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 T _d  = T _d (2 p ₁, ..., 2 p _d ) with even sides 2p ₁, ..., 2p _d 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 $\frac{\textstyle |{\sf V}({\sf G})|} {\textstyle 4} + \frac{\textstyle |{\sf \Omega|}} {\textstyle 12}$, 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 $\frac{\textstyle |{\sf V}({\sf G})|} {\textstyle 4}$; 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 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 ∑  _{j ∈ [2]} p _j and κ= 3, and (ii) the hypercube H _d 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.

Supported by the IST Program of the European Union under contract number 15964 (AEOLUS).