Fractional Top Trading Cycle

Transcription

1 Fractional Top Trading Cycle Jingsheng Yu Jun Zhang April 21, 2018 Abstract We extend Top Trading Cycle to two random assignment models: the fractional endowment exchange model and the weak-priority-based assignment model. The mechanism is individually rational, ordinally efficient and satisfies desirable fairness property. At every step, it lets agents point to most preferred objects and objects point to all of their owners. Within each absorbing set in the generated network, agents trade objects among themselves according to a closed Leontief input-out model. Absorbing sets can be regarded as the extension of the cycles in TTC. Keywords: Top Trading Cycle, random assignment, school choice JEL Classification: C71, C78, D71 Yu: School of Economics, Southwestern University of Finance and Economics, Chengdu, China; yujs@swufe.edu.cn. Zhang: Institute for Social and Economic Research, Nanjing Audit University, Nanjing, China; zhangjun404@gmail.com. Acknowledgments are to be added. 1

2 1 Introduction Shapley and Scarf (1974) propose the Top Trading Cycle (TTC) mechanism to solve the endowment exchange (EE) problem (also known as housing market problem). In the problem finite agents bring their distinct endowments to the market and want to trade with each other. TTC is a desirable algorithm to guide the trade: at every step, TTC first lets every remaining agent point to his most preferred remaining object and every remaining object point to its owner, then trades every cycle by letting agents in the cycle obtain the objects they point to. As an algorithm, it can be run by a simple computer code. As a function from the set of problems to the set of solutions, TTC finds the unique core assignment, and is the unique mechanism that satisfies individual rationality, strategy-proofness, and Pareto efficiency (Ma, 1994). These merits help TTC become one of the few important algorithms in the market design literature. Many papers have extended TTC to other models including, to name a few, on-campus housing allocation (Abdulkadiroğlu and Sönmez, 1999), school choice (Abdulkadiroğlu and Sönmez, 2003; Abdulkadiroǧlu et al., 2017), kidney exchange (Roth et al., 2004), and tuition and worker exchange (Dur and Ünver, 2015). In market design there is also a growing literature on random assignment problems. In a random assignment, agents obtain lotteries that specify the probabilities of obtaining each object. Fairness is the main motivation for the usage of random assignments. Important mechanisms in the literature include the Random Priority (RP) mechanism proposed by Abdulkadiroğlu and Sönmez (1998), the Probabilistic Serial (PS) mechanism proposed by Bogomolnaia and Moulin (2001), and the Fractional Deferred Acceptance (FDA) mechanism proposed by Kesten and Ünver (2015). Because of the good properties of TTC in deterministic environments, it is natural to ask whether this desirable algorithm can be used to solve random assignment problems. A proper extension of TTC remains unknown in the literature. Our contribution is answering this question. We study the fractional endowment exchange (FEE) problem, in which each agent owns a fractional amount of each object, but the total amount does not exceed one. It is a natural extension of the EE problem. Ideally, a TTC-like mechanism should also have discrete steps. At every step, every remaining agent points to his most preferred remaining object and every remaining object points to all of its remaining owners, and the mechanism specifies how to trade the generated network. Because an agent may own 2

3 fractions of multiple objects and multiple agents may own fractions of the same object, in the generated network there are often many cycles, and some cycles intersect with the others. This is different from TTC in which cycles are always disjoint. If we naively mimic the procedure of TTC, we need to propose a criterion to select cycles from the generated network to trade, and answer how much of objects is traded in each selected cycle. These questions are not easy to answer if we take fairness into account, which is a desideratum for random assignment mechanisms. Intuitively, there are two respects of fairness. First, if multiple agents point to the same object and each is involved in a cycle, a fair algorithm should give each agent a positive probability of obtaining the object. Second, if some amount of an object is traded at a step, then a fair algorithm should give each of its multiple owners at the step a positive probability of giving up his endowment of the object to obtain his preferred object. We illustrate these points by the following example. Example 1. Consider a problem that contains four agents {1, 2, 3, 4}, and three objects {a, b, c}. Agent 1 owns a and most prefers a. Agent 2 owns b and prefers a to c and c to b. Agent 3 owns 1/4b and most prefers c. Agent 4 owns 1/4c and most prefers b. The generated networks of a TTC-like mechanism are shown in Figure 1, in which the weight of an edge from an object to an agent denotes the amount of the object owned by the agent. a b 1/4 b 1/ /4 1/4 c Step one c Step two Figure 1: Generated networks of TTC-like mechanism. At step one, there are two cycles: a 1 a and b 3 c 4 b. If we trade both cycles simultaneously, then 2 would keep his b. This assignment is not fair for 2 since he prefers c to b and owns more fraction of b than 3, but he has no chance to 3

4 trade b with 4 to obtain his endowment of c. A fair algorithm should observe that in any case, 1 must obtain his endowment a. Then 2 will point to c and form a cycle with b. Therefore, only the first cycle should be traded. At step two, there are also two cycles: b 2 c 4 b and b 3 c 4 b. A fair algorithm should trade both cycles. Obviously, since 4 owns 1/4c, at most 1/4 of objects can be traded in total in both cycles. The question is how to divide the 1/4 quota between the two cycles. Although 2 owns more fraction of b than 3, 3 owns more than 1/8b so that equally diving the 1/4 quota between the two cycles is possible. Therefore, a fair algorithm should let 2 and 3 each lose 1/8b and obtain 1/8c. To better illustrate fairness, consider another case that 4 owns 2/3b. In this case, an efficient algorithm must let 2 obtain more fraction of c than 3. But a fair algorithm should let 2 and 3 have equal chance to use their respective endowments of 1/4b to trade with 4, to obtain 1/4c, then let 2 use his remaining endowment of b to trade with 4. In other words, the advantage of 2 should appear only when 4 has used up his endowment. Our Fractional TTC (FTTC) algorithm is not based on any criterion of selecting cycles. Instead, we find that trading endowments in the FEE problem implies that there is a natural extension of cycle called absorbing set, such that agents in an absorbing set must trade their endowments in the absorbing set only among themselves. Formally, an absorbing set is a subset of nodes in a directed network such that there is a path from every node to every other node within the set, but there is no path from any node in the set to any node outside of the set. In other words, an absorbing set is connected inside but isolated from the outside. So if an object is in an absorbing set, all of its owners are in the absorbing set; if an agent is in an absorbing set, his most preferred object is in the absorbing set. Remember that trading endowments means that an agent must obtain an amount of objects by losing an equal amount of endowments. This balance condition still holds for any group of agents. To see why an absorbing set has to trade endowments within itself, suppose some outside agent obtains a positive amount of some object in the absorbing set, since the agents in the absorbing set do not point to any outside object, the balance condition must fail to hold within the absorbing set. Therefore, at every step of our FTTC, we identify (disjoint) absorbing sets and trade each. The remaining question is how to trade the endowments in each absorbing set in a fair way. FTTC adopts a strong fair requirement in its procedure: if an amount of an object 4

