1. Introduction

Note: The original lecture on this material is available on streaming video at http://www.msri.Org/publications/ln/msri/2000/gametheory/landman/l/.

Wythoff's Game (also called Wythoff's Nim or simply Wyt) is an impartial 2-player game played with 2 piles of counters. Each player may, on their turn, remove any number of counters from either pile, or remove the same number of counters from both piles. The player removing the last counter wins. The set of winning positions (those for which the Sprague-Grundy value equals is well-known. However, no direct formula is known for computing the Sprague-Grundy value G(m,n) of an arbitrary position with m counters in one pile and n in the other; it appears necessary to compute all the G(i, j)-values for smaller games first.

It is clearly impossible to compute the n-th row of (that is, the sequence directly with a finite state machine. One reason is that the values grow without bound, and thus the number of bits needed to represent the value will eventually exceed the capacity of any FSM. It also appears at first that the FSM would have to remember an ever-growing number of values which have already been used. The following lemmas provide bounds on and allow us to overcome these obstacles.

1. What are Combinatorial Games?

Roughly speaking, the family of combinatorial games consists of two-player games with perfect information (no hidden information as in some card games), no chance moves (no dice) and outcome restricted to (lose, win), (tie,tie) and (draw, draw) for the two players who move alternately. Tie is an end position such as in tic-tac-toe, where no player wins, whereas draw is a dynamic tie: any position from which a player has a nonlosing move, but cannot force a win. Both the easy game of Nim and the seemingly difficult chess are examples of combinatorial games. And so is Go. The shorter terminology game, games is used below to designate combinatorial games.

2. Why are Games Intriguing and Tempting?

Amusing oneself with games may sound like a frivolous occupation. But the fact is that the bulk of interesting and natural mathematical problems that are hardest in complexity classes beyond NP, such as Pspace, Exptime and Expspace, are two-player games; occasionally even one-player games (puzzles) or even zero-player games (Conway's “Life“). Some of the reasons for the high complexity of two-player games are outlined in the next section. Before that we note that in addition to a natural appeal of the subject, there are applications or connections to various areas, including complexity, logic, graph and matroid theory, networks, error-correcting codes, surreal numbers, on-line algorithms and biology.

But when the chips are down, it is this “natural appeal” that compels both amateurs and professionals to become addicted to the subject. What is the essence of this appeal? Perhaps the urge to play games is rooted in our primal beastly instincts; the desire to corner, torture, or at least dominate our peers. An intellectually refined version of these dark desires, well hidden under the façade of a passion for local, national or international tournaments or for scientific research, is the consuming strive “to beat them all”, to be more clever than the most clever, in short — to create the tools to Math-master them all in hot combinatorial corafeat! Reaching this goal is particularly satisfying and sweet in the context of combinatorial games, in view of their inherent high complexity.

1. Introduction

The rules of Hex are extremely simple. Nevertheless, Hex requires both deep strategic understanding and sharp tactical skills. The massive game-tree search techniques developed over the last 30-40 years mostly for Chess (Adelson-Velsky, Arlazarov, and Donskoy 1988; Marsland 1986), and successfully used for Checkers (Schaeffer et al. 1996), and a number of other games, become less useful for games with large branching factors like Hex and Go. For a classic 11 x 11 Hex board the average number of legal moves is about 100 (compare with 40 for Chess and 8 for Checkers).

Combinatorial (additive) Game Theory provides very powerful tools for analysis of sums of large numbers of relatively simple games (Conway 1976; Berlekamp, Conway, and Guy 1982; Nowakowski 1996), and can be also very useful in situations, when complex positions can be decomposed into sums of simpler ones. This method is particularly useful for an analysis of Go endgames (Berlekamp and Wolfe 1994; Müller 1999).

Hex positions do not tend to decompose into these types of sums. Nevertheless, many Hex positions can be considered as combinations of simpler subgames. We concentrate on the hierarchy of these subgames and define a set of deduction rules, which allow to calculate values of complex subgames recursively, starting from the simplest ones. Integrating the information about subgames of this hierarchy, we build a far-sighted evaluation function, foreseeing the potential of Hex positions many moves ahead.

