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1 25 Improvement of Simulation Accuracy in Monte Carlo Daihinmin Players

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3 ,000 15,000 69% ii

4 Abstract Improvement of Simulation Accuracy in Monte Carlo Daihinmin Players Takamasa ZIBIKI In recent studies of Daihinmin, the Monte-Carlo method has been applied to Daihinmin players. Monte-Carlo player calculates an evaluation value of each meld by performing a lot of playouts, which are random simulations to the end of the game. The strength of Monte-Corlo players depends on the accuracy of the playouts. In this study, we took two approaches in order to improve the accuracy of the playouts of Daihinmin Monte-Carlo players. In the first approach, we estimate the opponents hands, from which playouts are performed, as close to the real ones as possible. Since Daihinmin is a game with imperfect information, it is impossible to know the opponents hands. The difference between the estimated hands and the real ones would become a factor that reduces the accuracy of playouts. If we can estimate the opponents hands more precisely, we can improve the accuracy of playouts. The second method is to approximate the play strategies in the playouts. The simplest Monte-Carlo method, called primitive Monte-Carlo, performs playouts just with random numbers. The difference between the random plays and the real ones would become a factor that reduces the accuracy of playouts. If we can mimic the play strategies of the players, we can improve the accuracy of playouts. In this study, we investigated the effects of these two approaches on Daihinmin iii

5 Monte-Carlo players. For the first approach, we compared the melds computed by Monte-Carlo players with different hand informations with the best hand computed by Monte-Carlo Tree Search player with perfect information. We found after this experiments that partial information of opponents hands may improve evaluation values but the effects on selecting the best hand are small. The two experiments were performed for the second approach. The first experiment investigated the performance of the evaluation function developed with a three-layer neural network. We investigated how often we can select the real melds played in the game records using the evaluation function. It was confirmed that the concordance rate of submission hand combination increases by increasing the training data to be used for learning, and concordance rate saturates over 15,000 training data. In the evaluation function learned from 15,000 training data, submitted hand role concordance rate for the unknown boards has become approximately 69%. The second experiment investigated the performance of the Monte-Carlo player applying the evaluation function. The proposed player selected the meld computed with the three-layer neural network in the playouts. We compared the proposed Monte-Carlo player with the primitive Monte- Carlo player. After the experiments, we confirmed that the Monte-Carlo player with the evaluation function is stronger than the primitive Monte-Carlo player. key words Daihinmin, Monte-Carlo Method, Monte-Carlo Tree Search, Three-Layer Neural Networks iv

6 v

7 vi

8 MCMP MCMP vii

9 VS Nakanaka VS Nakanaka VS paoonr VS paoonr viii

10 [21] [17, 20, 22] [19] 2 [10] 1

11

12 UEC UECda [23] [10, 13, 14] [13] [16] [15] [5] UECda snowl [19] 19 [19] 3

13 1.4 UCB1[3] [1, 19] [11] [4, 18] [2] 10,000 [17] [22] A 10 [20] [9] % 89%

14

15 2 2.1 UEC (UECda) [23]

16 2.2 Order Normal Lock_Suits H Last_Meld Single:H5 Put_Meld Single:H8 Player0 Yes C3 S7 H7 D7 H8 C8 SJ HQ DQ Player1 Yes S3 S4 H4 S5 S6 S8 S9 S10 CQ HA D2 Player2 Yes H6 HJ CJ SQ SK HK Player3 No C7 D8 H10 DK CA C2 Player4 No C9 DJ CK S2 H Player0 Player0 7

17 [12]

18 3.1 UCB1 Upper Confidence Bound [3] j X j n j n j c UCB1 X j + c 2 log n n j (3.1) c 2 [1]

19 , ,001 2, ,001 3, ,001 4, ,001 6, ,001 8, ,001 10, UECda UCB1 c ,

20 , ,001 4,000 4,001 6,000 3,001 4,000 4,001 6, ,000 11

21 3.2 MCT(root) { loop until end_flag { leaf <- select_downwards(root) leaf.n <- leaf.n + 1 if (expand_flag(leaf)) { leaf <- expand(leaf).first_child } board = playout(leaf.board) update_upwards(leaf, getvalue(board)) } return select_best_child(root) } [1] [1] 3.3 end_flag select_downwards playout expand_flag expand 1 evaluate 12

22 3.2 update_upwards UCB UCB1 100 max n 3.4 A P3 PASS [P , P , P , P , P ] ,

23 , ,001 20, ,001 30, ,001 40, ,001 50, ,001 60, ,001 70, ,001 80, ,001 90, , ,

24 ,000 15

25 i j ω i,j j υ j ω i,j I J υ j J ω i,j (1 i I, 1 j J) ω υ j (1 j J)

26 υ S(x) S(x) = e x 4.2 (, ) (0, 1) S(x) = 0 S(x) = 1 D(x) D(x) = x(1 x) S (x) = D(S(x)) p i f i (p) f i (p) ω i,j j g j (p, ω) ( I g j (p, ω) = S i=1 f i (p)ω i,j ) g j (p, ω) υ j 3 17

