Sequential, nonzero-sum "Blotto": Allocating defensive resources prior to attack

The strategic allocation of resources across multiple fronts has long been studied in the context of Blotto games in which two players simultaneously select their allocations. But many allocation problems are sequential. For example, a state trying to defend against a terrorist attack generally allocates some or all of its resources before the attacker decides where to strike. This paper studies the allocation problem confronting a defender who must decide how to distribute limited resources across multiple sites before an attacker chooses where to strike. Unlike many Blotto games which only have very complicated mixed-strategy equilibria, the sequential, nonzero-sum “Blotto” game always has a purestrategy subgame perfect equilibrium, the defender always plays the same pure strategy in any equilibrium, and the attacker’s equilibrium response is generically unique and entails no mixing. The defender minmaxes the attacker, and the attacker strikes the site among its best replies that minimizes the defender’s expected losses. Sequential, Nonzero-sum Blotto: Allocating Defensive Resources Prior to Attack Allocating resources against a strategic adversary is an old problem in game theory and has long been studied in the context of Colonel Blotto games (e.g., Borel 1921, Tukey 1949, Blackett 1958, Shubik and Weber 1981). Blotto games are two-person, zero-sum games in which the players simultaneously decide how to allocate limited resources across N independent battlefields. How well a player does on a particular battlefield depends on how much each player allocates to that front, and each player’s payoff in the game is the sum of its battlefield payoffs. However, some allocation problems are sequential. Following the attacks of 9/11, for example, the United States undertook a massive effort to protect its critical infrastructure and key assets. More generally, defenders often have to allocate some or all of their resources before an attacker decides where to strike. In the model below, a defender chooses an allocation prior to an attack. After observing the defender’s allocation, the attacker, e.g., a terrorist group, decides where to strike. (With additional assumptions the model can also be interpreted as one in which the attacker decides how to allocate a fixed amount of resources or effort.) The more the defender allocates to a site, the “harder” that site becomes and the less likely an attack is to succeed. The defender and attacker may value the sites differently so that the game will generally not be zero-sum.1 This sequential, nonzero-sum “Blotto” game has a generically unique equilibrium path and there is no mixing along it. Even though the game is not zero-sum, the defender’s equilibrium strategy is to minmax the attacker. The attacker, by contrast, acts in the way that is most favorable to the defender; it strikes the site among its best responses that minimizes the defender’s loss. The existence of a pure-strategy equilibrium path, much less the fact that it is the generically unique path, contrasts with the emphasis of recent work on Blotto games (e.g., Golman and Page 2006, Roberson 2006, Hart 2007). But this contrast is more 1 The Minimax Theorem trivially implies that that the equilibrium paths of the static and dynamic games are identical when the game is zero-sum.