Backgammon opening theory
Encyclopedia
The first moves of a backgammon
game are the opening moves, collectively referred to as the opening, and studied in the backgammon opening theory. Backgammon opening theory is not developed in as much detail as opening theory in chess, which has been widely studied. The reason for this is that following the first move in backgammon, there are 21 dice roll outcomes on each subsequent move, and many alternative plays for each outcome, making the tree of possible positions
in backgammon expand much more rapidly than in chess.
Despite the complications posed by this rapid branching of possibilities, over the course of many years, a consensus did develop amongst backgammon experts on what is the preferred opening move for each given roll. Following the emergence of self-trained
backgammon-playing neural networks
, the insights on what are the best opening moves have changed in some unexpected ways.
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There are no opening moves consisting of doubles, because at the start of the game, each player rolls one die. Whoever rolls higher moves first, using the numbers on the already-rolled dice. In the case of a tie, the players roll again.
In cases where no preferred play but only two or more alternative plays are given, these appear to be of equivalent strength within the statistical uncertainties of the simulations and no play could be singled out that is clearly superior.
The moves are captured in standard backgammon notation
. For instance, 8/5 denotes the move of a piece from the 8-point to the 5-point.
The general message that emerges from the above table can be summarised as follows: unless one can make a point, and with the exception of the running move 24/13 (in which, having thrown a 6-5, the player moves a piece from the 24-point to the 13-point; the lover's leap), a successful play is often the combination of splitting the 24-point and moving a checker from the 13-point. The latter move should be as small as possible (resulting in a builder close to the 13-point), unless the stack at the 8-point can be reached (resulting in an equal distribution between the 13-point and the 8-point). However, there are plenty of exceptions to this rule; for example, a 4-3 or 3-2 is often best played by moving two checkers from the 13 point.
The above opening moves which emerged from computer analysis demonstrate that a number of opening moves that were unquestioned for many decades are now considered suboptimal. One example is the move 13/11-13/8 on the roll 5-2. Although not a bad move, the alternative choice preferred by the analyses, 24/22-13/8, is now generally agreed upon to be optimal. In other cases, computer analysis has resulted in alternative strategies that were not seriously considered in the past. For instance, the opening move 8/2-6/2 for a roll of 6-4 was in the past greeted with disdain from experts (making the 2-point instead of trying for a higher home point), but turns out to be on average as effective as the usual plays (24/14 and 24/18-13/9).
In practice, an even more important influencing factor is the preferred style of the player. A player might have a strong preference for one out of a number of alternative opening plays that are on average as effective, because the character of the move (passive or aggressive) better suits his or her playing style.
Backgammon
Backgammon is one of the oldest board games for two players. The playing pieces are moved according to the roll of dice, and players win by removing all of their pieces from the board. There are many variants of backgammon, most of which share common traits...
game are the opening moves, collectively referred to as the opening, and studied in the backgammon opening theory. Backgammon opening theory is not developed in as much detail as opening theory in chess, which has been widely studied. The reason for this is that following the first move in backgammon, there are 21 dice roll outcomes on each subsequent move, and many alternative plays for each outcome, making the tree of possible positions
Game tree
In game theory, a game tree is a directed graph whose nodes are positions in a game and whose edges are moves. The complete game tree for a game is the game tree starting at the initial position and containing all possible moves from each position; the complete tree is the same tree as that...
in backgammon expand much more rapidly than in chess.
Despite the complications posed by this rapid branching of possibilities, over the course of many years, a consensus did develop amongst backgammon experts on what is the preferred opening move for each given roll. Following the emergence of self-trained
Reinforcement learning
Inspired by behaviorist psychology, reinforcement learning is an area of machine learning in computer science, concerned with how an agent ought to take actions in an environment so as to maximize some notion of cumulative reward...
backgammon-playing neural networks
Artificial neural network
An artificial neural network , usually called neural network , is a mathematical model or computational model that is inspired by the structure and/or functional aspects of biological neural networks. A neural network consists of an interconnected group of artificial neurons, and it processes...
