By Kulikov A. S.

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Therefore r = O(1). Summing (1) over all stars with associated partition W , we obtain cost(f ) ≤ cost(f )+cn, for some constant c. Remark that ) the social cost of any equilibrium is at least n − k. Hence, cost(f cost(f ) = O(n). 28 6 C. K. Thang Open Problems It would be interesting to close the gap between the lower and upper bounds for the social cost discrepancy. The price of anarchy is still to be studied. Just notice that it can be as large as Ω(n), as for the star graph and k = n − 1 players: The unique Nash equilibria locates all players in the center, while the optimum is to place every player on a distinct leaf.

Whereas most papers about these games [3,5] study the existence of a winning strategy, or computing the best strategy for a player, we study in this paper the Nash equilibria. Formally the discrete Voronoi game plays on a given undirected graph G(V, E) with n = |V | and k players. Every player has to choose a vertex (facility) from V , and every vertex (customer ) is assigned to the closest facilities. A player’s payoff is the number of vertices assigned to his facility. We define the social cost as the sum of the distances to the closest facility over all vertices.

If player 1 ≤ q ≤ m moves from vertex uijk to a vertex u then his gain can be at most 5d < Bc + c/m. But what can be his gain, if he moves to another vertex ui j k ? In case where i = i , j = j , k = k , ai c + aj c is smaller than 34 Bc because ai + aj + ak = B and ak > B/4. Since ak < B/2, and player q gains only half of 24 C. K. Thang it, his payoff is at most ai c + aj c + ak c/2 + c/m < Bc + c/m so he again cannot improve his payoff. The other cases are similar. Now we show that if there is a Nash equilibrium, then it corresponds to a solution of the 3-Partition instance.

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