Author: David Eppstein
Date: 11:40:55 09/28/98
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On September 28, 1998 at 09:09:28, Robert Hyatt wrote: >On September 27, 1998 at 23:13:51, David Eppstein wrote: >>I think a more principled approach would be to have an eval that gives you >>explicit and separate probabilities of various game-ending events: >>What are the probabilities that black or white wins by a quick checkmate or >>decisive combination? That black or white can force a repetition draw? >>That (if the game reaches an endgame) white or black will win there? >>The overall eval would then be a simple nonlinear combination of these >>probabilities (just the expected value of the overall game). > >the problem is that alpha/beta only will use *one* value. So you have to >figure out which one to use, or how to combine all of them into one large >score. And that can be a problem... I did say which one value to use (the sentence beginning "the overall eval"). Maybe it will be clearer if I spell it out more explicitly. First, to forestall people from saying that probabilities make no sense in a deterministic game, by probability I mean what fraction of positions with similar eval terms have a given outcome. I will assume that the eval is from the point of view of white (i.e. a bigger number is better like with the evals you have now). Let W,L,D be parameters specifying the numerical value of a win, loss, or draw. It would be normal to make these 1,0,-1 but you could instead make them be the number of rating points you would gain or lose, or change them according to your tournament situation and desired aggressiveness. Let p1 estimate the probability that, if the game reaches an endgame, white will win, and let p2 estimate the probability that, if the game reaches an endgame, black will win. These estimates would presumably be closely related to the usual material count and positional terms for pawn structure etc. Compute E = p1*W + p2*L + (1-p1-p2)*D. E is the expected value of the position, counting only endgame terms. Let p3 estimate the probability that (unless black does something first) white can force a winning combination (checkmate or decisive win of material) before reaching an endgame. Let p4 be the same estimate for black. Both of these estimates involve terms like king safety, number of active pieces, pawn storms, etc. Also, let p5 be the probability that white's attack happens before black's. Let p6 = p3*(p5 + (1-p4)*(1-p5)) and p7 = p4*((1-p5) + (1-p3)*p5). Compute M = p6*W + p7*L + (1 - p6 - p7)*E. M is the expected value of the position, combining the endgame eval E with terms measuring the strength of the middlegame attacks. Let p8 be an estimate of the probability that white can force a perpetual check and let p9 be the same estimate for black. These estimates involve terms like the ability of one side to give check and the exposure of the other side's king. If M < D compute X = p8*D + (1-p8)*M, else compute X = p9*D + (1-p9)*M. Use X as your evaluation function, as part of the usual alpha-beta routine.
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