An Efficient Approximation Algorithm for the Colonel Blotto Game
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In the storied Colonel Blotto game, two colonels allocate a and b troops, respectively, to k distinct battlefields. A colonel wins a battle if they assign more troops to that particular battle, and each colonel seeks to maximize their total number of victories. Despite the problem's formulation in 1921, the first polynomial-time algorithm to compute Nash equilibrium (NE) strategies for this game was discovered only quite recently. In 2016, <cit.> formulated a breakthrough algorithm to compute NE strategies for the Colonel Blotto game[To the best of our knowledge, the algorithm from <cit.> has computational complexity O(k^14max{a,b}^13)], receiving substantial media coverage (e.g. <cit.>, <cit.>, <cit.>). In this work, we present the first known ϵ-approximation algorithm to compute NE strategies in the two-player Colonel Blotto game in runtime O(ϵ^-4 k^8 max{a,b}^2) for arbitrary settings of these parameters. Moreover, this algorithm computes approximate coarse correlated equilibrium strategies in the multiplayer (continuous and discrete) Colonel Blotto game (when there are ℓ > 2 colonels) with runtime O(ℓϵ^-4 k^8 n^2 + ℓ^2 ϵ^-2 k^3 n (n+k)), where n is the maximum troop count. Before this work, no polynomial-time algorithm was known to compute exact or approximate equilibrium (in any sense) strategies for multiplayer Colonel Blotto with arbitrary parameters. Our algorithm computes these approximate equilibria by a novel (to the author's knowledge) sampling technique with which we implicitly perform multiplicative weights update over the exponentially many strategies available to each player.
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