Algorithms for rationalizability and CURB sets

Abstract

Significant work has been done on computational aspects of solving games under various solution concepts, such as Nash equilibrium, subgame perfect Nash equilibrium, correlated equilibrium, and (iterated) dominance. However, the fundamental concepts of rationalizability and CURB (Closed Under Rational Behavior sets have not, to our knowledge, been studied from a computational perspective. First, for rationalizability we describe an LP-based polynomial algorithm that finds all strategies that are rationalizable against a mixture over a given set of opponent strategies. Then, we describe a series of increasingly sophisticated polynomial algorithms for finding all minimal CURB sets, one minimal CURB set, and the smallest minimal CURB set. Finally, we give theoretical results regarding the relationships between CURB sets and Nash equilibria, showing that finding a Nash equilibrium can be exponential only in the size of the smallest CURB set. We show that this can lead to an arbitrarily large reduction in the complexity of finding a Nash equilibrium. On the downside, we also show that the smallest CURB set can be arbitrarily larger than the supports of the enclosed Nash equilibrium. Copyright © 2006, American Association for Artificial Intelligence (www.aaai.org). All rights reserved.