Download Computer Games - CSE, IIT Bombay

Group 1 :
Ashutosh
Pushkar
Ameya
Sudhir
From
CHECK
TO
CHECKMATE !
Motivation
 Game playing was one of the first tasks
undertaken in AI
 Study of games brings us closer to :
 Machines capable of logical deduction
 Machines for making strategic decisions
 Analyze the limitations of machines to human
thought process
 Games are an idealization of worlds
 World state is fully accessible
 Actions & outcomes are well-defined
Outline of the presentation
 Evaluation Functions
 Algorithms
 Deep Blue
 Conclusions from Deep Blue
 Conclusion
 References
Chess
 Neither too simple
 Nor too difficult for satisfactory solution
 Requires “thinking” for a skilled player
 Designing a chess playing program
 Perfect chess playing : INTRACTABLE
 Legal Chess : TRIVIAL
 Play tolerably good game : SKILLFULLY
Evaluation Function
Evaluation Function
 Utility function
 Whole game tree is explored
 computationally expensive task !!
 Estimates the expected utility of a state
 Evaluation functions
 cut off the exploration depth by estimating
whether a state will lead to a win or loss
Evaluation Function (cont.)
 A good evaluation function should
 not take too long
 Preserve ordering of the terminal states otherwise it
will lead to bad decision making
 Consider strategic moves that lead to long term
advantages
Evaluation Function (cont.)

Typically includes :

Material Advantage (difference in total material of both sides)
f (P) = 200(k – k’) + 9(q – q’) + 5(r – r’)
+ 3(b – b’) + (p – p’) + g(P) + h(P)
+…

Positions of pieces





Rook on open file
double rooks
rook on seventh rank etc. and their relative positions.
Pawn Formation
Mobility
Evaluation function is an attempt to write a mathematical
formula for intelligence 
Algorithms
Games as Search Problems
 Initial states

Where game starts
 Initial position in chess
 Successor function
 List of all legal moves from current position
 Terminal State
 Where the game is concluded
 Utility function
 Numeric value for all terminal states
Minimax Strategy
Minimax Strategy
 Optimal strategy
 Assumption : opponent plays his best possible
move.
 An option is picked which
 Minimizes damage done by opponent
 Does most damage to the opponent
 Idea:
 For each node find minimum minimax value
 Choose the node with maximum of such values
 This will ensure best value against most damage
done by opponent
Strategy of Minimax
 Opponent tries to reduce utility function’s
value
 For any move made by opponent in reply of
computer’s move, choose minimum reduced
value by opponent
 Find the move with maximum such value
Analysis
 Algorithm is complete for complete tree only
 Not best strategy against irrational opponent
 According to definition
 Time complexity :O(bm)
 b = max. no. of possible moves
 m = max. depth of tree
 In chess even in average case, b = 35 and
m = 100 => time exceeds practical limits
 # of states grows exponentially as per number of
moves played
α-β Pruning
α-β Pruning
 The problem of minimax search
 # of state to examine: exponential in number of
moves
 Returns same moves as minimax does
 Prunes away branches that can’t influence final
decision.
 α: the value of the best (highest) choice so far in
search of MAX
 β: the value of the best (lowest) choice so far in
search of MIN
 Order of considering successor
Algorithm for α-β Pruning
 Current highest β is found and assigned as α
 β is current lowest for α’s from that move
 For next possible node, while finding β, if some α
is found lower than current highest β:
 It will only give lesser value of final β
 S0, other α’s are not found for that node
 After calculating β for this node, α is replaced by
max(α,β for this node)
 In this way after all possible set of moves final
value of α is found
α-β Pruning (2)
If m is
better than
n for
Player, we
will never
get to n in
play
and just
prune it.
Analysis of α-β Pruning:
Improvement over minimax algorithm
 Does not affect final results
 Worthwhile to examine that successor first which
is likely to be best
 Time complexity: O(b(m/2))
 Effective branching factor = √b
 i.e. 6 rather than 35
 In case of random ordering :
 Total number of nodes examined is of the order
O(b^(3m/4))
Transposition table
 Dynamic programming
 Multiple paths to the same position
 Savings through memorization
 Use a hash table of evaluated positions
Iterative Deepening
 Sometime chess is played under a strict time
 Depth of search depend on time
 Use of Breadth first Search
 Advantage : program know which move was
best at previous level
Horizon Effect
 Problem with fixed
depth search
 Positive Horizon
Effect
 Negative horizon
effect
Quiescence Search
 Search till “quiet” position
 Quiet Position
 Doesn’t affect the current position so much
 Example : no capture of any piece, no check, no
pawn promotions/threats
State of the art : DEEP BLUE
Defeats Gary Kasparov
 Won a match in 1997
 Brute force computing power
 Massive, parallel architecture
 Special purpose hardware for chess
 Parameters of the evaluation function
 Learnt by studying many master games
 Different evaluation function for different
positions
 Utilized heavily loaded endgame databases
Humans vs. Computers
Humans
Computers
Lower Computational Speed
Higher
Errors Possible
Error Free
Tend to be instinctive
No instincts
Imaginative
None
High Learning Capabilities
Limited
Inductive
Not Inductive
Some intricacies of a chess
playing system
 Should not play the same sequence of moves again
 A player wins a match against the computer
 Starts playing the same sequence of moves
 Hence, a statistical element is required
 Opponent can learn the algorithm used by computer
 Hence, again the need for a statistical element
 Different game play during different phases
 Start Game
 Mid Game
 End Game
Conclusion
 Computer chess as a search problem
 Good enough decisions
 Simulation of “skill” by “knowledge”
 Limitations of computers to humans
 Future work :
 Better evaluation functions through learning
 Need for different AI techniques to play chess
References
 Claude E. Shannon: Programming a
Computer for Playing Chess, Philosophical
Magazine, Ser.7, Vol. 41, No. 314, March
1950.
 S.Russel & P. Norvig: Artificial Intelligence: A
Modern Approach 2/E, Prentice Hall, ISBN-10:
0137903952
 Wikipedia
Thank You