CS 8520: Artificial Intelligence

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CS 8520 Artificial Intelligence

• Adversarial Search
• Paula Matuszek
• Fall, 2005

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Search in an Adversarial Environment

• Iterative deepening and A useful for
  single-agent search problems
• What if there are TWO agents?
• Goals in conflict
• Adversarial Search
• Especially common in AI
• Goals in direct conflict
• IE GAMES.

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Games vs. search problems

• "Unpredictable" opponent ? specifying a move for
  every possible opponent reply
• Time limits ? unlikely to find goal, must
  approximate
• Efficiency matters a lot
• HARD.
• In AI, typically "zero sum" one player wins
  exactly as much as other player loses.

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Types of games

• Deterministic Chance
• Perfect Info Chess Monopoly
• Checkers
  Backgammon
• Othello
• Tic-Tac-Toe
• Imperfect Info Bridge
• 
  Poker
• 
  Scrabble

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Tic-Tac-Toe

• Tic Tac Toe is one of the classic AI examples.
  Let's play some.
• Tic Tac Toe version 1.
• http//www.ourvirtualmall.com/tictac.htm
• Tic Tac Toe version 2.
• http//thinks.com/java/tic-tac-toe/tic-tac-toe.htm
  
• Try them both, at various levels of difficulty.
• What kind of strategy are you using?
• What kind does the computer seem to be using?
• Did you win? Lose?

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Problem Definition

• Formally define a two-person game as
• Two players, called MAX and MIN.
• Alternate moves
• At end of game winner is rewarded and loser
  penalized.
• Game has
• Initial State board position and player to go
  first
• Successor Function returns (move, state) pairs
• All legal moves from the current state
• Resulting state
• Terminal Test
• Utility function for terminal states.
• Initial state plus legal moves define game tree.

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Tic Tac Toe Game tree
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Optimal Strategies

• Optimal strategy is sequence of moves leading to
  desired goal state.
• MAX's strategy is affected by MIN's play.
• So MAX needs a strategy which is the best
  possible payoff, assuming optimal play on MIN's
  part.
• Determined by looking at MINIMAX value for each
  node in game tree.

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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

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Minimax algorithm
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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
• Even tic-tac-toe is much too complex to diagram
  here, although it's small enough to implement.

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Pruning the Search

• If you have an idea that is surely bad, don't
  take the time to see how truly awful it is. --
  Pat Winston
• Minimax exponential with of moves not feasible
  in real-life
• But we can PRUNE some branches.
• Alpha-Beta pruning
• If it is clear that a branch can't improve on the
  value we already have, stop analysis.

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a-ß pruning example
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a-ß pruning example
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a-ß pruning example
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a-ß pruning example
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a-ß pruning example
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Properties of a-ß

• Pruning does not affect final result
• Good move ordering improves effectiveness of
  pruning
• With "perfect ordering," time complexity
  O(bm/2)
• ? doubles depth of search which can be carried
  out for a given level of resources.
• A simple example of the value of reasoning about
  which computations are relevant (a form of
  metareasoning)

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Why is it called a-ß?

• a is the value of the best (i.e., highest-value)
  choice found so far at any choice point along the
  path for max
• If v is worse than a, max will avoid it
• ? prune that branch
• Define ß similarly for min

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The a-ß algorithm
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The a-ß algorithm
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"Informed" Search

• Alpha-Beta still not feasible for large game
  spaces.
• Can we improve on performance with domain
  knowledge?
• Yes -- if we have a useful heuristic for
  evaluating game states.
• Conceptually analogous to A for single-agent
  search.

