy of Top Trading Cycle(TTC) to incorporate randomization, Yu and Zhang (2020) propose the class of Frac-tional TTC mechanisms to solve random allocation problems efficiently and fairly. Theassumption of strict preferences in the paper restricts the application scope. This paperextends Fractional TTC to the full preference domain in which agents can be indiffer-ent between objects. Efficiency and fairness of Fractional TTC are preserved. As acorollary, we obtain an extension of the probabilistic serial mechanism in the houseallocation model to the full preference domain. Our extension does not require anyknowledge beyond elementary computation.
Keywords : fractional top trading cycle; fractional endowment; weak preferences; houseallocation; probabilistic serial
JEL Classification : C71, C78, D71 ∗ Acknowledgment is to be added. Section 5 of this paper subsumes Zhang’s earlier working paper circu-lated as “A simple solution to the random assignment problem on the full preference domain.” † Yu: School of Economics & China Center for Behavioral Economics and Finance, Southwestern Univer-sity of Finance and Economics, 555 Liutai Road, Chengdu, 611130, China. Email: [email protected] ‡ Zhang: Institute for Social and Economic Research, Nanjing Audit University, 86 Yushan West Road,Nanjing, 211815, China. Email: [email protected]
Introduction
Efficiency and fairness are two desiderata in market design. In many allocation problems,fairness requires the use of randomization because of indivisibility of resources. Throughallocating probabilities, we restore symmetry between agents and get a measure of fair-ness. Among the few successful matching mechanisms, Gale’s Top Trading Cycle (TTC;Shapley and Scarf, 1974) is best known for being efficient to solve deterministic allocationproblems. By trading all efficiency-enhancing cycles, TTC excludes any further Pareto im-provement over its assignment. However, TTC is no longer efficient when randomizationis incorporated into its procedure through randomizing endowments or randomly breakingpriority ties. To solve this problem, Yu and Zhang (2020) propose the class of FractionalTop Trading Cycle mechanisms (FTTC). FTTC extends TTC to solve random allocationproblems efficiently and fairly. However, though the assumption of strict preferences in thatpaper covers many applications, there are still many environments where the natural prefer-ence domain is the full preference domain in which preferences can be weak. To maintainefficiency, preference ties cannot be broken arbitrarily before running FTTC. This paperextends FTTC to the full preference domain, and maintains its efficiency and fairness.On the strict preference domain, we present FTTC in the fractional endowment exchange(FEE) model, which is a direct extension of Shapley and Scarf’s housing market model. Inthe model an agent may own fractions of multiple objects, and an object may be ownedby multiple agents. The idea of trading cycles in TTC cannot be directly extended to themodel. In FTTC agents report favorite objects step by step as in TTC. Our innovation isto use linear equations to define how agents trade endowments at each step. The equationssatisfy a balanced trade condition, which requires that at each step the amount of favoriteobject obtained by each agent be equal to the amount of endowments lost by the agent.By connecting the equations to the closed Leontief input-outuput model (Leontief, 1941),we prove that the solution to the equations exists. This ensures that FTTC is well-defined.We add parameters to the equations to control fairness. The parameters determine how theowners of each object divide the right of using the object to trade with the others.A simple example can illustrate why preference ties cannot be arbitrarily broken. Let us For example, in the house allocation model, TTC with uniformly random endowments is equivalentto the Random Priority mechanism (Abdulkadiro˘glu and S¨onmez, 1998), which is not ex-ante efficient(Bogomolnaia and Moulin, 2001). See Bogomolnaia et al. (2005); Erdil and Ergin (2017) for arguments for why the full preference domainnaturally appears in many environments. i, j who own equal divisions of two objects a, b . Agent i is indifferentbetween a and b , but j strictly prefers a to b . If after breaking ties we let i strictly prefer a to b , then in the only individually rational assignment i, j will keep their endowments. But fortrue preferences, the only individually rational and efficient assignment is the one in which i obtains b and j obtains a .Our idea to solve weak preferences is to utilize endowment exchange in FTTC. At anystep, if an object has been used up by all of its owners in the trading process, but someagent who obtains an amount of the object finds that a remaining object is as good as theformer object, we let the agent label his consumption of the former object as an endowmentthat is available for trading. In the following trading process, the balanced trade conditionensures that when the agent loses an amount of the object from his consumption, he will becompensated by obtaining an equal amount of indifferent objects. The agent will withdrawthe label when there no longer exists an indifferent available object. In the above example,suppose at the first step i, j obtain their own endowments / a through self-trading. Afterthat, a is exhausted, but i labels his consumption / a as available for trading because thereremains an indifferent object b . At the second step, i demands b and j demands a . So i will not only obtain his endowment / b through self-trading, but also obtain / b throughexchanging endowments with j . We obtain the unique desirable assignment in the example.Of course, the example illustrates a simple case. In general cases, when an agent labelshis consumption of an object as available for trading, this may induce another agent to labelhis consumption of another object as available for trading, and so on until a chain appears.To maintain efficiency, it is crucial to find all such chains in the procedure of FTTC. So inthe definition of FTTC in this paper, we add a labeling stage at the beginning of each stepto find all such chains. Other than that, the definition remains almost same as on the strictpreference domain. We prove that FTTC