Variety-Seeking Jump Games on Graphs

TL;DR

This paper studies a class of graph-based jump games where agents seek to maximize neighborhood diversity, analyzing equilibrium existence, potential game conditions, and efficiency bounds under various graph structures and diversity measures.

Contribution

It identifies conditions under which the jump game is a potential game, proves equilibrium existence on specific graph classes, and establishes bounds on the price of anarchy for diversity metrics.

Findings

Jump games may have cycles but are potential games under specific conditions.

Equilibria always exist on trees, cylinders, and tori.

Tight bounds are provided for the price of anarchy regarding social welfare and diversity.

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).