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Set-Based Retrograde Analysis: Precomputing the Solution to 24-card Bridge Double Dummy Deals

Isaac Stone, Nathan R. Sturtevant, Jonathan Schaeffer

TL;DR

The setrograde algorithm performs a factor of $10^3$ fewer search operations than a standard retrograde algorithm, producing a database with a factor of $10^4$ fewer entries, which allows retrograde searching to reach unprecedented search depths.

Abstract

Retrograde analysis is used in game-playing programs to solve states at the end of a game, working backwards toward the start of the game. The algorithm iterates through and computes the perfect-play value for as many states as resources allow. We introduce setrograde analysis which achieves the same results by operating on sets of states that have the same game value. The algorithm is demonstrated by computing exact solutions for Bridge double dummy card-play. For deals with 24 cards remaining to be played ($10^{27}$ states, which can be reduced to $10^{15}$ states using preexisting techniques), we strongly solve all deals. The setrograde algorithm performs a factor of $10^3$ fewer search operations than a standard retrograde algorithm, producing a database with a factor of $10^4$ fewer entries. For applicable domains, this allows retrograde searching to reach unprecedented search depths.

Set-Based Retrograde Analysis: Precomputing the Solution to 24-card Bridge Double Dummy Deals

TL;DR

The setrograde algorithm performs a factor of fewer search operations than a standard retrograde algorithm, producing a database with a factor of fewer entries, which allows retrograde searching to reach unprecedented search depths.

Abstract

Retrograde analysis is used in game-playing programs to solve states at the end of a game, working backwards toward the start of the game. The algorithm iterates through and computes the perfect-play value for as many states as resources allow. We introduce setrograde analysis which achieves the same results by operating on sets of states that have the same game value. The algorithm is demonstrated by computing exact solutions for Bridge double dummy card-play. For deals with 24 cards remaining to be played ( states, which can be reduced to states using preexisting techniques), we strongly solve all deals. The setrograde algorithm performs a factor of fewer search operations than a standard retrograde algorithm, producing a database with a factor of fewer entries. For applicable domains, this allows retrograde searching to reach unprecedented search depths.

Paper Structure

This paper contains 26 sections, 4 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Overview
  4. State-Space Reduction
  5. Set-Space Reduction
  6. Generalization
  7. Querying
  8. Iteration
  9. Example 1: Generalization in Bridge
  10. Example 2: Low-Cost Iteration in Bridge
  11. Setrograde Analysis
  12. Implementation in Bridge
  13. Set Representation:
  14. Low-Cost Generalization:
  15. Low-Cost Querying:
  16. ...and 11 more sections

Figures (4)

  • Figure 1: $\spadesuit$98 wins two tricks regardless of the locations of lower cards in the perfect-information deal (left) and set of deals (right); x refers to any low card. Play proceeds clockwise starting with North (N) until each player has played one card (East, South, West). The highest card wins. The winning player is next-to-play.
  • Figure 2: Sets generated by replacing 0 or more low-rank cards with x's. East is on the lead.
  • Figure 3: Top: $\spadesuit$98 wins two tricks regardless of the locations of lower cards in each set depicted. East is next-to-play in all diagrams. Bottom: Compact representation of the union of the sets above.
  • Figure 4: Illustration of a shallow tree structure used to store setrograde databases. The descent condition for a state lookup is superimposed on the path from the root node to its first child.