From Battlefields to Elections: Winning Strategies of Blotto and Auditing Games

Mixed strategies are often evaluated based on the expected payoff that they guarantee. This is not always desirable. In this paper, we consider games for which maximizing the expected payoff deviates from the actual goal of the players. To address this issue, we introduce the notion of a (u, p)-maxmin strategy which ensures receiving a minimum utility of u with probability at least p. We then give approximation algorithms for the problem of finding a (u, p)maxmin strategy for these games. The first game that we consider is Colonel Blotto, a well-studied game that was introduced in 1921. In the Colonel Blotto game, two colonels divide their troops among a set of battlefields. Each battlefield is won by the colonel that puts more troops in it. The payoff of each colonel is the weighted number of battlefields that she wins. We show that maximizing the expected payoff of a player does not necessarily maximize her winning probability for certain applications of Colonel Blotto. For example, in presidential elections, the players’ goal is to maximize the probability of winning more than half of the votes, rather than maximizing the expected number of votes that they get. We give an exact algorithm for a natural variant of continuous version of this game. More generally, we provide constant and logarithmic approximation algorithms for finding (u, p)maxmin strategies. We also introduce a security game version of Colonel Blotto which we call auditing game. It is played between two players, a defender and an attacker. The goal of the defender is to prevent the attacker from changing the outcome of an instance of Colonel Blotto. Again, maximizing the expected payoff of the defender is not necessarily optimal. Therefore we give a constant approximation for (u, p)maxmin strategies. ∗This work was conducted in part while the authors were visiting the Simons Institute for the Theory of Computing. †University of Maryland, College Park. Supported in part by NSF CAREER award CCF-1053605, NSF BIGDATA grant IIS-1546108, NSF AF:Medium grant CCF-1161365, DARPA GRAPHS/AFOSR grant FA9550-12-1-0423, and another DARPA SIMPLEX grant. ‡TTI-Chicago and CMU. Supported in part by NSF CCF1331175. §Google Research, New York ¶Computer Science Division, UC Berkeley ‖CSAIL, MIT, Cambridge ∗∗Department of Statistics, UC Berkeley