remains to be individually rational and sd-efficient,and the conditions imposed on the parameters in FTTC to ensure various fairness axiomson the strict preference domain remain to work on the full preference domain.The literature since Hylland and Zeckhauser (1979) has studied random allocation inten-sively in the house allocation model. The model implicitly assumes that agents collectivelyown all objects, and no agent is favored over any other. So we regard the model as a specialcase of FEE in which agents own equal divisions of all objects. On the strict preferencedomain, we have shown that every FTTC conicides with a simultaneous eating algorithmof Bogomolnaia and Moulin (2001), and a subclass of FTTC that treats all agents equallycoincides the Probabilistic Serial mechanism (PS). This means that our definition of FTTC2n this paper subsumes an extension of PS to the full preference domain. The extension canbe described as an eating algorithm and maintains efficiency and fairness of PS. Its only de-viation from PS is that when an agent labels his consumption of an object as available for theothers to consume and some others are indeed consuming his consumption, we instantly in-crease his eating rate following the rule we call “you request my house - I get your rate”. Thisrule is the degeneration of the balanced trade condition to eating algorithms. An extremecase of weak preferences is the dichotomous preference domain. On such domain we showthat our extension of PS finds the egalitarian solution proposed by Bogomolnaia and Moulin(2004). Katta and Sethuraman (2006) propose an algorithm to find the egalitarian solutionthrough solving a network flow problem. By iteratively applying the algorithm, they obtainan extension of PS to the full preference domain. Comparing with their algorithm, ours doesnot require any knowledge beyond elementary computation.On the strict preference domain, we extend FTTC to school choice with coarse priorities.At each step, among remaining students, only those of highest priority at each school canuse the seats of the school to trade with the others, and for fairness we let them use equalfractions of the seats of the school to trade with the others. We can extend this mechanism tothe full preference domain as we did in the FEE model. We omit the details. We are differentfrom Erdil and Ergin (2017) who focus on stable matchings in two-sided matching with weakpreferences on both sides. Because FTTC reduces to TTC in the housing market model, ourdefinition of FTTC also subsumes an extension of TTC to the full preference domain. Theextension resembles Jaramillo and Manjunath (2012) and Alcalde-Unzu and Molis (2011),and thus can preserve strategy-proofness of TTC if parameters in its definition are properlychosen. We also omit the details.The rest of the paper is organized as follows. Section 2 presents the FEE model. Section3 briefly revisits the definition of FTTC on the strict preference domain, and then presentsour definition of FTTC on the full preference domain. Section 4 shows that efficiency andfairness of FTTC are preserved on the full preference domain. Section 5 applies FTTC to thehouse allocation model. FTTC finds the egalitarian solution on the dichotomous preferencedomain, and extends PS to the full preference domain. Yu and Zhang (2020) use this rate-adjusting rule to obtain an extension of PS to the house allocationwith existing tenants model. Each agent regards each object as either acceptable or unacceptable, and regards the objects in the sameclass as indifferent. Fractional Endowment Exchange Model A fractional endowment exchange (FEE) problem is a four-tuple ( I, O, % I , ω ) in which • I is a finite set of agents; • O is a finite set of objects; • % I = { % i } i ∈ I is the preference profile of agents; • ω = ( ω i,o ) i ∈ I,o ∈ O is the endowment matrix.For each agent i , ω i = ( ω i,o ) o ∈ O denotes i ’s endowments, with ω i,o ∈ [0 , being the amount(probability share) of o ∈ O owned by i . Let q o = P i ∈ I ω i,o denote the total amount of o ∈ O in the market, which is an integer. Each agent demands one object and his total amountof endowments is no more than one; that is, P o ∈ O ω i,o ≤ . Each agent has a preferencerelation % i over objects. % i is complete and transitive, but needs not to be strict. Let ≻ i and ∼ i respectively denote the asymmetric and the symmetric components of % i . The housingmarket problem is a special case of the FEE model if | I | = | O | and ω is a permutationmatrix. The house allocation problem can be regarded as a special case of the FEE modelif | O | = | I | and ω i,o = 1 / | I | for all i ∈ I and all o ∈ O .A lottery is a vector l ∈ ℜ | O | + such that P o ∈ O l o ≤ . A lottery l weakly (first-order)stochastically dominates another lottery l ′ for agent i , denoted by l % sdi l ′ , if P o ′ % i o l o ′ ≥ P o ′ % i o l ′ o ′ for all o ∈ O . If the inequality is strict for some o , l strictly stochastically dominates l ′ , denoted by l ≻ sdi l ′ . We denote by l ∼ sdi l ′ if l % sdi l ′ and l ′ % sdi l .An assignment is a matrix p = ( p i,o ) i ∈ I,o ∈ O ∈ ℜ | I |×| O | + such that P i ∈ I p i,o ≤ P i ∈ I ω i,o forall o ∈ O and P o ∈ O p i,o ≤ for all i ∈ I . Each p i,o is the amount of o assigned to i . The rowvector p i = ( p i,o ) o ∈ O is the lottery assigned to i . If all elements of p are integers, p is a deter-ministic assignment. The Birkhoff-von Neumann theorem and its generalization (Birkhoff,1946; Von Neumann, 1953; Kojima and Manea, 2010) guarantee that every assignment is aconvex combination of deterministic assignments. An assignment p is ex-post efficient if itcan be written as a convex combination of Pareto efficient deterministic assignments. Anassignment p strictly stochastically dominates another assignment p ′ , denoted by p ≻ sdI p ′ , if p i % sdi p ′ i for all i and p j ≻ sdj p ′ j for some j . An assignment p is sd-efficient if it is neverstrictly stochastically dominated. It is individually rational (IR) if p i % sdi ω i for all i ∈ I . IRimplies that P o ∈ O p i,o = P o ∈ O ω i,o for all i ∈ I . If an agent has more endowments than his demand, when his demand is satisfied in our mechanisms, hisresidual endowments can be inherited by the remaining agents. For simplicity we do not discuss inheritance.
