Gambits: Theory and Evidence

TL;DR

This paper develops a theoretical framework for chess gambits using Bellman optimality and empirically tests their effectiveness through analysis of expert human play and engine evaluations, revealing insights into decision-making and irrational biases.

Contribution

It introduces a Bellman-based decision-theoretic model for gambits and empirically validates it using engine and human data, advancing understanding of strategic disruption in chess.

Findings

Bellman Q-values effectively measure gambit strength.

Expert play often deviates from optimal Bellman paths.

Certain gambits show irrational skewness in decision-making.

Abstract

Gambits are central to human decision-making. Our goal is to provide a theory of Gambits. A Gambit is a combination of psychological and technical factors designed to disrupt predictable play. Chess provides an environment to study gambits and behavioral game theory. Our theory is based on the Bellman optimality path for sequential decision-making. This allows us to calculate the $Q$-values of a Gambit where material (usually a pawn) is sacrificed for dynamic play. On the empirical side, we study the effectiveness of a number of popular chess Gambits. This is a natural setting as chess Gambits require a sequential assessment of a set of moves (a.k.a. policy) after the Gambit has been accepted. Our analysis uses Stockfish 14.1 to calculate the optimal Bellman $Q$ values, which fundamentally measures if a position is winning or losing. To test whether Bellman's equation holds in play, we estimate the transition probabilities to the next board state via a database of expert human play. This then allows us to test whether the \emph{Gambiteer} is following the optimal path in his decision-making. Our methodology is applied to the popular Stafford and reverse Stafford (a.k.a. Boden-Kieretsky-Morphy) Gambit and other common ones including the Smith-Morra, Goring, Danish and Halloween Gambits. We build on research in human decision-making by proving an irrational skewness preference within agents in chess. We conclude with directions for future research.