5 is exhausted at a step, then each of its owners at the step should lose an equal amount of his endowment of the object. This requirement pins down FTTC. Specifically, we use an equation system to characterize this requirement. Interestingly, the equation system is an instance of the closed Leontief input-output model, which is a classical model developed by Wassily Leontief (Leontief, 1941) to provide a simple general equilibrium view of an economy. By the classical literature, the equation system has a unique solution, which tells us how each absorbing set is traded. If we run FTTC in Example 1, at step one, the cycle a 1 a is the only absorbing set. So FTTC trades it by letting 1 obtain a. At step two, FTTC lets 2 and 3 each lose 1/8b and obtain 1/8c. In the another case that 4 owns 2/3b, FTTC lets 2 and 3 each lose 1/4b and obtain 1/4c at step two, and lets 2 use his remaining endowment to trade with 4 at step three. To understand the properties of FTTC, in Section 4 we introduce an abstract procedure to formalize the notion of TTC-like mechanism. The assignment of the procedure is individually rational and ordinally efficient. By introducing some requirements on the procedure, we clarify how the fairness of the procedure is related to the fairness or other property of the assignment. If the procedure satisfies weak stepwise equal treatment of equals, which says that agents who are equal before a step obtain equal amounts of their common preferred object at the step, then it may not produce a weak core assignment. An implication is that FTTC may not produce a weak core assignment, in contrast with the core property of TTC in deterministic environment. If the procedure satisfies equalendowment equal treatment requirement on the procedure, which says that agents who own equal endowments before a step obtain equal amounts of their respective preferred objects at the step, its assignment must satisfy equal-endowment no envy (EENE). If the procedure satisfies equal-endowment-set equal treatment (EESET), which says that any two agents with an equal set of remaining endowments at a step obtain equal amounts of their respective preferred objects at the step, and an invariance property, then it is FTTC. In special class of FEE problems, FTTC satisfies a stronger fairness property than EENE. In Section 5 we study the incentive property of FTTC. Not surprisingly, FTTC is not strategy-proof. But we show FTTC is not weakly strategy-proof. This is caused by a fundamental incompatibility between individual rationality, ordinal efficiency, and weak strategy-proofness. Motivated by the success of some manipulable mechanisms that be- 5

6 come strategy-proof in large markets (Che and Kojima, 2010; Kojima and Manea, 2010; Kesten and Ünver, 2015; He et al., 2017), we also show that FTTC becomes asymptotically strategy-proof when the market is sufficiently large and preferences of agents are diverse. School choice is an important application field of TTC. Abdulkadiroğlu and Sönmez (2003) extend TTC to school choice by assuming that the priority rankings used by schools are strict. But in practice, schools often use coarse rankings of students. In Section 6 we extend FTTC to the weak-priority-based assignment problem. Our idea is regarding every remaining object as equally owned by the remaining agents in the highest priority tier of the object. Ordinal efficiency and fairness properties can be extended to this model. We also show that FTTC degenerates to existing mechanisms when the model degenerates to special cases. A remarkable observation is that FTTC coincides with PS in the house allocation problem. This observation provides a simple explanation of the connection between PS and TTC that first pointed out by Kesten (2009). To our best knowledge, Aziz (2015) is the only other paper that has studied the extension of TTC to FEE. In Section 7 we discuss his mechanism in detail. Simply speaking, his mechanism trades the generated network at every step by selecting a subset of cycles to trade. The mechanism is not fair since it violates the basic fairness property equal treatment of equals. We have some discussions in Section 8 and conclude in Section 9. The appendix contains proofs. Related literature Athanassoglou and Sethuraman (2011) is the first paper that studies the FEE problem. Their mechanism is an extension of the P S IR mechanism of Yılmaz (2010), which is a further extension of PS to the house allocation problem with existing tenants. P S IR minimizes the deviation from PS under the individuality rationality constraint, and the mechanism of Athanassoglou and Sethuraman (2011) maintains this feature in the FEE problem. The mechanism does not satisfy EENE. It satisfies a different fairness notion called no justified envy (NJE). Actually, the authors prove that if a mechanism satisfies ordinal efficiency and individual rationality, it cannot satisfy NJE and EENE simultaneously. The authors pose the existence of a mechanism satisfying ordinal efficiency, individual rationality and EENE, and a generalization of TTC to solve the FEE problem as two open questions. Our FTTC answers both questions. Zhang 6

