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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

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
Outline
  • Sum Games
  • Combinatorial Game Theory
  • Previous Strategies
  • New Strategies
  • Experimental Results
  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt

sum games
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

sum games cont

=

+

+

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 cont5
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 cont6
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 cont7
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 cont8
Sum Games (cont.)
  • Example III: Go (cont.)
    • Towards endgame, board becomes partitioned into a number of independent subgames

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sum games cont9
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 cont10
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

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sum games11
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

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outline12
Outline
  • Sum Games
  • Combinatorial Game Theory
  • Previous Strategies
  • New Strategies
  • Experimental Results
  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt

combinatorial game theory
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

ICCSE’06, Cairo, Egypt

combinatorial game theory cont
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 cont15
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 cont16
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 cont17
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 cont18
Combinatorial Game Theory (cont.)
  • Properties of hot games (cont.)
    • Thermograph

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combinatorial game theory cont19
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 cont20
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

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outline21
Outline
  • Sum Games
  • Combinatorial Game Theory
  • Previous Approximate Strategies
  • New Strategies
  • Experimental Results
  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt

previous approximate strategies for playing sum games
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 games23
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
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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outline25
Outline
  • Sum Games
  • Combinatorial Game Theory
  • Previous Approximate Strategies
  • New Strategies
  • Experimental Results
  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt

outline28
Outline
  • Sum Games
  • Combinatorial Game Theory
  • Previous Approximate Strategies
  • New Strategies
  • Experimental Results
  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt

experimental results
Experimental Results
  • Performance

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experimental results cont
Experimental Results (cont.)

Time Considerations

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outline31
Outline
  • Sum Games
  • Combinatorial Game Theory
  • Previous Approximate Strategies
  • New Strategies
  • Experimental Results
  • Conclusions and Future Work

ICCSE’06, Cairo, Egypt

conclusions
Conclusions

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future work
Future Work

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thank you
Thank You

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