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).
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Feldmann, R., Mavronicolas, M., Monien, B. (2009). Nash Equilibria for Voronoi Games on Transitive Graphs. In: Leonardi, S. (eds) Internet and Network Economics. WINE 2009. Lecture Notes in Computer Science, vol 5929. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10841-9_26
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DOI: https://doi.org/10.1007/978-3-642-10841-9_26
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