Endgames in bidding chess

TL;DR

This paper analyzes bidding chess, a variant where players bid for moves, and demonstrates that for certain game classes, each position has rational upper and lower values representing winning and losing thresholds, with exact calculations for three-piece endgames.

Contribution

It generalizes known results on Richman games to bidding chess and computes precise rational values for all three-piece endgames, revealing complex value structures.

Findings

All three-piece endgames have coinciding upper and lower values.

Values can have denominators up to 138 digits, indicating high complexity.

Values are rational numbers representing winning thresholds.

Abstract

Bidding chess is a chess variant where instead of alternating play, players bid for the opportunity to move. Generalizing a known result on so-called Richman games, we show that for a natural class of games including bidding chess, each position can be assigned rational upper and lower values corresponding to the limit proportion of money that Black (say) needs in order to force a win and to avoid losing, respectively. We have computed these values for all three-piece endgames, and in all cases, the upper and lower values coincide. Already with three pieces, the game is quite complex, and the values have denominators of up to 138 digits.