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Strategyproof Exchange with Multiple Private Endowments Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence Strategyproof Exchange with Multiple Private Endowments Taiki Todo, Haixin Sun, Makoto Yokoo Dept. of Informatics Kyushu University Motooka 744, Fukuoka, JAPAN ftodo, sunhaixin, [email protected] Abstract efficient and individually rational. Moreover, it is the only We study a mechanism design problem for exchange rule that satisfies those three properties (Ma 1994). Due to economies where each agent is initially endowed with these advantages, TTC has been attracting much attention a set of indivisible goods and side payments are not al- from both economists and computer scientists. lowed. We assume each agent can withhold some en- In this paper we consider exchange economies where each dowments, as well as misreport her preference. Under this assumption, strategyproofness requires that for each agent is endowed with a set of multiple goods, instead of agent, reporting her true preference with revealing all a single good. Under that model, Pareto efficiency, indi- her endowments is a dominant strategy, and thus implies vidual rationality, and strategyproofness cannot be simulta- individual rationality. Our objective in this paper is to neously satisfied (Sonmez¨ 1999). Therefore, one main re- analyze the effect of such private ownership in exchange search direction on exchanges with multiple endowments is economies with multiple endowments. As fundamental to achieve strategyproof rules that guarantee a limited notion results, we first show that the revelation principle holds of efficiency. In particular, Papai´ (2003) defined a class of under a natural assumption and that strategyproofness exchange rules and gave a characterization by strategyproof- and Pareto efficiency are incompatible even under the ness and individual rationality with some other weak effi- lexicographic preference domain. We then propose a ciency requirements. class of exchange rules, each of which has a correspond- ing directed graph to prescribe possible trades, and pro- Another important assumption in this work is that each vide necessary and sufficient conditions on the graph agent’s endowments are also private information, and agents structure so that they satisfy strategyproofness. reveal to an exchange rule their endowments, as well as their preferences. Since an exchange rule cannot exactly observe Introduction which agent really owns which goods, guaranteeing strat- The housing market problem, introduced by Shapley and egyproofness seems much harder than in a traditional case Scarf (1974), is a fundamental exchange model where each where ownerships are recognized. Indeed, in environments good is indivisible and monetary transfers are not allowed. with private (and multiple) goods, the possibility of various Precisely, there is a group of agents, each of whom is ini- manipulations via endowments has been pointed out (Postle- tially endowed with one good, say a house, and has a strict waite 1979; Atlamaz and Klaus 2007), such as hiding (or ordering, so-called a preference, over the set of all goods withholding). Furthermore, in some anonymous environ- owned by the agents. An exchange rule (aka., a mechanism) ments, splitting an account into multiple ones might also be takes the preferences revealed by the agents as input and de- problematic (Moulin 2008). termines which goods are traded. The model has been ap- In this paper we first focus on hiding manipulations as plied to various market environments, such as kidney ex- well as misreporting preferences, since they can be easily ¨ changes (Roth, Sonmez,¨ and Unver 2004) and on-campus done by only one agent without any side communication university housing markets (Chen and Sonmez¨ 2002). with other agents. We say an exchange rule is strategyproof One desirable property for a mechanism is strategyproof- if for every agent, reporting her true preference with reveal- ness, which requires that for every agent, the truth revela- ing all her endowments (i.e., not hiding anything) to the rule tion of her private information, say a preference over a set of is a dominant strategy. The main objective of this paper is houses, to the mechanism is a dominant strategy. From the to analyze the effect of such private ownership in exchange revelation principle, it is without loss of generality to focus economies with multiple endowments. More precisely, we on strategyproof direct revelation mechanisms if we are only clarify how the space of possible exchanges by strategyproof interested in exchange rules with dominant strategy equilib- rules shrinks due to the lack of information on ownership. ria. Fortunately, for the housing market problem, Gale’s Top- One of the most closely related works to this paper is Atla- Trading-Cycle (TTC) rule is strategyproof, as well as Pareto maz and Klaus (2007), which also investigates the effect of Copyright c 2014, Association for the Advancement of Artificial hiding manipulations. However, they only focus on hidings Intelligence (www.aaai.org). All rights reserved. and ignore any misreports of preferences. 