Why do popular mechanisms lack e ciency in random assignment. problems?

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1 Why do popular mechanisms lack e ciency in random assignment problems? Onur Kesten January 2007 Abstract We consider the problem of randomly assigning n indivisible objects to n agents. Recent research introduced a promising mechanism, the probabilistic serial that has superior e ciency properties than the most common real-life mechanism random priroity. On the other hand, mechanisms based on Gale s celebrated top trading cycles method have long dominated the indivisible goods literature (with the exception of the present context) thanks to their outstanding e ciency features. We present an equivalence result between the three kinds of mechanisms, that helps better understand why e ciency di erences among popular mechanisms might arise in random assignment problems. JEL Classi cation: C71; C78; D71; D78 Key words: Indivisible goods; Random priority; Probabilistic serial; Top trading cycles; Ordinal e ciency 1 Introduction The problem of assigning n indivisible objects to n agents where each agent receives exactly one object is usually known as the assignment problem (also referred as the house allocation problem). Typical examples are the assignment of houses to applicants, o ces to graduate students, tasks to workers, and so on. The deterministic approaches to this problem su er from the con ict between fairness and e ciency. Therefore a common tool is to introduce randomization to recover compatibility between the two requirements. In this richer setting, the problem takes the name the random assignment problem. A random assignment speci es a random allotment over objects for each agent. A mechanism is a systematic way of selecting a random assignment for any given random assignment problem. Tepper School of Business, Carnegie Mellon University 5000 Forbes Ave, Pittsburgh, PA 15213, USA; okesten@andrew.cmu.edu. I would like to thank the editor of the journal as well as two anonymous referees for useful suggestions. I also thank Tayfun Sönmez, Al Roth, Herve Moulin, Utku Unver, Mihai Manea, and Ozgur Yilmaz for helpful discussions. Any remaining errors are my own. 1

2 Probably the most common real-life mechanism of this problem is the random priority (RP): 1 a random ordering of agents is drawn from the uniform distribution, and agents, starting with the rst agent, are asked to successively choose their favorite objects from the available ones. Di ently put, RP is the induced average of n! serial dictatorship outcomes. An important advantage of RP is that it only requires ordinal preferences of agents over objects as opposed to cardinal preferences which are harder to work with and not very realistic 2 to expect agents to reveal. RP induces ex post e cient random assignments (i.e., puts weight only on Pareto e cient deterministic assignments), and moreover selects the central point within this set. Bogomolnaia and Moulin (2001) introduce and study a more appealing and an even stronger notion of e ciency than ex post e ciency which they call ordinal e ciency: a random assignment is ordinally e cient if it is not stochastically dominated by another random assignment. Surprisingly, widely-used RP may not always induce ordinally e cient outcomes. Bogomolnaia and Moulin (2001) characterize the entire set of ordinally e cient random assignments for any given random assignment problem. They propose an important contender to RP which selects the unique central point in the ordinally e cient set. They call this mechanism the probabilistic serial (PS). An increasingly popular resource allocation method in the indivisible goods literature is the so called top trading cycles (TTC) method. It is attributed to David Gale, and was rst proposed in the context of housing markets (Shapley and Scarf, 1974) where one seeks an optimal reallocation of objects (which are now endowments) among agents. The TTC method works as follows: each agent points to the agent who owns his favorite object. There is at least one cycle. Within each cycle the corresponding trades are carried out, and each agent in a cycle is removed. Next consider the new market, and apply the same method, and so on. This method yields the unique core allocation (Roth and Postlewaite, 1979) of the market which also coincides with the competitive equilibrium outcome (Shapley and Scarf, 1974). Because of its appealing e ciency and incentive features, a number of mechanisms based on the TTC method have been proposed and characterized for a variety of contexts which include the single and multiple assignment problems, 3 the school choice problem, 4 and the kidney exchange problem 5. Thus far the only adaptation of the TTC method to the present context is due to Abdulkadiro¼glu and Sönmez (1998). They introduce a mechanism called core from random endowments (CfRE): a random initial assignment of objects is drawn from the uniform distribution, and the TTC method is applied to 1 It is also referred as random serial dictatorship. Zhou (1990) is the rst to study RP in the literature. Hylland and Zeckhauser (1979) proposed the rst mechanism which is based on an adaptation of the competitive equilibrium concept to this context. This mechanism also requires information about the vnm indices of agents. 2 See for example Kagel and Roth (1995). 3 See for example, Svensson (1999), Abdulkadiro¼glu and Sönmez (1999), Pápai (2000), and Ehlers (2000). 4 The problem of assigning a set of students to a set of schools: see for example, Abdulkadiro¼glu and Sönmez (2003), Kesten (2005), and Abdulkadiro¼glu, Pathak, Roth, and Sönmez (2005). 5 The problem of reallocating kidneys from a group of donors to a group of recipients: see for example, Roth, Sönmez, and Unver (2004). 2

