Convergence and Hardness of Strategic Schelling Segregation

Abstract

The phenomenon of residential segregation was captured by Schelling's famous segregation model where two types of agents are placed on a grid and an agent is content with her location if the fraction of her neighbors which have the same type as her is at least τ , for some 0 < τ < 1. Discontent agents simply swap their location with a randomly chosen other discontent agent or jump to a random empty cell.We analyze a generalized game-theoretic model of Schelling segregation which allows more than two agent types and more general underlying graphs modeling the residential area. For this we show that both aspects heavily influence the dynamic properties and the tractability of finding an optimal placement. We map the boundary of when improving response dynamics (IRD), i.e., the natural approach for finding equilibrium states, are guaranteed to converge. For this we prove several sharp threshold results where guaranteed IRD convergence suddenly turns into the strongest possible non-convergence result: a violation of weak acyclicity. In particular, we show such threshold results also for Schelling's original model, which is in contrast to the standard assumption in many empirical papers. Furthermore, we show that in case of convergence, IRD find an equilibrium in O(m) steps, where m is the number of edges in the underlying graph and show that this bound is met in empirical simulations starting from random initial agent placements.

Swap Stability in Schelling Games on Graphs

“…For both types of games (swap and jump), Chauhan et al identify values of τ for which the best response dynamics of the agents leads to an equilibrium when the topology is a ring or a regular graph. Echzell et al (2019) strengthen these results and extend them to more than two agent types, as well as study the complexity of computing assignments that maximize the number of happy agents. Elkind et al (2019) consider a similar model with k types; however, they treat agents' location preferences differently from Chauhan et al Namely, in their model each agent is either stubborn (i.e., has a preferred location and is unwilling to move) or strategic (i.e., aims to maximize the fraction of her neighbors that are of her own type; this corresponds to setting τ = 1 in the model of Chauhan et.…”

“…For an introduction to the model and a survey of its many variants, we refer the reader to the book of Easley and Kleinberg (2010), and the papers by Brandt et al (2012) and Immorlica et al (2017). Besides the closely related papers by Chauhan, Lenzner, and Molitor (2018), Elkind et al (2019) and Echzell et al (2019), another work that is similar in spirit is a recent paper by Massand and Simon (2019), who study swap stability in games where a set of items is to be allocated among agents who are connected via a social network, so that each agent gets one item, and her utility depends on the items she and her neighbors in the network get; however, their results are not directly applicable to our setting. Also, Schelling games share a number of properties with hedonic games (Drèze and Greenberg 1980;Bogomolnaia and Jackson 2002), and in particular, with fractional hedonic games (Aziz et al 2019) and hedonic diversity games (Bredereck, Elkind, and Igarashi 2019).…”

“…We begin by discussing the existence of equilibria in swap games. The work of Echzell et al (2019) implies that when the topology is a regular graph, at least one equilibrium assignment is guaranteed to exist. Furthermore, using a potential function similar to the one proposed by Elkind et al (2019) for jump games, we can show that equilibria always exist when the topology is a graph of maximum degree 2; we omit the details due to space constraints.…”

“…For k-swap games with topology that is a δ-regular graph (in which all nodes have degree δ), we show an upper bound of 1 on the social price of stability. To this end we employ a potential function that is similar to the one defined by Chauhan, Lenzner, and Molitor (2018) and Echzell et al (2019) to show the existence of equilibria in such games.…”

“…So far, only a few papers have followed such a game-theoretic approach. In particular, Zhang (2004b) considered a game where the agents optimize a single-peaked utility function, and very recently, Chauhan, Lenzner, and Molitor (2018), and Echzell et al (2019) studied strategic settings that are closer to the original model of Schelling, but capture more than two agent types and richer graph topologies.…”