–
April 26, 2018
Resource allocation games are commonly used to model problems in many domains ranging from security to advertising. One of the most important resource allocation games is the Colonel Blotto game: Two players distribute a fixed budget of resources over multiple battlefields to maximize the aggregate value of battlefields they win, each battlefield being won by the player who allocates more resources to it. Despite its long-standing history and importance, the continuous version of the game---where players can choose any fractional allocation---still lacks a complete Nash equilibrium solution in its general form with asymmetric players' budgets and heterogeneous battlefield values.
In this work, we propose an approximate equilibrium for this general case. We construct a simple strategy (the independently uniform strategy) and prove that it is an epsilon-equilibrium. We give a theoretical bound on the approximation error in terms of the number of battlefields and players’ budgets which identifies precisely the parameters regime for which our strategy is a good approximation. We also investigate an extension to the discrete version (where players can only have integer allocations), for which we proposed an algorithm to compute very efficiently an approximate equilibrium. We perform numerical experiments that guarantee that we can "safely" use this strategy in practice. Our work extends the scope of application of Colonel Blotto games in several practical cases, especially with large games' parameters (e.g. in advertisements, voting, security, etc.,)