4e define four fairness axioms. From the weakest to the strongest, they are equal treat-ment of equals (ETE), equal-endowment no envy (EENE), bounded envy (BE), and envy-freeness (EF). ETE and EENE require fairness among “equal” agents. BE is proposed byYu and Zhang (2020) to require fairness among any agents. It requires that if an agent isenvied by another agent, then the envy is bounded by the former agent’s advantage in en-dowments. EF is the strongest axiom that eliminates envy between any two agents. It iscompatible with IR in special cases of the FEE model, but they are incompatible in generalcases.Formally, an assignment p satisfies • ETE if for all i, j ∈ I such that ω i = ω j and % i = % j , p i = p j ; • EENE if for all i, j ∈ I such that ω i = ω j , p i % sdi p j and p j % sdj p i ; • BE if for all i, j ∈ I , max o ∈ O (cid:2) P o ′ % i o p j,o ′ − P o ′ % i o p i,o ′ (cid:3) ≤ P o ∈ O : ω j,o >ω i,o (cid:0) ω j,o − ω i,o (cid:1) ; • EF if for all i, j ∈ I , p i % sdi p j and p j % sdj p i .We denote an FEE problem by its preference profile when the other elements are fixed.Let R denote the set of all complete and transitive preference relations. A mechanism ϕ finds an assignment ϕ ( % I ) for each % I ∈ R | I | . The lottery assigned to each i ∈ I in ϕ ( % I ) is denoted by ϕ i ( % I ) . A mechanism satisfies an efficiency or fairness axiom if its foundassignments satisfy the axiom.We say an agent i weakly manipulates a mechanism ϕ at % I by reporting % ′ i ∈ R\{ % i } if ϕ i ( % I ) % sdi ϕ i ( % ′ i , % − i ) . We say i strongly manipulates ϕ at % I by reporting % ′ i if ϕ i ( % ′ i , % − i ) ≻ sdi ϕ i ( % I ) . ϕ is (weakly) strategy-proof if it is never (strongly) manipulated. To understand our definition of FTTC on the full preference domain, we revisit the definitionon the strict preference domain. When preferences are strict, at each step of FTTC, eachremaining agent reports his unique favorite remaining object. A parameterized linear equa-tion system describes how agents trade endowments at each step. By solving the equations,we obtain the amount of favorite object each agent obtains and the amount of endowmentshe loses. Parameters in the equations control fairness.5e define some notations to describe the equations. These notations will also be usedin the next subsection. At the end of step d , let I ( d ) and O ( d ) denote the set of remainingagents and the set of remaining objects respectively ( I (0) = I and O (0) = O ); let ω ( d ) =( ω i,o ( d )) i ∈ I,o ∈ O denote the matrix of remaining endowments. At step d , each i ∈ I ( d − reports his favorite object among O ( d − , denoted by o i ( d ) . At step d , let x i ( d ) denote theamount of o i ( d ) assigned to i ∈ I ( d − , and let x o ( d ) denote the amount of o ∈ O ( d − assigned to all agents. So x o ( d ) is also the total amount of o lost by its owners from theirendowments at step d . We use a parameter λ i,o ( d ) denote the proportion of x o ( d ) that islost by i from his endowments. That is, i loses λ i,o ( d ) x o ( d ) of o at step d . We write allsuch parameters into a matrix λ ( d ) = (cid:0) λ i,o ( d ) (cid:1) i ∈ I ( d − ,o ∈ O ( d − , and call it ratio matrix . Thematrix controls how the owners of each object divide the right of using the object to tradewith the others at step d . For all i ∈ I ( d − and all o ∈ O ( d − , P i ∈ I ( d − λ i,o ( d ) = 1 and λ i,o ( d ) > only if ω i,o ( d − > . We use another parameter β i,o ( d ) to control the maximumamount of o ∈ O ( d − that each i ∈ I ( d − can lose at step d . So ≤ β i,o ( d ) ≤ ω i,o ( d − .We write them into another matrix β ( d ) = (cid:0) β i,o ( d ) (cid:1) i ∈ I ( d − ,o ∈ O ( d − , and call it quota matrix .At each step d , we solve the equations x o ( d ) = P i ∈ I ( d − o i ( d )= o x i ( d ) for all o ∈ O ( d − ,x i ( d ) = P o ∈ O ( d − λ io ( d ) x o ( d ) for all i ∈ I ( d − , (1)subject to the constraints λ i,o ( d ) x o ( d ) ≤ β i,o ( d ) for all i ∈ I ( d − and all o ∈ O ( d − . (2)The first equation of (1) is obtained by the definition of x , while the second equation of(1) describes balanced trade among agents. Denote the maximum solution to (1) subjectto (2) by x ∗ ( d ) . In Yu and Zhang (2020) we prove that, given λ ( d ) and β ( d ) , because thecoefficient matrix of (1) is stochastic (i.e., its every column sums to one), the solution x ∗ ( d ) at each step d exists. So FTTC is well-defined. The fact that agents obtain favorite objectsstep by step and they trade endowments in a balanced way straightforwardly implies thatFTTC is IR and sd-efficient. When λ ( d ) is properly chosen (as presented in Section 4),FTTC can satisfy any of ETE, EENE, and BF. With weak preferences, we can run FTTC after breaking preference ties. But as explainedin Introduction, any preference-independent tie-breaking rule can cause efficiency loss. Our6ethod is to let agents label some of their consumptions as endowments available for tradingwhen they find other available indifferent objects. The balanced trade condition ensures thatwhen they lose an amount of consumptions, they will be compensated by obtaining an equalamount of indifferent objects. We have briefly explain this method through a simple examplein Introduction. Below we use another example to explain it more clearly.
Example 1.
