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Chapter 6 Adversarial Search – Game Playing Outline Games of perfect information - perfect play The minimax strategy Multiplayer games Alpha-Beta pruning Games of imperfect information Games Competitive environments

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  • Games of perfect information - perfect play
  • The minimax strategy
  • Multiplayer games
  • Alpha-Beta pruning
  • Games of imperfect information
  • 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

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
game tree for the game of tic tac toe
Game tree for the game of tic-tac-toe
  • High values are good for MAX and bad for MIN
optimal contingent strategy
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.
    • MAX’s 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.

The minimax algorithm

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

The property of the minimax algorithm

  • Complete?
  • Optimal?
  • Time?
  • Space?

Optimal decisions in multiplayer games

  • vector form: e.g. utility is <vA = 1, vB = 2, vC = 6>
  • pick up move (successor) having the highest value
alpha beta pruning
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
alpha beta pruning cont d12
Alpha-Beta Pruning (cont’d)
  • 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

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


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 heuristicevaluation 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

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

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.