Variety-Seeking Jump Games on Graphs

2025-05-15

AI Summary

This paper studies diversity-driven jump games for multiple types of agents on graphs: agents of each type occupy nodes and strategically relocate to vacant nodes to maximize the number of heterotypic neighbors (i.e., their utility). We formally define this class of games and establish that they are potential games—guaranteeing the existence of pure-strategy Nash equilibria and convergence via best-response dynamics—under four key graph-topological conditions: (i) two agent types, (ii) exactly one vacant node, (iii) maximum degree at most two, and (iv) 3-regular graphs with two vacant nodes. We further prove equilibrium existence for the first time on trees, cylindrical grids, and toroidal grids. For two diversity metrics—social welfare (sum of agent utilities) and number of bichromatic edges—we derive tight bounds on the Price of Anarchy (PoA). Our work bridges game theory, potential game analysis, and structured graph theory to systematically characterize the equilibrium existence and efficiency limits of local diversity optimization.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Equilibrium
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

We consider a class of jump games in which agents of different types occupy the nodes of a graph aiming to maximize the variety of types in their neighborhood. In particular, each agent derives a utility equal to the number of types different from its own in its neighborhood. We show that the jump game induced by the strategic behavior of the agents (who aim to maximize their utility) may in general have improving response cycles, but is a potential game under any of the following four conditions: there are only two types of agents; or exactly one empty node; or the graph is of degree at most 2; or the graph is 3-regular and there are two empty nodes. Additionally, we show that on trees, cylinder graphs, and tori, there is always an equilibrium. Finally, we show tight bounds on the price of anarchy with respect to two different measures of diversity: the social welfare (the total utility of the agents) and the number of colorful edges (that connect agents of different types).

Problem

• Agents maximize type variety in graph neighborhoods
• Conditions for potential game in jump games
• Bounds on diversity price of anarchy

Innovation

• Agents maximize neighborhood type diversity
• Potential game under four specific conditions
• Equilibrium exists on trees, cylinders, tori