Nash Equilibria in Voronoi Games on Graphs

Abstract

In this paper we study a game where every player is to choose a vertex (facility) in a given undirected graph. All vertices (customers) are then assigned to closest facilities and a player's payoff is the number of customers assigned to it. We show that deciding the existence of a Nash equilibrium for a given graph is N P-hard which to our knowledge is the first result of this kind for a zero-sum game. We also introduce a new measure, the social cost discrepancy, defined as the ratio of the costs between the worst and the best Nash equilibria. We show that the social cost discrepancy in our game is Ω( n/k) and O( √ kn), where n is the number of vertices and k the number of players.

Multi-player Diffusion Games on Graph Classes

“…Some interesting generalizations of this model are the model by Tzoumas et al [12], who considered a more complex underlying diffusion process, and the model studied by Etesami and Basar [3], allowing each player to choose multiple vertices. Dürr and Thang [2] and Mavronicolas et al [7] studied so-called Voronoi games, which are closely related to our model (but not similar; there, players can share vertices). Recently, Ito et al [4] considered the competitive diffusion game on weighted graphs, including negative weights.…”

Infrastructure topology optimization under competition through cross-entropy

“…To give a few up to date illustrations, Albareda-Sambola et al consider multi period location problems where at each time instant the decision maker decides which existing facilities should be closed and where new facilities should be opened [1]. Dürr and Nguyen tackle the problem of placing facilities on the nodes of a metric network inhabited by a fixed number of autonomous self-interested agents [9], [32]. They provide bounds for the strategyproof mechanisms where none of the players can be better off by misreporting his facility locations.…”

“…They provide bounds for the strategyproof mechanisms where none of the players can be better off by misreporting his facility locations. The agents' choices and therefore the optimal facility locations are based solely on the distance between the agents and the available facilities [9], [32]. Borndörfer et al consider the problem of optimizing the spatial distribution of controls over a motorway in order to maximize the sum of the revenues generated by transit fees imposed to the drivers assuming evasion is possible [5].…”

A note on competitive diffusion through social networks

“…Note that some previous papers did consider the problem of choosing an influential set of users as an optimization problem (see, e.g., [6]), but not in a competitive game-theoretic setting. Other papers, which deal with Voronoi games on graphs, provide a game-theoretic study of a facility location problem that does not involve a diffusion process, where rather each vertex is assigned to the closest agent and the utility of an agent is the number of vertices assigned to it (see, e.g., [3,7]). …”