On Doubling in Tournament Backgammon
Norman Zadeh, Columbia University, 1977
Management Science, Vol. 23, No. 9, May 1977, pp 986–993

A strategy is determined which takes into account a player's skill, the score of the match, the value of the current game, and the possibility of winning a double-game (a gammon). The strategy described here has worked quite well in practice.

# Introduction

In tournament backgammon, the objective is to win a specified number of points V before one's opponent wins V. (V is typically anywhere from 11 to 25.) Each game is initially worth 1 point. During the course of a 1-point game, either player may double the stakes before rolling his dice. If A doubles to 2, then B may either concede 1 point to A and start a new game or agree to play the existing game for 2 points. If B elects to play on then only he may redouble. Suppose that B decides to play and eventually redoubles to 4. Then A may either give B  2 points or play on for 4. If A decides to continue then only he may redouble, etc. There is no limit to the number of doubles. However, once the value of any game exceeds V doubling becomes irrelevant.

# Gammons

It is possible for a player to win a gammon, or double-game, meaning that he wins  2K  points if the value of the game is  K.  We will not consider the possibility of a triple-game or "backgammon" because these are rare.

# The Crawford Rule

In tournament play the Crawford Rule is almost always used. This rule says that when either player reaches a score of  V − 1  the other player may not double for one game. He may double after that game (assuming he wins it). Thus, suppose A needs 3 and B needs 4. A wins a 2 game, se he needs 1 and B needs 4. During the next game B may not double. If he wins, he will need 3 unless he gammons A and may double during any remaining game. Note that it is to B's advantage to double immediately whenever possible when A needs only 1 point.

# Continuous Time Markov Approximation

Let  p(t)  denote the probability that player A will win a given game, assuming that neither player may double. If  p(t)  is a continous function of time, we call the game continuous. In backgammon,  p(t)  is not continuous since it is defined only on nonnegative integers and may jump considerably from one move to the next. We begin by assuming that  p(t)  is continuous and then make modifications to account for its actual discontinuity. The following lemma is from .

Lemma 1.  In a continuous game, let the probability of A winning be x, and let a, b > 0 be such that 0 ≤ x − a < x < x + b ≤ 1. Let E be the event that p(t) becomes x + b before it becomes x − a. Then the probability of E is a / (a + b).

Proof.  We have, by conditioning on whether p(t) first reaches x + b or x − a, x = p(E) · (x + b) + (1 − p(E))(x − a). This yields

p(E) =
 x − (x − a) (x + b) − (x − a)
=
 a a + b
.

Example 1.  Suppose that  x = 0.75.  Then the probability that A wins  (p = 1  before  p = 0)  is  0.75 / 1 = 0.75.

Example 2.  Suppose that  x = 0.6.  Then the probability that  x = 0.8  before  x = 0  is  0.6 / (0.6 + 0.2) = ¾.

Lemma 1 may be extended to any linear function of  p(t),  such as A's expectation. In other words, the probability that A's expectation rises by  b  before it falls by  a  is also  a / (a + b).

It is shown in  that in a continuous game A should double or redouble when  p = 0.8.  When A doubles, it does not matter whether B accepts or declines, his expectation in both cases is  K,  where  K  is the value of the game before the double. To see this, observe that if B accepts, he will own the doubling cube at  2K.  He will win by reaching  p = 0.2  before  p = 1.  Using Lemma 1, his probability of winning is  0.2 / (0.6 + 0.2) = ¼.  Therefore, his expectation is  ¼(2K) − ¾(2K) = −K.  This is exactly what he would lose if he did not accept the double.

Note that A's chances of winning when  p(t) = 0.8  are 0.8 assuming that neither player may double. If B owns the cube then A's chances are 0.75.

