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New Approximate Strategies for Playing Sum Games Based on Subgame TypesPowerPoint Presentation

New Approximate Strategies for Playing Sum Games Based on Subgame Types

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New Approximate Strategies for Playing Sum Games Based on Subgame Types

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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

- 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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=

+

+

- 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
- 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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- 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

- minimax search

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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

ICCSE’06, Cairo, Egypt

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

ICCSE’06, Cairo, Egypt

- 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 (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} }

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

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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

- Sente

- Temperature:

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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

- Compute simple properties of each subgame

ICCSE’06, Cairo, Egypt

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

ICCSE’06, Cairo, Egypt

- 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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- 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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- Maxthreat’s performance depends on the order in which
subgames are considered when the sum contains one or more sente gamesas shown

ICCSE’06, Cairo, Egypt

- 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

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

ICCSE’06, Cairo, Egypt

- Performance

ICCSE’06, Cairo, Egypt

Time Considerations

ICCSE’06, Cairo, Egypt

- 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