3 the resulting housing market. It turns out that RP and CfRE are equivalent (Abdulkadiro¼glu and Sönmez, 1998). Although for deterministic settings, all TTC based mechanisms are e cient, the only probabilistic TTC based mechanism CfRE, by this equivalence, may not induce ordinally e cient outcomes. In this paper we aim to understand why popular real-life mechanism RP and mechanisms based on adaptations of the popular TTC method (such as CfRE) may be short of e ciency in a random assignment environment. This, we believe, is a key question the answer to which may prove quite useful for indivisible good allocation problems in general. An alternative way to approach the random assignment problem is the following: initially all objects are collectively owned, and each agent a priori can be seen to have equal right over each object. This suggests that each agent should initially be endowed with a 1=n share of each object. Then this problem could be viewed similarly to a housing market where one can again apply the TTC method in a way that would allow agents to carry out the trades in their best interest. We call this class of mechanisms top trading cycles from equal division (TTCfED). Next we turn our attention to RP, and argue that its e ciency loss is closely related with the degree of of variation across individual serial dictatorship outcomes. To make this point, we introduce the following extension of the RP procedure: consider an extended problem with a k replica of resources and compute the random assignment which is the induced average of kn! serial dictatorship outcomes where for each ordering of agents the associated serial dictatorship is applied consecutively k times until all objects are exhausted. We call this class of mechanisms random priority (RP) [k]. Our main result is that the two class of mechanisms TTCfED and RP [k] coincide with PS when one restricts attention to single and pairwise cycles for the former class and when k is a multiple of (n!) n for the latter one (Theorem 1). 2 The Model Let N f1; 2; : : : ; ng denote the nite set of agents. Let H fh 1 ; h 2 ; : : : ; h n g denote the nite set of objects (say houses). We use the letters of the alphabet such as a; b; and x to represent typical elements of H. Each agent i 2 N is equipped with a complete, transitive, and strict preference relation < i over H: Let i denote the strict relation associated with < i : Let < (< i ) i2n : An assignment is a bijection : N! H; and can be represented as a permutation matrix (a n n matrix with entries 0 or 1 and exactly one non-zero entry per row and one per column). Let A denote the set of all assignments. An assignment is Pareto e cient if there is no 0 2 A such that 0 (i) < i (i) for all i 2 N, and 0 (i) i (i) for some i 2 N. 3

4 A random allotment is a probability distribution over H. Let H denote the set of all random allotments. A lottery L = P is a probability distribution over assignments where 2 [0; 1] and P = 1. A random assignment P = [p ix ] i2n; x2h is a bistochastic matrix where p ix 2 [0; 1] denotes the proability that agent i receives object x. Let P i be the random allotment of agent i at P; and let P i [x] be the random allotment of object x of agent i: Note that for all i 2 N and all x 2 H; p ix 2 [0; 1] and X j p jx = X y p iy = 1 Given an allocation represented by the permutation matrix (); the lottery L = P induces the random assignment P = P (): Given any random assignment, by the well-known Birkho -von Neumann theorem there is at least one lottery that induces it. A (random assignment) problem is given by a preference pro le < : A mechanism ' is a function that associates with each problem < a random assignment '(<): 2.1 Ordinal e ciency A random assignment is ex post e cient if it can be represented as a probability distribution over Pareto e cient assignments. A popular ex post mechanism is the random priority (RP) which is often used in real life: a random ordering of agents is drawn from the uniform distribution; the rst agent gets his favorite object, the second agent gets his favorite object among the remaining objects, and so on. Bogomolnaia and Moulin (2001) propose a more desirable and even stronger notion of e ciency than ex post e ciency. Given an agent i; a preference relation < i ; and two random assignments P and Q; P stochastically dominates Q if and only if for all i 2 N; all x 2 H; and all y 2 fz : z < i xg; X p iy X y y q iy A random assignment is ordinally e cient if it is not stochastically dominated by another random assignment. Surprisingly, the outcome of RP may not always be ordinally e cient (Bogomolnaia and Moulin, 2001). 6 If a lottery induces an ordinally ine cient random assignment, then it is clearly suboptimal to use such a lottery when agents are expected utility maximizers. 6 See Abdulkadiro¼glu and Sönmez (1998) for an interesting characterization of ordinally e cient random assignments. Also see McLennan (2002) for a result concerning the relationship between ordinal e ciency and ex ante e ciency. 4

