# Farsighted Collusion in Stable Marriage Problem

## TL;DR

This paper introduces a linear-time method to find a unique, farsightedly stable collusion matching in the Stable Marriage Problem, demonstrating its strength over traditional equilibrium concepts and simplifying previous complex results.

## Contribution

It presents a novel, simplified approach to identify a farsightedly stable collusion outcome in the stable marriage game, improving understanding of proposers' strategic cooperation.

## Key findings

- The optimal collusion matching is unique and can be found in linear time.
- This outcome is stronger than a Strong Nash Equilibrium.
- Prior results on proposers' collusion are often unrealistic.

## Abstract

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.