Stochastic asymmetric Blotto games: Some new results

Abstract

Stochastic Asymmetric Blotto Games: Some New Results John Duffy ∗ Abstract. Blotto games. Alexander Matros † We develop some new theoretical results for stochastic asymmetric Keywords: Colonel Blotto game, Contests, Resource Allocation, Lotteries. JEL Classification Nos. C72, C73, D72, D74. 1. Introduction The Colonel Blotto game (Borel 1921), is a two-player non-cooperative game in which players decide how to allocate their given resources across  battlefields. In Borel’s original version of this game, the player who allocates the most resources to any given battlefield wins that battlefield with certainty. The players’ objective function is either to maximize the sum of the value of the battlefields won, or to win a majority value of the  battlefields. In this paper we study stochastic asymmetric versions of the Blotto game under both of these objective functions. In an “asymmetric Blotto” game, the values of the  battlefields may differ from one another though these different values are common to all players. In the “stochastic asymmetric” version of the Blotto game, the deterministic rule for determining which player wins each battlefield is replaced by a lottery contest success function where the chances of winning a given battlefield are increasing with the amount of resources devoted to that battlefield. This stochastic lottery specification makes the payoff function continuous; as a result, if a Nash equilibrium exists, it is unique and in pure strategies, as opposed to the multiplicity of (typically mixed strategy) equilibria that arise in deterministic versions of the Blotto game. There are two main theoretical papers about stochastic asymmetric Blotto games. 1 The first one, Friedman (1958), considers two players who seek to maximize their expected total payoff. We show that Friedman’s result can be extended to any number of players. The second paper, Lake (1979), was the first to study the stochastic asymmetric “majority rule” Blotto game. 2 This version of the game is particularly relevant to understanding electoral competitions in two party systems, e.g., the electoral college system for electing the U.S. president. Lake studied only the case of equal budget constraints. We show that if players’ budgets are the same (as in Lake) or if they are sufficiently similar and the number of items (battlefields) is not too large, then resource allocation under the majority rule version of the stochastic, asymmetric Blotto game is proportional to the Banzhaf power index for each item, while more generally, resource allocation for a particular item will not be proportional to each item’s Banzhaf power index. Our findings thus generalize those of Lake (1979). Department of Economics, University of California, Irvine. Email: duffy@uci.edu. Moore School of Business, University of South Carolina and Lancaster University Management School. Email: alexander.matros@gmail.com See Kovenock and Roberson (2012) for a broader survey of the Blotto game literature. We discovered Lake’s (1979) paper only after we had completed our analysis. His proofs are different from ours, but his main result coincides with our prediction for the case of equal budgets. We thank Steve Brams for providing this reference.