cs 371: introduction to artificial intelligencecs.gettysburg.edu/~tneller/cs371/ppt/game-tree...
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CS 371: Introduction to
Artificial Intelligence
Game-Tree Search
© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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Game-Playing
• Introduction
• Minimax
• Alpha-beta pruning
• Expectiminimax
© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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Games as Search Problems
• Previously, we've looked at search problems with static
environments. (One agent affects the environment.)
• Now, we generalize just a bit and allow two agents to
affect the environment in turn. dynamic environment
• Previously, we've looked for a sequence of actions to a
goal state.
• Now, we're looking for a sequence of actions which
maximizes some utility measure regardless of how an
adversarial agent acts.
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A.I.M.A. text figures © Prentice Hall.
Used by permission.
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Example of Tree Search
• Search: Peg solitaire
– jump a peg over another
to empty space,
removing jumped peg
– Initial state: only one
space empty
– Goal state: only one
space occupied
– Find sequence of jumps
from initial state to goal
graphics from hoppers.puzzles.com
© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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Example of Game-Tree Search
• Search: Tic-Tac-Toe
– players place X and O in
turn.
– Initial state: empty 33 grid
– Goal state: three of a player's
symbol in a row
– Count win = +1, draw = 0,
loss = -1
– Find sequence of move
which maximizes utility
regardless of adversarial
play
x x
x
o o x
x x o
o x x
draw
0
…
… … …
© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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Example of Game-Tree Search
• Suppose you construct the complete tree of possible plays.
• Evaluate terminal states as (+1,0,-1)
• Evaluate non-terminal states as maximum/minimum of children evaluations for player X/O respectively.
• This propagation of evaluations is called minimax.
• Consider minimax on a subtree of possible tic-tac-toe plays…
© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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x
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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© Todd Neller.
A.I.M.A. text figures © Prentice Hall.
Used by permission.
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Another Minimax Example
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Minimax Decision-Making
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Perfect Decisions in Two-
Player Games • Problem definition: initial state, operators,
terminal test, utility (or payoff) function
• Given whole game tree, minimax yields perfect decisions*
• Minimax: minimum of the maximum of the minimum of the maximum of the…
*assuming adversary acts according to minimax importance of player modeling
• Can’t search whole game tree, so…
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Imperfect Decisions in Two-
Player Games
• Evaluate states passing cutoff-test according to
heuristic evaluation function
• Consider Chess
– enormous state space
– can't possibly search whole tree with current
computational limitations
• Must
– limit depth of search
– evaluate non-terminal nodes at limit
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• How would
you evaluate
these
positions?
• Material
advantage isn't
the whole
story.
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Heuristic Game-Play
• A good evaluation function
– returns actual value at terminal states,
– approximates actual value at non-terminal nodes, and
– isn't too computationally intensive
• Most attribute recent game-playing success to
better speed ("brute force") rather than better
evaluation (knowledge base)
• Still, most minimax search is pointless…
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Pruning Example
?
1 2 0 1 2 3 4 5
MAX
MIN
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Pruning Example
?
1 2 0 1 2 3 4 5
MAX
MIN
- ?
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Pruning Example
?
1 2 0 1 2 3 4 5
MAX
MIN
- ?
- ?
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A.I.M.A. text figures © Prentice Hall.
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Pruning Example
?
1 2 0 1 2 3 4 5
MAX
MIN
- ?
- ? 1
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Pruning Example
?
1
1
2 0 1 2 3 4 5
MAX
MIN
- ?
- ? 1
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A.I.M.A. text figures © Prentice Hall.
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Pruning Example
?
1
1
2 0 1 2 3 4 5
MAX
MIN
1 ?
- ? 1
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Pruning Example
?
1
1
2 0 1 2 3 4 5
MAX
MIN
1 ?
- ? 1 1 ?
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A.I.M.A. text figures © Prentice Hall.
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Pruning Example
?
1
1
2 0 1 2 3 4 5
MAX
MIN
1 ?
- ? 1 1 ? 0
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A.I.M.A. text figures © Prentice Hall.
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Pruning Example
?
1
1
2 0 1 2 3 4 5
MAX
MIN
1 ?
- ? 1 1 ? 0
Contradiction!
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Pruning Example
?
1
1
2 0 1 2 3 4 5
MAX
MIN
1 ?
- ? 1 1 ? 0
Therefore, play
will not rationally
proceed this way
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Pruning Example
?
1
1
2 0
1 2 3 4 5
MAX
MIN
1 ?
- ? 1 1 ? 0
Therefore, we can
prune unseen
nodes from
consideration.
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Pruning Example
?
1
1
2 0
1 2 3 4 5
MAX
MIN
1 ?
- ? 1 1 ? 0 0
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A.I.M.A. text figures © Prentice Hall.
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Pruning Example
1
1
1
2 0
1 2 3 4 5
MAX
MIN
1 ?
- ? 1 1 ? 0 0
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A.I.M.A. text figures © Prentice Hall.
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Pruning Guarantees
• While we search the tree, we
can keep track of guaranteed
maximum/minimum utilities if
play proceeds to each node.
• When we see a contradiction in
guarantees, we can prune
remaining children from further
consideration, because we've
proven a rational player will
never reach that node.
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Assuming Rationality? Ha!
• What if other player isn't rational?
• If evaluation is perfect, then one can always
do as well if not better against an irrational
player with rational play.
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Alpha-Beta Pruning
• Let , be local lower,upper bound guarantees
– "If play proceeds here, root will score at least ."
– "If play proceeds here, root will score at most ."
• Pruning thus according to and is called alpha-
beta pruning.
• Minimax search with alpha-beta pruning is
sometimes called alpha-beta search.
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Alpha-Beta Pruning Algorithm
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Alpha-Beta Pruning Exercise
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Alpha-Beta Pruning Exercise
(cont.)
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Games of Chance
Minimax doesn't help
here because dice are
random and impartial.
Backgammon
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Chance Nodes
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Expectiminimax
• A chance node is evaluated as follows
– the value of each child is multiplied times the probability of reaching that child
– these products are then summed.
• Disadvantages to this approach:
– branching factor of chance nodes can be large!
– no pruning allowed
– evaluation functions are hard!…
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Expectiminimax Evaluations
Note that the relative ordering of the leaf values are the same,
but the decision has changed! must approximate positive
linear transformation of likelihood of winning
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Why Minimax Isn't Always
Appropriate
What if these numbers are roughly approximate and evaluation
occasionally has significant error? © Todd Neller.
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