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Adversarial Search Chapter 6 Section 1 – 4 Games vs. search problems "Unpredictable" opponent  specifying a move for every possible opponent reply Time limits  unlikely to find goal, must approximate Solutions Search problems: a path Games: a strategy But how?

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adversarial search

Adversarial Search

Chapter 6

Section 1 – 4

games vs search problems
Games vs. search problems
  • "Unpredictable" opponent  specifying a move for every possible opponent reply
  • Time limits  unlikely to find goal, must approximate
  • Solutions
    • Search problems: a path
    • Games: a strategy
    • But how?
minimax
Minimax
  • Perfect play for deterministic games
  • Idea: choose move to position with highest minimax value = best achievable payoff against best play
  • E.g., 2-ply game:
properties of minimax
Properties of minimax
  • Complete? Yes (if tree is finite)
  • Optimal? Yes (against an optimal opponent)
  • Time complexity? O(bm)
  • Space complexity? O(bm) (depth-first exploration)
  • For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible
properties of
Properties of α-β
  • Pruning does not affect final result
  • Good move ordering improves effectiveness of pruning
  • With "perfect ordering," time complexity = O(bm/2)
  • A simple example of the value of reasoning about which computations are relevant (a form of meta-reasoning)
resource limits
Resource limits

Suppose we have 100 secs, explore 104 nodes/sec106nodes per move

Standard approach:

  • cutoff test:

e.g., depth limit (perhaps add quiescence search)

Explore nonquiescent positions further until quiescence

  • evaluation function

= estimated desirability of position

evaluation functions
Evaluation functions
  • For chess, typically linear weighted sum of features

Eval(s) = w1 f1(s) + w2 f2(s) + … + wn fn(s)

  • e.g., w1 = 9 with

f1(s) = (number of white queens) – (number of black queens), etc.

cutting off search
Cutting off search

MinimaxCutoff is identical to MinimaxValue except

  • Terminal? is replaced by Cutoff?
  • Utility is replaced by Eval

Does it work in practice?

bm = 106, b=35  m=4

4-step lookahead is a hopeless chess player!

  • 4-step ≈ human novice
  • 8-step ≈ typical PC, human master
  • 12-step ≈ Deep Blue, Kasparov