Home * Evaluation

		An evaluation function is used to heuristically determine the relative value of a position, i.e. the chances of winning. If we could see to the end of the game in every line, the evaluation would only have values of -1 (loss), 0 (draw), and 1 (win). In practice, however, we do not know the exact value of a position, so we must make an approximation. Beginning chess players learn to do this starting with the value of the pieces themselves. Computer evaluation functions also use the value of the material as the most significant aspect and then add other considerations.
Wassily Kandinsky, Schach-Theorie, 1937 ^[1]

Where to Start

The first thing to consider when writing an evaluation function is how to score a move in Minimax or the more common NegaMax framework. While Minimax usually associates the white side with the max-player and black with the min-player and always evaluates from the white point of view, NegaMax requires a symmetric evaluation in relation to the side to move. We can see that one must not score the move per se – but the result of the move (i.e. a positional evaluation of the board as a result of the move). Such a symmetric evaluation function was first formulated by Claude Shannon in 1949 ^[2] :

f(p) = 200(K-K')
       + 9(Q-Q')
       + 5(R-R')
       + 3(B-B' + N-N')
       + 1(P-P')
       - 0.5(D-D' + S-S' + I-I')
       + 0.1(M-M') + ...
 
KQRBNP = number of kings, queens, rooks, bishops, knights and pawns
D,S,I = doubled, blocked and isolated pawns
M = Mobility (the number of legal moves)

Here, we can see that the score is returned as a result of subtracting the current side's score from the equivalent evaluation of the opponent's board scores (indicated by the prime letters K' Q' and R'.. ).

Side to move relative

In order for NegaMax to work, it is important to return the score relative to the side being evaluated. For example, consider a simple evaluation, which considers only material and mobility:

materialScore = kingWt  * (wK-bK)
              + queenWt * (wQ-bQ)
              + rookWt  * (wR-bR)
              + knightWt* (wN-bN)
              + bishopWt* (wB-bB)
              + pawnWt  * (wP-bP)
 
mobilityScore = mobilityWt * (wMobility-bMobility)

return the score relative to the side to move (who2Move = +1 for white, -1 for black):

Eval  = (materialScore + mobilityScore) * who2Move

Linear vs. Nonlinear

Most evaluations terms are a linear combination of independent features and associated weights in the form of

$\displaystyle Eval = \sum_{i=1}^{n} {Fi * Wi}$

A function f is linear if the function is additive:

$f(a+b) = f(a) + f(b)$

and second if the function is homogeneous of degree 1:

$f(c*a) = c * f(a)$

It depends on the definition and independence of features and the acceptance of the axiom of choice (Ernst Zermelo 1904), whether additive real number functions are linear or not ^[3] . Features are either related to single pieces (material), their location (piece-square tables), or more sophisticated, considering interactions of multiple pawns and pieces, based on certain patterns or chunks. Often several phases to first process simple features and after building appropriate data structures, in consecutive phases more complex features based on patterns and chunks are used.

Based on that, to distinguish first-order, second-order, etc. terms, makes more sense than using the arbitrary terms linear vs. nonlinear evaluation ^[4] . With respect to tuning, one has to take care that features are independent, which is not always that simple. Hidden dependencies may otherwise make the evaluation function hard to maintain with undesirable nonlinear effects.

General Aspects

Basic Evaluation Features

Considering Game Phase

Game Phases
Opening
Middlegame
Endgame
Evaluation Discontinuity
Tapered Eval (a score is interpolated between opening and endgame based on game stage/pieces)

Miscellaneous

Publications

1949

Claude Shannon (1949). Programming a Computer for Playing Chess. pdf from The Computer History Museum

1950 ...

Eliot Slater (1950). Statistics for the Chess Computer and the Factor of Mobility, Proceedings of the Symposium on Information Theory, London. Reprinted 1988 in Computer Chess Compendium, pp. 113-117. Including the transcript of a discussion with Alan Turing and Jack Good
Alan Turing (1953). Chess. part of the collection Digital Computers Applied to Games, in Bertram Vivian Bowden (editor), Faster Than Thought, a symposium on digital computing machines, reprinted 1988 in Computer Chess Compendium, reprinted 2004 in The Essential Turing, google books

1960 ...

Israel Albert Horowitz, Geoffrey Mott-Smith (1960,1970,2012). Point Count Chess. Samuel Reshevsky (Introduction), Sam Sloan (2012 Introduction), Amazon ^[5]
Jack Good (1968). A Five-Year Plan for Automatic Chess. Machine Intelligence II pp. 110-115

1970 ...

Ron Atkin (1972). Multi-Dimensional Structure in the Game of Chess. In International Journal of Man-Machine Studies, Vol. 4
Ron Atkin, Ian H. Witten (1975). A Multi-Dimensional Approach to Positional Chess. International Journal of Man-Machine Studies, Vol. 7, No. 6
Gerard Zieliński (1976). Simple Evaluation Function. Kybernetes, Vol. 5, No. 3
Ron Atkin (1977). Positional Play in Chess by Computer. Advances in Computer Chess 1
David Slate, Larry Atkin (1977). CHESS 4.5 - The Northwestern University Chess Program. Chess Skill in Man and Machine (ed. Peter W. Frey), pp. 82-118. Springer-Verlag, New York, N.Y. 2nd ed. 1983. ISBN 0-387-90815-3. Reprinted (1988) in Computer Chess Compendium
Hans Berliner (1979). On the Construction of Evaluation Functions for Large Domains. IJCAI 1979 Tokyo, Vol. 1, pp. 53-55.

