Zero-Sum Stochastic Differential Blotto-Type Games on Graphs

Overview & Synopsis

We introduce in this paper a dynamic version of resource allocation games, in the setting of continuous-time two-player zero-sum stochastic differential games. The players control the drift of their output process on each battlefield, subject to a budget constraint. When both players have the same budget, using a dynamic programming approach, we explicitly characterize the Nash equilibrium strategies and the value function of the game, in terms of a PDE that admits a smooth solution. Our formulation is general enough to include various rewards, continuous or not, allowing different interpretations of the game. In particular, when the game is played on a graph, where each edge represents a battlefield, and where the reward depends on the spectrum of the graph, we are able to explicitly compute the value function of the game and characterize an equilibrium. When the players' budgets are different, we characterize the value function via a particular BSDE and we introduce a numerical algorithm to approximate the equilibrium strategies.