1. 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

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

3. 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

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

5. 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?

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

7. 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  (2006):
WHAT ARE “TRUE” POSITION VALUES!?
… and it is clearly not the game theoretical values what everyone seem to have in mind!

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

9. 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

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

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

12. 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

13. 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

14. 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

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

16. 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 , 2007.

17. 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 , 2006.

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

19. 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.

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

21. 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.

22. Thank you