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New Approximate Strategies for Playing Sum Games Based on Subgame Types

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**Outline**• Sum Games • Combinatorial Game Theory • Previous Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt**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.) • 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) ICCSE’06, Cairo, Egypt**Sum Games (cont.)**• Many games tend to decompose into a number of independent regions or subgames. • Examples: • Domineering • GO • Amazons ICCSE’06, Cairo, Egypt**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 = ICCSE’06, Cairo, Egypt**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 ICCSE’06, Cairo, Egypt**Sum Games (cont.)**• Example III: Go (cont.) • Towards endgame, board becomes partitioned into a number of independent subgames ICCSE’06, Cairo, Egypt**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 ICCSE’06, Cairo, Egypt**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**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**Outline**• Sum Games • Combinatorial Game Theory • Previous Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt**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 (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} ICCSE’06, Cairo, Egypt**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} } ICCSE’06, Cairo, Egypt**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: ICCSE’06, Cairo, Egypt**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 ICCSE’06, Cairo, Egypt**Combinatorial Game Theory (cont.)**• Properties of hot games (cont.) • Thermograph ICCSE’06, Cairo, Egypt**Combinatorial Game Theory (cont.)**• Thermographs of simple hot games of the form G={{A|B}{C|D}} temperature ICCSE’06, Cairo, Egypt**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**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**• 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**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**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 ICCSE’06, Cairo, Egypt**Outline**• Sum Games • Combinatorial Game Theory • Previous Approximate Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt**Outline**• Sum Games • Combinatorial Game Theory • Previous Approximate Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt**Experimental Results**• Performance ICCSE’06, Cairo, Egypt**Experimental Results (cont.)**Time Considerations ICCSE’06, Cairo, Egypt**Outline**• Sum Games • Combinatorial Game Theory • Previous Approximate Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt**Conclusions**ICCSE’06, Cairo, Egypt**Future Work**ICCSE’06, Cairo, Egypt**Thank You**ICCSE’06, Cairo, Egypt