A short way to the stability

TL;DR

This paper presents a concise method for deriving the core results of stability theory in bipartite markets, simplifying the process through the use of the desirability operator and ample contract systems.

Contribution

It introduces a short, uniform approach to stability theory, emphasizing the use of the desirability operator and ample systems to streamline existence and lattice results.

Findings

Simplified proof of stability existence and lattice results

Introduction of the desirability operator for uniformity

Reduction of complex bipartite problems to two-agent cases

Abstract

A longer and more correct title is `a short and direct path to the theory of stable contract systems in a bipartite market'. There is no new meaningful results in the article. It is dedicated to the presentation of a short method for obtaining the main body of stability theory: existence, polarization, and latticing. The brevity and uniformity are achieved through the use of the desirability operator (Section 1) and, most importantly, the successful notion of an ample system of contracts (Section 3). The use of the latter radically simplifies the problem of the existence of fixed points. The general bipartite problem (with many agents using Plott choice functions) is reduced easily by the aggregation to the case of two agents (see [2, 5]). Therefore, further, we restrict ourselves to the case of two agents (the Worker and the Firm) and a large set E of contracts between them.