N-dimensional Blotto game with heterogeneous battlefield values

This paper introduces the irregular N-gon solution, a new geometric method for constructing equilibrium distributions in the Colonel Blotto game with heterogeneous battlefield values, generalising known construction methods. Using results on the existence of tangential polygons, it derives necessary and sufficient conditions for the irregular N-gon method to be applied, given the parameters of a Blotto game. The method does particularly well when the battlefield values satisfy some clearly defined regularity conditions. The paper establishes the parallel between these conditions and the constrained integer partitioning problem in combinatorial optimisation. The properties of equilibrium distributions numerically generated using the irregular N-gon method are illustrated. They indicate that the realised allocations, weighted by battlefield value, are less egalitarian and depend more strongly on battlefield values than previously thought. In the context of the US presidential elections, the explicit construction of equilibria provides new insights into the relation between the size of a state and the campaign resources spent there by presidential candidates.