Nash equilibria for voronoi games on transitive graphs

Abstract

In a Voronoi game, each of κ≥2 players chooses a vertex in a graph G=〈V(G), E(G)〉. The utility of a player measures her Voronoi cell: the set of vertices that are closest to her chosen vertex than to that of another player. In a Nash equilibrium, unilateral deviation of a player to another vertex is not profitable. We focus on various, symmetry-possessing classes of transitive graphs: the vertex-transitive and generously vertex-transitive graphs, and the more restricted class of friendly graphs we introduce; the latter encompasses as special cases the popular d-dimensional bipartite torus Td =Td(2p1, ⋯, 2p d) with even sides 2p1, ⋯, 2pd and dimension d≥2, and a subclass of the Johnson graphs. Would transitivity enable bypassing the explicit enumeration of Voronoi cells? To argue in favor, we resort to a technique using automorphisms, which suffices alone for generously vertex-transitive graphs with κ=2. To go beyond the case κ=2, we show the Two-Guards Theorem for Friendly Graphs: whenever two of the three players are located at an antipodal pair of vertices in a friendly graph G, the third player receives a utility of |V(G)|/4+|ω|/12, where Ω is the intersection of the three Voronoi cells. If the friendly graph G is bipartite and has odd diameter, the utility of the third player is fixed to |V(G)|/4; this allows discarding the third player when establishing that such a triple of locations is a Nash equilibrium. Combined with appropriate automorphisms and without explicit enumeration, the Two-Guards Theorem implies the existence of a Nash equilibrium for any friendly graph G with κ=4, with colocation of players allowed; if colocation is forbidden, existence still holds under the additional assumption that G is bipartite and has odd diameter. For the case κ=3, we have been unable to bypass the explicit enumeration of Voronoi cells. Combined with appropriate automorphisms and explicit enumeration, the Two-Guards Theorem implies the existence of a Nash equilibrium for (i) the 2-dimensional torus T 2 with odd diameter ∑jε[2]Pj and κ=3, and (ii) the hypercube Hd with odd d and κ=3. In conclusion, transitivity does not seem sufficient for bypassing explicit enumeration: far-reaching challenges in combinatorial enumeration are in sight, even for values of κ as small as 3. © 2009 Springer-Verlag Berlin Heidelberg.