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1 a g e n t 10 a g e n t a g e n t / / a g e n t a g e n a g e n t a g e n t b l a c k w h i t e w h i t e w h i t e b l a c k b l a c k w h i t e 12-1 b l a c k 12-2 w h i t e b l a c k w h i t e a g e n t 12-2

n a g e n t a g e n t 1 2-1 b l a c k w h i t e w h i t

2 / / a g e n t z e ro - s u m a g e n t a g e n t a g e n t ( d r a n g h t ) Ti c - Ta c - To e B a c k g a m m o n M A X M I N M A X M A X M A M A X M I N M I 0k 2k 2k+ 1 M A X M I N /3 ns W h i t e S c h a e ffer [ S c h a e ffer & Lake 1996] M A X 10 9 S c h a e ff e r Lake

2 M A X M I N M A X M A X M A M A X M I N M I 0k 2k 2k+ 1 M A X M I N 10 40 1/3

3 120 M A X M I N M A X M I N M A X M I N M A X M I N M A X M I N M A X M I N M I N M A X M A X M A X 3 3 X O M A X XM I N OM A X 2 p e p p e p = M A X M I N M A X p e p = ( M I N p e p = p e p = 6 4 =

4 M A X 12-3 / / M A X M I N X O M I N M A X M A X M I N M A X 12-5 A M I N M A X M I N M A

5 122 MAX 12-4 MAX

6 α 12-5 A B C D M I N A M I N A A B C D B C D M A X M I N A A B A α β α 12-6 B C C 1 B 1 B 1 1 B B A B M A X M I N 1) M I N M A X M I N

7 124 M I N 2) M A X M I N M A X M A X M A X M I N n [Pearl 1984 P. 234]: A B n 1) n A B(n) = n n 1 n k n b n k 1 n M A X 2 2 ' 2 ) m a x [ A B(n k ; ) ] 3) ; 4) k = b n k+ 1 4') k = b ; k k ') m i n [ A B(n k ; ) ]. 3') ; n k+ 1 k k+ 1 2 ' ( =+1) MAX MAX 12-7

8 A B s + - s < M A X M I N X b d d b - M I N M A X N d [Slagle & Dixon 1969, Knuth & Moore 1975] - d - d/2 - d/2 - d - b b - P e a r l 4-4 / 3 b 3 [Pearl 1982b] [pearl ] - c h e s s C H E S S4. 5 [slate & Atkin 1977] K K d d M A X M I N

9 126 M A X M I N M A X - M A X M I N M A X M I N M I N a g e n t 12.6 B a c k g a m m o n M A X M I N 6 DICE MAX MAX MIN MIN DICE DICE MAX M I N M A X (e x p e c t i m a x i n g) M A X

6 B a c k g a m m o n 12-8 6 M A X M I N 6 DICE

10 M A X 12.7 [Zobrist 1970 Reitman & Wilcox 1979 K i e r u l f Ch e n & Nievergelt 1990] T D-G A M M O N [ Tesauro 1992 Tesauro 1995] S i g m o i d 4 p 1 p 2 p 3 p 4 v = p 1 +2 p 2 p 3 2 p 4 v v S c= TD-GAMMON

11 128 v t t v t+ 1 t+1 W t t W v t T D G A M M O N 3 t v t v t B i l l R o b e r t i e T D-G A M M O N 2. 1 [ Tesauro 1995] 12.8 [Russell & Wefald ] M G S S * M G S S 2 - B e r l i n e r s B* [Berliner 1979] [korf 1991] - [Huberman 1968 Bratko & Michie 1980] Arthur Samuel [Samuel 1959 Samuel 1967] S a m u e l A l b e r t a Jonathan Schaeff e r C H I N O O K [ S c h a e ff e r et al S c h a e ffer 1997] 1997 I B M G a r r y K a s p a r o v [Newborn 1996] Garry kasparov 1996 [McCarthy 1997] M c C a r t h y D ro s o p h i l a. [Michie 1966] (e x p e c t i m a x) [Lee & Mahajan 1988] O t h e l l o [ S c h r a u d o l p h D a y a n & Sejnowski 1994] [Pearl ] v t

1967] S a m u e l A l b e r t a Jonathan Schaeff e r C H I N O O K [ S c h a e ff e r et al.

12 M A X 1) 2) n n A B C

4.1 * / # 65 2015 6 65 43,500 5,548^ 147 428 13% 88.9 7,340 11,830 210,000 318,000 2,390 420 227 37% 113.4 1,235 216 205 19% 27.9 ** 7,340 1

4.1 * / # 65 2015 6 65 43,500 5,548^ 147 428 13% 88.9 7,340 11,830 210,000 318,000 2,390 420 227 37% 113.4 1,235 216 205 19% 27.9 ** 7,340 1 4.1 4.2 73% 13% 37% 19% 1 3%70 87% 4.1 4.3 17 60 2 4.4 43,500 65,000 5,1005,548 4.2 3,667 10,462 1 2 60 18 4.1 * / # 65 2015 6 65 2014-15 @ 43,500 5,548^ 147 428 13% 88.9 7,340 11,830 210,000 318,000 2,390