In a tournament game, the optimal doubling strategy for each player depends on the score of the match and the current value of the game. Let Ai(x1, x2) denote player i's chances of winning the match at the start of a new game when player 1 needs x1, player 2 needs x2, and x1 ≥ 2, x2 ≥ 2. When x1 or x2 = 1, it is possible that the player who is behind may not be able to double for one game because of the Crawford Rule. Let Bi(x1, x2) denote player i's chances when the underdog cannot double in the present game (but can in future ones). Let Bi(x1, x2) denote player i's chances when the underdog can double. Let y denote player 1's chances of winning a game and let gy denote his chances of winning a gammon. Then we have

 B1(1, 1) = y, B1(2, 1) = gy + (y − gy)y, B1(1, 1) = y, (1) B1(2, 1) = y  (because player 1 will double immediately), B1(3, 1) = gy + (y − gy)y  (because player 1 will double immediately), B1(4, 1) = gy + (y − gy)y  (because player 1 will double immediately),

and, in general,

 B1(x1, 1) = gyB1(x1 − 4, 1) + (1 − g)yB1(x1 − 2, 1),  and B1(x1, 1) = gyB1(x1 − 2, 1) + (1 − g)yB1(x1 − 1, 1). (2)

B1(1, x1) and B1(1, x1) are obtained from (1) and (2) by replacing y by 1 − y.

When the state of the match is (x1, x2, K) (K represents the value of the current game), there will be some point in the current game at which player 1's (2's) advantage will be such that he can double and it will not matter whether his opponent takes or not. This point is called the minimum take point for player 2 (1). Let D1(x1, x2, K) denote player 1's chances of winning at the minimum take point for player 2, assuming that neither player may double. (In nontournament play, assuming the game is continuous, Di(–, –, K) = 0.80 ∀K, and for i = 1, 2.)  Let D1(x1, x2, K) denote player 1's chances of winning at the minimum take point for player 2, given that his opponent may double. (In nontournament play  Di(–, –, K) = 0.75 ∀K, i, whether the game is continuous or not.)

If a player's chances of winning after doubling are D(x1, x2, K) let G1 · D(x1, x2, K) denote his chances of winning a gammon. G1 is a scalar which represents the fraction of wins that are gammons and will vary depending on the position. (In a straight race G1 = 0.) Let G2 · (1 − D(x1, x2, K)) denote the probability that the player accepting the double wins a gammon. In general, G1 > G2, since the player who accepts the double will often redouble instead of playing for the gammon.

When x1 > 2K and x2 > 2K, D1(x1, x2, K) is determined by the condition that player 1's probability of winning the match is the same whether player 2 takes or drops. Stating this mathematically, and letting D1 represent D1(x1, x2, K), we have

 G1D1 · A1(x1 − 4K, x2) + (1 − G1)D1 · A1(x1 − 2K, x2) + G2(1 − D1) · A1(x1, x2 − 4K) + (1 − G2)(1 − D1) · A1(x1, x2 − 2K) =  A1(x1 − K, x2). (3)

Solving (3) for D1 we obtain

 D1( G1A1(x1 − 4K, x2) + (1 − G1)A1(x1 − 2K, x2) − G2A1(x1, x2 − 4K) − (1 − G2)A1(x1, x2 − 2K) )  = A1(x1 − K, x2) − G2A1(x1, x2 − 4K) − (1 − G2)A1(x1, x2 − 2K), (4)

or  D1(x1, x2, K)  =

 A1(x1 − K, x2) − G2A1(x1, x2 − 4K) − (1 − G2)A1(x1, x2 − 2K) G1A1(x1 − 4K, x2) + (1 − G1)A1(x1 − 2K, x2) − G2A1(x1, x2 − 4K) − (1 − G2)A1(x1, x2 − 2K)
(5)

A similar expression may be determined for D2 and is  D2(x1, x2, K)  =

 A2(x1, x2 − K) − G2A2(x1 − 4K, x2) − (1 − G2)A2(x1 − 2K, x2) G1A2(x1, x2 − 4K) + (1 − G1)A2(x1, x2 − 2K) − G2A2(x1 − 4K, x2) − (1 − G2)A2(x1 − 2K, x2)
(6)

When x1 ≤ 2K and x2 > 2K, player 2 will redouble immediately after player 1 doubles and rolls since player 1 needs at most 2K points. Therefore, when x1 ≤ 2K and x2 > 2K, (5) should be adjusted by substituting 2K for all occurrences of K, except in the term A1(x1K, x2), which remains unchanged. The same adjustment should be made to (6) except for the term A2(x1, x2K).

