On the exhaustiveness of truncation and dropping strategies in many-to-many matching markets

Abstract

We consider two-sided many-to-many matching markets in which each worker may work for multiple firms and each firm may hire multiple workers. We study individual and group manipulations in centralized markets that employ (pairwise) stable mechanisms and that require participants to submit rank order lists of agents on the other side of the market. We are interested in simple preference manipulations that have been reported and studied in empirical and theoretical work: truncation strategies, which are the lists obtained by removing a tail of least preferred partners from a preference list, and the more general dropping strategies, which are the lists obtained by only removing partners from a preference list (i.e., no reshuffling). We study when truncation / dropping strategies are exhaustive for a group of agents on the same side of the market, i.e., when each match resulting from preference manipulations can be replicated or improved upon by some truncation / dropping strategies. We prove that for each stable mechanism, truncation strategies are exhaustive for each agent with quota 1 (Theorem 1). We show that this result cannot be extended neither to group manipulations (even when all quotas equal 1-Example 1), nor to individual manipulations when the agent's quota is larger than 1 (even when all other agents' quotas equal 1-Example 2). Finally, we prove that for each stable mechanism, dropping strategies are exhaustive for each group of agents on the same side of the market (Theorem 2), i.e., independently of the quotas.

Static versus dynamic deferred acceptance in school choice: Theory and experiment

“…Then, since the set of Nash equilibrium outcomes in the static game is equal to set of stable matchings whenever each school has exactly one seat (Gale and Sotomayor [16], Roth [27]), we can conclude that the sets of Nash equilibrium outcomes coincide for both implementations in our setting. Moreover, it is well known that the static mechanism is manipulable and that the set of "truncation" strategies, whereby the student removes a tail of least preferred schools from her true preference list, is strategically exhaustive in the sense that for each strategy a student may use, the induced match can be replicated or improved upon by some truncation of her true preferences (see, e.g., Jaramillo et al [21] and Roth and Vande Vate [28]). Nevertheless, in the dynamic game we can construct strategy profiles so that for all truncations of the true preferences, going down the list with respect to the truncation is not a best reply.…”

A many-to-many ‘rural hospital theorem’

“…In other words, for each S ⊆ S a , Ch(S, P a ) is agent a's most preferred subset of S according to P a . One easily verifies that Ch satisfies consistency (Alkan, 2002), 14 i.e., for each pair S, T ⊆ S a ,…”

Accomplice Manipulation of the Deferred Acceptance Algorithm

“…The impossibility result of Roth (1982) on the con ict between strategyproofness and stability has led to extensive follow-up research. Much of the earlier work in this direction focused on truncation manipulation (Gale and Sotomayor, 1985a;Roth and Rothblum, 1999;Coles and Shorrer, 2014;Jaramillo et al, 2014), where the misreported preference list is required to be a pre x of the true list. In the context of accomplice manipulation, however, the truncation manipulation problem becomes trivial.…”