Lecture 17 Games and Adversarial Search

42 Pages·2011·1.76 MB·English

by  Marco Chiarandini

Lecture 17
Games and Adversarial Search
Marco Chiarandini
DepartmentofMathematics&ComputerScience
UniversityofSouthernDenmark
SlidesbyStuartRussellandPeterNorvig
Introduction
Minimax
Course Overview α–βAlgorithm
StochasticGames
(cid:52) Introduction (cid:52) Learning
(cid:52) Artificial Intelligence (cid:52) Supervised
(cid:52) Intelligent Agents Decision Trees, Neural
(cid:52) Search Networks
Learning Bayesian Networks
(cid:52) Uninformed Search
(cid:52) Unsupervised
(cid:52) Heuristic Search
EM Algorithm
(cid:52) Uncertain knowledge and
(cid:52) Reinforcement Learning
Reasoning
(cid:52) Probability and Bayesian (cid:73) Games and Adversarial Search
approach (cid:73) Minimax search and
(cid:52) Bayesian Networks Alpha-beta pruning
(cid:52) Hidden Markov Chains (cid:73) Multiagent search
(cid:52) Kalman Filters (cid:73) Knowledge representation and
Reasoning
(cid:73) Propositional logic
(cid:73) First order logic
(cid:73) Inference
(cid:73) Plannning
2
Introduction
Minimax
Outline α–βAlgorithm
StochasticGames
♦ Games
♦ Perfect play
– minimax decisions
– α–β pruning
♦ Resource limits and approximate evaluation
♦ Games of chance
♦ Games with imperfect information
3
Introduction
Minimax
Outline α–βAlgorithm
StochasticGames
1. Introduction
2. Minimax
3. α–β Algorithm
4. Stochastic Games
4
Introduction
Minimax
Multiagent environments α–βAlgorithm
StochasticGames
Multiagent environments:
(cid:73) cooperative
(cid:73) competitive (cid:232) adversarial search in games
AI game theory (combinatorial game theory)
(cid:73) deterministic/stochastic
(cid:73) turn taking
(cid:73) two players
(cid:73) zero sum games = utility values equal and opposite
(cid:73) perfect/imperfect information
(cid:73) agents are restricted to a small number of actions described by rules
“Classical” (economic) game theory includes cooperation, chance, imperfect
knowledge, simultaneous moves and tends to represent real-life decision
making situations.
5
Introduction
Minimax
Types of Games α–βAlgorithm
StochasticGames
deterministic chance
chess, checkers, kalaha backgammon,
perfect information
go, othello monopoly
battleships,
imperfect information bridge, poker, scrabble
blind tictactoe
6
Introduction
Minimax
Games vs. search problems α–βAlgorithm
StochasticGames
“Unpredictable” opponent ⇒ solution is a strategy/policy
specifying a move for every possible opponent reply (cid:232) contingency strategy
Optimal strategy: the one that leads to outcomes at least as good as any
other strategy when one is playing an infallibile opponent
Search problem (cid:32) game tree
(cid:73) initial state: root of game tree
(cid:73) successor function: game rules/moves
(cid:73) terminal test (is the game over?)
(cid:73) utility function, gives a value for terminal nodes (eg, +1, -1, 0)
Terminology:
(cid:73) Two players called MAX and MIN.
(cid:73) MAX searches the game tree.
(cid:73) Ply: one turn (every player moves once) from “reply”. [A. Samuel 1959]
7
Introduction
Minimax
Game tree (2-player, deterministic, turnαs–β)Algorithm
StochasticGames
MAX (X)
X X X
MIN (O) X X X
X X X
XO X O X . . .
MAX (X) O
XO X X O XO . . .
MIN (O) X X
. . . . . . . . . . . .
XO X X O X X O X . . .
TERMINAL O X O O X X
O X XO X OO
Utility −1 0 +1
9
Introduction
Minimax
Measures of Game Complexity α–βAlgorithm
StochasticGames
(cid:73) state-space complexity: number of legal game positions reachable from
the initial position of the game.
an upper bound can often be computed by including illegal positions
Eg, TicTacToe:
39 =19.683
5.478 after removal of illegal
765 essentially different positions after eliminating symmetries
(cid:73) game tree size: total number of possible games that can be played:
number of leaf nodes in the game tree rooted at the game’s initial
position.
Eg: TicTacToe:
9!=362.880 possible games
255.168 possible games halting when one side wins
26.830 after removal of rotations and reflections
10
Introduction
Minimax
α–βAlgorithm
StochasticGames
11