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, dropping strategies are exhaustive for each group of agents on the same side of the market (Theorem 1), i.e., independently of the quotas. Then, we show that for each stable mechanism, truncation strategies are exhaustive for each agent with quota 1 (Theorem 2). Finally, we show that this result cannot be extended neither to individual manipulations when the agent's quota is larger than 1 (even when all other agents' quotas equal 1 -Example 1), nor to group manipulations (even when all quotas equal 1 -Example 2).

Coalition Manipulation of Gale-Shapley Algorithm

“…Remark 2. Note that this result is different from the exhaustiveness result in (Jaramillo, Kayı, and Klijn 2014), since ex-haustiveness only requires that ∀μ ∈ S A (P ), there exists a truncation manipulation such that the induced matching is weakly preferred by the manipulators.…”

“…Knuth, Motwani, and Pittel (1990) show that the number of different partner that a woman can have in all stable matchings is between 1 2 − ln n and (1 + ) ln n, where n is the number of men and is a positive constant. Jaramillo, Kayı, and Klijn (2014) study possible manipulations by the women in a many-to-many setting. They consider the socalled dropping strategies, where women are allowed to strategically remove some men in their true preference lists but cannot shuffle their lists.…”

Equilibria Under Deferred Acceptance: Dropping Strategies, Filled Positions, and Welfare (Equilibrios bajo el mecanismo de aceptaci´on diferida: Estrategias de eliminaci´on, posiciones ocupadas y bienestar)

“…Stability does not depend on the particular responsive extensions of the agents' preferences over individual acceptable partners. 7 A mechanism assigns a matching to each market. We assume that quotas are commonly known by the agents (because, for instance, the quotas are determined by law).…”

“…Since we focus on ϕ S , we assume that students are truthful and that hospitals are the only strategic agents. Henceforth we fix and suppress P S , which in particular leads 7 Note that the set of stable matchings does not depend on the agents' orderings of the (individual) unacceptable partners either.…”