Consider three agents { , , } and three objects { a, b, c } . Agents have equalendowments (1 / a, / b, / c ) and the following preferences: % % % { a, b } a ab cc c b • Step one: Agent 1 points to a, b . Agents 2 and 3 point to a . Suppose agents obtainequal amounts of favorite objects and lose equal amounts of endowments; in particular, obtains equal amounts of a, b . So after this step, 1 obtains (1 / a, / b ) , 2 and 3 eachobtain / a , and each agent loses (1 / a, / b ) . Object a is used up. • Step two: Because 1 is indifferent between a and b , and b has not been used up, 1 labelshis consumption of a as an endowment available for trading. So 1 points to b and 2,3point to a . Suppose 2,3 obtain equal amounts of a . Then each of 2,3 obtains / a ,and 1 obtains / b . Note that 1 loses his consumption / a to obtain an additional / b . His net consumption amount is / − / / , which is equal to that of 2,3. • Step three: Because ’s consumption of a has been exhausted at step two, 1,2 point to b and 3 points to c . Suppose agents obtain equal amounts of objects. Then 1,2 eachobtain / b , and obtains / c . • Step four: All agents point to c , and each obtains / c .The mechanism finds the following assignment, which is the unique IR, sd-efficient andenvy-free assignment in this example: / b / a / a / c / b / c / c
7s mentioned in Introduction, after an agent labels his consumption of some object asan endowment available for trading, it may induce a chain of the other agents also to labeltheir consumptions as endowments available for trading. Every chain will look like i m +1 → o m → i m → · · · → o → i → o , where o is an object that has been exhausted in the trading process, while the other objectsin the chain have been exhausted. For each k = 1 , . . . , m , i k is indifferent between o k − and o k , and labels his consumption of o k as a new endowment. The last agent i m +1 strictlyprefers o m to all remaining endowments and most prefers o m among all new endowments.Finding such chains are crucial for maintaining sd-efficiency of FTTC. FTTC on the full preference domainNotations : O ( d ) , ω ( d ) , x i ( d ) , and x o ( d ) are defined as in Section 3.1. Let p ( d ) = ( p i,o ( d )) i ∈ I,o ∈ O denote the assignment found by the end of step d . Step d ≥ : Every step consists of three stages.1. Labeling • Round 1: If any i ∈ I is indifferent between any o ∈ O ( d − and any o ′ ∈ O \ O ( d − with p i,o ′ ( d − > , label o ′ as an endowment and let o ′ point to i .Denote the set of such i by L ( d − . For each i ∈ L ( d − , let ˜ O i ( d − denotethe set of objects labeled by i . Let ˜ O ( d −
1) = ∪ i ∈ L ( d − ˜ O i ( d − . • Round 2: If any i ∈ I \ L ( d − is indifferent between any o ∈ ˜ O ( d − and any o ′ ∈ O \ [ O ( d − ∪ ˜ O ( d − with p i,o ′ ( d − > , label o ′ as an endowment andlet o ′ point to i . Denote the set of such i by L ( d − . For each i ∈ L ( d − , let ˜ O i ( d − denote the set of objects labeled by i . Let ˜ O ( d −
1) = ∪ i ∈ L ( d − ˜ O i ( d − .... • Round n: If any i ∈ I \ ∪ n − k =1 L k ( d − is indifferent between any o ∈ ˜ O n − ( d − and any o ′ ∈ O \ [ O ( d − ∪ ˜ O ( d − ∪ · · · ∪ ˜ O n − ( d − with p i,o ′ ( d − > , label o ′ as an endowment and let o ′ point to i . Denote the set of such i by L n ( d − .For each i ∈ L n ( d − , let ˜ O i ( d − denote the set of objects labeled by i . Let ˜ O n ( d −
1) = ∪ i ∈ L n ( d − ˜ O i ( d − . 8ince there are finite agents and finite objects, the above procedure must stop in finiterounds. Suppose it stops in n rounds. Let L ( d −
1) = ∪ nk =1 L k ( d − , ˜ O ( d −
1) = ∪ i ∈ L ( d − ˜ O i ( d − , and O ( d −
1) = O ( d − ∪ ˜ O ( d − . So O ( d − is the set ofobjects that are available in the trading process at step d .2. Pointing
Define I ( d −
1) = L ( d − ∪ { i ∈ I : P o ∈ O ω i,o ( d − > } to be the set of activeagents. These agents can join the trading process at step d . • Round 1: For every i ∈ I ( d − , if i ’s favorite objects among O ( d − includeobjects from O ( d − , let i point to all of his favorite objects from O ( d − .Denote the set of such agents by P ( d ) . • Round 2: For every i ∈ I ( d − \ P ( d ) , if i ’s favorite objects among O ( d − include objects from ˜ O ( d − , let i point to all of his favorite objects from ˜ O ( d − . Denote by the set of such agents by P ( d ) .... • Round m: For every i ∈ I ( d − \ [ ∪ m − k =1 P k ( d )] , if i ’s favorite objects among O ( d − include objects from ˜ O m − ( d − , let i point to all of his favorite objects from ˜ O m − ( d − . Denote the set of such agents by P m ( d ) .Since there are finite agents and finite objects, the above procedure must stop in finiterounds. Suppose it stops in m rounds. Then it must be that m ≤ n + 1 . For every i ∈ I ( d − , let A i ( d ) denote the set of objects pointed by i .3. Trading
We choose a ratio matrix λ ( d ) = (cid:0) λ i,o ( d ) (cid:1) i ∈ I ( d − ,o ∈ O ( d − and a quota matrix β ( d ) = (cid:0) β i,o ( d ) (cid:1) i ∈ I ( d − ,o ∈ O ( d − as we did on the strict preference domain. The only differenceis that now the ratio matrix includes parameters for ˜ O ( d − . For all o ∈ ˜ O ( d − ,we require that P i ∈ I ( d − λ i,o ( d ) = 1 and λ i,o ( d ) > only if o ∈ ˜ O i ( d − .Because an agent may point to several objects, we introduce another nonnegativematrix γ ( d ) = (cid:0) γ i,o ( d ) (cid:1) i ∈ I ( d − ,o ∈ O ( d − to control how each agent i divides his demandamong the objects in A i ( d ) . We call γ ( d ) division matrix and require that, for all i ∈ I ( d − , P o ∈ O ( d − γ i,o ( d ) = 1 , and γ i,o ( d ) > only if o ∈ A i ( d ) . Every agent will appear in at most one round of the Labeling stage. x ∗ ( d ) = ( x ∗ a ( d )) a ∈ I ( d − ∪ O ( d − be the maximum solution to the equation system: x o ( d ) = P i ∈ I ( d − o ∈ A i ( d ) γ i,o ( d ) x i ( d ) for all o ∈ O ( d − ,x i ( d ) = P o ∈ O ( d − λ i,o ( d ) x o ( d ) for all i ∈ I ( d − , (3)subject to the constraints λ i,o ( d ) x o ( d ) ≤ β i,o ( d ) for all i ∈ I ( d − and all o ∈ O ( d − ,λ i,o ( d ) x o ( d ) ≤ p i,o ( d − for all i ∈ I ( d − and all o ∈ ˜ O ( d − . (4)For all i ∈ I ( d − and all o ∈ O , let ω i,o ( d ) = ω i,o ( d − − λ i,o ( d ) x o ( d ) if o ∈ O ( d − , otherwise,and p i,o ( d ) = p i,o ( d − − λ i,o ( d ) x o ( d ) if o ∈ ˜ O i ( d ) ,p i,o ( d −