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Resource limits

• Suppose we have 100 secs, explore 104 nodes/sec?
  106 nodes per move
• Standard approach
• cutoff test
• e.g., depth limit (perhaps add quiescence search)
• evaluation function
• estimated desirability of position

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Evaluation function

• Evaluation function or static evaluator is used
  to evaluate the goodness of a game position.
• Contrast with heuristic search where the
  evaluation function was a non-negative estimate
  of the cost from the start node to a goal and
  passing through the given node
• The zero-sum assumption allows us to use a single
  evaluation function to describe the goodness of a
  board with respect to both players.
• f(n) gtgt 0 position n good for me and bad for
  you
• f(n) ltlt 0 position n bad for me and good for
  you
• f(n) near 0 position n is a neutral position
• f(n) infinity win for me
• f(n) -infinity win for you

DesJardins www.cs.umbc.edu/671/fall03/slides/c8-
9_games.ppt
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Evaluation function examples

• Example of an evaluation function for
  Tic-Tac-Toe
• f(n) of 3-lengths open for me - of
  3-lengths open for you
• where a 3-length is a complete row, column, or
  diagonal
• Alan Turings function for chess
• f(n) w(n)/b(n) where w(n) sum of the point
  value of whites pieces and b(n) sum of blacks
• Most evaluation functions are specified as a
  weighted sum of position features
• f(n) w1feat1(n) w2feat2(n) ...
  wnfeatk(n)
• Example features for chess are piece count,
  piece placement, squares controlled, etc.
• Deep Blue (which beat Gary Kasparov in 1997) had
  over 8000 features in its evaluation function

DesJardins www.cs.umbc.edu/671/fall03/slides/c8-
9_games.ppt
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Cutting off search

• MinimaxCutoff is identical to MinimaxValue except
• Terminal? is replaced by Cutoff?
• Utility is replaced by Eval
• Does it work in practice?
• For chess bm 106, b35 ? m4
• 4-ply lookahead is a hopeless chess player!
• 4-ply human novice
• 8-ply typical PC, human master
• 12-ply Deep Blue, Kasparov

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Deterministic games in practice

• Checkers Chinook ended 40-year-reign of human
  world champion Marion Tinsley in 1994. Used a
  precomputed endgame database defining perfect
  play for all positions involving 8 or fewer
  pieces on the board, a total of 444 billion
  positions.
• Chess Deep Blue defeated human world champion
  Garry Kasparov in a six-game match in 1997. Deep
  Blue searched 200 million positions per second,
  used very sophisticated evaluation, and
  undisclosed methods for extending some lines of
  search up to 40 ply.
• Othello human champions refuse to compete
  against computers, who are too good.
• Go human champions refuse to compete against
  computers, who are too bad. In go, b gt 300, so
  most programs use pattern knowledge bases to
  suggest plausible moves.

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Games of chance

• Backgammon is a two-player game with
  uncertainty.
• Players roll dice to determine what moves to
  make.
• White has just rolled 5 and 6 and has four legal
  moves
• 5-10, 5-11
• 5-11, 19-24
• 5-10, 10-16
• 5-11, 11-16
• Such games are good for exploring decision making
  in adversarial problems involving skill and luck.

DesJardins www.cs.umbc.edu/671/fall03/slides/c8-
9_games.ppt
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Decision-Making in Non-Deterministic Games

• Probable state tree will depend on chance as well
  as moves chosen
• Add "chance" notes to the max and min nodes.
• Compute expected values for chance nodes.

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Game Trees with Chance Nodes

• Chance nodes (shown as circles) represent random
  events
• For a random event with N outcomes, each chance
  node has N distinct children a probability is
  associated with each
• (For 2 dice, there are 21 distinct outcomes)
• Use minimax to compute values for MAX and MIN
  nodes
• Use expected values for chance nodes
• For chance nodes over a max node, as in C
• expectimax(C) ?i(P(di) maxvalue(i))
• For chance nodes over a min node
• expectimin(C) ?i(P(di) minvalue(i))

Min Rolls
Max Rolls
DesJardins www.cs.umbc.edu/671/fall03/slides/c8-
9_games.ppt
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Meaning of the evaluation function
A1 is best move
A2 is best move
2 outcomes with prob .9, .1

• Dealing with probabilities and expected values
  means we have to be careful about the meaning
  of values returned by the static evaluator.
• Note that a relative-order preserving change of
  the values would not change the decision of
  minimax, but could change the decision with
  chance nodes.
• Linear transformations are OK

DesJardins www.cs.umbc.edu/671/fall03/slides/c8-
9_games.ppt
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Summary

• Games are fun to work on!
• They illustrate several important points about AI
• perfection is unattainable ? must approximate
• good idea to think about what to think about