In Section 2 we introduce the game of Hex and its history. In Section 3 we discuss the concept of virtual connections. In Section 4 we introduce the AND and OR deduction rules. In Section 5 we show how to recursively calculate the hierarchy of virtual connections.

We also report some of the most interesting results here. Some of these are real surprises for human Chinese chess experts. For example, the aegp-aaee1 ending is a theoretical win, not as previously believed, a draw. Human analysis for this endgame over many years by top players has been proved to be wrong.

1. Introduction

Endgame databases have several benefits. First, the knowledge they provide about the game is perfect knowledge. Second, the databases, because of their complete knowledge about certain domains, are a useful background for Artificial Intelligence research, especially in machine learning. Third, the databases often provide knowledge beyond that achieved by humans, (and increasingly, beyond that achievable by humans). Many endgame databases in many games have been constructed since Ströhlein's pioneering work (Ströhlein, 1970). In Chess, Thompson (1986) generated almost all 5-men chess endgame databases and made them widely available in CD format. Databases construction also enable Gasser (1996) to solve the game of Nine Men's Morris. Perhaps the most impressive endgame database construction so far is Schaeffer's work (Schaeffer et al., 1994) on Checkers. His program created all 8 men endgame databases and comprised more than 440 billion (4.4 x 1011) positions. The databases played a very important role in Chinook's success.

We started our work on constructing Chinese chess endgame databases back in 1992. We have constructed many Chinese chess endgame databases since. We wanted our program to solve as many endgames as possible using only moderate hardware we had access to. So a major effort was made to improve the retrograde algorithm as well as examine ways to reduce the size of the database.

Armed with careful analysis to reduce the databases’ size and our fast, memory efficient retrograde algorithm, we were able to solve one class of very interesting Chinese chess endgames. In this class of endgames, one side has no attacking pieces left but have various defending pieces, while the other side has various attacking pieces.

Introduction

A great deal is known about the partial order structure of large subsets of games. See, for instance, [BCG82] [Con76] for a complete characterization of games generated by numbers, and infinitesimals such as and *n. Linear operators applied to these games of temperature zero can often leverage this characterization to apply to hot games, such as positions occurring in Go [BW94] and Domineering [Ber88] [Wol93]. Some general results are known about the group structure of games, including a complete characterization of the group generated by games born by day 3 [Moe91], but surprisingly little has been written about the overall structure of the partial-ordering of games. Here we prove that the games born by day n form a distributive lattice, but that the collection of all finite games do not form a lattice.

We assume the reader is already familiar with combinatorial game theory definitions as in [BCG82] or [Con76]. In particular, we assume knowledge of the definitions of a game [BCG82, p. 21], sums and negatives of games [BCG82, p. 33], and the standard partial ordering on games [BCG82, p. 34].

1. Easy Examples without Kos

Figure 1.1 shows a simple width-two-entrance 5-room position. Though it is easily analyzed by mathematicians, it is difficult even for strong Go players to determine the best move. Black's two options from Figure 1.1 are shown in Figures 1.2 and 1.3, and White's two options are shown in Figures 1.4 and 1.5.

1. Introduction

John Conway's Game of Life has fascinated and inspired many enthusiasts, due to the emergence of complex behavior from a very simple system. One of the many interesting phenomena in Life is the existence of gliders and spaceships: small patterns that move across space. When describing gliders, spaceships, and other early discoveries in Life, Martin Gardner wrote (in 1970) that spaceships “are extremely hard to find” [10]. Very small spaceships can be found by human experimentation, but finding larger ones requires more sophisticated methods. Can computer software aid in this search? The answer is yes - we describe here a program, gfind, that can quickly find large low-period spaceships in Life and many related cellular automata.