1) + γ i,o ( d ) x i ( d ) if o ∈ A i ( d ) ,p i,o ( d − otherwise.For all i ∈ I \ I ( d − , ω i ( d ) = ω i ( d − and p i ( d ) = p i ( d − .Let O ( d ) = { o ∈ O ( d −
1) : P i ∈ I ω i,o ( d ) > } . If O ( d ) is empty, stop the algorithm.Otherwise, go to step d + 1 .Because the coefficient matrix of the equation system (3) is still stochastic, the maximumsolution x ∗ ( d ) at each step exists. By choosing different parameter values, we obtain differentFTTC mechanisms on the full preference domain.To facilitate our discussion in remaining sections, define x ci ( d ) = X o ∈ ˜ O ( d − λ i,o ( d ) x o ( d ) ,x ni ( d ) = X o ∈ O ( d − λ i,o ( d ) x o ( d ) . In words, x ci ( d ) denotes the amount of consumptions that i loses at step d , and x ni ( d ) denotesthe amount of i ’s net consumption at step d . So x i ( d ) = x ci ( d ) + x ni ( d ) . By losing x ci ( d ) ofconsumptions, i obtains an equal amount x ci ( d ) of indifferent objects. What matters for i ’swelfare is the net consumption x ni ( d ) . 10 Efficiency and Fairness
We show that IR, efficiency and fairness of FTTC on the strict preference domain remain tohold on the full preference domain.Recall that at each step d , O ( d −
1) = O ( d − ∪ ˜ O ( d − is the set of objects availablefor trading. We prove a lemma stating that O ( d − weakly shrinks in the procedure ofFTTC. It means that once an object becomes unavailable for trading at some step, it remainsunavailable at following steps. This feature is crucial for maintaining desirable properties ofFTTC on the full preference domain. Lemma 1.
For any step d ≥ , O ( d ) ⊆ O ( d − .Proof. By definition, O ( d ) ⊆ O ( d − ⊆ O ( d − . So we only need to prove that ˜ O ( d ) ⊆ O ( d − . We enumerate the objects in ˜ O ( d ) to prove this result. Define E ( d ) = O ( d − \ O ( d ) to be the set of objects that are exhausted in the trading process at step d . In the labelingstage of step d + 1 , we know that there exists n ∈ N such that ˜ O ( d ) = ∪ nk =1 ˜ O k ( d ) . Base step.
For every o ′ ∈ ˜ O ( d ) , if o ′ ∈ E ( d ) , then o ′ ∈ O ( d − ⊆ O ( d − . If o ′ / ∈ E ( d ) , then o ′ / ∈ O ( d − . So o ′ is exhausted before step d . The fact that o ′ ∈ ˜ O ( d ) means that some agent i is indifferent between o ′ and a distinct object o ∈ O ( d ) . Since o ∈ O ( d ) ⊆ O ( d − , i ’s consumption of o ′ must be labeled as available for trading at step d . So o ′ ∈ O ( d − . Thus, ˜ O ( d ) ⊆ O ( d − . Inductive step.
Suppose for all ℓ = 1 , . . . , k − , ˜ O ℓ ( d ) ⊆ O ( d − . For every o ′ ∈ ˜ O k ( d ) , if o ′ ∈ E ( d ) , then o ′ ∈ O ( d − ⊆ O ( d − . If o ′ / ∈ E ( d ) , then o ′ / ∈ O ( d − . So o ′ is exhaustedbefore step d . The fact that o ′ ∈ ˜ O k ( d ) means that some agent i is indifferent between o ′ and a distinct object o ∈ ˜ O k − ( d ) . Since o ∈ ˜ O k − ( d ) ⊆ O ( d − , i ’s consumption of o ′ mustbe labeled as available for trading at step d . So o ′ ∈ O ( d − . Thus, ˜ O k ( d ) ⊆ O ( d − .By induction, for all ≤ ℓ ≤ n , ˜ O ℓ ( d ) ⊆ O ( d − . So ˜ O ( d ) ⊆ O ( d − .We prove that FTTC remains to be IR and sd-efficient. Proposition 1.
FTTC on the full preference domain is individually rational and sd-efficient.Proof. (IR) At every step d , i ’s net consumption stochastically dominates the endowmentshe loses. So IR is obvious.(Sd-efficiency) Suppose for some preference profile, the assignment found by some FTTCis not sd-efficient. Then there must exist k ≥ agents who need not be distinct, denoted by i , i , . . . , i k , and k objects in the lotteries they obtain, denoted by o , o , . . . , o k , such that11f the k agents trade an amount of the k objects in their consumptions as indicated by thefollowing cycle, none of them becomes worse off and some becomes strictly better off: i → o → i → o → i → · · · → o k → i k → o → i . By trading the cycle, i obtains an amount of o , i obtains an amount of o , and so on.Without loss of generality, assume that i is strictly better off. This means that i strictlyprefers o to o . Suppose in the FTTC procedure, i starts consuming o at step d . Thenit must be that o ∈ O ( d − and o / ∈ O ( d − . Consider agent i . Assume that i startsconsuming o at step d ′ . There are two cases: • If i strictly prefers o to o , then it must be that o ∈ O ( d ′ − and o / ∈ O ( d ′ − .Since o / ∈ O ( d − , by Lemma 1, it must be that d ′ < d , and so o / ∈ O ( d − . • If i is indifferent between o and o , since o / ∈ O ( d − , it must be that o / ∈ O ( d − ;otherwise, given o ∈ O ( d − , i should label his consumption of o as available.So in any case we get o / ∈ O ( d − . By applying the above arguments inductively to theremaining agents and objects in the cycle, we get o / ∈ O ( d − , which is a contradiction.Yu and Zhang (2020) present conditions on λ ( d ) to satisfy various fairness axioms. Weshow that they still ensure fairness on the full preference domain. Definition 1.
