1. September 8th 2005Advances in Computer Games 111 Enhancing Search Efficiency by Using Move Categorization Based on Game Progress in Amazons Yoshinori Higashiuchi Saga University, Japan Reijer Grimbergen Yamagata University, Japan

2. September 8th 2005Advances in Computer Games 112 Outline Amazons Move categories in Amazons Priority ordering of moves Adjusting category priorities using game progress Experimental results Conclusions and future work

3. September 8th 2005Advances in Computer Games 113 Amazons Two-player perfect information game 10  10 board with 4 Amazons each A move consists of Moving the Amazon like a chess queen Shooting an arrow like a chess queen Arrows stay on the board and can not be passed The player who can not move any of his Amazons loses

4. September 8th 2005Advances in Computer Games 114 Amazons Main feature: large number of legal moves 2176 moves in the initial position 479 moves on average Relatively new game No expert players No heuristics for good moves How to do a reasonable search in Amazons?

5. September 8th 2005Advances in Computer Games 115 Improving search in Amazons Step 1: improving efficiency of  -  search Searching good moves first Best move of previous iteration Killer moves History heuristic Etc… What are good moves in Amazons?

6. September 8th 2005Advances in Computer Games 116 Move categories in Amazons No.CategoryType# 1 Move blocks opponent Amazons0, 1, 2, 3 or more 4 2 Arrow blocks opponent Amazons0, 1, 2, 3 or more 4 3 Move adjacent to opponent AmazonTrue, false 2 4 Arrow adjacent to opponent AmazonTrue, false 2 5 Blocking a single Amazon in multiple waysTrue, false 2 6 Move previously blocked Amazon0, 1, 2 3 7 Move Amazon to which Amazon moved adjacentlyTrue, false 2 8 Move Amazon to which arrow was shot adjacentlyTrue, false 2 9 Block Amazon that moved on previous moveTrue, false 2 10 Move Amazon not blocking any opponent AmazonTrue, false 2

7. September 8th 2005Advances in Computer Games 117 Amazon and arrow block 3 opponent Amazons Move categories in Amazons

8. September 8th 2005Advances in Computer Games 118 Move categories in Amazons Amazon and arrow adjacent to opponent, also blocking twice

9. September 8th 2005Advances in Computer Games 119 Move categories in Amazons Moving the threatened Amazon, also blocking twice

10. September 8th 2005Advances in Computer Games 1110 Move categories in Amazons Blocking the Amazon that just moved with a free Amazon

11. September 8th 2005Advances in Computer Games 1111 Move categories in Amazons Moves can belong to different categories Total number of combinations: 6,144 Theoretically possible: 1,420 CatNo.12345678910Move 133FFT0FFFFD1-D7(G7) 212TTT1FFFFD10-I5(I4) 302FFF2TTFFJ4-H2(C7) 401FTF1FFTTD1-G1(D4)

12. September 8th 2005Advances in Computer Games 1112 P i : The realization probability of category i A i : Number of times a move from category i was played B i : Number of positions where a move from category i was possible P i : The realization probability of category i A i : Number of times a move from category i was played B i : Number of positions where a move from category i was possible Priority ordering of moves Importance of categories i i i B A P 

13. September 8th 2005Advances in Computer Games 1113 Priority ordering of moves Category RP Category RP Move blocked 0 Amazons 33.1 Arrow adjacent to opponent Amazon 72.3 Move blocked 1 Amazon 56.9 Multiple opponent block 36.2 Move blocked 2 Amazons 13.1 Blocking the previously moved Amazon 50.6 Move blocked 3 or more Amazons 6.6 Moving the Amazon that was blocked once by the previous move 49.0 Arrow blocked 0 Amazons 14.9 Moving the Amazon that was blocked twice by the previous move 33.9 Arrow blocked 1 Amazon 71.4 Moving Amazon to which Amazon moved adjacently 32.8 Arrow blocked 2 Amazons 17.2 Moving Amazon to which arrow was shot adjacently 34.5 Arrow blocked 3 or more Amazons 5.3 Moving a non-blocking Amazon 45.6 Move adjacent to opponent Amazon 52.3

14. September 8th 2005Advances in Computer Games 1114 Priority ordering of moves We use the 1,420 theoretically possible categories 11,000 games to find the realization probabilities Realization probabilities for our four example moves: CatNo.MoveRPOrder 1D1-D7(G7)32.1652 2D10-I5(I4)0.902371 3J4-H2(C7)0.158876 4D1-G1(D4)7.25025

15. September 8th 2005Advances in Computer Games 1115 Adjusting category priorities Step 2: adjusting the probabilities using game progress The strategic features of Amazons change as the game progresses Good moves change as the game progresses Probabilities must change as well Most basic progress measurement: move number Grouping probabilities in intervals of 8 moves

16. September 8th 2005Advances in Computer Games 1116 Blocking with the move

17. September 8th 2005Advances in Computer Games 1117 Blocking with the arrow

18. September 8th 2005Advances in Computer Games 1118 Adjusting category priorities Movno Category Number 1234 1 - 732.650.510.000.52 8 - 150.010.990.181.20 16 - 230.010.880.214.20 24 - 310.012.220.208.32 32 - 390.012.410.8018.39 40 - 470.011.762.5317.75 48 - 550.0117.650.0130.86 56 - 630.01 50.44 64 - 710.01 37.50 72 - 790.01 75.00 80 - 870.01 88 - 920.01

19. September 8th 2005Advances in Computer Games 1119 Experimental results Three program versions No Move Categories (NMC) No Game Progress (NGP) Game Progress (GP) Comparing search times to depth 3 Using 1,521 positions from 30 games Self-play experiments Matches of 100 games with 10, 30 and 60 seconds per move

20. September 8th 2005Advances in Computer Games 1120 Experimental results

21. September 8th 2005Advances in Computer Games 1121 Experimental results

22. September 8th 2005Advances in Computer Games 1122 Experimental results

23. September 8th 2005Advances in Computer Games 1123 Experimental results

24. September 8th 2005Advances in Computer Games 1124 Experimental results Matches between the different versions Match 10 seconds30 seconds60 seconds ResultSqDifResultSqDifResultSqDif NGP – NMC58 – 42+14660 – 40+9259 – 41+87 GP – NMC79 – 21+42872 – 28+43975 – 25+388 GP – NGP62 – 38+21470 – 30+25963 – 37+165

25. September 8th 2005Advances in Computer Games 1125 Experimental results Total self-play results NoVersion10 seconds30 seconds60 seconds 1G P141 – 59142 – 58138 – 62 2NGP 96 – 104 90 – 110 96 – 104 3NMC 63 – 137 68 – 132 66 – 134

26. September 8th 2005Advances in Computer Games 1126 Conclusions and future work Conclusions Using move categories is better than not using move categories Using game progress is better than not using game progress Future work Investigate different ways to decide realization probabilities Games against different programs Different representation of game progress Using game progress in other games