7 (2017a) proposes another extension of PS, denoted by P S E, to the house allocation problem with existing tenants. It is equivalent to regarding social endowments as equally owned by all agents and letting agents trade endowments in the spirit of TTC. We show that FTTC degenerates to P S E in the special model. Echenique et al. (2018) study the FEE problem by assuming that agents have cardinal utilities. They propose two new fairness notions and prove the existence of assignments that satisfy individual rationality, Pareto efficiency, and the new fairness notions. Their first fairness notion is an extension of NJE stated above. In the second, they first prove that the non-existence of Walrasian equilibrium with fractional endowments, as first pointed out by Hylland and Zeckhauser (1979), can be solved by adding a fractional exogenous budget. Then they develop a fairness notion based on the market values of agents endowments, so that if one agent envies the other, then the market value of the other agent s endowments must be higher. Their methodology and results are rather different from ours in this paper. Kesten and Ünver (2015) study the weak-priority-based assignment problem in the context of school choice. Their contribution is adapting the stability notion to the model and proposing the Fractional Deferred Acceptance (and Trading) mechanism. DA and TTC are two leading mechanisms for deterministic assignments in the literature. Therefore, our paper is another one that extends popular mechanisms from deterministic setting to random setting. Because of stability constraint, FDA and FDAT usually have efficiency loss relative to FTTC. This is similar to the efficiency loss of DA relative to TTC. An interesting difference between FTTC and FDA(T) is that when all agents have equal priorities at all objects, FTTC degenerates to PS, but neither FDA nor FDAT degenerate to PS. He et al. (2017) also study the weak-priority-based assignment problem by assuming that agents have cardinal utilities. They extend the Walrasian equilibrium mechanism proposed by Hylland and Zeckhauser (1979) by allowing prices to be priority-specific. The mechanism satisfies an ex-ante stability-like fairness notion. So it is different from FTTC that lets agents trade priorities. There are many other papers that characterize or extend TTC. To name a few, Morrill (2013); Dur (2013); Fujinaka and Wakayama (2017) respectively provide new characterizations of TTC by using axioms different from those of Ma (1994). Morrill (2015); 7

8 Hakimov and Kesten (2017) respectively propose adaptations of TTC to improve its equity property without losing its strategy-proofness. Pápai (2000) proposes an extension of TTC called Hierarchical Exchange Rule and provides a characterization. Pycia and Ünver (2017) further extend it to a new class of mechanisms called Trading Cycles. In the classical paper Shapley and Scarf (1974), the authors observe that the outcome of TTC in the EE problem is a competitive equilibrium allocation when the prices of objects are chosen according to the ordering of cycles being traded. Dur and Morrill (2017) extend this observation to the school choice setting. Recently, Leshno and Lo (2017) provide a new interpretation of TTC in school choice, which makes the role of priorities and their relation with assignments more clear. 2 Definitions 2.1 Model of Fractional Endowment Exchange A fractional endowment exchange (FEE) problem is a tuple (O, Q, I, I, ω) where O is a finite set of objects; Q = {q o } o O is the quota vector of objects, where q o Z ++ is the quota of o O; I is a finite set of agents. Every agent demands one object; I = { i } i I is the preference profile of agents, where i is the strict preference relation of i I over O, with the weak relation denoted by i. Let P be the set of all strict preference relations; ω = {ω io } i I,o O is the endowment matrix, where ω io [0, 1] is the amount (probability) of o O owned by i. The row vector ω i = (ω io ) o O is the endowment of i. We require that o O ω io 1 for all i I and i I ω io = q o for all o O. Agent i is an owner of object o if ω io > 0. When some elements of an FEE problem are known, we denote the problem by the remaining elements. We often denote a problem simply by ( I, ω), or by ( I ). Let M be the set of all FEE problems. 8

9 2.2 Random Assignment and Mechanism A random assignment is a matrix p = (p io ) i I,o O such that p io [0, 1], i I p io q o for all o O and o O p io 1 for all i I. Here p io is the amount (probability) of o obtained by i. The row vector p i = (p io ) o O is the lottery of i in p. If all elements of p are either zero or one, then p is a deterministic assignment. The Birkhoff-von Neumann theorem (Birkhoff, 1946; Von Neumann, 1953) states that every random assignment is a convex combination of deterministic assignments. From now on we call random assignments simply assignments as long as there is no confusion. We define supp(p i ) = {o O : p io > 0} to be the support of p i, and define p i = o O p io to be the total amount of objects in p i. A lottery p i first-order stochastically dominates another lottery p i for agent i, denoted by p i sd( i )p i, if o i o p io o i o p io for all o O. If the inequality holds strictly for some o, we say p i strictly dominates p i, denoted by p i ssd( i ) p i. An assignment p strictly dominates another assignment p, denoted by p ssd( I ) p, if p i sd( i )p i for all i and p j ssd( j ) p j for some j. An assignment p is ordinally efficient if it is not strictly dominated by any other assignment; is non-wasteful if for any i and any two objects o i o, if p io > 0, then j I p jo = q o ; is individually rational if p isd( i )ω i for all i. Individual rationality implies that p i = ω i. 1 An assignment p satisfies equal treatment of equals (ETE) if for any distinct i, j I such that ω i = ω j and i = j, we have p i = p j. That is, if two agents have equal endowments and equal preferences, then they obtain equal lotteries. An assignment p satisfies equal-endowment no envy (EENE) if for any distinct i, j I such that ω i = ω j, we have p i sd( i )p j and p j sd( j )p i. A subset of agents I I blocks an assignment p if by assigning their endowments only among themselves, every agent in I is weakly better off and some agent is strictly better off. If every agent in I is strictly better off, then I strongly blocks p. Formally, I I blocks p if there exists an assignment p such that i I p i = i I ω i, p isd( i )p i for all i I, and p j ssd( j ) p j for some j I ; I strongly blocks p if p i ssd( i )p i for all 1 Let o be the least preferred object of i among the objects he owns with a positive probability. Then p i sd( i )ω i requires that o p io io o ω io io = o O ω io. Therefore, o O p io o O ω io for all i. But we know that i I o O p io i I o O ω io. So it must be that o O p io = o O ω io for all i. 9

