A COMPUTER-ASSISTED STUDY OF GO ON M X N BOARDS* by Edward Thorp Mathematics Department University of California at Irvine Irvine, California and William E. Walden Computing Center Washington State University Pullman, Washington P 1 7V6Q 7-Aua..- _aX? a<_ <^A.rf. /XX" <=^r/if~**i iA^J^T^^M^ -cvaVlp. f-t^.t&tL. &Q fe-w. Asy(X cr/ Zifo- .^.yAy-i /frm<J<J! pen £X^ -Ovl^ . l^}-^JL ptZF/c. /2>£ V : Rp u> a-^-X : *~«^mU <=t<s~<3.<_~ Zytfy UF:^-UF:"- J..J.- lLA.^t'tCJ^yCT Ja^<2yLZA7a,,: UiUj Jy\jX~^J^L*^r2 _ ftSLs r //Vl.-^^^^y y * j « r * fyxv c s z>«2_ *■* aa: cx^y^ v-'/"X "/yx^.^y" L4J~)SV^~> /^-*-i-t^>-'L, _AIX <Av<^i. / it t r: /v <^ " A^ kA UcX^b, S £?>--;. CONTENTS 1. Introduction 1 2. Topology 3 3. Combinatorics 6 4. Rules H 5. Illustrative Game Trees for Tiny Boards 19 6. Evaluation of Tiny Boards 27 7. Heuristics 8. The Computer Program 9. Potential Uses of the Program 10. Miscellany Figures References A COMPUTER-ASSISTED STUDY OF GO ON M x N BOARDS ': by Edward Thorp and Wil liana Walden 1. Introduction: The game of Go is believed to have originated in China about four thousand years ago. It has been intensely cultivated in Japan for more than a thousand years and is now played throughout the world. For the history of Go, refer to Falkener (1961), or Smith (1956). Playing technique is discussed in Goodcll (1957), Lasher (i960), Morris (1951), Smith (1956), and Takagawa (1958). For those who worry about whether the study of games is useful, we remark that an interest in games of chance led Cardano, Fermat and Pascal to initiate the theory of probability. Providing a theoretical frame- work for poker was one of the objectives of Yon Neuman's theory of games. Recently one of the authors was faced with the problem of determining the optimum amount to bet on positive expectation situations in casino blackjack. Using results by Kelley (1956) and Breiman (1961), a theory of resource, allocation (synonyms are bet sizing and portfolio selection) was developed. This theory (Thorp 1969) supplants and refutes the theory of portfolio selection due to Markowitz (1959) which is generally accepted by economists. This research was supported in part by Grant AF-AFOSR 1113-66. 2 * We find Go a particularly promising game of skill to analyze. Tens of millions of people play it and it has been developed to an extremely high level, of skill. This means that computer analysis can be checked or tested again analysis by highly skilled human players. Also, the rules are few and simple, which suggests that the game may have significant theorems . There are several ways to study Go with the aid of a computer. Remus (1962) wrote a computer program, to learn "good" strategies and simulated the game on a smaller board. Another approach is to combine positional evaluation functions and tree computations, as has been done for chess. We study the game on an M X N board rather than the usual 19x19 board . First we give the complete tree calculation for tiny boards. Then we extend the analysis to larger boards. Previous studies of bridge and poker via tiny decks are, for instance, close in spirit to our approach. This paper continues our work in Thorp and Walden (1964). We suggest reading that paper before proceeding here. These approaches can be modified by combining man and machine. Our computer program, or others, can be used like a "super slide rule" to assist a human player in actual play. This might fit in with the project announced in Engelbart, 1968. One wonders why this isn't done in chess for instance. A contention sometimes made is that: two players who 3 * consult in a chess game don't generally do as well as the stronger player would have done alone. Whether or not this is true, we observe that with two consulting players, if there is much disparity, the abilities of the weaker player are nearly subsumed by those of the stronger player. Thus, the weaker player is redundant and only distracts the stronger player with fruitless discussions. With two equal players we might expect their different com- peting strategies to interfere with each other. But in a man-machine symbiosis, skills are complementary. Also, disputes do not arise: the man uses the machine when he wishes to, and not otherwise. The difference is that between calculating with a super slide rule and calculating with a committee. 