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Search with Random Leaf Values is of interest concerning playing strength, match statistics and search pathology. Randomized evaluation by adding noise concerns evaluation accuracy and evaluation error analysis - it might be used in introducing and learning new evaluation terms for various games or general game playing programs, or simply in randomizing or weakening engine play.
Japanese maple autumn leaves [1]

The Beal Effect

Random evaluation was first examined for the game of chess by Don Beal [2] and Martin C. Smith at the Advances in Computer Chess 7 conference at University of Limburg, July 1993, published in the ICCA Journal and conference proceedings [3], and further analyzed by Mark Levene and Trevor Fenner in 1995 [4] and 2001 [5]. Although using random numbers as "evaluation" results in random play with a one ply search (root-random), it was found that the strength of play rises rapidly with increased depth (lookahead-random) using a full-width minimax search. While a natural assumption is that lookahead on random numbers would also result in a random choice at the root as well, random evaluation would create a statistical preference for positions with large mobilty, and thus likely strong material [6].


To demonstrate this so called Beal Effect it is neccessary to consider awareness of terminal nodes where mate scores would favour deeper lookahead. Therefor root-random is replaced by lookahead-zero, performing a lookahead with the same search depth as lookahead-random, but non terminal leaves evaluated as zero, only tie-breaking at the root by a random number. Still a very weak player, a five ply search was already sufficient to win all of 100 games versus a random player.

Beal and Smith used following setup to automatically play the games: Draws by stalemate and four cases of insufficient material were recognized (KK, KNK, KBK, KNNK), but 50-move rule or threefold repetition discarded. Therefor games were limited to 200 moves and then WDL adjudicated by +=- material balance (which happend rarely) [7].

Further Experiments

Beal and Smith further applied random evaluations to components rather than the whole evaluation. They used material balance as dominating term plus a random number below one pawn unit. While a five ply search with random component only gained 63% over zero component, quiescing the material balance by exploring a capture tree in order to obtain the chess specific part of the evaluation, the random component gained 97% within the same search depth of five plies.

See also


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External Links


  1. ^ Acer palmatum subsp. matsumurae (Koidz) Ogata, image by 松岡明芳, November 22, 2006, CC BY 3.0, Wikimedia Commons, Autumn leaf color from Wikipedia
  2. ^ The term "Beal Effect" was coined by Robert Hyatt, see Re: To kick off some technical discussions by Robert Hyatt, OpenChess Forum, June 20, 2010
  3. ^ Don Beal, Martin C. Smith (1994). Random Evaluations in Chess. Advances in Computer Chess 7
  4. ^ Mark Levene, Trevor Fenner (1995). A Partial Analysis of Minimaxing Game Trees with Random Leaf Values. ICCA Journal, Vol. 18, No. 1
  5. ^ Mark Levene, Trevor Fenner (2001). The Effect of Mobility on Minimaxing of Game Trees with Random Leaf Values. Artificial Intelligence, Vol. 130, No. 1
  6. ^ Re: "random mover" chess programs by Harm Geert Muller, CCC, June 24, 2016
  7. ^ Don Beal, Martin C. Smith (1994). Random Evaluations in Chess. ICCA Journal, Vol. 17, No. 1

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