Learning Models of Opponent's Strategies in Game Playing. In
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row shows that MSTAR indeed beneeted using the extra knowledge about the opponent. In the second experiment all players searched to depth 4. The three players used the same evaluation function f(b; pl) = Mat(b; pl) ? 1 25 Tot(b) where Mat(b; pl) returns the material advantage of player pl and Tot(b) is the total number of pieces. However, while M knows its opponent's function, MM implicitly assumes that its opponent, o, uses the function f(b; o) = ?f(b; pl) = ?Mat(b; pl) + 1 25 Tot(b) = Mat(b; o)+ 1 25 Tot(b). Therefore, while OP prefers exchange positions, MM assumes that it prefers to avoid them. The following table shows the results obtained. Wins Draws Losses Points MM vs. OP 171 461 168 803 MSTAR vs. OP 290 351 159 931 This result is rather surprising. Despite the fact that all players used the same evaluation function, and searched to the same depth, the modeling program achieved a signiicantly higher score. We repeated the last experiment replacing the M 1?pass algorithm with and measured the amount of pruning. The bound used for pruning is jf 1 + f 0 j = jMat(b; pl) ? 1 25 Tot(b) + Mat(b; o) ? 1 25 Tot(b)j 2 25 Tot(b). While the average number of leaves-per-move for a search to depth 4 by M 1?pass was 723, managed to achieve an average of 190 leaves-per-move. achieved an average of 66 leaves-per-move. The last results raise an interesting question. Assume that we allocate a modeling player and an un-modeling opponent the same search resources. Is the beneet achieved by modeling enough to overcome the extra depth that the non-modeling player can search due to the better pruning? We have tested the question in the context of the above experiment. We wrote an iterative deepening versions of both and and let the two play against each other with the same limit on the number of leaves. Wins Draws Losses Points vs. OP 226 328 246 780 This result is rather disappointing. The beneet of modeling was outweighed by the cost of the reduced pruning. In general this is an example of the known tradeoo between knowledge and search. The question of when is it worthwhile to use the modeling approach remains open and depends mostly on the particular nature of the search tree and evaluation functions. Conclusions This paper describes a generalized version of …