Discrete Colonel Blotto games with two battlefields

The Colonel Blotto game is one of the most classical zero-sum games, with diverse applications in auctions, political elections, etc. We consider the discrete two-battlefield Colonel Blotto Game, a basic case that has not yet been completely characterized. We study three scenarios where at least one player’s resources are indivisible (discrete), and compare them with a 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, somewhat 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 resources than for the weaker player.