1980 ...

Helmut Horacek (1984). Some Conceptual Defects of Evaluation Functions. ECAI-84, Pisa, Elsevier
Peter W. Frey (1985). An Empirical Technique for Developing Evaluation Functions. ICCA Journal, Vol. 8, No. 1
Tony Marsland (1985). Evaluation-Function Factors. ICCA Journal, Vol. 8, No. 2, pdf
Jens Christensen, Richard Korf (1986). A Unified Theory of Heuristic Evaluation functions and Its Applications to Learning. Proceedings of the AAAI-86, pp. 148-152, pdf
Dap Hartmann (1987). How to Extract Relevant Knowledge from Grandmaster Games. Part 1: Grandmasters have Insights - the Problem is what to Incorporate into Practical Problems. ICCA Journal, Vol. 10, No. 1
Dap Hartmann (1987). How to Extract Relevant Knowledge from Grandmaster Games. Part 2: the Notion of Mobility, and the Work of De Groot and Slater. ICCA Journal, Vol. 10, No. 2
Bruce Abramson, Richard Korf (1987). A Model of Two-Player Evaluation Functions. AAAI-87. pdf
Kai-Fu Lee, Sanjoy Mahajan (1988). A Pattern Classification Approach to Evaluation Function Learning. Artificial Intelligence, Vol. 36, No. 1
Dap Hartmann (1989). Notions of Evaluation Functions Tested against Grandmaster Games. Advances in Computer Chess 5
Maarten van der Meulen (1989). Weight Assessment in Evaluation Functions. Advances in Computer Chess 5
Bruce Abramson (1989). On Learning and Testing Evaluation Functions. Proceedings of the Sixth Israeli Conference on Artificial Intelligence, 1989, 7-16.
Danny Kopec, Ed Northam, David Podber, Yehya Fouda (1989). The Role of Connectivity in Chess. Workshop on New Directions in Game-Tree Search, pdf

1990 ...

Bruce Abramson (1990). On Learning and Testing Evaluation Functions. Journal of Experimental and Theoretical Artificial Intelligence 2: 241-251.
Ron Kalnim (1990). A Positional Assembly Model. ICCA Journal, Vol. 13, No. 3
Paul E. Utgoff, Jeffery A. Clouse (1991). Two Kinds of Training Information for Evaluation Function Learning. University of Massachusetts, Amherst, Proceedings of the AAAI 1991
Ingo Althöfer (1991). An Additive Evaluation Function in Chess. ICCA Journal, Vol. 14, No. 3
Ingo Althöfer (1993). On Telescoping Linear Evaluation Functions. ICCA Journal, Vol. 16, No. 2 ^[6]
Alois Heinz, Christoph Hense (1993). Bootstrap learning of α-β-evaluation functions. ICCI 1993, pdf
Alois Heinz (1994). Efficient Neural Net α-β-Evaluators. pdf ^[7]
Peter Mysliwietz (1994). Konstruktion und Optimierung von Bewertungsfunktionen beim Schach. Ph.D. Thesis (German)
Don Beal, Martin C. Smith (1994). Random Evaluations in Chess. ICCA Journal, Vol. 17, No. 1
Yaakov HaCohen-Kerner (1994). Case-Based Evaluation in Computer Chess. EWCBR 1994
Michael Buro (1995). Statistical Feature Combination for the Evaluation of Game Positions. JAIR, Vol. 3
Peter Mysliwietz (1997). A Metric for Evaluation Functions. Advances in Computer Chess 8
Michael Buro (1998). From Simple Features to Sophisticated Evaluation Functions. CG 1998, pdf

2000 ...

Dan Heisman (2003). Evaluation Criteria, pdf from ChessCafe.com
Jeff Rollason (2005). Evaluation by Hill-climbing: Getting the right move by solving micro-problems. AI Factory, Autumn 2005 » Automated Tuning
Shogo Takeuchi, Tomoyuki Kaneko, Kazunori Yamaguchi, Satoru Kawai (2007). Visualization and Adjustment of Evaluation Functions Based on Evaluation Values and Win Probability. AAAI 2007
Omid David, Moshe Koppel, Nathan S. Netanyahu (2008). Genetic Algorithms for Mentor-Assisted Evaluation Function Optimization, ACM Genetic and Evolutionary Computation Conference (GECCO '08), pp. 1469-1475, Atlanta, GA, July 2008.
Omid David, Jaap van den Herik, Moshe Koppel, Nathan S. Netanyahu (2009). Simulating Human Grandmasters: Evolution and Coevolution of Evaluation Functions. ACM Genetic and Evolutionary Computation Conference (GECCO '09), pp. 1483 - 1489, Montreal, Canada, July 2009.
Omid David (2009). Genetic Algorithms Based Learning for Evolving Intelligent Organisms. Ph.D. Thesis.