Expert-driven genetic algorithms for simulating evaluation functions

Abstract

In this paper we demonstrate how genetic algorithms can be used to reverse engineer an evaluation function’s parameters for computer chess. Our results show that using an appropriate expert (or mentor), we can evolve a program that is on par with top tournament-playing chess programs, outperforming a two-time World Computer Chess Champion. This performance gain is achieved by evolving a program that mimics the behavior of a superior expert. The resulting evaluation function of the evolved program consists of a much smaller number of parameters than the expert’s. The extended experimental results provided in this paper include a report on our successful participation in the 2008 World Computer Chess Championship. In principle, our expert-driven approach could be used in a wide range of problems for which appropriate experts are available.

Appendix

1.1 A. Experimental setup

Our experimental setup consisted of the following resources:

• Falcon chess engine running under UCI protocol, and Crafty 19, Junior 9, Fritz 8, and Hiarcs 8 running as a native ChessBase engines.

• Encyclopedia of Chess Middlegames (ECM) test suite, consisting of 879 positions.

• Fritz 8 interface for automatic running of matches. Fritz opening book was used for all games.

• AMD Athlon 64 3200+ with 1 GB RAM and Windows XP operating system.

1.2 B. Elo rating system

The Elo rating system, developed by Arpad Elo, is the official system for calculating the relative skill levels of players in chess. The following statistics from the January 2009 FIDE rating list provide a general impression of the meaning of the Elo rating system:

• 21079 players have a rating above 2200 Elo.

• 2886 players have a rating between 2400 and 2499, most of whom have either the title of International Master (IM) or Grandmaster (GM).

• 876 players have a rating between 2500 and 2599, most of whom have the title of GM.

• 188 players have a rating between 2600 and 2699, all of whom have the title of GM.

• 32 players have a rating above 2700.

Only four players have ever had a rating of 2800 or above. A novice player is generally associated with rating values below 1400 Elo. Given the rating difference (RD) between player A and player B, the expected winning rate w (0 ≤ w ≤ 1) of player A is given by

$$ w = {\frac{1} {10^{-RD/400} + 1}}. $$

(1)

Given the winning rate of player A against player B (as is the case in our experiments), the expected rating difference between the two players can be derived from the above formula, i.e.,

$$ RD = -400 \log_{10}({\frac{1} {w}} - 1). $$

(2)

In addition, given the results of a series of N matches between two players, we can derive confidence intervals for their rating difference. Without loss of generality, let W, D, and L denote, respectively, the number of wins, draws, and losses of the first player. The mean score and standard deviation are given, respectively, by

$$ \overline{x} = {\frac{W + D/2} {N}}. $$

(3)

and

$$ s = \sqrt{{\frac{W \cdot (1 - \overline{x})^2 + D \cdot(0.5 - \overline{x})^2 + L \cdot \overline{x}^2} {N - 1}}}. $$

(4)

Note that $\overline{x}$ is essentially an estimate of the expected winning rate. Now, suppose that we are interested in computing, for example, the 95% confidence interval (which corresponds to ± two standard deviations) of the rating difference. For this we compute the lower and upper ends of the winning rate, i.e., $w_{lo} = \overline{x} - 2s$ and $w_{hi} = \overline{x} + 2s$. Substituting w _lo and w _hi in Eq. 2 we obtain the corresponding lower and upper ends of the 95% confidence interval of the rating difference. Given any confidence level, one can compute the corresponding RD confidence interval similarly to the above described steps.