1. Artificial Intelligence AIMA §5: Adversarial Search
Rodney Nielsen
Many / most of these slides were adapted from: AI MA, Russell and Norvig

2. Adversarial Search Games Perfect Play
Minimax algorithm
α-β pruning
Resource limits and approximate evaluation
Games of chance
Games of imperfect information

3. Games vs. Search Problems
Unpredictable Opponent  Solution is a strategy specifying a move for every possible opponent reply
Time limits  unlikely to find goal, must approximate

4. Types of Games Deterministic Chance Perfect Information
Chess, Checkers, Go, Othello
Backgammon, Monopoly
Imperfect Information
Battleship, Blind TicTacToe
Bridge, Poker, Scrabble, Nuclear War

5. Game tree (2-player, deterministic, turns)

6. Minimax Perfect play for deterministic, perfect-information games
Idea: choose move to position with highest minimax value = best achievable payoff against opponent’s best play
E.g., 2-ply game:

7. Minimax Algorithm

8. Properties of Minimax Complete? Optimal? Search type?
Yes, if tree is finite
Optimal?
Yes, against an optimal opponent.
Otherwise?
Search type?
Depth-first exploration
Complexity?
Time: O(bm)
Space: O(bm)
For chess, b≈35 for “reasonable” games
 Exact solution completely infeasible
But do we need to explore every path?

9. α-β Pruning

10. α-β Pruning X ≥ -∞ ≥ -∞ ≤ 3 ≤ 3 ≥ 3 ≥ 3 ≤ 3 ≤ 3 ≤ 2 ≥ 3 ≥ 3 ≤ 3 ≤ 2
≤ 14
≤ 3
≤ 2
≤ 2

11. Game Objective Function
Utility Functions
Defn: Zero-sum game

12. α-β Pruning Efficiency / Space and Time Complexity Move ordering
Killer moves
Transposition table
Heuristic evaluation function

13. Move Ordering X ≥ -∞ ≥ -∞ ≤ 3 ≤ 3 ≥ 3 ≥ 3 ≤ 3 ≤ 3 ≤ 2 ≥ 3 ≥ 3 ≤ 3 ≤ 2
≤ 14
≤ 3
≤ 2
≤ 2

14. Move Ordering Move ordering Transposition table Killer moves
Need to avoid repeated evaluation (as in state space searches)

15. Heuristic Evaluation Functions
Estimates the expected utility value
Instead of calculating exact utility
Based on probabilities
Chess example (two pawns versus one):
72% wins, 20% losses, and 8% draws
f = 0.72* * *½ = 0.76
Order terminal states same as utility function
Use exact utility at terminal states

16. Heuristic Evaluation Functions
Use features of non-terminal state to estimate utility
Creates equivalence classes
Often a weighted linear function of features
Eval(s) = w1f1(s) + w1f1(s) + … + wdfd(s)
Examples
Chess features ?
Sweep Fives features

17. Cutting Off Search if Cutoff-Test(state, depth) return Eval(state)
When should you cut off the search
Depth
Quiescence

18. Cutting Off Search When should you cut off the search Depth Quiescence
Horizon Effect

19. Forward Pruning Beam Search
Only explore best n moves past a certain depth

20. Search versus Lookup
Pre-compute and store the best move and its value for some states

21. Stochastic Games
What to do when future states are based on a stochastic process?
Calculate the expected value
Based on an exhaustive evaluation, or
Based on Monte Carlo simulation

22. Partially Observable Games
Belief States
Monte Carlo Simulations - again

23. State-of-the-Art
Chess
Checkers
Othello
Poker Go
Bridge?