10 i I. An assignment p is a core assignment if it cannot be blocked and is a weak core assignment if it cannot be strongly blocked. A mechanism is a function from the set of FEE problems to the set of assignments. For a problem ( I ) and a mechanism ϕ, ϕ( I ) is the assignment found by ϕ for the problem, and ϕ i ( I ) is the lottery i obtains. ϕ is called individually rational, ordinally efficient, and a (weak) core mechanism if ϕ( I ) is individually rational, ordinally efficient, and a (weak) core assignment for all ( I ). Agent i manipulates a mechanism ϕ at a preference profile I if there exists another preference relation i P\{ i } such that ϕ i ( I )sd( i )ϕ i ( i, i ) does not hold. If ϕ i ( i, i )ssd( i )ϕ i ( I ), we say i strongly manipulates ϕ. ϕ is weakly strategy-proof if it is not strongly manipulable and is strategy-proof if it is not manipulable. 2.3 Top Trading Cycle In the endowment exchange (EE) problem studied by Shapley and Scarf (1974), q o = 1 and ω io {0, 1} for all i I and all o O. Below is the procedure of TTC in solving the EE problem. Step d 1: Let every remaining agent point to his most preferred remaining object, and every remaining object point to its owner. There must be cycles. Trade all cycles by letting the agents in every cycle obtain the objects they point to. Then remove these agents with their assignments. 3 Fractional TTC At every step of FTTC, every remaining agent points to his most preferred remaining object and every remaining object points to all of its remaining owners. This generates a directed network. We denote the network by (V, E), where V is the set of nodes, consisting of a subset of agents and a subset of objects, and E V V is the set of directed edges. We say there is a path from a node v to another node v if there exists a sequence of nodes v 1, v 2,..., v z such that v points to v 1, v l points to v l+1 for all l = 1,..., z 1, and v z points to v. Definition. A subset of nodes V V is an absorbing set in (V, E) if (1) there is a 10

11 path from every node in V to every other node in V, and (2) there is no path from any node in V to any node not in V. Let A be the set of absorbing sets in the network. A must be nonempty and all absorbing sets must be disjoint. For every A A, every i A and every o A, the object pointed by i and the set of agents pointed by o must belong to A. Moreover, if an object points to any i A but does not belong to A, then it must not belong to any absorbing set. Similarly, if an agent points to any o A but does not belong to A, then he must not belong to any absorbing set. In our algorithm, the set of agents and objects that do not belong to any absorbing set are not involved in any trade at the step. For convenience, we denote by A o the set of agents in A that point to o, by A o the set of agents in A that pointed by o, by A i the set of objects that point to i, and by r io the remaining amount of o owned by i. Within each A, agents trade endowments only among themselves. The fairness of FTTC originates from the requirement that the owners of every object always lose equal amounts of the object if it is consumed at a step. Formally, let x i denote the amount of preferred object that every i A obtains, and x o denote the exhausted amount of every o A. Then x = (x a ) a A is the solution to the following equation system: x o = x i for all o A, (1) i A o x i = x o for all i A, (2) A o o A { i } xo / A o max : o A and i A o = 1. (3) r io Equation (1) is the definition of x o. Equation (2) requires that if x o of o is exhausted, then every i A o loses an equal amount x o A o of o. Since every agent obtains an amount of objects by giving up an equal amount of endowments, i obtains o A i x o A o of his preferred object. Equation (3) means that at every step FTTC trades as much of objects as possible in A until some agent s endowment of some object is exhausted. Theorem 1. The equation system (1)-(3) has a unique solution. We observe that the equation system characterizes a closed Leontief input-ouput model. Based on the classical literature we prove that the equation system has a unique solution. 11

12 We present the definition of FTTC below. At the end of every step d 1, let I(d) and O(d) be the set of remaining agents and the set of remaining objects respectively; let ω io (d) be remaining amount of o owned by i. Let I(0) = I, O(0) = O, and ω io (0) = ω io for all i I(0) and all o O(0). Fractional Top Trading Cycle Step d 1: Let every i I(d 1) point to his most preferred object in O(d 1). Let every o O(d 1) point to every i I(d 1) with ω io (d 1) > 0. This generates a network. In every absorbing set in the network, agents trade endowments according to the solution to the equation system (1)-(3). If O(d) is empty, stop the procedure. Otherwise, go to step d + 1. At every step, at least one agent s endowment of at least one object is exhausted. So FTTC has at most I O steps. Below we use an example to illustrate the procedure of FTTC. Example 2. Consider an FEE problem with I = {i 1, i 2, i 3, i 4, i 5 } and O = {o 1, o 2, o 3, o 4, o 5 }. Every object has one copy. The endowments and preferences of agents are as follows. o 1 o 2 o 3 o 4 o 5 i i i /4 1/4 i /4 1/4 i i1 i2 i3 i4 i5 o 3 o 4 o 4 o 1 o 3 o 4 o 3 o 3 o 4 o 5 o 1 o 1 o 1 o 3 o 1 o 2 o 2 o 5 o 5 o 2 o 5 o 5 o 2 o 2 o 4 At step 1 of FTTC, the generated network is shown in Figure 2. The nodes other than o 2, o 5 constitute the only absorbing set. By solving the equation system, we obtain the solution: x (1) = x i 1 (1) x i 2 (1) x i 3 (1) x i 4 (1) x i 5 (1) x o 1 (1) x o 2 (1) x o 3 (1) x o 4 (1) x o 5 (1) 1/3 1/3 2/3 2/3 1/6 2/ Therefore, i 3, i 4 s respective endowments of o 4 are exhausted. i 1, i 2 obtain 1/3o 3, 1/3o 4 respectively, and both lose 1/3o 1, i 3, i 4 obtain 2/3o 4, 2/3o 1 and lose 1/6o 3, o 4 respectively, and i 5 obtains 1/6o 3 and also loses 1/6o 3 in his endowment. 12

13 o 4 i 2 i 3 1/4 1/4 1/4 o 5 o 3 i 4 i 5 1/4 o 2 o 1 i 1 Figure 2: First step of a TTC-like mechanism in solving Example 2. i 2 i 3 1/4 o 5 1/12 1/4 1/6 o 3 i 4 i 5 1/12 1/3 o 2 o 1 1/6 i 1 Figure 3: Step 2 of FTTC in solving Example 2. At step 2, the generated network is shown in Figure 3. The nodes other than o 2, o 5 constitute the only absorbing set. In the figure we use solid lines to denote their edges. By solving the equation system, we obtain the solution: x (2) = x i 1 (2) x i 2 (2) x i 3 (2) x i 4 (2) x i 5 (2) x o 1 (2) x o 2 (2) x o 3 (2) x o 5 (2) 4 4 1/12 1/12 1/12 1/12 0 3/12 0 Therefore, i 3, i 4 s respective endowments of o 3 are exhausted. i 1, i 2 both obtain 4o 3 and lose 4o 1, i 3, i 4 obtain 1/12o 3, 1/12o 1 respectively and both lose 1/12o 3, i 5 obtains 1/12o 3 and also loses 1/12o 3 in his endowment. At step 3, the generated network is shown in Figure 4. The only absorbing set is {o 3, i 5 }. So i 5 obtains 1/4o 3. Then o 3 is exhausted. 13

