Game Playing

1
Game Playing

• 20003574 ???
• 20003439 ???

2
Overview

• Intro Games as search problems
• Perfect decisions in 2-person games
• Imperfect decisions
• Alpha-beta pruning
• Games that include an element of chance
• State-of-the-art game program
• Discussion

3
Intro Games as search problems

• Games
• Pure, abstract competition
• Environment
• Accessible, deterministic, small
• Action
• Small of well-defined actions

4

• Uncertainty
• Opponent
• Not like by throwing dice or by the weather
• Complexity
• Not enough time to calculate the exact consequence

5
Perfect decisions in 2-person games

• Two players, MAX MIN
• Search problem
• Initial state
• A set of operators
• Terminal test(terminal states)
• Utility function(payoff function)

6

• MAX wants to find a strategy that will lead to a
  winning terminal state regardless of what MIN
  does.

7

• Minimax algorithm
• Generate the whole game tree
• Apply the utility function to the terminal
• Back-up values
• In root, MAX chooses highest value

8
(No Transcript)
9
(No Transcript)
10

• Complete?
• Yes(tree is finite)
• Optimal?
• Yes(against optimal opponent)
• Time complexity O(bm)
• Space complexity O(bm)

11
Imperfect decision

• No time to build complete search tree
• Evaluation functions
• Replace the utility function
• Estimates the expected utility value from a given
  position(chess material value)
• Desirable properties
• Must agree with the utility function
• Must not take too long to evaluate
• Must accurately reflect the chance of winning

12
Alpha-beta pruning

• Pruning
• Pruning does not affect final result
• Alpha-beta pruning
• Good move ordering improves effectiveness of
  pruning
• Asymptotic time complexity
• O((b/log b)d)
• With perfect ordering, time complexity
• O(bd/2)

13
(No Transcript)
14
(No Transcript)
15
Games That Include an Element of Chance

• Many games mirror unpredictability by including a
  random element
• Ex) backgammon, ???

16
Backgammon
17
Game tree for a backgammon
18
Game tree for a backgammon (contd)

• Chance nodes
• Branches leading from each chance node denote the
  possible dice rolls
• Labelled with the roll and the chance that it
  will occur
• Expectimax value of C
• Expectimin value

19
Position evalution in games with chance nodes

• For minimax, any order-preserving transformation
  of the leaf values
• does not affect the choice of move
• With chance node, some order-preserving
  transformation of the leaf values
• does affect the choice of move

20
Position evalution in games with chance nodes
(contd)
21
Complexity of expectiminimax

• The expectiminimax considers all the possible
  dice-roll sequences
• It takes O(bmnm)
• , where n is the number of distinct rolls
• Whereas, minimax takes O(bm)
• Problems
• The extra cost compared to minimax is very high
• Alpha-beta pruning is much less effective

22
State-of-the-art Game Programs for Chess

• Chess
• computer would beat the human champion
• computer power, rather than complex strategies
• Ratings of human and machine chess champions

23
State-of-the-art Game Programs for Chess (Contd)

• Deep Blue
• Defeated Garry Ksaparov in 1997 match
• (2 wins 3 draws 1 lose)
• 32 P2SC Processors
• capable of searching 50 to 100 billion positions
  within three minutes
• 1000 times faster than its predecessor, Deep
  Thought
• Match clips
• http//www.research.ibm.com/deepblue/home/html/cli
  ps.html (multimedia clip)
• http//www.insidechess.com/events/kasparov.html
• (text)

24
State-of-the-art Game Programs for the other
games

• Checker
• Arthur Samuel (1952)
• official world champion Chinook
• Othello, Backgammon, Go

25
Discussion

• Divorced from the main stream AI
• Grand Prix motor racing --- car industry
• game --- AI
• Minimax
• works well when the leaf node evaluation is
  perfect
• misleading
• remedy --- probability distribution
• problem --- Siblings are highly correlated.

26
Discussion(contd)

• Alpha-beta prunning
• may generate a large, useless tree.
• needs a concept of utility of a node expansion
• meta-reasoning