An FTTC satisfies(1) stepwise equal treatment of equals (stepwise ETE) if at every step d , ω i ( d −
1) = ω j ( d − , ˜ O i ( d −
1) = ˜ O j ( d − and A i ( d ) = A j ( d ) = ⇒ λ i ( d ) = λ j ( d ) .(2) stepwise equal-endowment equal treatment (stepwise EEET) if at every step d , ω i ( d −
1) = ω j ( d −
1) = ⇒ λ i,o ( d ) = λ j,o ( d ) for all o ∈ O ( d − .(3) bounded advantage if at every step d , ω i,o ( d − ≥ ω j,o ( d −
1) = ⇒ λ i,o ( d ) ≥ λ j,o ( d ) and ω i,o ( d ) ≥ ω j,o ( d ) . Proposition 2. (1) An FTTC satisfying stepwise ETE satisfies ETE;(2) An FTTC satisfying stepwise EEET satisfies EENE;(3) An FTTC satisfying bounded advantage satisfies BE. roof. (1) For any two agents i, j with ω i = ω j and % i = % j , stepwise ETE implies that atevery step, i and j label the same set of consumptions as available, point to the same set offavorite objects, and obtain equal consumptions. So ETE is satisfied.(2) For any two agents i, j , at any step d , if ω i ( d −
1) = ω j ( d − , by stepwise EEET, λ i,o ( d ) = λ j,o ( d ) for all o ∈ O ( d − . It implies that ω i ( d ) = ω j ( d ) and x ni ( d ) = x nj ( d ) . So i, j have equal amounts of net consumptions at step d and their remaining endowments arestill equal after step d . Now if i, j have equal endowments (i.e., ω i = ω j ), then i, j must haveequal amounts of net consumptions throughout the procedure of FTTC. Because at everystep i, j point to their respective favorite objects, in the found assignment there must be noenvy between them.(3) Let p denote the assignment found by any FTTC satisfying bounded advantage. Forany distinct i, j ∈ I , let o ∗ be the solution to max o ∈ O (cid:2) P o ′ % i o p j,o ′ − P o ′ % i o p i,o ′ (cid:3) . Let d bethe earliest step after which all objects in { o ∈ O : o % i o ∗ } become unavailable. Thatis, { o ∈ O : o % i o ∗ } ∩ O ( d ) = ∅ and { o ∈ O : o % i o ∗ } ∩ O ( d − = ∅ . By Lemma 1, { o ∈ O : o % i o ∗ } ∩ O ( d ′ ) = ∅ for all d ′ ≥ d . So, X o % i o ∗ p i,o = d X d ′ =1 x ni ( d ′ ) = X o ∈ O (cid:0) ω i,o − ω i,o ( d ) (cid:1) , X o % i o ∗ p j,o ≤ d X d ′ =1 x nj ( d ′ ) = X o ∈ O (cid:0) ω j,o − ω j,o ( d ) (cid:1) . For all o ∈ O such that ω i,o ≥ ω j,o , bounded advantage implies that for all ≤ d ′ ≤ d , ω i,o ( d ′ ) ≥ ω j,o ( d ′ ) and λ i,o ( d ′ ) ≥ λ j,o ( d ′ ) . So, ω j,o − ω j,o ( d ) = d X d ′ =1 λ j,o ( d ′ ) x o ( d ′ ) ≤ d X d ′ =1 λ i,o ( d ′ ) x o ( d ′ ) = ω i,o − ω i,o ( d ) , or equivalently, (cid:0) ω j,o − ω j,o ( d ) (cid:1) − (cid:0) ω i,o − ω i,o ( d ) (cid:1) ≤ . For all o ∈ O such that ω i,o < ω j,o , bounded advantage implies that for all ≤ d ′ ≤ d , ω i,o ( d ′ ) ≤ ω j,o ( d ′ ) and λ i,o ( d ′ ) ≤ λ j,o ( d ′ ) . In particular, ω i,o ( d ) ≤ ω j,o ( d ) . So, (cid:0) ω j,o − ω j,o ( d ) (cid:1) − (cid:0) ω i,o − ω i,o ( d ) (cid:1) ≤ ω j,o − ω i,o . X o % i o ∗ p j,o − X o % i o ∗ p i,o ≤ X o ∈ O (cid:20)(cid:0) ω j,o − ω j,o ( d ) (cid:1) − (cid:0) ω i,o − ω i,o ( d ) (cid:1)(cid:21) = X o ∈ O : ω i,o ≥ ω j,o (cid:20)(cid:0) ω j,o − ω j,o ( d ) (cid:1) − (cid:0) ω i,o − ω i,o ( d ) (cid:1)(cid:21) + X o ∈ O : ω i,o <ω j,o (cid:20)(cid:0) ω j,o − ω j,o ( d ) (cid:1) − (cid:0) ω i,o − ω i,o ( d ) (cid:1)(cid:21) ≤ X o ∈ O : ω i,o <ω j,o (cid:0) ω j,o − ω i,o (cid:1) . So p satisfies BE.As examples, Yu and Zhang (2020) present three fair FTTC and connect their fairnessmotivations to classical solution rules in the bankruptcy problem. All of the three FTTCsatisfy bounded advantage, and thus BE. One of them is called equal-FTTC and denoted by T e . Its idea is to let the remaining owners of each object at each step use equal amounts ofthe object to trade with the others. Formally, it uses the following parameters: λ ei,o ( d ) = | j ∈ I ( d −
1) : ω j,o ( d − > | if ω i,o ( d − > , if ω i,o ( d −
1) = 0 ,β ei,o ( d ) = ω i,o ( d − . Another is called proportional-FTTC and denoted by T p . Its idea is to let the remainingowners of each object at each step use amounts proportional to their endowments of theobject to trade with the others. Formally, it uses the following parameters: λ pi,o ( d ) = ω i,o ( d − P j ∈ I ( d − ω j,o ( d − ,β pi,o ( d ) = ω i,o ( d − . The third FTTC, omitted here, can be found in our other paper. To extend them tothe full preference domain, we can choose the parameters λ ( d ) = (cid:0) λ i,o ( d ) (cid:1) i ∈ I ( d − ,o ∈ ˜ O ( d − and γ ( d ) arbitrarily. The extensions will still satisfy bounded advantage. Because of theirintuitive fairness, they are appealing candidates in applications when market designers wantto choose an FTTC. At each step of the third FTTC, among the remaining owners of each object only those who own themost amount of object can use the object to trade with the others, and the amount they can use is no morethan the difference between the most amount and the second most amount. House allocation
In the house allocation model, a number of objects O are to be assigned to an equal num-ber of agents I . In Yu and Zhang (2020) we regard the model as a special case of FEE inwhich agents own equal divisions of all objects. We have explained that, on the strict prefer-ence domain, every FTTC coincides with a simultaneous eating algorithm (SEA) defined byBogomolnaia and Moulin (2001), and every FTTC satisfying stepwise EEET coincides withPS. This means that in the house allocation model, every FTTC satisfying stepwise EEETdefined in this paper (e.g., T e and T p ) is an extension of PS to the full preference domain.Because agents have equal endowments, such FTTC satisfies envy-freeness. Proposition 3.