Among the interesting new patterns found by gf ind are the “weekender” 2c/7 spaceship in Conway's Life (Figure 1, right), the “dragon” c/6 Life spaceship found by Paul Tooke (Figure 1, left), and a c/7 spaceship in the Diamoeba rule (Figure 2, top). The middle section of the Diamoeba spaceship simulates a simple one-dimensional parity automaton and can be extended to arbitrary lengths. David Bell discovered that two back-to-back copies of these spaceships form a pattern that fills space with live cells (Figure 2, bottom). The existence of infinite-growth patterns in Diamoeba had previously been posed as an open problem (with a $50 bounty) by Dean Hickerson in August 1993, and was later included in a list of related open problems by Gravner and Griffeath [11]. Our program has also found new spaceships in well known rules such as HighLife and Day&Night as well as in thousands of unnamed rules.

Introduction

Combinatorial game theory [Berlekamp et al. 1982, Conway 1976] has discovered a fascinating array of mathematical structures that provide explicit winning strategies for many positions in a wide variety of games. The theory has been most successful on those games that tend to decompose into sums of smaller games. After assigning a game-theoretic mathematical value to each of the summands, these values are added to determine the value of the entire position. This approach has provided powerful new insights into a wide range of popular games, including Go [Berlekamp and Wolfe 1994], Dots-and-Boxes [Berlekamp 2000a], and even certain endgames in Chess [Elkies 1996].

Some positions on two-dimensional board games breakup into one-dimensional components, and the values of these components often have their own interesting structures. Examples include Blockbusting [Berlekamp 1988], Toads and Frogs [Erickson 1996] and Amazons [Berlekamp 2000b]. The same is true for Konane as suggested by previous analyses of one-dimensional positions [Ernst 1995 and Scott 1999].

Konane is an ancient Hawaiian game similar to checkers. It is played on an 18x18 board with black and white stones placed in an alternating fashion so that no two stones of the same color are in adjacent squares. Two adjacent pieces adjacent to the center of the board are removed to begin the game. A player moves by taking one of his stones and jumping, in the horizontal or vertical direction, over an adjacent opposing stone into an empty square. The jumped stone is removed. A player can make multiple jumps on his turn but cannot change direction mid turn. The first player who cannot make a move loses.

Our goal was to find values for all possible 1 x n Konane positions that consist of three solid patterns each separated by a single space, which we call an “almost solid pattern with two spaces”. We assumed that the positions were far away from the edges of the board so that there was no interference.

1. Introduction

The game of Quadraphage, invented by R. Epstein (see [3] and [4]), pits two players against each other on a (generalized) m x n chess board. The Chess player possesses a single chess piece such as a King or a Knight, which starts the game on the center square of the board (or as near as possible if mn is even); his object is to move his piece to any square on the edge of the board. The Go player possesses a large number of black stones, which she can play one per turn on any empty square to prevent the chess piece from moving there; her object is to block the chess piece so that it cannot move at all. (Thus the phrase “a large number” of black stones can be interpreted concretely as mn - 1 stones, enough to cover every square on the board other than the one occupied by the chess piece.) These games can also be called Chessgo, or indeed Kinggo, Knight go, etc. when referring to the game played with a specific chess piece.

Quadraphage can be played with non-conventional chess pieces as well; in fact, if we choose the chess piece to be an “angel” with the ability to fly to any square within a radius of 1000, we encounter J. Conway's infamous angel-vs.-devil game [2]. In this paper we consider the case where the chess piece is S. Golomb's Duke, a Fairy Chess piece that is more limited than a king, in that it moves one square per turn but only in a vertical or horizontal direction. In this game of Dukego, we will call the Chess player D and the Go player G.

Berlekamp, Conway, and Guy analyzed the game of Dukego in [1], drawing upon strategies developed by Golomb.

1. Introduction

Go is an ancient game which has been played for several millennia throughout Asia. Although playable rules are relatively simple (tournament rules can include many technicalities), the game is strategically very challenging. Go is replacing Chess as the pure strategy game of choice to serve as a test-bed for artificial intelligence ideas [9] [7] [6].

