1571 CS Introduction to AI Lecture 12
Adversarial search
Milos Hauskrecht [e-mail] 5329 Sennott Square
CS 1571 Intro to AI
M. Hauskrecht
Announcements
Homework assignment 4 is out Programming and experiments Simulated annealing + Genetic algorithm Competition Course web page: http://www.cs.pitt.edu/~milos/courses/cs1571/
CS 1571 Intro to AI
M. Hauskrecht
1
Search review
Search Path search Configuration search Optimality Finding a path versus finding the optimal path Finding a configuration satisfying constraints versus finding the best configuration
CS 1571 Intro to AI
M. Hauskrecht
Game search
Game-playing programs developed by AI researchers since the beginning of the modern AI era Programs playing chess, checkers, etc (1950s) Specifics of the game search: Sequences of players decisions we control Decisions of other player(s) we do not control Contingency problem: many possible opponents moves must be covered by the solution Opponents behavior introduces an uncertainty in to the game We do not know exactly what the response is going to be Rational opponent maximizes it own utility (payoff) function
CS 1571 Intro to AI M. Hauskrecht
2
Types of game problems
Types of game problems: Adversarial games: win of one player is a loss of the other Cooperative games: players have common interests and utility function A spectrum of game problems in between the two:
Adversarial games Fully cooperative games
we focus on adversarial games only!!
CS 1571 Intro to AI M. Hauskrecht
Example of an adversarial 2 person game: Tic-tac-toe
Player 1 (x) moves first
Loss
Draw
Win
M. Hauskrecht
CS 1571 Intro to AI
3
Game search problem
Game problem formulation: Initial state: initial board position + info whose move it is Operators: legal moves a player can make Goal (terminal test): determines when the game is over Utility (payoff) function: measures the outcome of the game and its desirability Search objective: find the sequence of players decisions (moves) maximizing its utility (payoff) Consider the opponents moves and their utility
CS 1571 Intro to AI M. Hauskrecht
Game problem formulation (Tic-tac-toe)
Objectives: Player 1: maximize outcome Player 2: minimize outcome Operators Initial state
Terminal (goal) states Utility: -1 0 1
M. Hauskrecht
CS 1571 Intro to AI
4
Minimax algorithm
How to deal with the contingency problem? Assuming that the opponent is rational and always optimizes its behavior (opposite to us) we consider the best opponents response Then the minimax algorithm determines the best move
MAX 3
MIN
3
2
2
3
12
8
2
4
6
14
5
2
CS 1571 Intro to AI
M. Hauskrecht
Minimax algorithm. Example
MAX
MIN
MAX
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
5
Minimax algorithm. Example
MAX
MIN
MAX
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
Minimax algorithm. Example
MAX
MIN
MAX
4
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
6
Minimax algorithm. Example
MAX
MIN
MAX
4
6
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
Minimax algorithm. Example
MAX
MIN
4
MAX
4
6
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
7
Minimax algorithm. Example
MAX
MIN
4
MAX
4
6
2
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
Minimax algorithm. Example
MAX
MIN
4
MAX
4
6
2
9
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
8
Minimax algorithm. Example
MAX
MIN
4
MAX
4
6
2
9
3
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
Minimax algorithm. Example
MAX 5
MIN
4
2
5
MAX
4
6
2
9
3
5
7
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
9
Minimax algorithm
CS 1571 Intro to AI
M. Hauskrecht
Complexity of the minimax algorithm
We need to explore the complete game before tree making the decision b Complexity: ?
m
-1
0
1
CS 1571 Intro to AI
M. Hauskrecht
10
Complexity of the minimax algorithm
We need to explore the complete game tree before making the decision
b
Complexity:
O (b m ) m
-1
0
1
Impossible for large games Chess: 35 operators, game can have 50 or more moves
CS 1571 Intro to AI M. Hauskrecht
Solution to the complexity problem
Two solutions: 1. Dynamic pruning of redundant branches of the search tree identify a provably suboptimal branch of the search tree before it is fully explored Eliminate the suboptimal branch Procedure: Alpha-Beta pruning 2. Early cutoff of the search tree uses imperfect minimax value estimate of non-terminal states (positions)
CS 1571 Intro to AI
M. Hauskrecht
11
Alpha beta pruning
Some branches will never be played by rational players since they include sub-optimal decisions (for either player)
CS 1571 Intro to AI
M. Hauskrecht
Alpha beta pruning. Example
MAX
MIN
MAX
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
12
Alpha beta pruning. Example
MAX
MIN
MAX
4
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
Alpha beta pruning. Example
MAX
MIN
4
MAX
= 4
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
13
Alpha beta pruning. Example
MAX
MIN
4
MAX
= 4
6
!!
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
Alpha beta pruning. Example
MAX
4
MIN
= 4
MAX
= 4
6
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
14
Alpha beta pruning. Example
MAX
4
MIN
= 4
MAX
= 4
6
2
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
Alpha beta pruning. Example
MAX
4
MIN
= 4
2
MAX
= 4
6
= 2
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
15
Alpha beta pruning. Example
MAX
4
!!
MIN
= 4
2
MAX
= 4
6
= 2
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
Alpha beta pruning. Example
MAX
4
MIN
= 4
2
MAX
= 4
6
= 2
5
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
16
Alpha beta pruning. Example
MAX
4
MIN
= 4
2
5
MAX
= 4
6
= 2
= 5
4
3
6
2
2
1
9
5
3
1
5
4
7
5
CS 1571 Intro to AI
M. Hauskrecht
Alpha beta pruning. Example
M...