In the house allocation model, every FTTC satisfying stepwise EEET is anextension of PS to the full preference domain. The extension is sd-efficient and envy-free.
Actually, every FTTC satisfying stepwise EEET can be described as an SEA that deviatesfrom PS in two respects. First, agents label their consumptions as available for others toconsume when they find indifferent available objects in the market. Second, when an agent’slabeled consumption is being consumed by the others, his eating rate is instantly increased,following the rule we call “ you request my house - I get your rate ”. Formally, at any time t ∈ [0 , , any agent i ’s eating rate is defined to be s i ( t ) = 1 + X o ∈ ˜ O i ( t ) (cid:2) λ i,o ( t ) X j ∈ I : o ∈ A j ( t ) γ j,o ( t ) s j ( t ) (cid:3) . In words, i ’s eating rate is increased by an amount that is equal to the total rate at whichhis labeled consumptions are being consumed by the others.The dichotomous preference domain is an extreme case of weak preferences. On suchdomain, all Pareto efficient deterministic assignments assign the same number of objectsto agents, and sd-efficiency coincides with ex-post efficiency for random assignments. Eachagent’s welfare in an assignment is simply measured by the amount of acceptable objects heobtains. Bogomolnaia and Moulin (2004) propose an efficient and fair welfare distributioncalled egalitarian solution . Its idea is to maximize agents’ total welfare and at the sametime equalize their welfare as much as possible. Katta and Sethuraman (2006) show thatthe egalitarian solution can be found by an algorithm that computes a lexicographicallyoptimal flow in a network. By iteratively applying the algorithm, Katta and Sethuramanpropose an extension of PS to the full preference domain. In the next subsection we provethat any FTTC satisfying stepwise EEET finds the egalitarian solution on the dichotomous15reference domain. This clarifies the relation between our extension of PS with Katta andSethuraman’s. The merit of our extension is that it remains to have an SEA description,and it does not require any knowledge beyond elementary computation. For every i ∈ I , let C i denote the set of i ’s acceptable objects. For every nonempty Y ⊆ I and nonempty O ′ ⊂ O , define Γ( Y, O ′ ) = (cid:0) ∪ i ∈ Y C i (cid:1) ∩ O ′ to be the set of objects from O ′ that are acceptable to at least one agent in Y . Givenany economy, the Gallai-Edmonds Decomposition Lemma (Bogomolnaia and Moulin, 2004)states that it can be decomposed into three subproblems. In the first subproblem there is aperfect match between objects and agents so that each agent obtains an acceptable object.In the second subproblem there are oversupply of acceptable objects for any subset of agents,so every agent can also obtain an acceptable object. But in the third subproblem there isshortage of acceptable objects. BM’s egalitarian solution is proposed to solve the thirdsubproblem. So we restrict attention to economies belonging to the third type. We assumethat every object is acceptable to at least one agent (that is, Γ( I, O ) = O ), and for everynonempty O ′ ⊂ O , |{ i ∈ I : C i ∩ O ′ = ∅}| > | O ′ | .The egalitarian solution is defined through finding a sequence of bottleneck sets of agents.The first bottleneck set is defined to be X ∗ = arg min Y ⊆ I | Γ( Y, O ) || Y | . (5)When there are multiple solutions to the above problem, let X ∗ be the solution of largestcardinality. Γ( X ∗ , O ) are assigned to X ∗ fairly such that each i ∈ X ∗ obtains | Γ( X ∗ ,O ) || X ∗ | ofacceptable objects.The second bottleneck set is found in the same way as (5) among the remaining agents Z = I \ X ∗ and remaining objects P = O \ Γ( X ∗ , O ) . In general, the k -th bottleneck set isdefined to be X ∗ k = arg min Y ⊆ Z k − | Γ( Y, P k − ) || Y | . When both sides are agents as in the model of Bogomolnaia and Moulin (2004), the second and the thirdsubproblems are symmetric. The union of two solutions is still a solution. So X ∗ is unique. X ∗ k be the solution of largest cardinality. In the egal-itarian solution, every i ∈ X ∗ k obtains | Γ( X ∗ k ,P k − ) || X ∗ | of acceptable objects. Let X ∗ , X ∗ , . . . , X ∗ m be the sequence of bottleneck sets.We prove that the above sequence of bottleneck sets is implicitly found in the procedureof any FTTC satisfying EEET. Recall that O ( d ) = O ( d − ∪ ˜ O ( d ) and O (0) = O . ByLemma 1, O ( d ) ⊂ O ( d − for every step d . Let d , d , . . . , d n be the sequence of steps inthe procedure of FTTC such that, for all k = 1 , . . . , n , O ( d k − \ O ( d k ) = ∅ .That is, d is the first step after which some objects (i.e., O \ O ( d ) ) become unavailable fortrading at following steps. The other steps are interpreted similarly. d n is the last step afterwhich all objects are unavailable. We prove that each O ( d k − \ O ( d k ) is a bottleneck setdefined by Bogomolnaia and Moulin (2004). Lemma 2. m = n , and for all k = 1 , . . . , nO ( d k − \ O ( d k ) = Γ( X ∗ k , P ∗ k − ) , where P ∗ = O and P ∗ k = P ∗ k − \ Γ( X ∗ k , P ∗ k − ) = O ( d k ) .Proof. We prove the lemma by induction.
