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General search techniques without common knowledge for imperfect-information games, and application to superhuman Fog of War chess

Brian Hu Zhang, Tuomas Sandholm

TL;DR

Oscuro is presented, the first superhuman AI for FoW chess, which introduces advances to search in imperfect-information games, enabling strong, scalable reasoning.

Abstract

Since the advent of AI, games have served as progress benchmarks. Meanwhile, imperfect-information variants of chess have existed for over a century, present extreme challenges, and have been the focus of decades of AI research. Beyond calculation needed in regular chess, they require reasoning about information gathering, the opponent's knowledge, signaling, etc. The most popular variant, Fog of War (FoW) chess (a.k.a. dark chess), has been a major challenge problem in imperfect-information game solving since superhuman performance was reached in no-limit Texas hold'em poker. We present Obscuro, the first superhuman AI for FoW chess. It introduces advances to search in imperfect-information games, enabling strong, scalable reasoning. Experiments against the prior state-of-the-art AI and human players -- including the world's best -- show that Obscuro is significantly stronger. FoW chess is the largest (by amount of imperfect information) turn-based zero-sum game in which superhuman performance has been achieved and the largest zero-sum game in which imperfect-information search has been successfully applied.

General search techniques without common knowledge for imperfect-information games, and application to superhuman Fog of War chess

TL;DR

Oscuro is presented, the first superhuman AI for FoW chess, which introduces advances to search in imperfect-information games, enabling strong, scalable reasoning.

Abstract

Since the advent of AI, games have served as progress benchmarks. Meanwhile, imperfect-information variants of chess have existed for over a century, present extreme challenges, and have been the focus of decades of AI research. Beyond calculation needed in regular chess, they require reasoning about information gathering, the opponent's knowledge, signaling, etc. The most popular variant, Fog of War (FoW) chess (a.k.a. dark chess), has been a major challenge problem in imperfect-information game solving since superhuman performance was reached in no-limit Texas hold'em poker. We present Obscuro, the first superhuman AI for FoW chess. It introduces advances to search in imperfect-information games, enabling strong, scalable reasoning. Experiments against the prior state-of-the-art AI and human players -- including the world's best -- show that Obscuro is significantly stronger. FoW chess is the largest (by amount of imperfect information) turn-based zero-sum game in which superhuman performance has been achieved and the largest zero-sum game in which imperfect-information search has been successfully applied.

Paper Structure

This paper contains 35 sections, 1 theorem, 14 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Challenges in imperfect-information games such as Fog of War chess
  3. Description of our AI agent Obscuro and the new algorithms therein
  4. Step 1: Generating the initial game tree at the beginning of a turn
  5. Step 2: Equilibrium computation
  6. Step 3: Expanding the game tree
  7. Step 4: Repeat
  8. Step 5: Selecting a move
  9. Experimental evaluation
  10. Ablations
  11. Other experiments
  12. Conclusions and future research
  13. Rules of FoW chess
  14. Notation and Preliminaries
  15. Notation and Preliminaries
  16. ...and 20 more sections

Key Result

Theorem 1

For any given $\epsilon > 0$, the average strategy profile $(\bar{x}, \bar{y})$ in one-sided GT-CFR eventually converges to an $\epsilon$-Nash equilibrium of any finite two-player zero-sum $\Gamma$.Technically, $(\bar{x}, \bar{y})$ is only a partial strategy in $\Gamma$, since it does not specify ho

Figures (12)

  • Figure 1: Two FoW chess positions in the same common-knowledge set. (A) position after moves 1. Nc3 g5 2. Nh3 d5; (B) position after moves 1. Nf3 e5 2. h3 Qh4. The boxed squares mark pieces visible to the opponent.
  • Figure 2: An example game tree, to illustrate KLUSS. The box ($\square$) is a chance node. Dotted lines connect nodes in the same infoset.
  • Figure 3: Visualization of time scaling of Obscuro. The $y$-axis is relative to the playing strength of Obscuro with 5 seconds per move.
  • Figure 4: The game tree from \ref{['fig:example2']}, now with some nodes labeled, which will be referenced in the text.
  • Figure 5: FoW chess position illustrating the existence of large infosets and common-knowledge sets. A full explanation is given in the text.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Claim 1
  • proof
  • Claim 2
  • proof