Diversity-seeking swap games in networks

TL;DR

The paper studies diversity-seeking swap games on networks, introducing three utilities $U_b$, $U_\{\\#\\}$, and $U_\\tau$ and four global diversity measures $DOI$, $CE$, $NV$, $EV$. It proves the swap games are potential games, ensuring convergence to equilibria, and derives asymptotically tight PoA bounds for each measure on general graphs, plus topology-specific refinements for cycles, cylinders, and tori; it also provides exact constants in several cases and constructs graphs $G^*$ that realize the bounds. Complementing theory, the authors perform simulations from random and segregated initial placements, showing that segregation is largely removed under diversity-seeking behavior, though achieving strong diversity like high neighborhood variety and evenness remains challenging. The results illuminate how local, diversity-oriented incentives translate into global diversity patterns and offer guidance for designing mechanisms to promote diverse neighborhoods while highlighting remaining gaps in achieving uniform diversity across all measures.

Abstract

Schelling games use a game-theoretic approach to study the phenomenon of residential segregation as originally modeled by Schelling. Inspired by the recent increase in the number of people and businesses preferring and promoting diversity, we propose swap games under three diversity-seeking utility functions: the binary utility of an agent is 1 if it has a neighbor of a different type, and 0 otherwise; the difference-seeking utility of an agent is equal to the number of its neighbors of a different type; the variety-seeking utility of an agent is equal to the number of types different from its own in its neighborhood. We consider four global measures of diversity: degree of integration, number of colorful edges, neighborhood variety, and evenness, and prove asymptotically tight or almost tight bounds on the price of anarchy with respect to these measures on both general graphs, as well as on cycles, cylinders, and tori that model residential neighborhoods. We complement our theoretical results with simulations of our swap games starting either from random placements of agents, or from segregated placements. Our simulation results are generally consistent with our theoretical results, showing that segregation is effectively removed when agents are diversity-seeking; however strong diversity, such as measured by neighborhood variety and evenness, is harder to achieve by our swap games.

Figures (5)

• Figure 1: Non-achievable and achievable instances.
• Figure 2: The $\delta$-regular graph $G^*$ for $t\ge 3$.
• Figure 4: The first row is for random input, the second row for Schelling input. From left to right, the average $\mathop{\mathrm{DOI}}\nolimits, \mathop{\mathrm{CE}}\nolimits, \mathop{\mathrm{NV}}\nolimits, \mathop{\mathrm{EV}}\nolimits$ (normalized) values of the equilibrium assignment reached by the swap games under $U_b$, $U_\#$, and $U_\tau$ are plotted. The values of the corresponding diversity measures present already in the inputs are also plotted.
• Figure 5: The first row is for random inputs, the second row is for Schelling inputs. From left to right, the average fraction of agents with 1, 2, 3, and 4 colorful edges incident on them at equilibrium under $U_b$, $U_\#$, and $U_\tau$, as well as the value in the input are plotted.
• Figure 6: The first row is for random inputs, the second row is for Schelling inputs. From left to right, the average fraction of agents with 1, 2, 3, and 4 types of agents in their neighborhood at equilibrium under $U_b$, $U_\#$, and $U_\tau$, as well as the value in the input are plotted.

Theorems & Definitions (39)

• Corollary 2.1
• proof
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Corollary 3.4
• Lemma 3.5