Go was proved PSPACE-hard by Lichtenstein and Sipser [8], and was later proved EXPTIME-complete (Japanese rules) by Robson [12]. More recently, Crâşmaru and Tromp [5] proved that Go positions called ladders are PSPACE-complete. Yedwab conjectured that the Go endgame is hard [14]. The endgame occurs when each local area of play has a polynomial size canonical game tree and is insulated from all other local areas by live stones. A combinatorial game theorist will recognize this as a sum of simple local positions.

Morris succeeded in proving that sums of small local game trees are PSPACE-complete [11]. Yedwab restricted the games to be of depth 2 [14], and Moews showed that sums of games with only three branches are NP-hard [10] [1, p. 109]. (Each of Moew's games is of the form {a || b | c} where a, b and c are integers.) Since Yedwab's and Moews’ Go-like game trees depend upon scores which are exponential in magnitude (yet polynomial in the number of bits of the scores), they did not translate to polynomial sized Go positions.

Berlekamp and Wolfe show how to analyze certain one-point Go endgames [1], Some of these endgame positions have values which are linearly related to dyadic rationals of the form. Since these endgame positions are polynomial in size in the number of bits of the numerator and denominator of x, their techniques can be combined with those of Yedwab and Moews to prove that the Go endgame is PSPACE-hard.

Robson observed that in Japanese rules (where repetitions of a recent position are forbidden), Go is easily seen to be in EXPTIME. However, he conjectures that according to Chinese rules (where any previous position is forbidden), Go is EXPSPACE-complete [13]. This paper fails to resolve his conjecture, but the techniques may be applicable.

We have retained the numbering from the list of unsolved problems given on pp. 183-189 of AMS Proc. Sympos. Appl. Math. 43(1991), called PS AM 43 below, and on pp. 475-491 of this volume's predecessor, Games of No Chance, hereafter referred to as GONC. This list also contains more detail about some of the games mentioned below. References in brackets, e.g., Ferguson [1974], are listed in Fraenkel's Bibliography later in this book; WW refers to

Elwyn Berlekamp, John Conway and Richard Guy, Winning Ways for your Mathematical Plays, Academic Press, 1982. A.K.Peters, 2000.

and references in parentheses, e.g., Kraitchik (1941), are at the end of this article.

1. Subtraction games are known to be periodic. Investigate the relationship between the subtraction set and the length and structure of the period. The same question can be asked about partizan subtraction games, in which each player is assigned an individual subtraction set. See Fraenkel and Kotzig [1987].

See also Subtraction Games in WW, 83-86, 487-498 and in the Impartial Games article in GONC. A move in the game S(s1, s2, s3,…) is to take a number of beans from a heap, provided that number is a member of the subtraction-set, {s1, s2, s3,…}. Analysis of such a game and of many other heap games is conveniently recorded by a nim-sequence,

meaning that the nim-value of a heap of h beans is nh, h = 0, 1, 2, …, i.e., that the value of a heap of h beans in this particular game is the nimbernh. To avoid having to print stars, we say that the nim-value of a position is n, meaning that its value is the nimber *n.

1. Introduction

Domineering or Crosscram is a game invented by Goran Andersson and introduced to the public in [1]. Two players, say Vera and Hepzibah, have vertical and horizontal dominoes respectively. They start with a board consisting of some subset of the square lattice and take turns placing dominoes until one of them can no longer move. For instance, the 2 x 2 board is a win for the first player, since whoever places a domino there makes another space for herself while blocking the other player's moves.

A beautiful theory of combinatorial games of this kind, where both players have perfect information, is expounded in [2; 3]. Much of its power comes from dividing a game into smaller subgames, where a player has to choose which subgame to make a move in. Such a combination is called a disjunctive sum. In Domineering this happens by dividing the remaining space into several components, so that each player must choose in which component to place a domino.

Each game is either a win for Vera, regardless of who goes first, or Hepzibah regardless of who goes first, or the first player regardless of who it is, or the second regardless of who it is. These correspond to a value G which is positive, negative, fuzzy, or zero, i.e.,or. (By convention Vera and Hepzibah are the left and right players, and wins for them are positive and negative respectively.) However, we will often abbreviate these values as G = V, H, 1st, or 2nd. We hope this will not confuse the reader too much.

