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Chapter 6 Adversarial Search Game Playing

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Title: Chapter 6 Adversarial Search Game Playing


1
Chapter 6 Adversarial Search
Game Playing
2
Outline
  • Games of perfect information - perfect play
  • The minimax strategy
  • Multiplayer games
  • Alpha-Beta pruning
  • Games of imperfect information

3
Games
  • Competitive environments
  • goals of two agents are in conflict adversarial
    search
  • Perfect play
  • deterministic and fully observable
  • turn-taking actions of two players (agents)
    alternate
  • zero-sum the utility values at the end of the
    game are equal and opposite (adversarial)
  • e.g., chess, winner (1) and loser (-1)
  • Types of games

4
Define game as a search problem
  • initial state
  • the board position, the player to move, etc.
  • successor function
  • generates a list of (move, state) pairs
  • terminal test
  • decides when the game is over
  • terminal states states when the game has ended.
  • utility function
  • gives a numeric value for the terminal states.
  • zero-sum games
  • game tree
  • defined by the initial state and the legal moves
    for each side

5
Game tree for the game of tic-tac-toe
  • High values are good for MAX and bad for MIN

6
Optimal contingent strategy
  • Optimal strategy
  • leads to outcomes at least as good as any other
    strategy when one is playing a infallible
    opponent infeasible in practice.
  • 2-ply game
  • the tree is one move deep, consisting of two
    half-moves, each of which is a ply.
  • MAXs moves in the states resulting from every
    possible response by MIN
  • minimax value of a node the utility of being the
    corresponding state
  • MAX (MIN) prefers to move to a state of maximum
    (minimum) value.
  • minimax decision at the root.

7
The minimax algorithm
  • computes the minimax decision from the current
    state
  • recursion proceeds down to the leaves
  • minimax values are backed up

8
The property of the minimax algorithm
  • Complete?
  • Optimal?
  • Time?
  • Space?

9
Optimal decisions in multiplayer games
  • vector form e.g. utility is ltvA 1, vB 2, vC
    6gt
  • pick up move (successor) having the highest value

10
Alpha-Beta Pruning
  • compute the minimax decision without looking at
    every node
  • pruning away branches that cannot possibly
    influence the final decision
  • Alpha value of best choice for MAX
  • Beta value of best choice for MIN

11
Alpha-Beta Pruning (contd)
12
Alpha-Beta Pruning (contd)
  • MINIMAX-VALUE (root) max(min(3,12,8), min(2, x,
    y), min(14,5,2))
  • max(3, min(2,x,y), 2)
  • max (3, z, 2)
    where z 2
  • 3
  • the value of the root (minimax decision) is
    independent of the values of the pruned leaves x
    and y.
  • depends on the order in which the successors are
    examined

13
How good is the Alpha-Beta pruning?
e.g., try captures first, then threats, then
forward moves, and then backward moves
effective branching factor becomes
14
Imperfect decisions
  • Moves must be made in a reasonable (minutes)
    amount of time
  • Using Alpha-Beta pruning, the depth is still not
    practical if we insist on reaching the terminal
    states
  • should cut off the search earlier by applying a
    heuristic evaluation function to states
  • evaluation function estimates the utility of the
    position
  • use cut off test instead of terminal test
  • turning nonterminal nodes into terminal leaves

15
How to design good evaluation functions?
  • Requirements
  • order the terminal states in the same way as the
    true utility function
  • must not take too long
  • chances of winning
  • uncertain about the final outcomes because of the
    cut off
  • categories or equivalence classes of states
  • the states have the same values, leading to wins,
    losses, or draws
  • the value of evaluation function should reflect
    the proportion of states with each outcome wins
    (72), losses (20), or draws (8)
  • weighted average (expected) value
  • requires experience and too many categories

16
How to design good evaluation functions?
  • In practice
  • computes separate numerical contributions from
    each feature and then combines them to find the
    total value
  • material value for each piece,
  • e.g., pawn 1, knight/bishop 3, rook 5, queen
    9
  • weighted linear function
  • nonlinear combinations of features if the
    contribution of each feature is depends on values
    of the other features.

17
Deterministic games in practice
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