Blotto on the Ballot: A Ballot Stuffing Blotto Game

Abstract

We consider the following Colonel Blotto game between parties $P_1$ and $P_A.$ $P_1$ deploys a non negative number of troops across $J$ battlefields, while $P_A$ chooses $K,$ $K < J,$ battlefields to remove all of $P_1$'s troops from the chosen battlefields. $P_1$ has the objective of maximizing the number of surviving troops while $P_A$ wants to minimize it. Drawing an analogy with ballot stuffing by a party contesting an election and the countermeasures by the Election Commission to negate that, we call this the Ballot Stuffing Game. For this zero-sum resource allocation game, we obtain the set of Nash equilibria as a solution to a convex combinatorial optimization problem. We analyze this optimization problem and obtain insights into the several non trivial features of the equilibrium behavior. These features in turn allows to describe the structure of the solutions and efficient algorithms to obtain then. The model is described as ballot stuffing game in a plebiscite but has applications in security and auditing games. The results are extended to a parliamentary election model. Numerical examples illustrate applications of the game.