27 4.2 h(p, υ, ω) J h(p, υ, ω) = S g j (p, ω)υ j j=1 4.2 q m p m M P E(P, υ, ω) E(P, υ, ω) = M S (h(p m, υ, ω) h(q m, υ, ω)) m=1 E(P, υ, ω) υ ω υ ω η υ ω (υ, ω E(P, υ, ω) ) = (υ, ω) η (υ, ω) υ j α i,j (p, υ, ω) = D(h(p, υ, ω))g j (p, ω) E(P, υ, ω) υ j M = D(E(P, υ, ω)) (α i,j (p m, υ, ω) α i,j (q m, υ, ω)) m=1 18

28 4.2 ω i,j β i,j (p, υ, ω) = D(h(p, υ, ω))υ j D(g j (p, ω))f i (p) E(P, υ, ω) ω i,j M = D(E(P, υ, ω)) (β i,j (p m, υ, ω) β i,j (q m, υ, ω)) m=1 19

29 5 1, , M M 1 M 1 20

30 M 1 M T 2.1 Player1 D2 Player2 HK Player3 C2 Player4 H2 Player1 S3 Player2 H6 Player3 C7 Player4 C9 Player1 S3 D2 Player2 H6 HK Player3 C7 C2 Player4 9 H2 T 100, ,000 T

31 5.1 M M M M M M 129 / / / 126 M 133 / / / 145 M 127 / / / 137 M 131 / / / 146 M 135 / / /

32 %

33 Player1 Player2 Player3 Player4 Player0 2.1 H8 H7,HQ,PASS 4 Player0 Player

34 JOKER , , MCMP Ave Max Min 95% 25

35 MCMP MCMP ,000 69% 1, MCMP 15 3, MCMP 15 19, MCMP 50 3, MCMP 50 27, ,

36 6.2 4% ,000 27,000 71% UECda Monte EVM EVH MonteEVM MonteEVH Sample Nakanaka UECda UECda 27

37 6.2 paoonr UECda 5 Monte 1,000 UCB1 1,000 EVM ,000 1,000 EVH ,245 1,000 MonteEVM ,000 1,000 Monte MonteEVH

38 6.2 3,245 1,000 Monte MonteEVM MonteEVH Monte EVM EVH MonteEVM MonteEVH Nakanaka UECda Nakanaka 5 Nakanaka Sample Monte EVM EVH Nakanaka 6.3 MonteEVM Nakanaka MonteEVM Nakanaka Nakanaka MonteEVM 2012 UECda paoonr MonteEVM Nakanaka paoonr2 6.4 MonteEVM Nakanaka 1 1 MonteEVM 2 2 Nakanaka MonteEVM Monte 29

39 VS Nakanaka Nakanaka Sample Sample Sample Nakanaka Monte EVM EVH MonteEVM MonteEVH VS Nakanaka Nakanaka Nakanaka Sample Nakanaka Monte EVM EVH MonteEVM MonteEVH VS paoonr2 paoonr2 Sample Sample Sample MonteEVM VS paoonr2 MonteEVM paoonr2 paoonr2 Sample MonteEVM ,245 30

40 ,000 15,000 69% 50 27,

41 32

42 4 TA

43 [1] L. Kocsis and C. Szepesvári. Bandit Based Monte-Carlo Planning, 17th European Conference on Machine Learning (ECML 2006), Lecture Notes in Computer Science 4212, pp (2006). [2] M. Buro. Improving heuristic mini-max search by supervised learning. Artificial Intelligence, Artificial Intelligence 134(1 2), pp (2002). [3] P. Auer, N. Cesa-Bianchi and P. Fischer. Finite-time Analysis of the Multi-armed Bandit problem. Machine Learning, Vol. 47, pp (2002). [4],. Ms. Pac-Man., Vol. 52, No. 12, pp (2011). [5],,.., Vol MPS-96, No. 4, pp.1 3 (2013). [6].. 12, pp (2007). [7],,,.. 8, pp (2003). [8],.. 13, pp (2008). [9],,.. 12, pp (2007). [10],.. (GI), Vol GI-25, No. 3, pp.1 4 (2010). [11],.. 34

44 , Vol. 50, No. 11, pp (2009). [12],,,,.., pp (1998). [13],,. UEC snowl. 2 UEC (2011). [14],.. 1 UEC (2010). [15],.. (BIO), Vol BIO-27, No. 31, pp. 1 4 (2011). [16],.. (MPS), Vol MPS-85, No. 9, pp. 1 6 (2011). [17].. 11, pp (2006). [18],,. UCT. (GI), Vol GI-24, No. 5, pp. 1 5 (2010). [19]. :., Vol. 49, No. 6, pp (2008). [20]. GPS. co.jp/avs/documents/pdf/seminar\ event/conf/19/1-3.pdf (2013). [21].. c.ooco.jp/entcog/contents/lecture/date/5-yoshizoe.pdf (2008). [22] 2 HUMAN VS COMPUTER, denousen2013/ (2013). [23]. UEC, / (2012). 35

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