14 i 2 i 3 1/4 o 5 1/4 1/8 o 3 i 4 i 5 1/4 o 2 o 1 1/8 i 1 Figure 4: Step 3 of FTTC in solving Example 2. i 2 i 3 1/4 o 5 1/8 1/4 o 1 i 4 i 5 1/8 o 2 i 1 Figure 5: Step 4 of FTTC in solving Example 2. At step 4, the generated network is shown in Figure 5. Note that although i 5 and o 5 form a cycle, it is not an absorbing set. The only absorbing set is {i 1, i 2, o 1 }. So i 1 and i 2 each obtain 1/8o 1. Then o 1 is exhausted. i 2 i 3 1/4 o 2 1/4 o 5 i 5 i 1 i 4 Figure 6: Step 5 of FTTC in solving Example 2. 14

15 At step 5, the generated network is shown in Figure 6. There are two absorbing sets {i 1, i 2, o 2 } and {i 3, i 4, i 5, o 5 }. In the first absorbing set, i 1, i 2 each obtain o 2. In the second absorbing set, FTTC first lets all agents obtain 1/4o 5, then lets i 5 obtain the remaining o 5 at the next step. The assignment found by FTTC is shown in Table 1. Note that there is no envy between i 1 and i 2, and between i 3 and i 4. o 1 o 2 o 3 o 4 o 5 i 1 1/8 3/8 0 0 i 2 1/8 4 1/3 0 i /12 2/3 1/4 i 4 3/ /4 i Table 1: The assignment found by FTTC for Example 2 4 Properties of FTTC To explain the origination of the properties of FTTC, we present an abstract procedure called general trading mechanism (GTM): at every step, it lets every remaining agent report his most preferred remaining object, and specifies each agent s consumption and his remaining endowments after the step. As long as an algorithm has a procedure in the abstract form, we say it is a GTM. Formally, let I(d), O(d), ω i (d) have same definitions as in Section 3. At step d, let o i (d) be the most preferred object of i among O(d 1); let x i (d) be the amount of o i (d) obtained by i, and x o (d) be the exhausted amount of o; if x o (d) > 0, let λ io (d) be the proportion of the exhausted amount of i s endowment of o in the whole exhausted amount of o. Then GTM has the following procedure. General Trading Mechanism Step d 1: Every i I(d 1) reports his most preferred object o i (d). procedure chooses {λ io (d) [0, 1] : i I(d 1), o O(d 1)} subject to the The 15

16 constraint that λ io (d) = 0 if ω io (d 1) = 0, and i I(d 1) λ io(d) = 1 if x o (d) > 0. Then {x a (d)} a I(d 1) O(d 1) is a solution to the following equation system: x o (d) = x i (d) = max i I(d 1):o i (d)=o o O(d 1) x i (d) for all o O(d 1), (4) λ io (d)x o (d) for all i I(d 1), (5) { } λio (d)x o (d) ω io (d 1) : o O(d 1) and ω io(d 1) > 0 = 1. (6) Let ω io (d) = ω io (d 1) λ io (d)x o (d); I(d) = {i I(d 1) : o O ω io(d) > 0}; O(d) = {o O(d 1) : i I(d) ω io(d) > 0}. If O(d) is empty, stop the procedure. Otherwise, go to step d + 1. FTTC is a GTM with λ io (d) = 1 j I(d 1):ω jo (d 1)>0 if ω io (d 1) > 0. Every GTM is ordinally efficient since it can be seen a simultaneous eating algorithm (Bogomolnaia and Moulin, 2001). Every GTM is also individually rational since at every step, every agent s consumption weakly dominates his exhausted endowment. Proposition 1. Every GTM is ordinally efficient and individually rational. Below we introduce several fairness requirements on the procedure of GTM. They help us understand how a fair procedure affects the properties of its assignment. Definition. A GTM satisfies (1) weak stepwise ETE if ω i (d 1) = ω j (d 1) and o i (d) = o j (d) = x i (d) = x j (d), and satisfies stepwise ETE if we further have ω i (d) = ω j (d). (2) equal-endowment equal treatment (EEET) if ω i (d 1) = ω j (d 1) = x i (d) = x j (d) and ω i (d) = ω j (d); (3) equal-endowment-set equal treatment (EESET) if supp(ω i (d 1)) = supp(ω j (d 1)) = x i (d) = x j (d) and ω i (d 1) ω i (d) = ω j (d 1) ω j (d). 16

17 Stepwise ETE is an application of the ETE idea to every step of GTM. Weak stepwise ETE is weaker in that it only requires agents who are equal before a step have equal consumption at the step, but does not require they still have equal endowments after the step. EEET strengthens stepwise ETE by requiring agents with equal endowments before a step have equal amount of consumption at the step and equal remaining endowments after the step. EESET further strengthens EEET by requiring that agents who own an equal set of endowments before a step have equal amount of consumption and lose equal endowments at the step. FTTC satisfies EESET. Proposition 2. (1) Every GTM that satisfies weak stepwise ETE is not a weak core mechanism. (2) Every GTM that satisfies stepwise ETE satisfies ETE. (3) Every GTM that satisfies EEET satisfies EENE. In the EE problem TTC always finds the unique core assignment. Here we prove that in the FEE problem, if a GTM satisfies weak stepwise ETE, it may not find a weak core assignment even though such assignments exist. In the above proposition we do not provide the implication of EESET because it is hard to define it in terms of agents endowments and final assignments. Suppose two agents own an equal set of endowments at the very beginning. When none of them have used up their endowments of any object, EESET guarantees that they obtain equal amounts of their respective preferred objects at every step so that they do not envy each other. However, once they do not own an equal set of endowments at some step, their consumption can be different and in the final assignment one of them may envy the other. However, in a special class of FEE problems in which every agent owns only one fractional object, EESET implies an envy-freeness property between any two agents who own the same object. But more important is that, in such problems EESET exactly captures the fairness requirement of FTTC. In general problems, agents may have overlapping but not equal set of endowments. FTTC requires that such agents have equal chance to trade their common endowments with others, but this requirement is not captured by EESET. Formally, a problem M = (O, Q, I, I, ω) is simple if supp(ω i ) = 1 for every i I. For any lottery p i (O) and any number a (0, p i ), we define p i [a] = {p i (O) : p io p io for all o O, and p i = a}. That is, p i [a] is the set of sub-lotteries subsumed by p i such that the total amount of 17