For x1 or x2 ≤ 2K, we have  D1(x1, x2, K) = D1(x1, x2, K),   since no additional doubles will be made after the first one, except possibly an automatic redouble made by the player who is behind. (The equation  D1 = D1  is still correct for this case.)

Suppose now that  x1 > 2Kx2 > 2K.  Recall that  p(t)  denotes player 1's chances of winning at time  t  assuming there is no cube. If player 1 doubles at  p(t) = D1(x1x2K),  he will have to reach  p(t) = 1  to win before his opponent reaches  p(t) = 1 − (D2(x1x2, 2K) + Δ).  (The Δ is a correction factor to take into account the fact that the game is not continuous, and is a function of the board position.)

Therefore, using Lemma 1, and recalling that  D1(x1, x2, K)  represents player 1's chances of winning after doubling at p(t) = D1(x1x2K),  we have

D1(x1, x2, K)  =
 D1(x1, x2, K) − (1 − (D2(x1, x2, 2K) + Δ)) D2(x1, x2, 2K) + Δ
,  or

 D1(x1, x2, K)  =  (D2(x1, x2, 2K) + Δ)(D1(x1, x2, K) − 1) + 1. (7)

A similar expression may be obtained for D2(x1x2K).

If both players are equally skillful, the probability that player 1 reaches a doubling position first when the match state is  (x1, x2)  is, by Lemma 1,

 D2(x1, x2, 1) + Δ − 0.5 (D1(x1, x2, 1) + Δ − 0.5) + (D2(x1, x2, 1) + Δ − 0.5)
,
(8)

where  Δ  is again a correction factor to account for discontinuities.

The probability that player 2 reaches a doubling position first is

 D1(x1, x2, 1) + Δ − 0.5 (D1(x1, x2, 1) + Δ − 0.5) + (D2(x1, x2, 1) + Δ − 0.5)
.
(9)

When either player doubles, we will assume that his opponent drops. This assumption is accurate since the doubling points were determined by the condition that the probability of winning is the same whether one drops or takes.

Therefore, using (8) and (9), we have  A1(x1, x2) =

 (D2(x1, x2, 1) + Δ − 0.5) A1(x1 − 1, x2) + (D1(x1, x2, 1) + Δ − 0.5) A1(x1, x2 − 1) D1(x1, x2, 1) + Δ + D2(x1, x2, 1) + Δ − 1
.
(10)

A similar expression may be written for  A2(x1, x2) .

(1)–(10) and their counterparts for player 2 were combined recursively to determine  A1(x1, x2), D1(x1, x2, K), D1(x1, x2, K), D2(x1, x2, K),  and  D2(x1, x2, K)  for  K =  1, 2, 4, 8, 16, and for  0 ≤ x1 ≤ 25 ,  0 ≤ x2 ≤ 25 .

In order to do this, values for  g, G1, G2, Δ  and  Δ  had to be determined. The following table was computed by taking  g = 0.25, G1 = 0.25, G2 = 0.15, Δ = 0.08, Δ = 0.06 . These values were selected based on our playing experience plus some work on running games .1

1 In , we show that the effective doubling point in a straight race with 40 pips to go is approximately 0.90, which would mean that  Δ  in a straight race should be  ≈ 0.10.  We take  Δ = 0.08  because gammons are being considered and the interval between the doubling points is smaller than would be the case in a straight race.

