Cooperative Games in Game Theory
Stable matching is an important concept in cooperative game theory, especially in the context of matching problems. The goal of stable matching is to find a stable matching between two groups of agents, where each agent has a preference over the agents in the other group. Stability means that there is no pair of agents that prefer each other to their current partners. The concept of stable matching was first introduced by Gale and Shapley in 1962, and their algorithm for finding stable matching is now known as the Gale-Shapley algorithm.
To illustrate stable matching, consider the classic example of matching medical students to hospitals. Suppose that there are n students and n hospitals. Each student has a preference ranking over the hospitals, and each hospital has a preference ranking over the students. The goal is to find a stable matching between the students and hospitals, where each student is matched to a hospital and each hospital is matched to a student.
The Gale-Shapley algorithm works as follows. First, each student applies to their most preferred hospital. Each hospital then selects the most preferred student among those who have applied. If multiple hospitals prefer the same student, the student chooses their most preferred hospital among those that have made offers. Each rejected student applies to their next preferred hospital, and the process repeats until a stable matching is found.
There are many applications of stable matching beyond the medical student-hospital example, such as matching job seekers to employers or matching organ donors to recipients. The Gale-Shapley algorithm has been shown to always produce a stable matching, and furthermore, the stable matching that it produces is optimal for one of the groups. For example, in the medical student-hospital example, the stable matching produced by the Gale-Shapley algorithm is optimal for the students in the sense that no student can be matched to a hospital that is lower on their preference ranking than the one they are matched to.
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