Monte-Carlo Go has recently improved to compete with the best Go programs [Coulom 2007; Gelly et al. 2006; Gelly and Silver 2007]. We are interested in the use of Monte-Carlo methods when there are independent games. In such cases it might be interesting to analyze the games independently instead of considering them as a unified game.

Section 2 describes related works Section 3 presents the Monte-Carlo algorithms we have tested. Section 4 details experimental results. Section 5 concludes.

In this section we expose related works on Monte-Carlo Go. We first explain basic Monte-Carlo Go as implemented in Gobble in 1993. Then we address the combination of search and Monte-Carlo Go, followed by the UCT algorithm, and previous works on the approximation of temperature.

The first Monte-Carlo Go program is Gobble [Brueg-mann 1993]. It uses simulated annealing on a list of moves. The list is sorted by the mean score of the games where the move has been played. Moves in the list are switched with their neighbor with a probability dependent on the temperature. The moves are tried in the games in the order of the list. At the end, the temperature is set to zero for a small number of games. After all games have been played, the value of a move is the average score of the games it has been played in first. Gobble-like programs have a good global sense but lack of tactical knowledge. For example, they often play useless Ataris, or try to save captured strings.

In this instalment of the Games Bibliography, Richard Nowakowski joined as a coauthor. Unlike previous instalments, this one is restricted, mainly, to new entries of the last few years that did not appear in previous versions, such as in “Games of No Chance 3” and the Electronic J. of Combinatorics (Surveys). We apologize profusely that due to lack of time, this version does not include all the new game papers we are aware of, and also lacks some unifying editing and polishing.

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.

Before reading this article one can consider the following Amazons position:

In Amazons there are a lot of hot positions: the players want to play to gain some territory. However, the position shown is not of this kind. If we use [Siegel 2011] to analyze it, this position proves to have value *3 + *2 = *. We will attempt to analyze the options of such a position and construct nimbers in partizan games. For instance, *3 = {0, *, *2 | 0, *, *2}, however, when we study the players options with [Siegel 2011] we see that the options are not {0,*,*2}. In a partizan game we have a bigger number of possible options to construct a nimber than in an impartial game. For instance, we know that a game like { ↑ | 0} has value * too. When we think about higher stars the number of possibilities is just gigantic. So it’s important to make some mathematical considerations to classify the games that “can act as nimbers”. In this article, we prove some useful results about the construction of nimbers and show some interesting examples in Amazons.

Combinatorial game theory has been applied to many kinds of existing games and has produced many excellent results. In the case of the game of Go, applications of CGT have been focused on endgames [Berlekamp and Wolfe1994; Berlekamp 1996; Müller et al. 1996; Nakamura and Berlekamp 2003; Spight 2003] and eyespace values [Landman 1996] so far. But it can be applied to any situations that involve counting. Recently, we developed a new genre of application of CGT to Go, that is, to count liberties in capturing races [Nakamura 2003; Nakamura 2009; Nakamura 2006].

Capturing races, or semeai is a particular kind of life and death problem in which two adjacent opposing groups are each fighting to capture the opponent’s group. A player’s strength in Go depends on their skills in winning capturing races as well as opening and endgame skills. In order to win a complicated capturing race, various techniques in counting liberties, taking away the opponent’s liberties, and extending self-liberties, are required in addition to broad and deep reading. Human expert players usually count liberties for each part of the blocks involved in semeai, sum them, and decide the outcome. A position of capturing races can also be decomposed into independent subpositions, as in the cases of endgames and eyespaces, and we can apply CGT to analyse the capturing races. We propose a method of analysing capturing races that have no shared liberty or have only simple shared liberties, and then, using combinatorial game values of external liberties, give an evaluation formula to find the outcome of the capturing races.

MAZE was introduced in [Albert et al. 2007], but apart from a few scattered observations, nothing was known about the values of the game. In the original article, MAZE is played on a rectangular grid oriented 45◦ to the horizontal.

The token starts at the top of the board and highlighted edges are walls that may not be crossed. Left is allowed to move a token any number of cells in a southwesterly direction and Right is allowed to move similarly in a southeasterly direction. However, for ease of referring to specific places in the position, we re-orient the sides parallel to the page so that Left moves downward and Right moves to the right; see Figure 1. One interesting feature is that any number of consecutive Left (Right) moves also can be accomplished in one move. This feature had been noted in several games, including HACKENBUSH strings [Berlekamp et al. 2001], and given the name of option-closed in [Nowakowski and Ottaway 2011], a reference we henceforth abbreviate as [NO]. Siegel [2011] notes that the partial order of option-closed games born on day n forms a planar lattice.

In two-player combinatorial games, the last player to move either wins (normal play) or loses (misère play). Traditionally, normal play games have garnered more attention due to the group structure which arises on such games. Less work has been done with games played under the misère play convention, Just as in normal play, misère games can be placed in equivalence classes, where two games G and H are equivalent if the outcome class of G + K is the same as the outcome class of H+K for all games K. However, Conway showed that, unlike in normal play, these misère equivalence classes are sparsely populated, making the analysis of misère games under such equivalence classes far less useful than their normal play counterparts [ONAG]. Even though these equivalence classes are sparse, Conway developed a method, called genus theory, for analyzing impartial games played under the misère play convention [Allen 2006; WW; ONAG]. For years, this was the only universal tool available for those studying misère games.