18 objects in every sub-lottery is a. Definition. A mechanism ϕ satisfies respecting-endowment no envy (RENE) if, for any problem M and any two distinct agents i, j with supp(ω i ) = supp(ω j ) and ω i ω j, we have ϕ i (M)sd( i )p j for every p j ϕ j (M)[ ω i ] That is, if we allow i to choose any sub-lottery from j s lottery as long as the total amount of objects does not exceed the total amount of his endowments, then i still prefers his own lottery. RENE is stronger than EENE. Proposition 3. In simple problems, (1) a GTM satisfies EESET if and only if it is FTTC, and (2) FTTC satisfies RENE. For every general problem M = (O, Q, I, I, ω), we can construct a simple problem M = (O, Q, Ī, Ī, ω) such that Ī = i I{i o } o supp(ωi ), io = i, ω ioo = ω io, and ω ioo = 0 if o o. That is, every i o has the same preferences with i and holds ω io of o. We call M the decomposed problem of M. FTTC satisfies the property that the assignment found for any M is equivalent to the assignment found for M in the sense that the lotteries of every i is equal to the sum of the lotteries of {i o } o supp(ωi ). We call this property decomposition invariance. Definition. (1) A mechanism ϕ satisfies decomposition invariance if for all M and all i I, ϕ i (M) = i o:o supp(ω i ) ϕ i o ( M). (2) A GTM ϕ satisfies strong decomposition invariance if for all M, all step d and all i I(d 1), x i (d) = i o:o supp(ω i ) x i o (d) and ω i (d) = i o:o supp(ω i ) ω i o (d). Theorem 2. (1) A GTM satisfies decomposition invariance and EESET only if it is equivalent to FTTC. (2) A GTM satisfies strong decomposition invariance and EESET if and only if it is FTTC. As a corollary, we obtain the following properties of FTTC. Corollary 1 (Properties of FTTC). (1) As a function, FTTC is individually rational, ordinally efficient, and satisfies EENE in general problems and RENE in simple problems. But FTTC is not a weak core mechanism. (2) As an algorithm, FTTC satisfies strong decomposition invariance and EESET. 18

19 5 Incentive A desirable incentive property of TTC is strategy-proofness. It ensures that no agent can obtain a better assignment by misreporting preferences. However, we show that FTTC is even not weakly strategy-proof. Example 3 (Strong manipulability). Consider an FEE problem in which I = {i 1, i 2, i 3, i 4, i 5 } and O = {o 1, o 2, o 3, o 4, o 5 }. Every object has one copy. The preferences and endowments of agents are as follows: o 1 o 2 o 3 o 4 o 5 i i i i i i1 i2 i3 i4 i5 o 3 o 5 o 1 o 2 o 5 o 1 o 1 o 4 o 4 o 3 o 2 o 2 o 2 o 1 o 1 o 4 o 3 o 3 o 3 o 2 o 5 o 4 o 5 o 5 o 4 If all agents report true preferences, FTTC finds the assignment p shown below. However, if i 1 reports the preference relation i 1 : o 1, o 3, o 2, o 4, o 5, then FTTC finds the assignment p. Since p i 1 ssd( i1 ) p i1, FTTC is strongly manipulated by i 1. p o 1 o 2 o 3 o 4 o 5 i i i i i p o 1 o 2 o 3 o 4 o 5 i i i i i In TTC, every agent is involved in a cycle only once. But in FTTC, an agent is often involved in absorbing sets multiple times. So he may strongly manipulate FTTC by exploring his best trading chances across steps. Aziz (2015) proves that strong manipulation is inevitable in the presence of individual rationality and ordinal efficiency. Proposition 4 (Aziz, 2015). There is no mechanism that satisfies individual rationality, ordinal efficiency, and weak strategy-proofness. 19

20 It is well known that some non-strategy-proof mechanisms turn out to be strategyproof if the market grows sufficiently large. In the rest of this section we consider the incentive property of FTTC in large markets. For any problem M = (O, Q, I, I, ω), we define a sequence of economies (M [n] ) n=1 such that M [n] = (O, Q [n], I [n], I [n], ω [n] ) satisfies the following conditions: (1) Q [n] = nq. (2) I [n] = n I. (3) ω [n] = n ω, which means that the rows of ω [n] are obtained by copying the rows of ω by n times. (4) For all ˆω Ω = {ˆω : ˆω = ω i for some i I} and all P, define A [n] (ˆω, ) = {i I[n] : ω [n] i = ˆω, i = }. I [n] There exists A [ ] (ˆω, ) (0, 1) for all ˆω Ω and all P such that A [n] (ˆω, ) A [ ] (ˆω, ) as n. That is, in M [n], every object o O has nq o copies, and every i I has n copies who have equal endowments with i. When n is large enough, the preferences of agents in M [n] are diverse such that, for every endowment type ˆω Ω and every preference type P, the proportion of agents who hold ˆω and have preferences is positive and converges to A [ ] (ˆω, ). In this sense we say M [n] converges to M [ ] We use (u io ) o O to denote the von Neumann-Morgenstern utilities of agent i. So o i o if and only if u io > u io, and u i (p i ) = o O u iop io is the expected utility of agent i when obtaining lottery p i. Definition. A mechanism ϕ is asymptotically strategy-proof if, for any ε > 0, there exists n Z ++ such that for any n > n, if any i I [n] reports i P\{ i } in the economy M [n], then ( u i ϕi ( i, i ) ) ( < u i ϕi ( I [n]) ) + ε. That is, the utility gain from unilateral misreporting is bounded by ε. Proposition 5. FTTC is asymptotically strategy-proof, and strategy-proof in the limit economy M [ ]. 20