TABLE  1
Results for a Match Between Two Players of Equal Skill

Top number in each box is player 1's chances of winning. Left (right) bottom number is minimum take point for player 2 (player 1). Take points have been normalized to include gammon equity (see text).

 PointsNeededbyPlayer2 15 50 81  81 14 50 81  81 46 81  82 13 50 81  81 46 81  82 42 80  82 12 50 81  81 46 81  82 42 80  82 38 80  83 11 50 81  81 46 80  82 41 80  83 37 80  83 33 79  84 10 50 81  81 45 81  82 41 80  82 37 79  83 33 79  84 29 79  84 9 50 81  81 45 80  82 40 80  83 36 79  84 32 79  84 28 78  85 25 78  86 8 50 81  81 45 81  81 40 80  82 35 79  83 31 78  84 27 78  85 24 78  85 21 77  86 7 50 81  81 45 79  83 39 80  83 34 79  84 30 78  85 26 77  86 23 77  87 20 77  87 17 76  88 6 50 80  80 44 80  82 38 78  83 33 79  84 29 78  85 25 71  85 21 77  86 18 76  87 15 76  87 13 75  88 5 50 82  82 43 79  83 37 78  85 32 77  87 27 77  87 23 77  88 20 75  89 17 75  89 14 74  90 12 75  90 10 74  91 4 50 78  78 42 80  79 36 79  80 30 77  82 25 76  85 21 76  86 17 76  86 15 75  87 12 75  89 10 74  89 8 75  90 7 74  91 3 50 78  78 43 73  83 35 77  85 29 76  87 24 73  91 19 74  93 16 73  94 13 75  94 10 72  95 8 74  96 7 71  97 5 74  97 4 71  98 2 50 69* 69* 40 71  75* 32 69  82* 25 75  79* 20 75  82* 16 73  82* 12 74  85* 10 73  83* 7 75  85* 6 72  84* 5 74  86* 4 72  85* 3 74  86* 2 72  85* 1 50 –  – 31 –  – 25 –  – 18 –  – 16 –  – 11 –  – 9 –  – 6 –  – 5 –  – 4 –  – 3 –  – 2 –  – 2 –  – 1 –  – 1 –  – 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Points Needed by Player 1 * Entry does not take gammon equity into account because of score. Figure given represents probability of winning the game.

Table 1 may be read as follows. The  x,  respectively  y,  coordinate of Table 1 gives the number of points player 1, respectively player 2, needs to win. For each pair  (x, y),  three numbers are given. The top number gives player 1's chances of winning the match  (A1(x, y)).  The lower left number gives a modification of  D1(x, y, 1)  which takes gammon equity into account. Recall that when player 1 has a probability  D1(x, y, 1)  of winning, his expectation is considered to be  2G1D1 + (1 − G1)D1 − (1 − G2)(1 − D1) − 2G2(1 − D1),  which when  G1 = 0.25  and  G2 = 0.15  equals  2.40D1 − 1.15.  The number appearing in the lower left which we call  D′1(x, y, 1)  gives the probability player 1 would need in a straight race to have an expectation equal to  2.40D1 − 1.15.  Since player 1's expectation in a straight race  = 2D′1(x, y, 1) − 1,  we have  = D′1(x, y, 1) = 1.2 D1(x, y, 1) − 0.075.  The number in the lower right is  D′2(x, y, 1).    D′1  and  D′2  are similar modifications of  D1  and  D2  which take gammon equity into account.

