On the existence of stable contract systems

TL;DR

The paper extends Gale–Shapley stability to generalized contract systems with two agents and Plott choice functions, clarifying when stable systems exist. It develops two complementary existence paradigms—ample systems and modest systems—each realized through fixed-point dynamics—$D_F(WB)\subseteq B$ and $Q\subseteq W(D_F(Q))$—and connects them via Desirability operators $D_C$. A key technical device is the Desirability operator, enabling reconstruction of choice functions from desirability and providing a unified lens on stability. Collectively, the work broadens stability results beyond bipartite graphs, offering dual, non-constructive and constructive routes to stable contract systems with clear implications for generalized matching and contract design.

Abstract

In 1962, Gale and Shapley \cite{GS} introduced the concept of stable marriages and proved their existence. Since then, the statement of the stability problem has been highly generalized. And a lot of proofs has emerged for the existence in these more general statements. It's time to review them and identify the similarities and differences. First, we will briefly discuss the classical case, because the existence proofs in the general case grew out of it. Or rather, from the idea of "deferred acceptance". When the best of the proposed contracts is temporarily retained until a better offer is received.