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2015-12-02

Abishek Gupta, University of Illinois at Urbana-Champaign, Tamer Başar, University of Illinois at Urbana-Champaign, Galina Schwartz, University of California, Berkeley. 2014. A Three-Stage Colonel Blotto Game: When to Provide More Information to an Adversary. 5th International Conference on Decision and Game Theory for Security (GameSec 2014).

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.

Abishek Gupta, University of Illinois at Urbana-Champaign, Galina Schwartz, University of California, Berkeley, Cedric Langbort, University of Illinois at Urbana-Champaign, S. Shankar Sastry, University of California, Berkeley, Tamer Başar, University of Illinois at Urbana-Champaign. 2014. A Three-stage Colonel Blotto Game with Applications to Cyberphysical Security. American Control Conference .

We consider a three-step three-player complete information Colonel Blotto game in this paper, in which the first two players fight against a common adversary. Each player is endowed with a certain amount of resources at the beginning of the game, and the number of battlefields on which a player and the adversary fights is specified. The first two players are allowed to form a coalition if it improves their payoffs. In the first stage, the first two players may add battlefields and incur costs. In the second stage, the first two players may transfer resources among each other. The adversary observes this transfer, and decides on the allocation of its resources to the two battles with the players. At the third step, the adversary and the other two players fight on the updated number of battlefields and receive payoffs. We characterize the subgame-perfect Nash equilibrium (SPNE) of the game in various parameter regions. In particular, we show that there are certain parameter regions in which if the players act according to the SPNE strategies, then (i) one of the first two players add battlefields and transfer resources to the other player (a coalition is formed), (ii) there is no addition of battlefields and no transfer of resources (no coalition is formed). We discuss the implications of the results on resource allocation for securing cyberphysical systems.