New Approximate Strategies for Playing Sum Games Based on Subgame Types

1. 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

2. Outline • Sum Games • Combinatorial Game Theory • Previous Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt

3. 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

4. = + + 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

5. Sum Games (cont.) • Many games tend to decompose into a number of independent regions or subgames. • Examples: • Domineering • GO • Amazons ICCSE’06, Cairo, Egypt

6. 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

7. 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

8. Sum Games (cont.) • Example III: Go (cont.) • Towards endgame, board becomes partitioned into a number of independent subgames ICCSE’06, Cairo, Egypt

9. 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

10. 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

11. 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

12. Outline • Sum Games • Combinatorial Game Theory • Previous Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt

13. 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

14. 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

15. 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

16. 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

17. 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

18. Combinatorial Game Theory (cont.) • Properties of hot games (cont.) • Thermograph ICCSE’06, Cairo, Egypt

19. Combinatorial Game Theory (cont.) • Thermographs of simple hot games of the form G={{A|B}{C|D}} temperature ICCSE’06, Cairo, Egypt

20. 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

21. Outline • Sum Games • Combinatorial Game Theory • Previous Approximate Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt

22. 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

23. 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

24. 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

25. Outline • Sum Games • Combinatorial Game Theory • Previous Approximate Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt

26. ICCSE’06, Cairo, Egypt

27. ICCSE’06, Cairo, Egypt

28. Outline • Sum Games • Combinatorial Game Theory • Previous Approximate Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt

29. Experimental Results • Performance ICCSE’06, Cairo, Egypt

30. Experimental Results (cont.) Time Considerations ICCSE’06, Cairo, Egypt

31. Outline • Sum Games • Combinatorial Game Theory • Previous Approximate Strategies • New Strategies • Experimental Results • Conclusions and Future Work ICCSE’06, Cairo, Egypt

32. Conclusions ICCSE’06, Cairo, Egypt

33. Future Work ICCSE’06, Cairo, Egypt

34. Thank You ICCSE’06, Cairo, Egypt