A Three-Stage Colonel Blotto Game: When to Provide More Information to an Adversary

In this paper, we formulate a three-player three-stage Colonel Blotto game, in which two players fight against a common adversary. We assume that the game is one of complete information, that is, the players have complete and consistent information on the underlying model of the game; further, each player observes the actions taken by all players up to the previous stage. The setting under consideration is similar to the one considered in our recent work [1], but with a different information structure during the second stage of the game; this leads to a significantly different solution. In the first stage, players can add additional battlefields. In the second stage, the players (except the adversary) are allowed to transfer resources among each other if it improves their expected payoffs, and simultaneously, the adversary decides on the amount of resource it allocates to the battle with each player subject to its resource constraint. At the third stage, the players and the adversary fight against each other with updated resource levels and battlefields. We compute the subgame-perfect Nash equilibrium for this game. Further, we show that when playing according to the equilibrium, there are parameter regions in which (i) there is a net positive transfer, (ii) there is absolutely no transfer, (iii) the adversary fights with only one player, and (iv) adding battlefields is beneficial to a player. In doing so, we also exhibit a counter-intuitive property of Nash equilibrium in games: extra information to a player in the game does not necessarily lead to a better performance for that player. The result finds application in resource allocation problems for securing cyber-physical systems.