5 3 Three mechanisms 3.1 Probabilistic Serial Bogomolnaia and Moulin (2001) provide an interesting characterization of the set of ordinally e cient random assignments for any given random assignment problem. Based on this result they propose a new mechanism, called the probabilistic serial (PS), 7 8 that always chooses the unique central point in this set point. The PS outcome is computed via the following algorithm: The Probabilistic Serial (PS) Algorithm Think of each object as an in nitely divisible good. Each agent eats away (representing his allotment of that object) from his favorite object at the same speed until the object is completely exhausted. Allotting p ix units of object x to agent i simply means that agent i gets object x with probability p ix : Step 1: Each agent eats away from his favorite object at the same speed. Stop when an object is completely exhausted. Step k, 2 k n: Consider the remaining objects with the remaining units of them. Each agent eats away from his favorite object at the same speed. Stop when an object is completely exhausted. Example 1: Let N = f1; 2; 3; 4; 5; 6g and H = fa; b; c; d; e; fg: Preferences are as follows: < 1 : a b c d < 3 : b a c d < 2 : a b d c < 4 : b c d a Step 1: Agents 1 and 2 eat away from object a; while 3 and 4 eat away from object b: We stop when objects a and b get completely exhausted. At this point each one of agents 1 and 2 has eaten 1=2 units of object a; and each one of 3 and 4; 1=2 units of object b: Step 2: Agents 1; 3; and 4 eat away from object c; while 2 eats away from object d: We stop when object c gets completely exhausted when each one of 1; 3; and 4 has eaten 1=3 units of it. At this point agent 2 has eaten 1=3 units of object d: Step 3: All agents eat away from the 2=3 remaining units of object d: We stop when object d gets completely exhausted when each agent has eaten 2=12 units of it. The overall random assignment that the PS algorithm yields is given in the following matrix: 7 Cres and Moulin (1998) introduce PS for the case where each agent has the same ranking of objects. Bogomolnaia and Moulin (1999) o er a characterization for the same context. Katta and Sethuraman (2004) extend PS to the preference domain where indi erence among alternatives is allowed. Chun and Koh (2005) independently o er a decentralized algorithm based on a ow of probabilities among agents. 8 In fact, one can obtain the complete set of ordinally e cient random assignments simply by varying the speed at which an agent can eat away from his favorite object at a given time (Bogomolnaia and Moulin, 2001). 5

