A Three-Stage Colonel Blotto Game with Applications to Cyberphysical Security Abhishek Gupta Galina Schwartz CLangbort S. Shankar Sastry Tamer Basar
Abhishek Gupta, Galina A. Schwartz, Shankar Sastry · 2014
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. I. INTRODUCTION The Colonel Blotto game models a scenario in which two players having certain resource levels fight over a finite number of battlefields. The players decide on the amount of resource they deploy on each battlefield in order to maximize their payoffs. The case of players with symmetric resources and three battlefields was solved by Borel and Ville in (1) in 1938. Gross and Wagner (2) generalized the result of symmetric resources to an arbitrary number of battlefields; they also derived a Nash equilibrium of the game when there are two battlefields and asymmetric resources among the players. Un- til 2006, the two-player asymmetric-resource Colonel Blotto game with more than two battlefields remained unsolved.