On Bandwidth-Limited Sum-of-Games Problems

Abstract

Game theory typically considers two types of imperfect information when analyzing conflicts: 1) chance moves and 2) information sets. This correspondence paper considers games where players compete on many fronts without having sufficient bandwidth to collect the game trees describing all the conflicts. The player therefore needs to prioritize between subgames without having detailed information about the subgames. We address this problem by using two combinatorial game theory tools: 1) surreal numbers and 2) thermographs. We consider the global conflict as a sum-of-games problem, which is known to be intractable (PSPACE complete). Known combinatorial game theory heuristics can analyze surreal-number encodings of games using thermographs to find solutions that are within a known constant offset of the optimal solution. To apply the combinatorial game theory to this domain, we first integrate chance moves into surreal-number encodings of games by showing that the expected values of surreal numbers are surreal numbers. We then show that, of the three commonly used sum-of-games heuristics, only hotstrat is applicable to this domain. Simulations compare solutions found using hotstrat to solution approaches used by other researchers on a similar problem (maximin, maximax, and mean). The simulations show that the hotstrat solutions dominate the existing approaches.