Dominated Actions in Imperfect-Information Games
Sam Ganzfried
TL;DR
The paper addresses the challenge of reducing imperfect-information extensive-form games via dominated actions. It introduces robust definitions of strictly and weakly dominated actions at information sets and shows these can be identified in polynomial time using sequence-form linear programs, even for n-player games through an opponent-merge transformation. It proves that iterated removal of dominated actions can be performed in polynomial time and demonstrates substantial practical reductions in a two-player No-Limit Hold'em benchmark, indicating strong preprocessing benefits for Nash equilibrium computation. The work suggests significant potential for scalability gains in larger, multi-player imperfect-information games and points to future refinements and related results in fast equilibrium computation.
Abstract
Dominance is a fundamental concept in game theory. In normal-form games dominated strategies can be identified in polynomial time. As a consequence, iterative removal of dominated strategies can be performed efficiently as a preprocessing step for reducing the size of a game before computing a Nash equilibrium. For imperfect-information games in extensive form, we could convert the game to normal form and then iteratively remove dominated strategies in the same way; however, this conversion may cause an exponential blowup in game size. In this paper we define and study the concept of dominated actions in imperfect-information games. Our main result is a polynomial-time algorithm for determining whether an action is dominated (strictly or weakly) by any mixed strategy in n-player games, which can be extended to an algorithm for iteratively removing dominated actions. This allows us to efficiently reduce the size of the game tree as a preprocessing step for Nash equilibrium computation. We explore the role of dominated actions empirically in "All In or Fold" No-Limit Texas Hold'em poker.