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Seminar on. Game Playing in AI. Definitionâ€¦. Game. Game playing is a search problem defined by: 1. Initial state 2. Successor function 3. Goal test 4. Path cost / utility / payoff function. Types of Games.

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**Seminar on**Game Playing in AI**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 University College of Engineering**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. University College of Engineering**Characteristics of game playing**• Unpredictable Opponent. • Time Constraints. University College of Engineering**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 University College of Engineering**Game Tree**• Tic – Tac – Toe Game Tree Ref: www.uni-koblenz.de/~beckert/ Lehre/Einfuehrung-KI-SS2003/folien06.pdf University College of Engineering**MAX**University College of Engineering**MAX cont..**University College of Engineering**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. University College of Engineering**MINIMAX Tree..**Ref: www.uni-koblenz.de/~beckert/ Lehre/Einfuehrung-KI-SS2003/folien06.pdf University College of Engineering**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)) else for all children of u: v1, .. vn ; return max (Minimax_value(v1),..., Minimax_value(vn)) } University College of Engineering**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) University College of Engineering**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? University College of Engineering**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 University College of Engineering**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 University College of Engineering**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 University College of Engineering**Alpha Beta Pruning Cont..**Ref: www.uni-koblenz.de/~beckert/ Lehre/Einfuehrung-KI-SS2003/folien06.pdf University College of Engineering**Alpha Beta Pruning Cont..**Ref: www.uni-koblenz.de/~beckert/ Lehre/Einfuehrung-KI-SS2003/folien06.pdf University College of Engineering**Alpha Beta Pruning Cont..**Ref: www.uni-koblenz.de/~beckert/ Lehre/Einfuehrung-KI-SS2003/folien06.pdf University College of Engineering**Alpha Beta Pruning Cont..**Ref: www.uni-koblenz.de/~beckert/ Lehre/Einfuehrung-KI-SS2003/folien06.pdf University College of Engineering**Alpha Beta Pruning Cont..**Ref: www.uni-koblenz.de/~beckert/ Lehre/Einfuehrung-KI-SS2003/folien06.pdf University College of Engineering**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! University College of Engineering**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 University College of Engineering**Applications**• Entertainment • Economics • Military • Political Science University College of Engineering**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. University College of Engineering**References**• 'Artificial Intelligence: A Modern Approach' (Second Edition) by Stuart Russell and Peter Norvig, Prentice Hall Pub. • http://www.cs.umbc.edu/471/notes/pdf/games.pdf • http://l3d.cs.colorado.edu/courses/AI-96/sept23glecture.pdf • Theodore L. Turocy, Texas A&M University, Bernhard von Stengel, London School of Economics "Game Theory" CDAM Research Report Oct. 2001 • http://www.uni-koblenz.de/~beckert/Lehre/Einfuehrung-KI-SS2003/folien06.pdf • http://ai-depot.com/LogicGames/MiniMax.html University College of Engineering

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