Expected Outcome Model of Two Player Games by Bruce Abramson

By Bruce Abramson

The Expected-Outcome version of Two-Player video games offers with the expected-outcome version of two-player video games, within which the relative advantage of game-tree nodes, instead of board positions, is taken into account. the anomaly of static assessment and the issues it generates within the seek approach are tested and the advance of a domain-independent static evaluator is described.

Comprised of 8 chapters, this ebook starts off with an outline of the reason for the mathematical examine of video games, by way of a dialogue on a few earlier synthetic intelligence (AI) examine efforts on game-trees. the following part opens with the definition of a nodes expected-outcome worth because the anticipated worth of the leaves underneath it. The expected-outcome version is printed, paying specific recognition to the expected-outcome worth of a game-tree node. This version was once carried out on a few small models of tic-tac-toe and Othello. The ebook additionally offers effects that supply powerful help for either the validity of the expected-outcome version and the rationality of its underlying assumptions.

This monograph is meant for experts in AI and desktop technological know-how.

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If the leaf values are generated randomly using different distribution functions, there is no reason to assume that similar variances should occur. In a real game, however, trees are built and leaf values are assigned according to the rules of the game. , one move) [33]. Since the subtrees beneath them are generated by a consistent rule set, the distributions will probably be quite similar. Differences among their means, then, are much more likely to occur than differences among their variances, and thus the assumptions necessary to vindicate expected-outcome users are reasonable.

For any z-score, the value of can be calculated. A quantile-based evaluation function inputs its relevant parameter zQ into the two Z-l functions, (one for each distribution being considered), receives as output the values Xl == Z1 1 (zQ) and X2 == Z2"l(ZQ), and chooses the best move. Lemma 1: If two evaluators, Qa-outcome and Qc-outcome, recommend different moves, then there is some z-score y, ZQa. < Y < ZQc' for which the two leaf distributions share a common x-intercept. Proof: If a function inputting ZQa.

B) If pi < (1 — Ξ&), then as d —> oo, Td{pi) = Fdfa) = · · · = Fd{Vb) = 0. Since all of the moves are LOSSes, they are all equally "optimal". Recall t h a t Td(p) is defined as the percentage of all d e p t h d trees whose leaves are assigned a 1 with probability p t h a t result in a forced win for M A X . According to (a) and (b), then, for any ρχ φ Ξ^, t h e percentage of forced-win trees approaches either 0 or 1 as d grows to oo. ,mi > be (d,6,/)-trees Expected-Outcome with identical branching factors and depths, and binomially distributed leaf values with expected-outcome values of pi > p2 > · · · > p&, respectively, where all roots converge.

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