Discrete Colonel Blotto Games with Two Battlefields

The Colonel Blotto Game is one of the most classical zero-sum games, with diverse applications in wars, auctions, political elections and anti-terrorists etc. We consider the discrete two-battlefield Colonel Blotto Game, a basic case that has not been completely characterized yet. We study three variant scenarios where at least one player's resources are indivisible (discrete), and compare them with the benchmark scenario where the resources of both players are arbitrarily divisible (continuous). We present the equilibrium values for all three scenarios, and provide a complete equilibrium characterization for the scenario where both players' resources are indivisible. Our main finding is that, more or less surprisingly, the distinction between continuous and discrete strategy spaces generally has no effect on players' equilibrium values. In some special cases, however, the larger continuous strategy space when resources are divisible does bring the corresponding player a higher equilibrium value than when resources are indivisible, and this effect is more significant for the stronger player who possesses more resource than for the weaker player