This tutorial introduces Nash equilibrium which is the most important equilibrium concept in non-cooperative games. In 2-player zero-sum normal form games, we can compute Nash equilibria in polynomial time in the number of pure strategies available to players. However, in complicated games in which players take turns choosing actions, the number of pure strategies becomes large. We explain algorithms that can compute Nash equilibria in complicated card games such as poker. On the other hand, it has not been shown whether Nash equlibria in 2-player general-sum games can be found in polynomial time. The concepts related to decision problems are not appropriate in order to discuss the computational complexity of finding Nash equilibria, since the existence of them has been already proven. Thus, we introduce the classes PPAD and PPAD-complete to characterize the complexity of computing Nash equilibria.
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