Base case.
We first prove that O ( d − \ O ( d ) = Γ( X ∗ , O ) . Let p denote the assignmentfound by any FTTC satisfying EEET, and p ( d ) denote the assignment found by the end ofstep d . Define X = { i ∈ I : p i,o ( d ) > for some o ∈ O ( d − \ O ( d ) } . Because all objectsin O ( d − \ O ( d ) are assigned to agents at the end of step d and they are no longer availableafter step d , it must be that O ( d − \ O ( d ) ⊂ Γ( X , O ) . Suppose there exist i ∈ X and o ∈ C i such that o / ∈ O ( d − \ O ( d ) . Then, o ∈ O ( d ) . But it implies that i should label hisconsumption of the objects in O ( d − \ O ( d ) as available at the beginning of step d + 1 ,which is a contradiction. So, O ( d − \ O ( d ) = Γ( X , O ) . Define t = | Γ( X , O ) || X | . Because no objects become unavailable before step d , no agent changes the objects hepoints to before step d . Then stepwise EEET implies that agents obtain equal amounts of17et consumptions at every step. So at the end of step d , every agent must obtain a totalamount t of acceptable objects.For any nonempty Y ⊂ X , Γ( Y, O ) ⊆ Γ( X , O ) = O ( d − \ O ( d ) . So, | Γ( Y, O ) || Y | = t . For any nonempty Y ⊂ I such that Y \ X = ∅ , it must be that for every j ∈ Y \ X , C j ∩ O ( d ) = ∅ , since otherwise j ∈ X . Because every i ∈ Y obtains an amount t ofacceptable objects at the end of step d and every j ∈ Y \ X has acceptable objects that arestill available after step d , it must be that | Γ( Y, O ) || Y | > t . So X = arg min Y ⊂ I | Γ( Y, O ) || Y | , and X is the solution of largest cardinality. It means that X = X ∗ . Induction step.
Suppose for all k = 2 , . . . , ℓ ( ℓ ≤ n − ), O ( d k − \ O ( d k ) = Γ( X ∗ k , P ∗ k − ) .We prove that O ( d ℓ +1 − \ O ( d ℓ +1 ) = Γ( X ∗ ℓ +1 , P ∗ ℓ ) where P ∗ ℓ = O ( d ℓ ) = O ( d ℓ +1 − . Define Z ℓ = I \ ( ∪ ℓk =1 X ∗ k ) , and X ℓ +1 = { i ∈ I : p i,o ( d ℓ +1 ) > for some o ∈ O ( d ℓ +1 − \ O ( d ℓ +1 ) } .Because all objects in O \ O ( d ℓ +1 − are assigned to the agents in ∪ ℓk =1 X ∗ k , it must be that X ℓ +1 ⊂ Z ℓ and O ( d ℓ +1 − \ O ( d ℓ +1 ) ⊂ Γ( X ℓ +1 , O ( d ℓ +1 − .Suppose there exist i ∈ X ℓ +1 and o ∈ C i such that o ∈ O ( d ℓ +1 ) . Then it implies that i should label his consumption of the objects in O ( d ℓ +1 − \ O ( d ℓ +1 ) as available at thebeginning of step d ℓ +1 + 1 , which is a contradiction. So O ( d ℓ +1 − \ O ( d ℓ +1 ) = Γ( X ℓ +1 , O ( d ℓ +1 − . Then we can use similar arguments as in the base case to prove that X ℓ +1 = arg min Y ⊂ Z ℓ | Γ( Y, O ( d ℓ +1 − || Y | , and X ℓ +1 is the solution of largest cardinality. It means that X ℓ +1 = X ∗ ℓ +1 .With Lemma 2, the following proposition is immediate. Proposition 4.
On the dichotomous preference domain, any FTTC satisfying stepwiseEEET finds the egalitarian solution of Bogomolnaia and Moulin (2004). Bogomolnaia and Moulin (2001) shows the sd-inefficiency of the Random Priority mech-anism (RP). On the dichotomous preference domain, RP will proceed as follows. It firstgenerates an ordering of agents uniformly at random, and then lets agents in the orderingsequentially choose favorite assignments. Every agent chooses a set of favorite assignmentsfrom those being chosen by previous agents. Because sd-efficiency coincides with ex-postefficiency, RP becomes desirable: It is sd-efficient and strategy-proof, and seems to fair be-cause the ordering of agents is uniformly random. However, noting that this randomizationis preference-independent, when it interacts with preferences through letting agents pickingobjects (assignments), the fairness of the final assignment becomes not transparent. This isalso true for TTC with randomized endowments or randomized priorities. Example 2 showsthat RP does not find the egalitarian solution on the dichotomous preference domain.
Example 2.
Consider five agents { , , , , } and three objects { o , o , o } . Agents’ di-chotomous preferences are shown by letting every agent point to all of his acceptable objectsin the following graph. o o o Any FTTC satisfying stepwise EEET finds an egalitarian assignment: / o / o / o / o / o / o But RP finds the following assignment: / o / o / o / o / o / o / o Comparing with the RP assignment, the egalitarian assignment equalizes agents’ welfare asmuch as possible. Bogomolnaia and Moulin (2004) prove that any such mechanism is group strategy-proof, meaning thatno group of agents can jointly manipulate the mechanism. eferences Abdulkadiro ˘glu, A. and T. S ¨onmez (1998): “Random serial dictatorship and the corefrom random endowments in house allocation problems,”
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