Influence of Search Depth on Position Evaluation heuristic search Matej Guid, Ivan Bratko Faculty of Computer and Information Science University of Ljubljana, Slovenia The Fifteenth International Conference on Advances in Computer Games 2017 ACG 2017: Leiden, Netherlands, July 3-5, 2017

The Turing Test for Chess Engines White to move. What would you play? What would computers play?

Outline IMPACT ON PRACTICE AND THEORY OF GAME PLAYING Why minimaxed heuristic evaluations should not respect the minimax relation? Empirical evidence IMPACT ON PRACTICE AND THEORY OF GAME PLAYING searching to variable depths revisited decision changes with deeper search computer analysis of chess games detecting fortresses in chess

The Minimax Relation ... the principal variation ... ... ... ... ... ... ... ... ... ... ... ... ... ... game-theoretic values of positions respect the minimax relation

Do chess engines approximate some unknown true position value? Chess Engines and Position Evaluation Heuristic values are supposed to reflect goodness of a position… or position’s “worth”, “merit”, “strength”, “quality”, or “promise”. Do chess engines approximate some unknown true position value?

“True” Values in Chess Games chess players accepted the centipawn as the unit used as a measure of advantage +1.15 according to Houdini

WHAT ARE “TRUE” POSITION VALUES!? “True” Values in Heuristic Search Papers several examples of heuristic values approaching “true” values in the literature Luštrek, M. et al. "Is real-valued minimax pathological?." Artificial Intelligence 170.6-7 (2006): 620-642. WHAT ARE “TRUE” POSITION VALUES!? … and it is clearly not the game theoretical values what everyone seem to have in mind!

Searching Deeper ... the principal variation ... ... ... ... ... ... ... ... ... ... ... ... ... ... Should minimaxed heuristic values respect the minimax relation? Do minimaxed heuristic values approximate the true position value?

A Curious Phenomenon EXPERIMENT RYBKA 2.1c 32-bit (one of the strongest engines in 2006) assigned the same heuristic evaluations to all winning positions in this elementary endgame (+4.92) this phenomenon only occurred at fixed search depths of 4 or more plies EXPERIMENT 100 randomly chosen mate-in-16 positions black player defending optimally (using tablebases) the program searched to 4, 5, … , 12 plies even at 12-ply search the program often failed to deliver checkmate (!) however, 2-ply search (with variable evaluations) checkmated in 100% cases

+ = _ Progress Achieving Play the winning player should be increasing the advantage ≠ advantage-preserving play + evaluation = _ depth

Experiment a large number of chess positions was divided into six groups based on evaluations qualitatively similar results were obtained with different chess engines

Experiment with Won Positions backed-up (minimaxed) evaluations tend to increase with depth of search the evaluations of better moves on average increase more rapidly

Experiment with Balanced Positions backed-up (minimaxed) evaluations tend to approach to zero with depth of search as before, the positions were chosen based on the evaluation at the highest depth

Outline IMPACT ON PRACTICE AND THEORY OF GAME PLAYING Why minimaxed heuristic evaluations should not respect the minimax relation? Empirical evidence IMPACT ON PRACTICE AND THEORY OF GAME PLAYING searching to variable depths revisited decision changes with deeper search computer analysis of chess games detecting fortresses in chess

?? Search to Variable Depth Revisited White to move. Which move is stronger? a) 40. a5-a6 +3.89 at depth 15 b) 40. Nc7-e6 +5.29 at depth 26 “higher value and more reliable search depth” ?? Evaluations obtained at different depths are not directly comparable!

Decision Changes with Deeper Search in positions with decisive advantage there are far less decision changes with depth decision changes with depth evaluation changes with depth Less decision changes with deeper search in positions with advantage for one side are due to bigger differences in evaluations. Guid M., Bratko I. Factors Affecting Diminishing Returns for Searching Deeper. ICGA Journal, Vol. 30, No. 2, pp. 75-84, 2007.

x Computer Analysis of Chess Games the scores of computer analysis, reflecting the champions‘ performance based on the evaluation of individual moves Should the analysis be: time-limit based fixed-depth based ? x Guid M., Bratko I. Computer Analysis of World Chess Champions. ICGA Journal, Vol. 29, No. 2, pp. 65-73, 2006.

Detecting Fortresses in Chess White to play and draw. How could chess engines discover the path to draw?

Experiment: Detecting Fortresses in Chess go deep experiment with fortresses from Dvoretsky’s Chess Endgame Manual Fortress: minimaxed evaluations cease to increase or decrease with deeper search.

Fortresses Are Not a Rare Occurrence in Chess Karsten Müller: Knight vs bishop - the eternal duel. ChessBase, July 4, 2017.

Conclusions IMPACT ON PRACTICE AND THEORY OF GAME PLAYING Game-theoretic values of positions respect the minimax relation, but minimaxed heuristic evaluations should not respect it! They should change with deeper search. Heuristic evaluation functions of chess engines do have this property that enables progress achieving play. IMPACT ON PRACTICE AND THEORY OF GAME PLAYING Evaluations obtained at different depths are not directly comparable! Less decision changes with deeper search in positions with advantage for one side are due to bigger differences in evaluations. In assessing players’ performance, computer analysis of chess games should be fixed-depth (not time-limit) based. In the positions that could be regarded as fortresses, minimaxed evaluations cease to increase or decrease with deeper search.

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