21 6 Weak-priority-based Assignment Problem 6.1 Extension of FTTC Abdulkadiroğlu and Sönmez (2003) extend TTC to the school choice problem by assuming that schools have strict priority rankings of students. This extension, often still called TTC, maintains the good properties of TTC. But in practice, the priority rankings used by schools are often weak (Ergin, 2008; Kesten and Ünver, 2015). That is, many students belong to the same priority tier at a school. The current method to solve this problem is letting schools randomly break priority ties. However, this method can result in efficiency loss. Consider the extreme case that all students are in the same priority tier at all schools. If we draw the priority ranking from the uniform distribution, TTC coincides with RP. But we know RP has efficiency loss (Bogomolnaia and Moulin, 2001). In this section we propose an extension of FTTC to solve the school choice problem with weak priorities. We still call it FTTC, and show it is ordinally efficient. Formally, a weak-priority-based assignment (WPA) problem is a tuple (O, Q, I, I, O ) where O, Q, I, I are defined same as before, and O = ( o ) o O is the profile of priority rankings. o is the weak priority ranking used by o, which is a complete and transitive relation on I. o and o are the asymmetric and symmetric components of o respectively. At every step, every remaining object points to each of the remaining agents who are in the highest priority tier at the object. Different from the FEE model, here we assume the agents in the highest priority tier own equal fractions of the remaining amount of the object. So in the procedure we track the residual amount of every object and the residual demand of every agent. At any step, if r i (0, 1] is the residual demand of agent i and r o is the residual amount of object o, then in every absorbing set A, x = (x a ) a A, which has same definition as before, is the solution to the following equation system: x o = x i for all o O, (7) i A o x i = x o for all i I, (8) A o o A { i } xa max : a A = 1. (9) r a By the proof of Theorem 1, we know the solution is unique. Then FTTC is defined as follows. 21

22 FTTC for Weak-priority-based Assignment Problem Step d 1: Let every i I(d 1) point to his most preferred object in O(d 1). Let every o O(d 1) point to each of the agents in I(d 1) who have highest priority at o. This generates a network. In every absorbing set, let agents trade objects according to the solution to the equation system (7)-(9). If O(d) is empty, stop the procedure. Otherwise, go to step d Properties As before, FTTC is ordinally efficient. We say FTTC satisfies no envy towards lower priority (NELP) if, for any two agents such that i o j for all o O, we have p i sd( i )p j where p i, p j are the lotteries of i, j respectively. NELP implies equal-priority no envy (EPNE): i o j for all o O implies p i sd( i )p j and p j sd( j )p i. By adapting the proofs of our previous propositions, we can easily prove that FTTC satisfies NELP. Proposition 6. For WPA problems, FTTC is ordinally efficient and satisfies no envy towards lower priority. The priority structure of WPA is rich enough to accommodate many special cases. In the literature different mechanisms have been proposed for these special cases. Below we show that FTTC degenerates to some of these mechanisms in these special cases. We say a WAP problem is a (1) strict priority problem if o is strict for all o O; (2) house allocation problem if q o = 1 for all o O, and i o j for all i, j I; (3) house allocation problem with existing tenants if q o = 1 for all o O, and there is a bijection π from some O O to some I I such that (3.1) for all o O\O, i o j for all i, j I; (3.2) for all o O, π(o) o j for all j I\{π(o)}, and i o j for all i, j I\{π(o)}. Proposition 7. (1) In strict priority problems, FTTC coincides with TTC; (2) In house allocation problems, FTTC coincides with PS; 22

23 (3) In house allocation problems with existing tenants, FTTC coincides with P S E of Zhang (2017a). It is interesting to compare the second result with Kesten (2009). Kesten proves that in house allocation problems, PS is equivalent to a probabilistic version of TTC. Specifically, we can interpret a house allocation problem as an FEE problem in which all agents own equal fractions of all objects. To mimic the procedure of TTC, Kesten decomposes every agent into multiple pseudo-agents who hold the fractional endowments of the agent, then lets pseudo-agents trade fractional endowments. Because many pseudoagents hold equal endowments, Kesten carefully controls the trading process by allowing only non-redundant cycles to trade. This makes the algorithm a little complicated. In particular, agents may obtain different amounts of their respective preferred objects at a step, and after some steps agents may no longer have equal endowments. So the algorithm is not step-by-step equivalent to PS. By contrast, at every step of FTTC, all agents are pointed by all objects and obtain equal amounts of their respective preferred objects. So the procedure of FTTC is step-by-step equivalent to that of PS. In this sense, our result provides a new and simple perspective on the equivalence between PS and TTC. 7 Comparison with Aziz (2015) Aziz (2015) proposes another generalization of TTC to solve FEE problems. We denote it by A-TTC to differentiate it with FTTC. A-TTC is based on the idea of decomposition invariance defined in Section 4. That is, A-TTC finds an assignment for any FEE problem by solving the corresponding decomposed problem. So the procedure of A-TTC is only well-defined for simple FEE problems. Aziz considers both strict preferences and weak preferences. In this section we only compare the strict-preference version of A-TTC with FTTC. For every problem M = (O, Q, I, I, ω), recall that the decomposed problem is M = (O, Q, Ī, Ī, ω) where Ī = {i o} i I,o supp(ωi ), io = i, ω ioo = ω io, and ω ioo = 0 if o o. Aziz calls {i o } o supp(ωi ) the subagents of i. Every step of A-TTC proceeds as follows: 1. Let every subagent i o point to i s most preferred object, and every object o point to the subagents who own positive fractions of it. 23

