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Adversarial Search Chapter 6 Section 1 – 4 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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adversarial search

Adversarial Search

Chapter 6

Section 1 – 4

search in an adversarial environment
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.
games vs search problems
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.
types of games
Types of games

Deterministic Chance

Perfect Info Chess, Monopoly

Checkers Backgammon



Imperfect Info Bridge



tic tac toe
  • Tic Tac Toe is one of the classic AI examples. Let's play some.
  • Tic Tac Toe version 1.
  • Tic Tac Toe version 2.
  • 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?
problem definition
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.
optimal strategies
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.
  • 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:
properties of minimax
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.
pruning the search
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.
properties of
Properties of α-β
  • 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)
why is it called
α 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 α, max will avoid it

 prune that branch

Define β similarly for min

Why is it called α-β?
informed search
"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.
resource limits
Resource limits

Suppose we have 100 secs, explore 104 nodes/sec106nodes per move

Standard approach:

  • cutoff test:

e.g., depth limit (perhaps add quiescence search)

  • evaluation function

= estimated desirability of position

evaluation function
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) >> 0: position n good for me and bad for you
    • f(n) << 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


evaluation function examples
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 Turing’s function for chess
    • f(n) = w(n)/b(n) where w(n) = sum of the point value of white’s pieces and b(n) = sum of black’s
  • Most evaluation functions are specified as a weighted sum of position features:

f(n) = w1*feat1(n) + w2*feat2(n) + ... + wn*featk(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


cutting off search
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, b=35  m=4

4-ply lookahead is a hopeless chess player!

  • 4-ply ≈ human novice
  • 8-ply ≈ typical PC, human master
  • 12-ply ≈ Deep Blue, Kasparov
deterministic games in practice
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 searches 200 million positions per second, uses 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 > 300, so most programs use pattern knowledge bases to suggest plausible moves.
games of chance
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.


decision making in non deterministic games
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.
game trees with chance nodes
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))






meaning of the evaluation function
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


  • 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