Tolerance is Necessary for Stability: Single-Peaked Swap Schelling Games

TL;DR

This paper introduces a game-theoretic model for residential segregation where agents prefer mixed neighborhoods, showing that tolerance is crucial for stable equilibria and analyzing the efficiency of these equilibria.

Contribution

It models non-monotone utility functions in Schelling games, demonstrating the necessity of tolerance for equilibrium existence and providing bounds on equilibrium quality.

Findings

Tolerance is necessary for equilibrium existence on certain graphs.

Bounds on Price of Anarchy and Price of Stability are established.

Price of Stability remains constant on bipartite and almost regular graphs.

Abstract

Residential segregation in metropolitan areas is a phenomenon that can be observed all over the world. Recently, this was investigated via game-theoretic models. There, selfish agents of two types are equipped with a monotone utility function that ensures higher utility if an agent has more same-type neighbors. The agents strategically choose their location on a given graph that serves as residential area to maximize their utility. However, sociological polls suggest that real-world agents are actually favoring mixed-type neighborhoods, and hence should be modeled via non-monotone utility functions. To address this, we study Swap Schelling Games with single-peaked utility functions. Our main finding is that tolerance, i.e., agents favoring fifty-fifty neighborhoods or being in the minority, is necessary for equilibrium existence on almost regular or bipartite graphs. Regarding the quality of equilibria, we derive (almost) tight bounds on the Price of Anarchy and the Price of Stability. In particular, we show that the latter is constant on bipartite and almost regular graphs.