1. Game Playing in AI by: Gaurav Phapale 05 IT 6010
Seminar on
Game Playing in AI
by:
Gaurav Phapale
05 IT 6010
11/6/2018
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Definition…. Game
Game playing is a search problem defined by:
1. Initial state
2. Successor function
3. Goal test
4. Path cost / utility / payoff function
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Types of Games
Perfect Information Game: In which player knows all the possible moves of himself and opponent and their results. E.g. Chess.
Imperfect Information Game: In which player does not know all the possible moves of the opponent. E.g. Bridge since all the cards are not visible to player.
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4. Characteristics of game playing
Unpredictable Opponent.
Time Constraints.
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5. Typical structure of the game in AI
2- person game
Players alternate moves
Zero-sum game: one player’s loss is the other’s gain
Perfect information: both players have access to complete information about the state of the game. No information is hidden from either player.
No chance (e.g. using dice) involved
E.g. Tic- Tac- Toe, Checkers, Chess
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Game Tree
Tic – Tac – Toe Game Tree
Ref: Lehre/Einfuehrung-KI-SS2003/folien06.pdf
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MAX
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MAX cont..
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MINIMAX..
2 players.. MIN and MAX.
Utility of MAX = - (Utility of MIN).
Utility of game = Utility of MAX.
MIN tries to decrease utility of game.
MAX tries to increase utility of game.
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MINIMAX Tree..
Ref: Lehre/Einfuehrung-KI-SS2003/folien06.pdf
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11. Assignment of MINIMAX Values
Minimax_value(u)
{   //u is the node you want to score
 if u is a leaf
return score of u;
else
if u is a min node
for all children of u: v1, .. vn ;
return min (Minimax_value(v1),..., Minimax_value(vn))
return max (Minimax_value(v1),..., Minimax_value(vn)) }
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MINIMAX Algorithm..
Function MINIMAX-DECISION (state) returns an operator
For each op in OPERATORS[game] do
VALUE [op] = MINIMAX-VALUE (APPLY (op, state), game)
End
Return the op with the highest VALUE [op]
Function MINIMAX-VALUE (state, game) returns a utility value
If TERMINAL-TEST (state) then
Return UTILITY (state)
Else If MAX is to move in state then
Return the highest MINIMAX-VALUE of SUCCESSORS (state)
Else
Return the lowest MINIMAX-VALUE of SUCCESSORS (state)
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Properties of MINIMAX
Complete: Yes, if tree is finite
Optimal: Yes, against an optimal opponent.
Time: O(bd) (depth- first exploration)
Space: O(bd) (depth- first exploration)
b: Branching Factor
d: Depth of Search Tree
Time constraints does not allow the tree to be fully explored.
How to get the utility values without exploring search tree up to leaves?
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Evaluation Function
Evaluation function or static evaluator is used to evaluate the ‘goodness’ of a game position.
The zero-sum assumption allows us to use a single evaluation function to describe the goodness of a position with respect to both players.
E.g. f(n) is the evaluation function of the position ‘n’. Then,
– f(n) >> 0: position n is good for me and bad for you
– f(n) << 0: position n is bad for me and good for you
– f(n) near 0: position n is a neutral position
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15. Evaluation Function cont..
One of the evaluation function for Tic- Tac- Toe can be defined as:
f( n) = [# of 3- lengths open for me] - [# of 3- lengths open for you]
where a 3- length is a complete row, column, or diagonal
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Alpha Beta Pruning
At each MAX node n, alpha(n) = maximum value found so far
At each MIN node n, beta(n) = minimum value found so far
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17. Alpha Beta Pruning Cont..
Ref: Lehre/Einfuehrung-KI-SS2003/folien06.pdf
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18. Alpha Beta Pruning Cont..
Ref: Lehre/Einfuehrung-KI-SS2003/folien06.pdf
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19. Alpha Beta Pruning Cont..
Ref: Lehre/Einfuehrung-KI-SS2003/folien06.pdf
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20. Alpha Beta Pruning Cont..
Ref: Lehre/Einfuehrung-KI-SS2003/folien06.pdf
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21. Alpha Beta Pruning Cont..
Ref: Lehre/Einfuehrung-KI-SS2003/folien06.pdf
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22. Effectiveness of Alpha Beta Pruning.
Worst-Case
branches are ordered so that no pruning takes place
alpha-beta gives no improvement over exhaustive search
Best-Case
each player’s best move is the left-most alternative (i.e., evaluated first)
In practice often get O(b(d/2)) rather than O(bd)
e.g., in chess go from b ~ 35 to b ~ 6
this permits much deeper search in the same amount of time
makes computer chess competitive with humans!
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23. Iterative Deepening Search.
IDS runs alpha-beta search with an increasing depth-limit
The “inner” loop is a full alpha-beta search with a specified depth limit m
When the clock runs out we use the solution found at the previous depth limit
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Applications
Entertainment
Economics
Military
Political Science
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Conclusion
Game theory remained the most interesting part of AI from the birth of AI. Game theory is very vast and interesting topic. It mainly deals with working in the constrained areas to get the desired results. They illustrate several important points about Artificial Intelligence like perfection can not be attained but we can approximate to it.
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References
'Artificial Intelligence: A Modern Approach' (Second Edition) by Stuart Russell and Peter Norvig, Prentice Hall Pub.
Theodore L. Turocy, Texas A&M University, Bernhard von Stengel, London School of Economics "Game Theory" CDAM Research Report Oct. 2001
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Thank you……
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