Limited lookahead in imperfect-information games

Christian Kroer,Tuomas Sandholm

doi:10.1016/j.artint.2019.103218

Christian Kroer, Tuomas Sandholm

Open Access

https://doi.org/10.1016/j.artint.2019.103218

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Journal: Artificial Intelligence	Publication Date: Mar 19, 2020
Citations: 5	License type: publisher-specific-oa

Affiliation: Columbia University, Carnegie Mellon University

Abstract

Limited lookahead has been studied for decades in perfect-information games. We initiate a new direction via two simultaneous deviation points: generalization to imperfect-information games and a game-theoretic approach. We study how one should act when facing an opponent whose lookahead is limited. We study this for opponents that differ based on their lookahead depth, based on whether they, too, have imperfect information, and based on how they break ties. We characterize the hardness of finding a Nash equilibrium or an optimal commitment strategy for either player, showing that in some of these variations the problem can be solved in polynomial time while in others it is PPAD-hard, NP-hard, or inapproximable. We proceed to design algorithms for computing optimal commitment strategies—for when the opponent breaks ties favorably, according to a fixed rule, or adversarially. We then experimentally investigate the impact of limited lookahead. The limited-lookahead player often obtains the value of the game if she knows the expected values of nodes in the game tree for some equilibrium—but we prove this is not sufficient in general. Finally, we study the impact of noise in those estimates and different lookahead depths.

Highlights

Limited lookahead has been a central topic in AI game playing for decades
We design algorithms for finding an optimal strategy to commit to for the rational player. We focus on this rather than equilibrium computation because the latter seems nonsensical in this setting: the limited-lookahead player determining a Nash equilibrium strategy would require her to reason about the whole game for the rational player’s strategy, which rings contrary to the limited-lookahead assumption
As mentioned in the introduction, we focus on commitment strategies rather than Nash equilibria because Player l playing a Nash equilibrium strategy would require that player to reason about the whole game for the opponent’s strategy

Summary

Introduction

Limited lookahead has been a central topic in AI game playing for decades. To date, it has been studied in singleagent settings and perfect-information games— in well-known games such as chess, checkers, Go, etc., as well as in random game tree models [Korf, 1990; Pearl, 1981; Pearl, 1983; Nau, 1983; Nau et al, 2010; Bouzy and Cazenave, 2001; Ramanujan, Sabharwal, and Selman, 2010; Ramanujan and Selman, 2011]. As is typical in the literature on limited lookahead in perfect-information games, we derive our results for a twoagent setting. Our results extend immediately to one rational player and more than one limited-lookahead player, as long as the latter all break ties according to the same scheme (statically, favorably, or adversarially—as described later in the paper). As in the literature on lookahead in perfect-information games, a potential weakness of our approach is that we require knowing the evaluation function h (but make no other assumptions about what information h encodes). As in the perfect-information setting, this can lead to the rational exploiter being exploited

Extensive-form games

Model of limited lookahead

Complexity

Algorithms

Experiments

Conclusions and future work