, the insights on what are the best opening moves have changed in some unexpected ways.
Preferred opening moves
The table below summarises the most commonly preferred moves, for each of the 15 possible opening rolls, as selected by detailed computer simulations, referred to as "rolloutsRollout (backgammon)
A rollout is an analysis technique for backgammon positions and moves. A rollout consists in playing the same position many times and recording the results. The balance of wins and losses is used to evaluate the equity of the position. This used to be done by hand, but it is now done mostly by...
".
There are no opening moves consisting of doubles, because at the start of the game, each player rolls one die. Whoever rolls higher moves first, using the numbers on the already-rolled dice. In the case of a tie, the players roll again.
In cases where no preferred play but only two or more alternative plays are given, these appear to be of equivalent strength within the statistical uncertainties of the simulations and no play could be singled out that is clearly superior.
The moves are captured in standard backgammon notation
Backgammon notation
Backgammon notation is a means for recording backgammon games, developed by Paul Magriel in the 1970s. The common way of describing the movement of checkers involves numbering the points around the board from 24 to 1 as depicted in Figure 1....
. For instance, 8/5 denotes the move of a piece from the 8-point to the 5-point.
| Roll | Preferred play | Common alternatives | |
|---|---|---|---|
| 2-1 | 13/11, 6/5 | 24/23, 13/11 | — |
| 3-1 | 8/5, 6/5 | — | — |
| 4-1 | 24/23, 13/9 | 13/9, 6/5 | — |
| 5-1 | 24/23, 13/8 | 13/8, 6/5 | — |
| 6-1 | 13/7, 8/7 | — | — |
| 3-2 | 24/21, 13/11 | 13/11, 13/10 | — |
| 4-2 | 8/4, 6/4 | — | — |
| 5-2 | 24/22, 13/8 | 13/11, 13/8 | — |
| 6-2 | 24/18, 13/11 | 13/5 | — |
| 4-3 | 13/10, 13/9 | 24/21, 13/9 | 24/20, 13/10 |
| 5-3 | 8/3, 6/3 | — | — |
| 6-3 | 24/18, 13/10 | 24/15 | — |
| 5-4 | 24/20, 13/8 | 13/9, 13/8 | 24/15 |
| 6-4 | 24/18, 13/9 | 8/2, 6/2 | 24/14 |
| 6-5 | 24/13 | — | — |
The general message that emerges from the above table can be summarised as follows: unless one can make a point, and with the exception of the running move 24/13 (in which, having thrown a 6-5, the player moves a piece from the 24-point to the 13-point; the lover's leap), a successful play is often the combination of splitting the 24-point and moving a checker from the 13-point. The latter move should be as small as possible (resulting in a builder close to the 13-point), unless the stack at the 8-point can be reached (resulting in an equal distribution between the 13-point and the 8-point). However, there are plenty of exceptions to this rule; for example, a 4-3 or 3-2 is often best played by moving two checkers from the 13 point.
The above opening moves which emerged from computer analysis demonstrate that a number of opening moves that were unquestioned for many decades are now considered suboptimal. One example is the move 13/11-13/8 on the roll 5-2. Although not a bad move, the alternative choice preferred by the analyses, 24/22-13/8, is now generally agreed upon to be optimal. In other cases, computer analysis has resulted in alternative strategies that were not seriously considered in the past. For instance, the opening move 8/2-6/2 for a roll of 6-4 was in the past greeted with disdain from experts (making the 2-point instead of trying for a higher home point), but turns out to be on average as effective as the usual plays (24/14 and 24/18-13/9).
Influencing factors
The consensus on opening moves is generally extended only to money play, meaning that these plays optimise the expected payout. However, a context different from money play can influence the choice of opening move and may very well tip the balance towards a play that is seen to be slightly inferior. Towards the end of a match play, one such important factor can be the state of the match.In practice, an even more important influencing factor is the preferred style of the player. A player might have a strong preference for one out of a number of alternative opening plays that are on average as effective, because the character of the move (passive or aggressive) better suits his or her playing style.