Chapter 6 Adversarial Search Game Playing

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Chapter 6 Adversarial Search
Game Playing
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Outline

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

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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

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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

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Game tree for the game of tic-tac-toe

• High values are good for MAX and bad for MIN

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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.

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The minimax algorithm

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

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The property of the minimax algorithm

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

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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

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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

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Alpha-Beta Pruning (contd)
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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

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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
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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

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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

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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
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• weighted linear function
• nonlinear combinations of features if the
  contribution of each feature is depends on values
  of the other features.

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Deterministic games in practice