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 \(\mathcal{NP}\)-hard. 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 \(\Omega(\sqrt{n/k})\) and \(O(\sqrt{kn})\), where n is the number of vertices and k the number of players.
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Dürr, C., Thang, N.K. (2007). Nash Equilibria in Voronoi Games on Graphs. In: Arge, L., Hoffmann, M., Welzl, E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75520-3_4
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DOI: https://doi.org/10.1007/978-3-540-75520-3_4
Publisher Name: Springer, Berlin, Heidelberg
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