Nash Equilibria for Voronoi Games on Transitive Graphs

• Authors: Rainer Feldmann;Marios Mavronicolas;Burkhard Monien
• Venue: WINE '09 Proceedings of the 5th International Workshop on Internet and Network Economics
• Year: 2009

In a Voronoi game, each of 驴 驴 2 players chooses a vertex in a graph G = 驴V(G), E(G) 驴. 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 T d = T d (2 p 1, ..., 2 p d ) with even sides 2p 1, ..., 2p d and dimension d 驴 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 驴= 2.To go beyond the case 驴= 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 $\frac{\textstyle |{\sf V}({\sf G})|} {\textstyle 4} + \frac{\textstyle |{\sf \Omega|}} {\textstyle 12}$, where 驴 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 $\frac{\textstyle |{\sf V}({\sf G})|} {\textstyle 4}$; 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 驴= 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 驴= 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 驴 j 驴 [2] p j and 驴= 3, and (ii) the hypercube H d with odd d and 驴= 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 驴 as small as 3.