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# New Approximate Strategies for Playing Sum - PowerPoint PPT Presentation

New Approximate Strategies for Playing Sum Games Based on Subgame Types Authored by: Manal M. Zaky Cherif R. S. Andraos Salma A. Ghoneim Presented by: Manal M. Zaky Outline Sum Games Combinatorial Game Theory Previous Strategies New Strategies Experimental Results

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### New Approximate Strategies for Playing SumGames Based on Subgame Types

Authored by:

Manal M. Zaky

Cherif R. S. Andraos

Salma A. Ghoneim

Presented by:

Manal M. Zaky

• Sum Games

• Combinatorial Game Theory

• Previous Strategies

• New Strategies

• Experimental Results

• Conclusions and Future Work

ICCSE’06, Cairo, Egypt

• Let G1 ,...,Gn represent n games

• Playing in the sum game

G = G1 +...+Gn

consists of picking a component game Gi and making a move in it

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+

+

Sum Games (cont.)

• Example I: NIM

• Several heaps of coins

• In his turn, a player selects a heap, and removes any positive number of coins from it, maybe all

• Goal

• Take the last coins

• Example with 3 piles: (3,4,5)

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• Many games tend to decompose into a number of independent regions or subgames.

• Examples:

• Domineering

• GO

• Amazons

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• Example II: Domineering

• In his turn a player places a domino on the board:

• Blue places them vertically

• Red places them horizontally

• Goal

• Place the last domino

• Example game

=

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• Example III: Go

• The standard Go board is 19X19; games are also played on 13X13 and 9X9.

• The Go board begins empty. One player uses the black stones and the other uses the white stones.

• Black always goes first. Players take turns placing one stone on the board.

• Once a stone is placed on the board, it is never moved unless it is captured

• Game ends when both players agree that there are no more moves to be played.

• Goal

• surround more territory than the opponent

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• Example III: Go (cont.)

• Towards endgame, board becomes partitioned into a number of independent subgames

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• Full game: high branching factor, long game

• Local game: low branching factor, short game

Challenge: how to combine local analyses

To achieve near optimal results

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• Tool for Local Game Search:

• minimax search

• Unable to consider successive moves by same player

• Cannot be used to find best global sequence

• Combinatorial game theory(CGT)

is used to perform the search due to its ability to represent a game as a sum of independent subgames

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• CGT

• Deals with partitioned game

• Local analysis

• Search time exponential insize ofsubproblems

Minimax

• Considers the sum game as one unit

• Full board evaluation

• Search time exponential in size of thefull problem

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• Sum Games

• Combinatorial Game Theory

• Previous Strategies

• New Strategies

• Experimental Results

• Conclusions and Future Work

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• Developed by Conway, Berlekamp and Richard K. Guy in the 1960s

• A combinatorial game is any two player perfect information game satisfying the following conditions:

• Alternating moves

• Player who cannot move loses

• no draws

• No random element

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• Combinatorial game theory (cgt) provides abstract definition of combinatorial games

• A game position is defined by sets of follow-up positions for both players (Left, Right)

G={GL|GR}={L1,L2,L3|R1,R2}

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• Examples:

• The simplest game is the ‘zero game’ in which no player has a move:

0 = { | } with GL, GR empty

• The game 1 = {0 | } = { { | } | } represents one free move for Left

• Similarly, The game -1 = { | 0 } = { | { | } } represents one free move for Right

• G = { {14 | 10} | {7|3} }

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• Hot Game

• A game in which each player is eager to play

• A hot game is not a number

• Example of a hot game:

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• Properties of Hot Games:

• Temperature:

• Measures urgency of move

• Type:

• Sente

• A sente move by a player implies a severe threat follow-up forcing the opponent to answer locally. This leaves the original player free to play where he chooses, thereby controlling the flow of the game.

• Double Sente

• is a move which is sente for either player

• Gote

• a move which loses the initiative, since it need not be answered by the opponent, thus giving him Sente

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• Properties of hot games (cont.)

• Thermograph

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• Thermographs of simple hot games of the form G={{A|B}{C|D}}

temperature

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• Approximate Strategies to Play Sum Games based on CGT

• Compute simple properties of each subgame

• Thermograph

• Temperature

• Type

• Make global decision based on one or more of these properties

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• Sum Games

• Combinatorial Game Theory

• Previous Approximate Strategies

• New Strategies

• Experimental Results

• Conclusions and Future Work

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Previous Approximate Strategies for Playing Sum Games

• ThermoStrat

• Graphical determination of the best subgame based on the compound thermograph of the sum

• MaxMove

• Compute the width of the thermograph at t=0 for each subgame

• Play in subgame with maximum value

• HotStrat:

• Compute temperature of each subgame

• Play in hottest subgame

• MaxThreat

• Choose the best subgame by comparing them two by two using minimax

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Previous Approximate Strategies for Playing Sum Games

• Performance of Approximate Strategies Compared to Optimal

• Thermostrat is always good. For subgames with different types gives same result as optimal in 90% of the cases and slightly less in others.

• The performance of each of Hotstrat and Maxmove is highly dependent on the types of subgames participating in the sum.

• MaxMove strategy gives the same result as ThermoStrat for the pattern with only reverse sente games. Performs badly for the rest.

• HotStrat shows very low performance in dealing with patterns that contain reverse sente subgames alone or when combined with only sente games but very good otherwise.

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Previous Approximate Strategies for Playing Sum Games - Performance

• Maxthreat’s performance depends on the order in which

subgames are considered when the sum contains one or more sente gamesas shown

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• Sum Games

• Combinatorial Game Theory

• Previous Approximate Strategies

• New Strategies

• Experimental Results

• Conclusions and Future Work

ICCSE’06, Cairo, Egypt

• Sum Games

• Combinatorial Game Theory

• Previous Approximate Strategies

• New Strategies

• Experimental Results

• Conclusions and Future Work

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Experimental Results

• Performance

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Time Considerations

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• Sum Games

• Combinatorial Game Theory

• Previous Approximate Strategies

• New Strategies

• Experimental Results

• Conclusions and Future Work

ICCSE’06, Cairo, Egypt

ICCSE’06, Cairo, Egypt

ICCSE’06, Cairo, Egypt

ICCSE’06, Cairo, Egypt