24 2. If there exist subagents who point to their endowments (i.e., self-cycles), let those subagents obtain their endowments, then remove them. Repeat (1) and (2) until there are no self-cycles. 3. In the remaining network, every subagent points to one object and is pointed by one object. But every object may point to multiple subagents. To select cycles, A-TTC chooses an exogenous ranking of agents and lets every object only point to its highest-ranked owner. Then every subagent or object is involved in at most one cycle such that all cycles are disjoint. A-TTC trades all such cycles simultaneously. In every cycle subagents obtain as much of the objects they point to as possible until some subagent s endowment is exhausted. A-TTC is a GTM. So it is individually rational and ordinally efficient. However, A-TTC is not fair. It satisfies neither ETE nor stepwise ETE. It is neither a weak core mechanism. These statements are proved by applying A-TTC to solve the example we use to prove Proposition 2. Recall that in the example agents have the following endowments and preferences. Every object has one copy. o 1 o 2 o 3 o 4 i i i i i1 i2 i3 i4 o 2 o 1 o 1 o 1 o 1 o 2 o 2 o 2 o 4 o 4 o 3 o 4 o 3 o 3 o 4 o 3 A-TTC solves the example as follows. First, let every agent i be represented by subagents {i o } o supp(ωi ) with i o holding i s endowment of o. Second, choose an exogenous ranking of agents. Without loss of generality, let the ranking be i 4 i 3 i 2 i 1. Lastly, run the procedure of A-TTC as follows: Step 1. Let i 1o1, i 1o3 point to o 2 ; i 2o1, i 2o3, i 3o2, i 3o4, i 4o2, i 4o4 point to o 1 ; o 1 point to i 1o1, i 2o1 ; o 2 point to i 3o2, i 4o2 ; o 3 point to i 1o3, i 2o3 ; o 4 point to i 3o4, i 4o4. There is a self-cycle i 2o1 o 1 i 2o1. Trade this cycle by letting i 2o1 obtain o 1, then remove i 2o1. In the remaining graph there are two cycles, i 1o1 o 2 i 3o2 o 1 i 1o1 and i 1o1 o 2 i 4o2 o 1 i 1o1. By the exogenous ordering, only the second cycle is traded. Then i 1o1, i 4o2 obtain o 2, o 1 respectively, and are removed. 24

25 From this step we can see that A-TTC does not satisfy stepwise ETE. Although i 3o2 and i 4o2 hold equal endowments and have equal preferences, they have different consumption at this step. Step 2. Because o 1 is exhausted, all remaining subagents point to o 2. There is a selfcycle i 3o2 o 2 i 3o2. After trading this cycle, i 3o2 obtains o 2 and is removed. Because o 2 is exhausted, let i 1o3, i 2o3, i 4o4 point to o 4, and let i 3o4 point to o 3. There is again a self-cycle i 4o4 o 4 i 4o4. After trading the cycle, i 4o4 obtains o 4 and is removed. In the remaining graph there are two cycles, i 1o3 o 4 i 3o4 o 3 i 1o3 and i 2o3 o 4 i 3o4 o 3 i 2o3. By the exogenous ordering, only the second cycle is traded. Then i 2o3, i 3o4 obtain o 4, o 3 respectively, and are removed. From this step we again see that A-TTC does not satisfy stepwise ETE. Step 3. By letting i 1o3 obtains o 3. point to o 3 we have the last self-cycle. By trading it, i 1o3 The table below shows the assignment found by A-TTC. From the procedure we have seen that A-TTC does not satisfy stepwise ETE. If we let i 3, i 4 have the equal preference relation o 1 o 2 o 3 o 4, the first step does not change. So A-TTC neither satisfies ETE. A-TTC is not a weak core mechanism because the assignment is strongly blocked by the coalition {i 1, i 3 } if they exchange their endowments. o 1 o 2 o 3 o 4 i i i i Here we point out a mistake in Aziz s paper. Aziz proves a theorem stating that A-TTC is a weak core mechanism. His proof is based on a lemma, which states that a weak core assignment in the decomposed problem is also a weak core assignment in the original problem. This lemma is incorrect. The reason is that the blocking incentive of an agent in the original problem is not equal to the sum of the blocking incentives of its subagents in the decomposed problem. In the above example, the assignment of A-TTC is a weak core assignment in the decomposed problem: subagent i 1o1 does not want to 25

26 form a blocking coalition with i 3o2 by exchanging their endowments because i 1o1 obtains o 2 in the assignment and cannot be better off; similarly, i 3o4 does not want to form a blocking coalition with i 1o3 by exchanging their endowments because i 3o4 obtains o 3 in the assignment. However, in the original problem, i 1, i 3 want to block the assignment by exchanging their endowments. 8 Discussion 8.1 A general impossibility result on weak core mechanism In Section 4, we use an example to prove that every GTM satisfying weak stepwise ETE is not a weak core mechanism. In the example all agents prefer o 1, o 2 to o 3, o 4. This is a case of tiered preferences. In our proof we separately find the assignments of o 1, o 2 and the assignments of o 3, o 4. We can do this because GTM has discrete steps that satisfy a separability property. Now we formalize this property and present a general impossibility result on the existence of weak core mechanism. Formally, we say a preference profile I is tiered with respect to {O 1, O 2,..., O l } if {O 1, O 2,..., O l } is a partition of O such that every agent prefers every object in O k to every object in O k+1 for all k {1,..., l 1}. We use I Ok to denote the restriction of agents preferences to the objects in O k. We define Q Ok and ω Ok similarly. Definition. A mechanism ϕ satisfies separability if, for any two FEE problems M = (O, Q, I, I, ω) and M = (O, Q, I, I, ω) such that both I and I are tiered with respect to a common partition {O 1,..., O l }, I Ok = I Ok = ϕ io (M) = ϕ io (M ) for all o O k {O 1,..., O l }. That is, as long as agents preferences over the objects in a tier are same in the two problems, the assignments of the objects in the tier are same in the two problems. In particular, if a mechanism has the property that, for any problem M with tiered preferences, its assignment can be equivalent found by first separately finding the assignment for each problem M k in which only O k is present, and then putting the separated assignments together, then the mechanism satisfies separability. That is, ϕ(m) = l ϕ(m k ) where M k = (O k, Q Ok, I, I Ok, ω Ok ). k=1 26