6 a b c d 1 1=2 0 1=3 1=6 2 1= = =2 1=3 1= =2 1=3 1=6 3.2 Top Trading Cycles from Equal Division An increasingly popular and well-known allocation method is Gale s top trading cycles (TTC). It was rst proposed in the context of a housing market (Shapley and Scarf, 1974). In such a market each agent is initially assigned (endowed with) a di erent object. The TTC method is applied to such a market as follows: each agent points to the agent who is assigned his favorite object. Since the number of agents is nite, there is at least one cycle. Within each cycle the corresponding trades are carried out (i.e., each agent in a cycle is allotted the object that was assigned to the agent he is pointing to), and each agent in every cycle is removed. Next consider the new market, and apply the same method, and so on. This method yields the unique core allocation (Roth and Postlewaite, 1979) of the market which also coincides with the competitive equilibrium outcome (Shapley and Scarf, 1974). It is also the unique Pareto e cient, individually rational, and strategy-proof method (Ma, 1996). In the recent literature various mechanisms that employ the TTC method have been proposed and characterized for a number of contexts (such as the assignment, school choice, and kidney exchange problems) as promising alternatives to real-life practises (see the references listed in footnotes 3, 4, and 5).. Abdulkadiro¼glu and Sönmez (1998) propose a rst adaptation of the TTC method to the random assignment problem. They introduce a mechanism called core from random endowments (CfRE): a random initial assignment of objects is drawn from the uniform distribution, and the TTC method is applied to the resulting housing market. Quite interestingly, RP and CfRE turn out to be equivalent (Abdulkadiro¼glu and Sönmez, 1998). In this paper we argue that the TTC method can indeed be used to achieve a stronger form of e ciency than ex post e ciency. For this purpose we o er an alternative way to utilize the TTC in the present context: initially all resources are collectively owned, and each agent a priori has equal right over each object. This suggests that each agent should initially be entitled to any object with probability 1=n: Then one can still apply a simple version of the TTC method in which (1) agents are allowed to trade equal units of objects along each cycle, and (2) only single and pairwise cycles are considered. 9 We call this mechanism top trading cycles from equal division (TTCfED), and its outcome can be calculated via the following algorithm: 9 The fact that initially objects are homogenously assigned among agents makes larger cycles irrelevant from an e ciency point of view. 6

7 [1/4] (1,a) [1/4] (1,b) (3,a) [1/4] (3,b) [1/4] [1/4] (2,a) [1/4] (2,b) (4,a) [1/4] (4,b) [1/4] Figure 1: Step 1 of TTCfED for Example 1. A two-sided arrow represents a pairwise cycle. The Top Trading Cycles Algorithm from Equal Division (TTCfED) Think of each agent as consisting of n pseudo-agents each of whom is initially assigned 1=n units of a di erent object. An agent s random allotment of a particular object is the sum of all the units of that object he is allotted. Consider any agent i. Step k, 1 k n: Each pseudo-agent of agent i points to the pseudo-agent(s) who is (are) assigned a positive unit of agent i s favorite object. More precisely: if a pseudo-agent of agent i is already assigned a positive unit of agent i s favorite object, then he points to himself, and forms a self-cycle; otherwise he points to those pseudo-agents each of whom is assigned a positive unit of his favorite object. There is at least one cycle. (Moreover, a pseudo-agent may even participate in multiple cycles.) Next the corresponding trades are performed: (i) self-cycles: agent i is allotted all the units assigned to his pseudo-agent who forms the self-cycle, and that pseudo-agent is removed; (ii) pairwise trading cycles: within each cycle a pseudo-agent forms, he trades equal units of the object he is assigned for equal units of the object the pseudo-agent he is pointing to is assigned until the point there is a pseudo-agent who has no units (of the object he is assigned) left to trade. At this point all trades stop, and all pseudo-agents without any object to trade are removed. Example 1 (Cont d): We now compute the outcome of TTCfED for the assignment problem given in Example 1. Initially each agent is assigned 1/4 units of each one of the four objects, and each agent i consists of 4 pseudo agents: (i; a); (i; b); (i; c); and (i; d): Step 1i: The favorite object of agents 1 and 2 is a; while that of 3 and 4 is b: Each agent s pseudo agent (whose favorite object is di erent than the one he is assigned) points to each pseudo-agent who is assigned 1=4 units of his favorite object (Figure 1). We rst identify self-cycles: each one of the pseudo-agents (1; a); (2; a); (3; b); and (4; b) forms a self-cycle, and is removed. Hence, each corresponding agent is allotted 1=4 units of his favorite object. Step 1ii: Next we identify the pairwise trading cycles: each one of (1; b) and (2; b) forms a cycle with each one of (3; a) and (4; a): Recall that each pseudo agent is assigned 1=4 units of some object. We next identify 7