805 Our Results We investigate exchange problems with mul- Our Model tiple private endowments. In our model an exchange rule is There is a set of agents N = f1; : : : ; ng and a set of in- strategyproof if, for each agent, reporting her true preference divisible goods K in the world. Each agent i 2 N is en- with revealing all her endowments is a dominant strategy. dowed with a set of goods, or endowments, wi ⊂ K. Let The definition implies individual rationality, since hiding all w = (wi)i2N be an endowment distribution to N, satisfy- endowments is equivalent to not participating. We prove that S ing i2N wi ⊆ K and wi \ wj = ? for any pair i; j 2 N. the revelation principle holds under a natural assumption, The endowment distribution is chosen, by nature, from the which is regarded as a variation of Yu’s result (Yu 2011). set W of all possible endowment distributions. We also show that strategyproofness and Pareto efficiency Each agent i 2 N has a linear ordering, or preference, are incompatible, even if we focus on a “smallest” prefer- Ri, over the set of all possible bundles L ⊆ K. Let R ence domain called the lexicographic preference domain. denote a set of all admissible preferences, or a preference 0 0 We next introduce a class of exchange rules called the domain. Given Ri 2 R and a pair L; L ⊆ K, let LRiL 0 agreement cycles rules. They are motivated from segmented denote that L is weakly preferred to L at Ri. We assume trading cycles rules (Papai´ 2003), so that each of them is preferences are strict, i.e., for any pair L; L0 6= L, either 0 0 defined based on a corresponding directed graph called a LPiL or L PiL, where Pi indicates the strict component of 0 0 0 one-for-one trading possibility graph. The performances of Ri. Therefore, LRiL but not LPiL implies L = L . We these rules are fairly better than trivial strategyproof rules. also assume fkgPi? for any i 2 N, any Ri 2 R, and any Under each preference domain, we provide a necessary and k 2 K. Let R = (Ri)i2N denote a preference profile of N sufficient condition on the structure of a one-for-one trading and R−i = (Rj)j6=i denote a preference profile of N n fig. possibility graph for the exchange rule to be strategyproof. In summary, each agent i has two private information, We also show that these rules are split-proof if and only if wi ⊂ K and Ri 2 R. We refer to θi = (wi;Ri) 2 Θw;i as they are strategyproof, where split-proofness requires that i’s type, where Θw;i denotes the set of all reportable types for each agent, using only one identity is a dominant strat- of i when an endowment distribution w is chosen. More pre- egy, even if it can use multiple identities and join an ex- cisely, for a given w, let W (w; i) denote the set of reportable change rule multiple times under them. S goods s.t. i2N W (w; i) = K, W (w; i)\W (w; j) = ? for 1 any pair i; j 2 N, and W (w; i) ⊇ wi for any i 2 N . With W (w;i) Related Works The Shapley-Scarf housing market (Shap- this notation, Θw;i := 2 ×R. Note that at this moment ley and Scarf 1974) has various applications, such as room- there is no restriction on revealed goods w^i of i, as long as mate problems and kidney exchanges. Ma (1994) charac- they are in W (w; i). Let θ 2 Θw = ×i2N Θw;i be a type terized TTC for the problem by strategyproofness, Pareto profile of N and θ−i = (θj)j6=i 2 Θw;−i be a type profile of efficiency, and individual rationality. Roth and Postlewaite agents except i, where Θw;−i = ×j6=iΘw;j. (1977) showed that it always chooses the unique core assign- We then formally define a set of possible assignments. ment. A quite similar model is assignment problems (Papai´ When a set of goods L ⊆ K is reported by agents, let XL 2000), where there are no initial endowments and monetary be a set of the possible assignments of goods to the agents, assignment x 2 X S x = L transfers are still not allowed. s.t. any L satisfies i2N i and xi \ xj = ? for any pair i; j 2 N. A natural extension of the Shapley-Scarf model is one ^ where each agent is allowed to initially have multiple goods. An exchange rule ' maps a reported type profile θ = ^ When at least one agent initially has more than one good, no (w ^i; Ri) 2 Θw into a possible assignment x 2 Xw^, where S ^ rule is strategyproof, Pareto efficient, and individually ra- w^ = i2N w^i.
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