In this paper, we find who wins Domineering on all rectangles, cylinders, and tori of width 2, 3, 5, and 7. We also obtain bounds on boards of width 4, 7, 9, and 11, and partial results on many others. We also comment briefly on toroidal and cylindrical boards.

Go thermography is more complex than thermography for classical combinatorial games because of loopy games called kos. In most situations, go rules forbid the loopiness of a ko by banning the play that repeats a position. Because of the ko ban one player may be able to force her opponent to play elsewhere while she makes more than one play in the ko, and that fact gives new slopes to the lines of ko thermographs. Each line of a thermograph is associated with at least one line of orthodox play [Berlekamp 2000, 2001]. Multiple kos require a new definition of thermograph, one based on orthodox play in an enriched environment, rather than on taxes or on composing thermographs from the thermographs of the followers [Spight 1998]. Orthodox play is optimal in such an environment.

Reading a Thermograph

Many go terms have associations with thermographs. They are not defined in terms of thermographs, but I will be using them in talking about the game, and it will be helpful to be able to visualize the thermographs when I do. The inverse of the slope of a thermographic line indicates the net number of local plays. If one player makes 2 local plays while her opponent makes only plays in the environment, the slope will be plus or minus 1/2. The color of a line indicates which player can afford to play locally at a certain temperature. The player does not necessarily wish to play at that temperature, but she can do so without loss.

In the simple thermograph in Figure 1, the black mast at the top extends upward to infinity. It indicates a region of temperature in which neither player can afford to play locally without taking a loss. The mast starts at temperature, t = 2, which we call the temperature of this game, at the top of the hill. It also indicates the local count (or mast value), which is —5. Just below the top of the hill, the blue line of the Left wall indicates that Black (or Left) will not be unhappy to make a local play in this region of temperature. And it shows what the local score at each temperature would be when Black plays first. Similarly, the red line of the Right wall indicates an initial play by White (or Right).

1. Introduction

In a broad class of games that have been studied in the literature, two players, A and *B, alternately color the edges of a graph G in red and in blue respectively. In the achievement game the objective is to create a monochromatic subgraph isomorphic to a given graph F. In the avoidance game the objective is, on the contrary, to avoid creating such a subgraph. Both the achievement and the avoidance games have strong and weak versions. In the strong version A and 23 both have the same objective. In the weak version B just plays against A, that is, tries either to prevent A from creating a copy of F in the achievement game or to force such creation in the avoidance game. The weak achievement game, known also as the Maker-Breaker game, is most studied [4; 1; 13]. Our paper is motivated by the strong avoidance game [7; 5] where monochromatic F-subgraphs of G are forbidden, and the player who first creates such a subgraph loses.

The instance of a strong avoidance game with complete graphs G = K6 and F = K3 is well known under the name SIM [15]. Since for any edge bicoloring of K6 there is a monochromatic K3 , a draw in this case is impossible. It is proven in [12] that a winning strategy in SIM is available for B. A few other results for small graphs are known [7].

Introduction to Loony Endgames and Controlled Value

The reader is assumed to be familiar with the game of Dots and Boxes. An excellent introduction can be found in [WW], [Nowakowski] or [D&B].

Some results about this game are more easily described in terms of the dual game, called Strings-and-Coins [D&B, Chapter 2]. This game is played on a graph G, whose nodes are called coins and whose branches are called strings. In this graph, the ends of each string are attached to two different coins or to a coin and the ground, which is a special uncapturable node. The valence of any coin is the number of strings attached to it. It has long been known that typical endgames reach a loony stage in which no coin has valence 1, and every coin of valence 2 is part of a chain of at least 3 such coins or of a loop of at least 4 such coins. When such a position occurs, the player who has just completed his turn is said to be in control. Let's call him Right. All moves now available to his opponent, called Left, are of the type called loony.