Paths to stable allocations

TL;DR

This paper investigates the dynamics of reaching stable allocations in bipartite graphs, showing that random better response algorithms always converge efficiently, while best response algorithms may require exponential time, except in correlated markets.

Contribution

It provides the first analysis of uncoordinated response dynamics in stable allocations, demonstrating polynomial convergence for better responses and exponential complexity for best responses in general.

Findings

Random better response algorithms converge with probability one.

Best response algorithms may require exponential time in general.

In correlated markets, random best responses converge in expected polynomial time.

Abstract

The stable allocation problem is one of the broadest extensions of the well-known stable marriage problem. In an allocation problem, edges of a bipartite graph have capacities and vertices have quotas to fill. Here we investigate the case of uncoordinated processes in stable allocation instances. In this setting, a feasible allocation is given and the aim is to reach a stable allocation by raising the value of the allocation along blocking edges and reducing it on worse edges if needed. Do such myopic changes lead to a stable solution? In our present work, we analyze both better and best response dynamics from an algorithmic point of view. With the help of two deterministic algorithms we show that random procedures reach a stable solution with probability one for all rational input data in both cases. Surprisingly, while there is a polynomial path to stability when better response strategies are played (even for irrational input data), the more intuitive best response steps may require exponential time. We also study the special case of correlated markets. There, random best response strategies lead to a stable allocation in expected polynomial time.