Because of space limitations, we were not able to give all the results obtained by the program. For example, for the pair  (4, 3)  we have

 D1(4, 3, 1) = 0.670, D1(4, 3, 2) = 0.602, D1(4, 3, 4) = 1.0, D2(4, 3, 1) = 0.756, D2(4, 3, 2) = 0.820, D2(4, 3, 4) = 1.0, D1(4, 3, 1) = 0.633, D1(4, 3, 2) = 0.602, D1(4, 3, 4) = 1.0, D2(4, 3, 1) = 0.642, D2(4, 3, 2) = 0.820, D2(4, 3, 4) = 1.0, D′1(4, 3, 2) = 0.647, D′2(4, 3, 2) = 0.909, D′1(4, 3, 1) = 0.685, D′2(4, 3, 2) = 0.695, D′1(4, 3, 2) = 0.647, D′2(4, 3, 2) = 0.909.

It is useful to observe that  D1(4, 3, 2) = A1(2, 3),  and  D2(4, 3, 2) = A2(1, 4).  Also,  D′1(4, 3, 1) = 0.73,  which is considerably less than  D′2(4, 3, 1) = 0.83,  even though  D1(4, 3, 1) ≈ D2(4, 3, 1).  This is because  D′1(4, 3, 2) = 0.647,  while  D′2(4, 3, 2) = 0.909.  In other words, player 1 can couble earlier when the cube is at 1 because player 2 must wait a long time to double back when the cube is at 2.

# Analysis When the Players Differ in Skill

The analysis for this situation is more complicated and we strive for a good approximate solution. Suppose that player 1 is the better player. Intuitively, the better player should travel from  p = 0.5  to  p = 1  with less effort, i.e., with effort proportional to  0.5 / (1 + B),  rather than  0.5,  where  B > 0.  If we assume that the better player improves his position with less effort, while the inferior player requires the same effort as before, then a reasonable expression for the probability of the good player moving from  x  to  x + b  before reaching  xa  would be

 a a + b / (1 + B)
.
(11)

Suppose instead that we assume that the better player moves more easily (with effort proportional to  1 / (1 + α) ) whereas the inferior player moves with greater effort (proportional to  1 / (1 − α) ). A natural expression for the probability of player 1 reaching  x + b  before  xa  would then be

 a / (1 − α) b / (1 + α) + a / (1 − α)
.
(12)

This expression equals

 a a + b(1 − α) / (1 + α)
,

in other words it equals (11) when  (1 + α) / (1 − α) = 1 + B.  Thus, without loss of generality we will only consider formulas similar to (11).

To take the skill factor into account, (10) now becomes  A1(x1, x2) =

 (D2(x1, x2, 1) + Δ2 − 0.5) A1(x1 − 1, x2) + (D1(x1, x2, 1) + Δ1 − 0.5) A1(x1, x2 − 1) (D2(x1, x2, 1) + Δ2 − 0.5) + (D1(x1, x2, 1) + Δ1 − 0.5) / (1 + B)
,
(13)

where  Δ2 > Δ1,  since the inferior player presumably makes less accurate doubles.

# Effect of Skill on Minimum Take Points

A skillful player should be able, in general, to double earlier because his chances in most positions are greater than they would be against an equal player. Similarly, in a continuous game the weaker player should in general wait longer before doubling.

## Determination of New Formulas for  D1(x1, x2, K)  and  D2(x1, x2, K)

We will assume that player 1 is the better player, and that his advantage is measured by  1 + B.  The  x1, x2  in  D1(x1, x2, K)  and  D2(x1, x2, K)  will be suppressed. In other words, we will let  D1(K)  denote the minimum take point for player 2 when player 1 doubles from  K  to  2K.