2. Topology: The game of Go is played on a rectangular board marked with two mutually perpendicular sets of parallel lines. The standard board has 19 lines in each direction. This produces 361 points of intersection. The two players, Black and White, move alternately beginning with Black. A move consists either of placing a piece Atone) on a vacant point of inter- section, or of passing. The object of the game is to capture stones and to surround territory. Two vertices are adjacent if they are on the same horizontal, or vertical line and there are no vertices between them. Classify vertices 4 into three types: black, white and vacant. Vertices v and w are connected if there is a chain (v., .... v ) of vertices of the same type such that v = v , 1 n l v is adjacent to v. , , 1-i § n I. and v =w. The v. need not be clis- i J l+l - n l tinct. In particular, we allow v= v for 1-i = n. Connectedness is an i equivalence relation (i. o. , it is reflexive, symmetric, and transitive). Thus, it partitions the board into disjoint equivalence classes. Call these equivalence classes groups. We will, distinguish between groups of stones and groups of spaces. If the vertices adjacent to a vertex in a given group are of the same type, that vertex is in the interior of the group. If the adjacent vertices are not of the same type, that vertex is a boundary point of the group. A vacant vertex adjacent to a black or to a white stone is a breathing space or degree of freedom (I. J. Good, 1965) for the group to which that stone belongs. Rule 1 ( Capture). If a group of stones has exactly one breathing space and the other player is permitted to move there (he may not be, because of the Ko rule or a rule prohibiting all cycles of even length, then if he does so, he captures that group. It is removed from the board and set aside. At the end of the game players receive one point for each captive. Rule 2 (Suicide is illegal). If a player's group of stones has exactly one breathing space, a move there by the player (which would deprive his own group of breathing space and cause it to be removed, i. c. , suicide) is illegal. If a single empty vertex is completely surrounded by stones of 5 the opposing color, a player can move there only if the move results in the removal of one or more of the adjacent men. (Otherwise, the move creates a group of one stone with no breathing space, which causes it to be removed, i. c. , suicide. ) A chain (v , v , . , v ) of vertices such that v. is adjacent to 12 n i v for 1-i = n - 1 is a path, If v= v and w= v , v and w are joined j — l n by path (v , v ). A group G surrounds a subset S of vertices the . n if (1) for each v in S, every path joining v to the edge of the board contains a member of G and (2) if II is any other group with this property every path joining S to H intersects G If an isolated, empty vertex v is surrounded by a group G of stones, then v is a special case of what is called an "eye". Note that a group of stones with two or more such isolated empty vertices cannot be captured because Rule 2 forbids a move to one isolated vacant inter- section while another remains. Any group with two eyes is impregnable. This principle of two eyes is fundamental in the game. There are other kinds of impregnable groups too; all such are called living. Figure I shows a Black group on a 1 X 3 board with two eyes. This is the smallest board which can have a group of stones with two eyes Isolated single vertices can be replaced by larger connected groups of vacant vertices in the above discussion, and each of these groups of vacant vertices is called an eye. However, if the group of empty vertices is large enough, the opponent may be able to build a living group inside. 6 In this case, he may or may not be able to prevent the creation of an eye. In practice this ambiguity about whether a surrounded group of vacant vertices is an eye does not seem to arise: we know of no example where the opponent, can destroy an eye by filling it with a living group. Of more immediate practical interest is the question of which eyes can be converted into two eyes if (a) the player whose group it is has the move, or (b) his opponent has the move. For instance, the Black player in Figure 2 can make two eyes if it is his turn by moving to the point x.