Idempotent Method for Dynamic Games and Complexity Reduction in Min-Max Expansions

In recent years, idempotent methods (specifically, max-plus methods) have been developed for solution of nonlinear control problems. It was thought that idempotent linearity of the associated semigroup was required for application of these techniques. It is now known that application of the max-plus distributive property allows one to apply the max-plus curse-of-dimensionality-free approach to stochastic control problems. Here, we see that a similar, albeit more abstract, approach can be applied to deterministic game problems. The main difficulty is a curse-of-complexity growth in the computational cost. Attenuation of this effect requires finding reduced-complexity approximations to min-max sums of max-plus affine functions. We demonstrate that this problem can be reduced to a pruning problem.