The formulas for  D1(K)  and  D2(K)  are the same as before and are given by (5) and (6). To obtain the new formula for  D1(K),  we note that when  p = D1(K),  player 1 must travel a distance  1 − D1(K)  to win before his opponent travels a distance  D1(K) − (1 − D2(2K) − Δ2).  Thus we have

D1(K)     =
 D1(K) − (1 − D2(2K) − Δ2) D1(K) − (1 − D2(2K) − Δ2) + (1 − D1(K)) / (1 + B)

=
 (D1(K) + D2(2K) + Δ2 − 1)(1 + B) (D1(K) + D2(2K) + Δ2 − 1)(1 + B) + 1 − D1(K)

=  1 −
 1 − D1(K) (D1(K) + D2(2K) + Δ2 − 1)(1 + B) + 1 − D1(K)
,

(14)

so

 (D1(K) − 1) ((D1(K) + D2(2K) + Δ2 − 1) (1 + B) + 1 − D1(K))  =  D1(K) − 1 ,

or

1 + (D1(K) − 1) ((D2(2K) + Δ2 − 1) (1 + B) + 1)
=  D1(K) ((1 − D1(K)) (1 + B) + D1(K) − 1 + 1) ,

hence

 1 + (D1(K) − 1) (D2(2K) + Δ2 + B(D2(2K) + Δ2 − 1))  =  D1(K) (1 + B(1 − D1(K))) ,

so

D1(K)  =
 1 + (D1(K) − 1) (D2(2K) + Δ2 + B(D2(2K) + Δ2 − 1)) 1 + B(1 − D1(K))
.
(15)

Performing a similar analysis and noting that

1 − D2(K)  =
 1 − D2(K) 1 − D2(K) + (D2(K) − (1 − D1(2K) − Δ1)) / (1 + B)
,

we obtain

D2(K)  =  1 +
 (D1(2K) + Δ1) (D2(K) − 1) BD2(K) + 1
.
(16)

To incorporate the above formulas into a computer program, we assumed that the more skillful player would be a 6 to 4 favorite in a game with no cube, and that his expectation would approximately double in a money game with a cube (his expectation would be approximately 0.4 of a point per game). These assumptions allowed us to compute  B  in the following manner.

In a money game,  D1(K) = D2(K) = 0.75.  Substituting ¾ for  D1(K)  and  D2(K)  in (15) and (16), and taking  Δ1 = 0.05, Δ2 = 0.07, Δ1 = 0.07, Δ2 = 0.09,  we were then able to combine formulas (13), (15) and (16) to determine that value of  B  which made player 1's expectation in a money game equal to 0.4. Upon running the program with different parameter values, it became clear that the guesses of  Δ1, Δ2 Δ1, Δ2  were relatively unimportant compared to the assumption that player 1's expectation would be 0.4. In other words, the numbers in Table 2 are not affected significantly by small changes in  Δ1, Δ2 Δ1, Δ2,  but are affected significantly by a change in player 1's expectation.

TABLE  2
Results for a Match in which Player 1 Has an expectation of
0.4 of a Point per Game (in Nontournament Play)

Top number in each box is player 1's chances of winning. Left (right) bottom number is minimum take point for player 2 (player 1). Take points have been normalized to include gammon equity (see text).