In [Plambeck 2009; 2005; Plambeck and Siegel 2008; Siegel 2006; 2015b], many results regarding impartial misère games have been achieved. These results were obtained by taking a game, restricting the universe in which that game was played, and obtaining its misère quotient. However, while, as Siegel [2015a] says “a partizan generalization exists”, few results have been obtained regardingthe structure of the misère quotients which arise from partizan games.

For a game G ={GL|GR}, we define Ḡ = {GR|GL}. Those familiar with normal play will notice that under the normal play convention rather than Ḡ, we would generally write̶G. In normal play, this nomenclature is quite sensible as G+(-G)=0 [Albert et al. 2007], giving us the Tweedledum–Tweedledee principle; the second player can always win the game G+(-G) by mimicking the move of the first player, but in the other component. However, in misère play, not only does the Tweedledum–Tweedledee strategy often fail, G + Ḡ is not necessarily equivalent to 0. For example, *2+*2=*2+*2 is not equivalent to 0 [Allen 2006; WW]. However, having the property that G+Ḡ is equivalent to 0 is much desired, as it gives a link to which partizan misère games may behave like their normal counterparts.

The Rat game is played on 3 piles of tokens by 2 players who play alternately. Positions in the game are denoted throughout in the form (x, y, z), with 0 ≤ x ≤ y ≤ z, and moves in the form (x, y, z)→ (u, v, w), where of course also 0 ≤ u ≤ v ≤ w (see below). The player first unable to move—because the position is (0, 0, 0)—loses; the opponent wins.

Disjunctive compounds of short combinatorial games have been studied for many years under a variety of assumptions. A structure theory for normal-play impartial games was established in the 1930s by the Sprague–Grundy theorem [Grundy 1939; Sprague 1935; 1937]. Every such game G is equivalent to a Nim-heap, and the size of this heap, known as the nim value of G, completely describes the behavior of G in disjunctive sums. The Sprague–Grundy theorem underpins virtually all subsequent work on impartial combinatorial games.

Decades later, Conway generalized the Sprague–Grundy theorem in two directions [Berlekamp et al. 2003; Conway 2001]. First, he showed that every partizan game G can be assigned a value that exactly captures its disjunctive behavior, and this value is represented by a unique simplest form for G. Conway’s game values are partizan analogues of nim values, and his simplest form theorem directly generalizes the Sprague–Grundy theorem.

Conway also introduced a misère-play analogue of the Sprague–Grundy theorem. He showed that every impartial game G is represented by a unique misère simplest form [Conway 2001]. Unfortunately, in misère play such simplifications tend to be weak, and as a result the canonical theory of misère games is less useful in practice than its normal-play counterparts.

In each case—normal-play impartial, normal-play partizan, and misère-play impartial—the identification of simplest forms proved to be a key result, at once establishing a structure theory and opening the door to further investigations. In this paper, we prove an analogous simplest form theorem for the misère-play partizan case. The proof integrates techniques drawn from each of Conway’s advances, together with a crucial lemma from [Mesdal and Ottaway 2007].

Hex is well-known for the simplicity of its rules and the complexity of its play. Nash’s strategy-stealing argument shows that a winning strategy for the first player exists, but finding such a strategy is intractable by current methods on large boards. It is not known, for instance, whether the center hex is a winning first move on an odd size board. The development of artificial intelligence for Hex is a notoriously rich and challenging problem, and has been an active area of research for over thirty years [Davis 1975/76; Nishizawa 1976; Anshelevich 2000; 2002a; 2002b; Cazenave 2001; Rasmussen and Maire 2004], yet the best programs play only at the level of an intermediate human [Melis and Hayward 2003]. Complete analysis of Hex is essentially intractable; the problem of determining which player has a winning strategy from a given board position is PSPACE-complete [Reisch 1981], and the problem of determining whether a given empty hex is dead, or irrelevant to the outcome of the game, is NP-complete [Björnsson et al. 2007]. Some recent research has focused on explicit solutions for small boards [Hayward et al. 2004; Hayward et al. 2005], but it is unclear whether such techniques will eventually extend to the standard 11x11 board.

Hex was invented independently by Piet Hein [1942] and John Nash [1952]. The game is played by two players, Black and White, on a board with hexagonal cells. The players alternate turns, colouring any single uncoloured cell with their colour. The winner is the player who creates a path of her colour connecting her two opposing board sides. See Figure 1.

Hein and Nash observed that Hex cannot end in a draw [Hein 1942; Nash 1952]: exactly one player has a winning path if all cells are coloured [Beck et al. 1969]. Also, an extra coloured cell is never disadvantageous for the player with that colour [Nash 1952]. For n x n boards, Nash showed the existence of a first-player winning strategy [1952]; however, his proof reveals nothing about the nature of such a strategy. For 8x8 and smaller boards, computer search can find all winning first moves [Hayward et al. 2004; Henderson et al. 2009]. For the 9x9 board, Yang found by human search that moving to the centre cell is a winning first move.

This paper is about the complexity of combinatorial games. Its main contribution is showing constructively that a large class of games whose complexity was hitherto unknown and its best winning strategy was exponential, is actually solvable in polynomial time.

Sprouts is a two-player pencil-and-paper game invented in 1967 in the University of Cambridge by John Conway and Michael Paterson [Gardner 1967]. The game starts with p spots and players alternately connect the spots by drawing curves between them, adding a new spot on each curve drawn. A new curve cannot cross or touch any existing one, leading necessarily to a planar graph. The first player who cannot play loses.