1. Introduction to Artificial Intelligence CS 438 Spring 2008 Today –AIMA, Ch. 6 –Adversarial Search Thursday –AIMA, Ch. 6 –More Adversarial Search The “Luke Arm”: embedded intelligence

2. Why is game playing an interesting AI task? Techniques used in game playing agents can be used in other problem solving tasks –Elements of uncertainty Search space is too large to look at every possible consequence Having an unpredictable opponent Many games have a random element –Real-time decision making –Learning environment

3. Game Agents vs Human Champions 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. Checkers – Chinook –ended 40-year-reign of human world champion Marion Tinsley in 1994. Used a pre-computed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions. Poker - Polaris –After two thousand hands and countless 'flops', 'rivers', and 'turns', two elite poker players, Phil "The Unabomber" Laak and Ali Eslami, have narrowly defeated Polaris. Othello - Logistello Scrabble - Quackle

4. Two Player Games Optimal Decisions State Space Definition –Initial state Board position and an indication of who goes first –Set of operators All legal moves a play can make –Terminal (goal) test Test to determine when the game is over –Utility function Assigns a numeric value for the outcome of the game –Chess: +1 (win), 0 (draw), -1 (loss) –Backgammon: +192 to -192 For multi-round games

5. Optimal Decisions Min Max (minimax) Search –Always assume that your opponent is going to make the move that puts you in the worst possible situation, and conversely improves their own position as much as possible. best achievable payoff against best play.

6. Minimax Search Tree

7. Minimax Steps 1.Generate search tree in depth first manner 2.Apply utility function to each terminal node 3.Use utility of terminal nodes to determine the utility of the node above If the level above is your move (max) choose the maximum value of the leaf nodes If the level above is your opponents move (min) choose minimum value of leaf nodes 4.Continue going back-up the tree assigning the utility value to each parent in a similar fashion 5.Once all of the states have been examined the utility of the best move will be assigned to the root.

8. Minimax 4-ply game:

9. Minimax algorithm

10. Properties of minimax Complete? Yes (if tree is finite) Optimal? Yes (against an optimal opponent) Time complexity? O(b m ) Space complexity? O(bm) (depth-first exploration) For chess, b ≈ 35, m ≈100 for "reasonable" games  exact solution completely infeasible

11. Resource limits A move must be make in a time limit that does not allow the agent to search down to the terminal nodes –Suppose we have 100 secs, explore 10 4 nodes/sec  10 6 nodes per move Standard approach: –cutoff test: depth limit (perhaps add quiescence search) –evaluation function: Eval(s) estimated desirability of position

12. Evaluation functions Eval(s) –A numeric value indicating how good the chances of winning the game are from state s –Should be consistent with a utility function for the game Largest value is a win, lowest value is a loss –Applied to the last level of state expanded For chess, typically linear weighted sum of features Eval(s) = w 1 f 1 (s) + w 2 f 2 (s) + … + w n f n (s) –Weights are adjusted to improve play or new features could be added.

15. Hmmm, could you apply a similar idea to the peg board puzzle?