Colonel Blotto games with a head start

This paper studies Colonel Blotto games with two battlefields where one player has a head start in the form of additional troops on one of the battlefields. Such games arise naturally in marketing, electoral competition, and military conflict. Sion and Wolfe (1957) have shown that, if the strategy space is continuous, a mixed-strategy Nash equilibrium need not exist. Therefore, we consider a finite approximation. Using the iterated elimination of (weakly) dominated strategies, we identify an equilibrium for all parameter constellations and discuss its uniqueness properties. In equilibrium, resource decisions are typically not uniform but tend to concern units that roughly correspond in size to multiples of the head start. Moreover, competition takes the form of a hide-and-seek game, where the favorite tries to outguess the number of units that the underdog commits to the balanced battlefield. Somewhat unexpectedly, equilibrium payoffs of finite approximations of the Sion-Wolfe game accumulate around precisely three values. We also discuss the relation to the model with heterogeneous budgets but no head start.