8 [1/4] (1,c) [1/4] (1,d) (2,d) [1/4] [1/4] (3,d) (2,c) [1/4] [1/4] (3,c) [1/4] (4,d) (4,c) [1/4] Figure 2: Step 2 of TTCfED for Example 1. the cycle(s) in which trade ends rst. Each of the four pseudo-agents participates in two cycles, and all trades end at the same time. Within each trading cycle of this step pseudo-agents exchange 1=8 units of their favorite objects for 1=8 units of the objects they are assigned. Hence, each one of 1 and 2 is allotted 2=8 units of a; and each one of 3 and 4 is allotted 2=8 units of b: All the pseudo-agents in these trading cycles are removed. Step 2i: At the end of step 1 no pseudo-agent with any positive units of object a or b remains. Now the favorite object of 1; 3; and 4 is c; while that of 2 is d: Each agent s each remaining pseudo-agent points to each pseudo-agent who is assigned a positive unit of his favorite object (Figure 2). We rst identify self-cycles: each one of (1; c); (3; c); and (4; c) forms a self-cycle, and is removed. Each corresponding agent is allotted 1=4 units of c: Similarly, (2; d) forms a self-cycle, and agent 2 is allotted 1=4 units of d: Step 2ii: Pairwise trading cycles: each one of (1; d); (3; d); and (4; d) forms a cycle with (2; c): Trade stops when within each trading cycle the corresponding pseudo-agents exchange 1=12 units of their favorite objects for 1=12 units of the objects they are assigned. Hence, each one of 1; 3; and 4 is allotted 1=12 units of c; and 2 is allotted 3=12 units of c. At this point each of (1; d); (3; d); and (4; d) is left with 1=6 units of d: Pseudo-agent (2; c) is removed. Step 3: The only remaining pseudo agents are (1; d); (3; d); and (4; d) each with 1=6 units. Each one of 1; 3; and 4 is allotted 1=6 units of d through the corresponding self-cycles. The overall random assignment is the same as the one obtained earlier for the PS mechanism. Remark 1: At any given step of the TTCfED algorithm, any two pseudo-agents who are assigned the same object must have equal units of that object. Indeed, consider any two pseudo-agents (i; x) and (i 0 ; x) of some step t n; with i; i 0 2 N and x 2 H : if (i; x) forms a trading cycle with some (unique) pseudo-agent of some agent j 2 Nnfi; i 0 g at some step t 0 < t; then (i 0 ; x) also must be forming a trading cycle with a pseudo-agent of agent j at step t 0 : Since the amount of trade within each trading cycle at a given step is the same, after the trades are carried out, the two pseudo-agents have to have equal units of x at the beginning of step t. 8

9 3.3 Random Priority [k] The random priority (RP) mechanism when applied to the problem in Example 1 induces an ordinally ine cient assignment (see the matrix below). A closer examination of this example shows why: The serial dictatorship (SD) associated with the ordering of agents allots agent 2 object b (the favorite object of 3), whereas the SD associated with the ordering of agents allots agent 3 object a (the favorite object of 2), causing e ciency loss in the induced random assignment. We argue that this phenomenon occurs because, loosely speaking, each particular SD con nes agents into a tight local competition which may result in a suboptimal overall assignment when put into a global perspective by the RP outcome. To better see this, let us consider the following relaxation of the procedure: for any given ordering f of agents, to obtain the outcome induced by f, suppose that we now consecutively apply SD f to a new problem obtained by introducing (same number of) extra copies of each object. Suppose for Example 1 that we were to introduce one extra copy of each object, and apply SD f twice consecutively until all objects are allocated. When f is , these two consecutive applications yield the allocations (1$a, 2$a, 3$b, 4$b) & (1$c, 2$d, 3$c, 4$d) in order. Suppose further that we were to do the same exercise for each of the 24 possible orderings, and now take the average of 2 24 SD outcomes to nd an overall random assignment. The resulting outcomes when one introduces one and two extra copies of each object are shown below: a b c d a b c d a b c d 1 11=24 1=12 1=4 5= =24 1= =24 3 1=12 5=12 1=3 1= =12 5=12 1=6 1 1=2 0 1=3 1=6 2 1= = =2 1=3 1= =2 1=3 1=6 1 35=72 2=72 23=72 12= =72 2= =72 3 2=72 34=72 24=72 12= =72 25=72 13=72 RP RP [2] RP [3] Observe that RP [2] is ordinally e cient whereas RP [3] su ers smaller e ciency loss than the RP outcome. We call the mechanism whose outcome is the random assignment that is induced as the average of kn! serial dictatorship outcomes where for each ordering of agents the associated SD is applied consecutively k times to a k replica of resources (until all objects are exhausted) as the random priority (RP) [k]. 4 Main Result Surprisingly, the outcomes of PS, TTCfED, and RP [2] for the random assignment problem given in Example 1 are the same. 10 In fact, this is not a mere coincidence. 10 Further, for PS and TTCfED mechanisms, the outcomes are not the same stepwise. For instance, at the end of step 2, PS allots 1=2 units of object d to agent 2, whereas TTCfED allots him 1=3 units of this object. Also, under PS, the amount of object any two agents are allotted at a given step is the same, which again is not true under TTCfED. 9

