Farsighted Collusion in Stable Marriage Problem
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The Stable Marriage Problem, as proposed by Gale and Shapley, considers producing a bipartite matching between two equally sized sets of boys (proposers) and respectively girls (acceptors), each member having a total preference order over the other set, such that the outcome is stable. In this paper we consider the Game directly induced by this problem and analyze the case when proposers collude. We present a linear time method for determining the unique optimal collusion matching which is farsightedly stable, under the following assumptions: (i) the sole utility in the Game is the rank of the match in own preference list (in particular, proposers are indifferent as to how other proposers fare); (ii) proposers make proposals iff farsightedly such plays would strictly improve their own outcome (thus proposers cooperate by refraining from making proposals which can only harm others, but not strictly help them; also, they cannot make concessions to others which harm themselves). We argue that this optimal outcome is actually stronger than a Strong Nash Equilibrium - no alternative feasible coalition exists which can offer at least one member a strictly better outcome under these assumptions.We also show why some prior results pertaining to collusion of proposers do not always yield a realistic outcome. The results in this paper are an independent rediscovery of results by Jun Wako (2010), derived in a simpler fashion and phrased such that less jargon is employed.
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