# Limited Lookahead in Imperfect-Information Games

## TL;DR

This paper explores the strategic implications of limited lookahead in imperfect-information games, analyzing computational complexity, designing algorithms for optimal strategies, and experimentally assessing the impact of lookahead limitations and noise.

## Contribution

It introduces a game-theoretic framework for limited lookahead in imperfect-information games, characterizes computational hardness, and develops algorithms for optimal commitment strategies.

## Key findings

- Limited lookahead often suffices to determine game value with known node values.
- Computational complexity varies; some problems are polynomial-time solvable, others are PPAD-hard or NP-hard.
- Noise and depth of lookahead significantly affect strategic outcomes.

## 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.