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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 Sum Games Based on Subgame Types

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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


Outline l.jpg

Outline

  • Sum Games

  • Combinatorial Game Theory

  • Previous Strategies

  • New Strategies

  • Experimental Results

  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt


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Sum Games

  • 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

ICCSE’06, Cairo, Egypt


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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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Sum Games (cont.)

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

  • Examples:

    • Domineering

    • GO

    • Amazons

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Sum Games (cont.)

  • Example II: Domineering

    • Start with the board empty

    • 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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Sum Games (cont.)

  • 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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Sum Games (cont.)

  • Example III: Go (cont.)

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

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Sum Games (cont.)

  • 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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Sum Games (cont.)

  • 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

ICCSE’06, Cairo, Egypt


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Sum Games

  • 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

ICCSE’06, Cairo, Egypt


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Outline

  • Sum Games

  • Combinatorial Game Theory

  • Previous Strategies

  • New Strategies

  • Experimental Results

  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt


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Combinatorial Game Theory

  • 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 (cont.)

  • 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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Combinatorial Game Theory (cont.)

  • 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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Combinatorial Game Theory (cont.)

  • 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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Combinatorial Game Theory (cont.)

  • 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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Combinatorial Game Theory (cont.)

  • Properties of hot games (cont.)

    • Thermograph

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Combinatorial Game Theory (cont.)

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

temperature

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Combinatorial Game Theory (cont.)

  • 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

ICCSE’06, Cairo, Egypt


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Outline

  • Sum Games

  • Combinatorial Game Theory

  • Previous Approximate Strategies

  • New Strategies

  • Experimental Results

  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt


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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

ICCSE’06, Cairo, Egypt


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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.

ICCSE’06, Cairo, Egypt


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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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Outline

  • Sum Games

  • Combinatorial Game Theory

  • Previous Approximate Strategies

  • New Strategies

  • Experimental Results

  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt


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Outline

  • Sum Games

  • Combinatorial Game Theory

  • Previous Approximate Strategies

  • New Strategies

  • Experimental Results

  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt


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Experimental Results

  • Performance

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Experimental Results (cont.)

Time Considerations

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Outline

  • Sum Games

  • Combinatorial Game Theory

  • Previous Approximate Strategies

  • New Strategies

  • Experimental Results

  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt


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Conclusions

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Future Work

ICCSE’06, Cairo, Egypt


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Thank You

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