Endgame studies have long served as a tool for testing human creativity and intelligence. We find that they can serve as a tool for testing machine ability as well. Two of the leading chess engines, Stockfish and Leela Chess Zero (LCZero), employ significantly different methods during play. We use Plaskett’s Puzzle, a famous endgame study from the late 1970s, to compare the two engines. Our experiments show that Stockfish outperforms LCZero on the puzzle. We examine the algorithmic differences between the engines and use our observations as a basis for carefully interpreting the test results. Drawing inspiration from how humans solve chess problems, we ask whether machines can possess a form of imagination. On the theoretical side, we describe how Bellman’s equation may be applied to optimize the probability of winning. To conclude, we discuss the implications of our work on artificial intelligence (AI) and artificial general intelligence (AGI), suggesting possible avenues for future research.
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$$ Q $$‐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$$ Q $$‐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 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.
Chess is not a game. Chess is a well-defined form of computation. You may not be able to work out the answers, but in theory, there must be a solution, a right procedure in any position--John von Neumann
How likely is it that Magnus Carlsen will achieve his goal of a 2900 Elo rating? At what level of play does Magnus have a reasonable chance of reaching the 2900 goal? These two questions are of great current interest to Magnus and the chess community. The probabilistic properties of Elo's rating system are well known, and together with a Brownian motion model of rating evolution, we use simulation‐based methods to address these questions. Our model assesses that Magnus has a 4.5% chance of reaching 2900 if he continues his 2020–2022 level of play. However, this increases dramatically to 80$$ 80 $$% chance if he can repeat his hot streak performance of 2019 which is not an easy undertaking. The probabilities are intimately related to Elo's choice K$$ K $$‐factor used for grandmaster chess play. Finally, we conclude with a discussion of the policy issues involved with the choice of K$$ K $$‐factor.
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