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

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Outline

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

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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

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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

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Sum Games

• CGT
• Deals with partitioned game
• Local analysis
• Search time exponential in size of subproblems

• Minimax
• Considers the sum game as one unit
• Full board evaluation
• Search time exponential in size of the full
  problem

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Outline

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

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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)
• GGLGRL1,L2,L3R1,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 73

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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
  GABCD

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

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Outline

• 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 t0 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

• Maxthreats performance depends on the order in
  which
• subgames are considered when the sum contains
  one or more sente games as shown

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Outline

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

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Outline

• 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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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

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Conclusions
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Future Work
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Thank You