On the parameterized complexity of manipulating Top Trading Cycles
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We study the problem of exchange when 1) agents are endowed with heterogeneous indivisible objects, and 2) there is no money. In general, no rule satisfies the three central properties Pareto-efficiency, individual rationality, and strategy-proofness Sonmez1999. Recently, it was shown that Top Trading Cycles is -hard to manipulate FujitaEA2015, a relaxation of strategy-proofness. However, parameterized complexity is a more appropriate framework for this and other economic settings. Certain aspects of the problem - number of objects each agent brings to the table, goods up for auction, candidates in an election consandlang2007, legislative figures to influence christian2007complexity - may face natural bounds or are fixed as the problem grows. We take a parameterized complexity approach to indivisible goods exchange for the first time. Our results represent good and bad news for TTC. When the size of the endowments k is a fixed constant, we show that the computational task of manipulating TTC can be performed in polynomial time. On the other hand, we show that this parameterized problem is [1]-hard, and therefore unlikely to be fixed parameter tractable.
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