Lossy stochastic game abstraction with bounds

Abstraction followed by equilibrium finding has emerged as the leading approach to solving games. Lossless abstraction typically yields games that are still too large to solve, so lossy abstraction is needed. Unfortunately, prior lossy game abstraction algorithms have no guarantees on solution quality. We developed a framework that enables the design of lossy game abstraction algorithms with guarantees on solution quality. It simultaneously handles state and action abstraction. We define a measure of reward approximation error and transition probability error achieved by state and action abstraction in stochastic games such that the regret of the equilibrium found in the abstract game when implemented in the original, unabstracted game is upper-bounded by a function of those measures. We then develop the first lossy game abstraction algorithms with bounds on solution quality. Both of them work level-by-level up from the end of the game. One of the algorithms is greedy and the other is an integer linear program. We also prove that the abstraction problem is NP-complete (even with just action abstraction, 2 agents, and a 1-step game), but point out that this does not mean that the game abstraction problems that occur in practice cannot be solved quickly.