 PointsNeededbyPlayer2 15 100 –  – 99 88* 70 99 99  70 98 94  72 97 93  72 96 90  73 94 89  74 93 88  75 91 87  76 89 87  77 86 86  77 84 85  78 81 85  78 79 84  79 76 84  79 14 100 –  – 99 89* 73 99 98  72 98 93  73 96 92  73 95 90  74 93 89  74 91 88  75 89 87  76 86 86  77 84 85  77 81 85  78 78 84  79 75 84  79 72 83  80 13 100 –  – 99 88* 70 98 98  70 97 92  72 95 92  73 93 89  74 91 88  75 89 87  76 86 86  77 83 85  78 80 85  78 77 84  79 74 84  79 71 83  80 68 82  81 12 99 –  – 99 89* 73 97 97  72 96 92  73 94 91  73 91 89  74 89 88  75 86 87  76 83 86  77 80 85  78 77 84  78 74 84  79 70 83  80 67 82  80 64 82  81 11 99 –  – 98 87* 71 96 96  70 94 90  73 92 90  74 89 88  75 86 87  76 83 86  78 80 85  78 76 84  79 73 83  80 69 83  80 66 82  81 62 82  81 59 81  82 10 99 –  – 97 88* 73 95 95  73 92 90  73 89 89  74 86 87  76 83 87  77 79 85  78 76 84  78 72 83  79 68 83  79 64 82  80 61 82  81 57 81  81 53 81  81 9 98 –  – 96 87* 72 93 95  72 90 89  74 86 89  76 83 87  77 79 85  78 75 84  79 71 83  80 67 83  81 63 82  81 59 81  82 55 81  83 51 80  83 48 80  84 8 97 –  – 94 89* 72 91 95  72 87 89  73 83 88  75 78 87  76 74 85  77 70 83  80 66 83  79 61 82  80 57 82  81 53 81  81 49 80  82 45 80  83 42 79  84 7 96 –  – 92 85* 72 88 92  72 83 86  75 79 87  77 74 84  78 69 83  80 64 82  81 60 81  82 55 81  82 51 80  83 47 80  84 43 79  85 39 79  86 36 78  86 6 95 –  – 89 85* 73 84 88  73 79 83  75 73 85  77 68 83  78 63 82  79 58 81  81 53 80  81 48 80  82 44 79  83 40 79  84 36 78  85 33 78  85 29 77  86 5 91 –  – 85 82* 75 78 86  77 73 82  79 67 83  81 61 82  82 55 79  84 50 79  86 46 78  87 41 79  87 37 77  88 33 77  88 30 76  89 26 77  89 23 75  90 4 89 –  – 79 86* 65 71 85  69 65 82  74 58 82  76 52 82  77 46 79  79 41 79  81 37 79  82 32 78  83 29 77  84 25 77  86 22 76  86 19 76  88 17 76  88 3 84 –  – 72 79* 69 64 79  76 57 76  82 50 78  85 44 78  86 38 75  89 33 76  91 29 75  93 25 76  93 22 73  94 19 76  95 16 73  96 14 75  96 12 73  98 2 78 –  – 62 72*66* 53 73 71* 45 73 78* 38 76 76* 31 78 78* 27 75 78* 22 77 82* 19 75 80* 16 77 82* 13 74 81* 11 76 83* 9 74 82* 8 76 84* 6 74 83* 1 60 –  – 42 –  – 36 –  – 28 –  – 25 –  – 19 –  – 17 –  – 13 –  – 11 –  – 9 –  – 8 –  – 6 –  – 5 –  – 4 –  – 3 –  – 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Points Needed by Player 1 * Entry does not take gammon equity into account because of score. Figure given represents probability of winning the game.

Table 2 may be read in the same fashion as Table 1. For example, the three numbers in the box corresponding to (4, 3) indicate that when player 1 needs 4 and player 2 needs 3, player 1's chances are 57%, that player 2's minimum take point in a straight race is 76%, and that player 1's minimum take point is 82%. This does not contradict the fact that the underdog should generally double with weaker positions. The assumption is that player 2 plays poorly, so that when player 1 takes with a position that would give him an 18% chance against a good player (with no cube), his actual chances are 39.7%,2 whereas when player 2 takes with a position that would give him a 24% chance against an equal player (with no cube), his actual chances are 20.3%.3, 4

2 This number is  1 − D′2(4, 3, 1).
3 This number is  1 − D′1(4, 3, 1).
4 The author would like to thank Gary Kobliska for helpful comments.
1. Keeler, E. and Spencer, J., "Optimal Doubling in Backgammon," Operations Research, Vol. 23, No. 6 (Nov.–Dec. 1975), pp. 1063–1071.
2. Zadeh, N., "Computation of Optimal Poker Strategies," Operations Research, (July–August, 1977).
3. ———, Winning Poker Systems, Prentice-Hall, Englewood Cliffs, N.J., 1974.
4. ——— and Kobliska, G., "On Optimal Doubling in Backgammon," Management Science, Vol. 23, No. 8 (April, 1977).