Nash Equilibria for Voronoi Games on Transitive Graphs

In a Voronoi game, each of κ ≥ 2 players chooses a vertex in a graph G = 〈V(G),E(G)〉. Theutility of a player measures her Voronoi cell: the set of vertices that are closest to her chosenvertex than to that of another player; each vertex contributes uniformly to the utilities of playerswhose Voronoi cells the vertex belongs to. In a Nash equilibrium, unilateral deviation of a playerto another vertex is not profitable;. so, the existence of a Nash equilibrium is determined fromthe cardinalities of Voronoi cells. We focus on various, symmetry-possessing classes of transitivegraphs: the vertex-transitive and generously vertex-transitive graphs, and the more restricted classof friendly graphs we introduce; the latter encompasses as special cases the popular d-dimensionalbipartite torus Td = Td(2p1, . . . ,2pd) with even sides 2p1, . . . ,2pd and dimension d ≥ 2 (the d-dimensional hypercube Hd being a special case), and a subclass of the Johnson graphs.How easily would transitivity enable bypassing the explicit enumeration of Voronoi cells? Toargue in favor, we resort to a technique using automorphisms, which suffices alone for generouslyvertex-transitive graphs with κ = 2.To go beyond the case κ = 2, we show the (perhaps surprising) Two-Guards Theorem forFriendly Graphs: whenever two of the three players are located at an antipodal pair of vertices ina friendly graph G, the third player receives a utility of|V(G)|4 +|Ω|12 , where Ω is the intersectionof the three Voronoi cells. If the friendly graph G is bipartite and has odd diameter, the utility ofthe third player is fixed to|V(G)|4 ; this allows discarding the third player when establishing thatsuch a triple of locations is a Nash equilibrium. Combined with appropriate automorphisms andwithout explicit enumeration, the Two-Guards Theorem implies the existence of a Nash equilibriumfor 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 where κ = 3, we have been unable to bypass the explicit enumeration of Voronoi cells.Combined with appropriate automorphisms and explicit enumeration, the Two-Guards Theoremimplies the existence of a Nash equilibrium for (i) the 2-dimensional torus T2 with odd diameter∑j∈[2] pj and κ = 3, and (ii) the hypercube Hd with odd d and κ = 3. In conclusion, transitivitydoes not seem sufficient for bypassing explicit enumeration: far-reaching challenges in combinatorialenumeration are in sight, even for values of κ as small as 3.