f1(s) = (number of white queens) (number of black queens), etc. Other features which ... Othello: human champions refuse to compete against computers, who are ...
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What does it means?
Involves Animal Behave
A Two-player zero-sum discrete finite deterministic game of perfect information is a quintuplet:
( S , I , N , T , V ) where:
+1 for MAX (=win)
-1 for MIN (=loose)
0 for a draw.
For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible
b … branching factor m … maximum number of moves
· perfect information5
Pruning possible!α-β pruning example
We see: possibility to prune depends on the order of processing the successors!
doubles possible depth of search doable in the same time
prune that branch
Suppose we have 100 secs, explore 104 nodes/sec106nodes per move
even with pruning not possible to explore the whole search space e.g. for chess!
e.g., depth limit (perhaps add quiescence search)
= estimated desirability of position
Eval(s) = w1 f1(s) + w2 f2(s) + … + wn fn(s)
w1 = 9 with
f1(s) = (number of white queens) – (number of black queens), etc.
Other features which could be taken into account: number of threats, good structure of pawns, measure of safety of the king.
MinimaxCutoff is identical to MinimaxValue except
Does it work in practice?
bm = 106, b=35 m=4
4-ply lookahead is a hopeless chess player!
2-player algorithms (minimax, -, cutoff-eval) can be extended to multi-player in a straightforward way:
Expectiminimax next slide
Chance nodes have certain probabibilities.
where P(s) is the probability of reaching s (e.g.
probability of rolling a certain number with the dice)
What makes Game theory interesting in practice?