Backgammon is Hard R. Teal Witter1[0000−0003−3096−3767] NYU Tandon, Brooklyn NY 11201, USA [email protected] Abstract. We study the computational complexity of the popular board game backgammon. We show that deciding whether a player can win from a given board configuration is NP-Hard, PSPACE-Hard, and EXPTIME- Hard under different settings of known and unknown opponents' strate- gies and dice rolls. Our work answers an open question posed by Erik Demaine in 2001. In particular, for the real life setting where the oppo- nent's strategy and dice rolls are unknown, we prove that determining whether a player can win is EXPTIME-Hard. Interestingly, it is not clear what complexity class strictly contains each problem we consider because backgammon games can theoretically continue indefinitely as a result of the capture rule. Keywords: Computational complexity · Backgammon 1 Introduction Backgammon is a popular board game played by two players. Each player has 15 pieces that lie on 24 points evenly spaced on a board. The pieces move in opposing directions according to the rolls of two dice. A player wins if they are the first to move all of their pieces to their home and then off the board. The quantitative study of backgammon began in the early 1970's and algo- rithms for the game progressed quickly. By 1979, a computer program had beat the World Backgammon Champion 7 to 1 [2]. This event marked the first time a computer program bested a reigning human player in a recognized intellec- tual activity. Since then, advances in backgammon programs continue especially through the use of neural networks [11,17,18]. On the theoretical side, backgammon has been studied from a probabilistic perspective as a continuous process and random walk [10,19]. However, the com- putational complexity of backgammon remains an open problem two decades arXiv:2106.16172v1 [cs.CC] 30 Jun 2021 after it was first posed [5]. One possible explanation (given in online resources) is that the generalization of backgammon is unclear. From a complexity standpoint, backgammon stands in glaring contrast to many other popular games. Researchers have established the complexity of nu- merous games including those listed in Table 1 but we are not aware of any work on the complexity of backgammon. In this paper, we study the computational complexity of backgammon. In order to discuss the complexity of the game, we propose a natural generalization of backgammon. Inevitably, though, we have to make arbitrary choices such as 2 R.T. Witter Table 1. A selection of popular games and computational complexity results. Game Complexity Class Tic-Tac-Toe PSPACE-Complete [12] Checkers EXPTIME-Complete [14] Chess EXPTIME-Complete [6] Bejeweled NP-Hard [7] Go EXPTIME-Complete [13] Hanabi NP-Hard [1] Mario Kart PSPACE-Complete [3] the number or size of dice in the generalized game. Nonetheless, we make every effort to structure our reductions so that they apply to as many generalizations as possible. There are two main technical issues that make backgammon particularly challenging to analyze. The first is the difficulty in forcing a player into a specific move. All backgammon pieces follow the same rules of movement and there are at least 15 unique combinations of dice rolls (possibly more for different generalizations) per turn. For other games, this problem has been solved by more complicated reductions and extensive reasoning about why a player has to follow specified moves [4]. In our work, we frame the backgammon problem from the perspective of a single player and use the opponent and dice rolls to force the player into predetermined moves. The second challenge is that the backgammon board is one-dimensional. Most other games with computational complexity results have at least two dimensions of play which creates more structure in the reductions [8]. We avoid using mul- tiple dimensions by carefully picking Boolean satisfiability problems to reduce from. We show that deciding whether a player can win is NP-Hard, PSPACE- Hard, and EXPTIME-Hard for different settings of known or unknown dice rolls and opponent strategies. In particular, in the setting most similar to the way backgammon is actually played where the opponent's strategy and dice rolls are unknown, we show that deciding whether a player can win is EXPTIME-Hard. Our work answers an open problem posed by Demaine in 2001 [5]. In Section 2, we introduce the relevant rules of backgammon and general- ize it from a finite board to a board of arbitrary dimension. In Section 3, we prove that deciding whether a player can win even when all future dice rolls and the opponent's strategy are known is NP-Hard. In Section 4, we prove that deciding whether a player can win when dice rolls are known and the opponent's strategy is unknown is PSPACE-Hard. Finally in Section 5, we prove that de- ciding whether a player can win when dice rolls and the opponent's strategy are unknown is EXPTIME-Hard. Backgammon is Hard 3 2 Backgammon and its Generalization We begin by describing the rules of backgammon relevant to our reductions. When played in practice, the backgammon board consists of 24 points where 12 points lie on Player 1's side and 12 points lie on Player 2's side. However, without modifying the structure of the game, we will think of the board as a line of 24 points where Player 1's home consists of the rightmost six points and Player 2's home consists of the leftmost six points. Figure 1 shows the relationship between the regular board and our equivalent model. Player 1 moves pieces right according to dice rolls while Player 2 moves pieces left. The goal is to move all of one's pieces home and then off the board. Player 2’s Home Player 1’s Home Player 2’s Home Player 1’s Home Fig. 1. Backgammon board in normal play (top); equivalent board `unfolded' (bottom). Players move their pieces by taking turns rolling dice. On their roll, a player may move one or more pieces `forward' (right for Player 1 and left for Player 2) by the numbers on the dice provided that the new points are not blocked. A point is blocked if the opponent has at least two pieces on it. The turn ends when either the player has moved their pieces or all moves are blocked. Note that a player must always use as many dice rolls as possible and if the same number appears on two dice then the roll `doubles' so a player now has four moves (rather than two) of the number. If only one of a player's pieces is on a point, the opponent may capture it by moving a piece to the point. The captured piece is moved off the board and must be rolled in from the opponent's home before any other move may be made. This sets back the piece and can prove particularly disadvantageous if all of the points in the opponent's home are blocked. 4 R.T. Witter The obvious way to generalize the backgammon board used in practice is to concatenate multiple boards together, keeping the top right as Player 1's home and the bottom right as Player 2's home without loss of generality. In the line interpretation we use, we can equivalently view this procedure as adding more points between the respective homes. The rules we described above naturally extend. But there is ambiguity in the generalization regarding the dice: How many dice are in a generalized board? How large are these dice? What happens to the doubling rule? Without loss of generality, we consider only two six-sided dice in our reductions. The effect of additional dice or additional numbers can be nullified by rolling `blocked' numbers and restricting the rolls of larger dice to six or less. We retain the doubling rule for the two dice we use in the reduction. 3 NP-Hardness In this section, we show that determining whether a player can win against a known opponent's strategy and known dice rolls (KSKR) is NP-Hard. We begin with formal definitions of Backgammon KSKR and the NP-Complete problem we reduce from. Definition 1 (Backgammon KSKR). The input is a configuration on a gen- eralized backgammon board, a complete description of the opponent's strategy, and all future dice rolls both for the player and opponent. The problem is to de- termine whether a player can win the backgammon game from the backgammon board against the opponent's strategy and with the specified dice rolls. We do not require that the configuration is easily reachable from the start state. However, one can imagine that given sufficient time and collaboration between two players, any configuration is reachable using the capture rule. An opponent's strategy is known if the player knows the moves the opponent will make from all possible positions in the resulting game. Notice that such a description can be very large. However, in our reduction, we limit the number of possible positions by forcing the player to make specific moves and predeter- mining the dice rolls. Therefore the reduction stays polynomial in the size of the 3SAT instance. Definition 2 (3SAT). The input is a Boolean expression in Conjunctive Nor- mal Form (CNF) where each clause has at most three variables. The problem is to determine whether a satisfying assignment to the CNF exists. Given any 3SAT instance, we construct a backgammon board configura- tion, an opponent strategy, and dice rolls so that the solution to Backgammon KSKR yields the solution to 3SAT.