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Expected Work Search: Combining Win Rate and Proof Size Estimation

Owen Randall, Martin Müller, Ting Han Wei, Ryan Hayward

TL;DR

Expected Work Search (EWS) introduces a unifying solver for perfect-information two-player games by merging win-rate estimation with proof-size estimation to minimize the expected computational work ($EW$). The method defines recursive $EW$ metrics for loss and win scenarios, and uses a principled, backpropagation-driven ordering to guide selection and expansion. Empirical results on Go and Hex show that EWS outperforms traditional solvers, solving the empty 8×8 Hex board in under four minutes and achieving the first 5×5 Go solution under positional superko rules. The work highlights the value of integrating and balancing proof-size and win-rate information, while demonstrating robustness to limited domain knowledge and potential for further improvements with domain-specific refinements and learned estimation models.

Abstract

We propose Expected Work Search (EWS), a new game solving algorithm. EWS combines win rate estimation, as used in Monte Carlo Tree Search, with proof size estimation, as used in Proof Number Search. The search efficiency of EWS stems from minimizing a novel notion of Expected Work, which predicts the expected computation required to solve a position. EWS outperforms traditional solving algorithms on the games of Go and Hex. For Go, we present the first solution to the empty 5x5 board with the commonly used positional superko ruleset. For Hex, our algorithm solves the empty 8x8 board in under 4 minutes. Experiments show that EWS succeeds both with and without extensive domain-specific knowledge.

Expected Work Search: Combining Win Rate and Proof Size Estimation

TL;DR

Expected Work Search (EWS) introduces a unifying solver for perfect-information two-player games by merging win-rate estimation with proof-size estimation to minimize the expected computational work (). The method defines recursive metrics for loss and win scenarios, and uses a principled, backpropagation-driven ordering to guide selection and expansion. Empirical results on Go and Hex show that EWS outperforms traditional solvers, solving the empty 8×8 Hex board in under four minutes and achieving the first 5×5 Go solution under positional superko rules. The work highlights the value of integrating and balancing proof-size and win-rate information, while demonstrating robustness to limited domain knowledge and potential for further improvements with domain-specific refinements and learned estimation models.

Abstract

We propose Expected Work Search (EWS), a new game solving algorithm. EWS combines win rate estimation, as used in Monte Carlo Tree Search, with proof size estimation, as used in Proof Number Search. The search efficiency of EWS stems from minimizing a novel notion of Expected Work, which predicts the expected computation required to solve a position. EWS outperforms traditional solving algorithms on the games of Go and Hex. For Go, we present the first solution to the empty 5x5 board with the commonly used positional superko ruleset. For Hex, our algorithm solves the empty 8x8 board in under 4 minutes. Experiments show that EWS succeeds both with and without extensive domain-specific knowledge.
Paper Structure (19 sections, 12 equations, 2 figures, 4 tables, 2 algorithms)

This paper contains 19 sections, 12 equations, 2 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Expected Work Search
  4. Expected Work
  5. Selection
  6. Expansion
  7. Backpropagation
  8. Implementation
  9. Win Rate Estimation
  10. Expected Work initialisation
  11. Generic Refinements
  12. Experiments
  13. The 6$\times$6Go Dataset
  14. Go Results
  15. Solving Small Square Go Boards
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Basic diagram of the structure of Expected Work Search.
  • Figure 2: Comparing EWS against Go-Solver, EWS without win rate estimation (EWS-WR), and EWS without proof size estimation (EWS-PS). The y-axis is a log scale of the time EWS took to solve the positions in seconds, and the x-axis is the time it took the other algorithms to solve the same positions. Data points below the diagonal dashed line took less time for EWS to solve than the comparison algorithm.