10 Theorem 1: The following mechanisms are equivalent: 11 (i) Probabilistic serial (ii) Top trading cycles from equal division (iii) Random priority [(n!) n ] The above equivalences all hint on a common intuition: for random assignment problems mechanisms tend to gain e ciency as they better utilize the richer probabilistic setup so as to mitigate any mishaps arising from the indivisibility of resources. The fact that we get an exact equivalence could be because the two anonymous mechanisms, TTCfED and RP [k], are perfectly synchronized with PS when they pick the right intrinsic parameters: for TTCfED, synchronization occurs when selected cycles are single and pairwise; for RP [k] this occurs when resources are replicated by a multiple of (n!) n : We close with a nal remark on RP [k]. Recall our earlier discussion about the (ordinal) ine ciency of RP. To follow up, when RP [k] is ordinally e cient, every ordering of agents induce the same k allocations for the k-replica of the initial resources. This, in turn, ensures global optimality of the local serial dictatorship outcomes. Proof of Theorem 1: We rst show (i)=(ii) and next that (i)=(iii). Fix a preference pro le <. Proof of (i)=(ii): Step 1: Comparison of the two algorithms. We rst claim that whichever favorite object an agent has at any given step of the PS algorithm, he has the same favorite object at the same step of the TTC algorithm (from equal division). For a given step t with 1 t n; let H t P S denote the set of objects which are still available at the end of step t of the PS algorithm, and H t T T C the set of objects which are still available at the end of step t of the TTC algorithm. Then it is su cient to show that H t P S = Ht T T C for all t with 1 t n: (However, it is not true [and therefore we do not claim] that the amount available of an object at the end of a given step is the same under the two algorithms.) Let us denote by k t x the number of agents each of whom has x as his favorite object at step t of the PS algorithm, and by l t x the same number corresponding to step t of the TTC algorithm. For each step of the PS algorithm, partition N into groups of agents according to their favorite objects. Let k t = max x2h k t x with 1 t n: Similarly, let l t = max x2h l t x with 1 t n: We prove our claim by induction. Let t = 1: Clearly, k 1 x = l 1 x for all x 2 H: If k 1 = l 1 = n; then this means under the PS algorithm all n agents eat away from the same object at step 1, and each gets 1=n units of it. For the same case, under the TTC algorithm each agent forms a self-cycle for the same object, and 11 For a given problem <, the necessary number of replications for part (iii) to still hold, which now depends on <, can be shown to be smaller than (n!) n : Note that RP [2] for the problem in Example 1 also coincides with PS. 10

