An Improved Exponential-Time Approximation Algorithm for Fully-Alternating Games Against Nature

• Published in: Proceedings / annual Symposium on Foundations of Computer Science pp. 1081 - 1090
• Main Author: Drucker, Andrew
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01.11.2020
• Subjects: Approximation algorithms; Artificial intelligence; Complexity theory; decision-making under uncertainty; Force; Games; games against Nature; interactive proofs; Runtime; Uncertainty
• ISSN: 2575-8454
• DOI: 10.1109/FOCS46700.2020.00104

"Games against Nature" [1] are two-player games of perfect information, in which one player's moves are made randomly (here, uniformly); the final payoff to the non-random player is given by some bounded-value function of the move history. Estimating the value of such games under optimal play, and computing near-optimal strategies, is an important goal in the study of decision-making under uncertainty, and has seen significant research in AI and allied areas [2], with only experimental evaluation of most algorithms' performance. The problem's PSPACE-completeness does not rule out nontrivial algorithms. Improved algorithms with theoretical guarantees are known in various cases where the payoff function F has special structure, and Littman, Majercik, and Pitassi [3] give a sampling-based improved algorithm for general F, for turn-orders which restrict the number of non-random player strategies. We study the case of general F for which the players strictly alternate with binary moves-for which the approach of [3] does not improve over brute force. We give a randomized algorithm to approximate the value of such games under optimal play, and to execute near-optimal strategies. Our algorithm, while exponential-time, achieves exponential savings over brute-force, and certifies a lower bound on the game value with exponentially small additive expected error. (On the downside, the constant of improvement in the base of the runtime exponent is tiny, and the algorithm uses exponential space.) Our algorithm is recursive, and bootstraps a "base case" algorithm for fixed-size inputs. The method of recursive composition used, the specific base-case guarantees needed, and the steps to establish these guarantees are interesting and, we feel, likely to find uses beyond the present work.