11 again gets 1=n units of it. If k 1 1 = l 1 1 6= n; then for the PS algorithm, any object that gets exhausted at step 1 solves: min x2h 1 k 1 x (1) For the TTC algorithm, the object whose trade ends rst at step 1 solves: min x2h 1 nl 1 x (2) At the end of step 1 of the TTC algorithm, any agent whose favorite object is di erent than a minimizer of (2) has no units of that minimizer left. Any agent whose favorite object is a minimizer of (2) also has no units of that mimizer left since he is allotted all 1=n units of it through the self-cycle he forms. Hence, there can not remain any units of any minimizer of (2) at the end of step 1. All other objects are still available (possibly with fewer units). Then, since any solution to (1) is also a solution to (2), we have HP 1 S = H1 T T C : Now suppose that for any step 1 < t < t 0 ; we have HP t S = Ht T T C. This implies that kt = l t for all t t 0 : Let EP A t > 0 denote the amount of object eaten away by any agent at step t with 1 t n; of the PS algorithm. Let T P C t 0 denote the amount of object traded in any cycle at step t with 1 t n; of the TTC algorithm. We consider two cases: Case 1. For all t t 0 ; k t = l t 6= n (i.e., at each step there are at least two agents with di erent favorite objects): We claim that EP A t = nt P C t for any t t 0 : Note that EP A 1 = 1=k 1 and EP A 1 = 1=nl 1. So the claim clearly holds for t = 1: The amount of object that gets eaten away by any agent at step t t 0 of the PS algorithm is determined by: EP A t = min x2h t 1 P S 1 kxep 1 A 1 kx t 1 EP A t 1 kx t (3) where the term in the numerator is the amount of object x available at the beginning of step t; and k t x is the number of agents who eat away from object x at step t. Any object that gets exhausted at the end of step t is a minimizer of the problem in (3). All other objects in H t 1 P S are still available (possibly with fewer units as compared to the previous step). Similarly, the amount of object that is tradeded in any cycle at step t t 0 of the TTC algorithm is determined by: T P C t = min x2h t 1 T T C 1=n l 1 xt P C 1 l t 1 x T P C t 1 l t x (4) 11

12 where the term in the numerator (by Remark 1) is the amount of object x assigned to each one of those pseudo-agents who still have positive units of x at the beginning of step t (more precisely: each such agent is initially assigned 1=n units of object x; and at each step s he forms l s x trading cycles in each one of which he trades T P C s units of x for his favorite object available at that step) and k t x is the number of agents who form trading cycles to trade object x at step t. Since for any step t < t 0, H t P S = Ht T T C by the induction hypothesis, we have kt x = l t x for any object x 2 H t 1 P S = Ht 1 T T C and any step t t0. Then since EP A 1 = nt P C 1 ; and since any minimizer of the problem in (3) is also a minimizer of the problem in (4) for any step t t 0 ; it follows that EP A t = nt P C t for any t t 0 : For a given step t t 0 ; let x t be a solution to the minimization problems in (3) and (4). Using a similar argument as before, there can not remain any units of x t at the end of step t, and all other objects in H t 1 T T C are still available (possibly with fewer units as compared to the previous step). Thus, we conclude that H t0 P S = Ht0 T T C : Case 2. There is s t 0 such that k s = l s = n : Let s t 0 be the earliest step with k s = l s = n: Let x s be the favorite object of every agent at step s of both algorithms. Under both algorithms, at the end of step s object x s gets exhausted, and there is no change in the available units of all other objects. More precisely, since for any x 2 H; both k t x and l t x are non-decreasing in t as long as object x is still available, at the end of step s there has to be exactly a total of one unit available from each remaining object. Thus, HP s S = Hs T T C : Now we have a smaller version of the initial problem, and for any remaining step until step t 0 ; we can iteratively re-visit cases 1 and 2 to nally conclude that H t0 P S = Ht0 T T C : Step 2: Establishing the equivalence. By step 1 the objects (not necessarily the units) that are available at every step of both algorithms are the same. Therefore each agent has the same favorite object at any given step of both algorithms. We show that the two mechaisms select the same random assignment. Take any agent i 2 N and any object x 2 H. Suppose agent i eats away from object x from step t to step t 0 ; t 0 t of the PS algorithm. By step 1 this means object x is his favorite object from step t to step t 0 of the TTC algorithm as well. Note rst that for any step s such that t s < t 0 ; we have k s 1 6= n because otherwise object x would be completely exhausted at the end of step s. We consider two cases: Case 1. k t0 1 6= n : Agent i s random allotment of object x under the PS mechanism is given by: X P S i (<)[x] = EP A s (5) Agent i s random allotment of object x under the TTC mechanism is given by: t 0 s=t 12

13 T T C i (<)[x] = ( 1 n Xt 1 X kxt s P C s ) + (n kx)t s P C s (6) t 0 s=1 s=t where the rst term is the amount of x agent i is allotted through the self-cycle at step t (his initial assignment of x minus the amount of x he trades with other agents until step t), and the second term is the amount of x agent i is allotted through the trading cycles from step t to step t 0 (at any step s with t s t 0 ; agent i forms a trading cycle with each one of those agents who have a di erent favorite object at step s; making a total of n k s x trading cycles). Note that by case 1 of step 1, for any s with t s t 0 ; we have EP A s = nt P C s : Then rearranging (6), we have: T T C i (<)[x] = ( 1 n X X kxt s P C s ) + EP A s (6 ) t 0 t 0 s=1 s=t Since object x gets completely exhausted at the end of step t 0 ; there is no agent with a positive amount of x at the end of step t 0 : This means the rst term on the righthand side of (6 ) is zero establishing the equivalence of (5) and (6). Case 2. k t0 1 = n : Agent i s random allotment of object x under the PS mechanism is again given by (5). Agent i s random allotment of object x under the TTC mechanism is now given by: T T C i (<)[x] = ( 1 n Xt 1 tx 0 1 kxt s P C s ) + (n kx)t s P C s (7) s=1 s=t where we now do not include any terms into (7) for step t 0 since agent i does not participate in any trading cycles at this step (the only cycles at this step are the self-cycles formed by those agents whose favorite object is x only at this step of the algorithm.). The amount of object x allotted to agent i at step t 0 of the PS algorithm is given by: EP A t0 = 1 P t 0 1 s=1 ks xep A s n (8) Inserting (8) in (5) and rearranging, we have: P S i (<)[x] = ( 1 n t 1 X 0 1 tx 0 1 k s n xep A s ) + EP A s (9) s=1 s=t Using EP A s = nt P C s for all s with t s < t 0 by case 1 of step 1, we have 13

14 P S i (<)[x] = ( 1 n 1 Xt 1 tx 0 1 k s n xep A s ) + (n kx)t s P C s (9 ) s=1 s=t Finally, for any s with 1 s t 1; we have either EP A s = nt P C s (again by case 1 of step 1); or T P C s = 0: If T P C s = 0 for some s; with 1 s t 1; then k s x = 0: Then for any s with 1 s t 1; k s xep A s = nk s xt P C s : Inserting this into (9 ), we get the equivalence with (7). Proof of (i)=(iii): Take an ordering f of agents. RP f is the random allotment induced by all allocations obtained from (n!) n consecutive applications of SD f to the given problem when there are initially [(n!) n ] copies of each object. We next de ne step 1 of this procedure to be those rst consecutive applications of SD f at the end of which one runs out of all copies of some object for the rst time. Similarly, de ne step t to be those rst consecutive applications of SD f since step t 1 at the end of which one runs out of all copies of some object for the rst time. Note that the procudure has at most n steps. Denote by m t0 x the number of agents each of whom has x as his favorite object at application t 0 of SD f. Let m 1 = max x2h m 1 x: Note that for each x 2 H; m t0 x is the same for all t 0 (n!)n m 1 2 Z: Let M t x denote the number of agents whose favorite object at step t is x. (Note that it is well-de ned by the de nition of a step.) For a given step t with 1 t n; let H t RP denote the set of objects which are still available at the end of step t. Any object whose all copies run out at the rst step solves: (n!) n min x2h m 1 x Similarly, any object whose all copies run out at step t solves the following equation, and its solution is the number of applications of SD f during step t. NSD t = min x2h t 1 RP (n!) n MxNSD 1 1 Mx t 1 NSD t 1 Mx t (10) where the term in the numerator is the number of copies of object x available at the beginning of step t: Note that for each x 2 H; Mx t 2 f1; 2; : : : ; ng for any t n: But then, by our choice of the number of initial copies of each object, NSD t 2 Z for every t n: Take any agent i 2 N and any object x 2 H. Suppose object x is agent i s favorite object from step t to step t 0 ; t 0 t: Then the contribution of (n!) n consecutive applications of SD f to RP i (<)[x] is given by 1=(n!) n P t 0 s=t NSDs (Eq. 11). Since RP i (<)[x] is the average of n! contributions (one for each ordering of agents), and f was arbitrary, comparing equations (10) and (11) 14

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