Match Equity 
Evaluating Match Equity

From Backgammon Times, Volume 2, Number 1, Winter 1982. 
When a match is coming to a finish and the cube may put one or both players over the top, a cube decision is by far the most important play a player has to make during the match. Even if there is only a 2% swing on the right decision, this is more significant than the swing on any play. To illustrate, let's suppose that in the first game of a 15 point match you make a play which decreases your chances of winning the game by 10%. Such a play would be an incredible blunder; the average mistake usually costs only 2 or 3 percent in equity for the game. The swing between being ahead 1–0 and being behind 1–0 is approximately 10%. Consequently, even such a blunder in an early game costs only about 1% in match equity.
In order to assess cube decisions late in the match, it is important to know what your match equity is at the potential scores. The following table is the consensus of independent analyses by Bill Robertie, Danny Kleinman, and myself. It may not be 100% accurate, but the fact that we all agreed to within 1% on most of the figures indicates that none of the results is likely to be very far off.
1  2  3  4  5  6  7  8  
1  50  70  75  83  85  90  91  94 
2  30  50  59  70  76  82  86  89 
3  25  41  50  60  67  74  79  84 
4  17  30  40  50  58  66  72  78 
5  15  24  33  42  50  58  64  70 
6  10  18  26  34  42  50  58  64 
7  9  14  21  28  36  42  50  57 
8  6  11  16  22  30  36  43  50 
The numbers at the top and left side of the table indicate the number of points to go for each player, and the numbers in the center of the table indicate the equity (in percent) for the corresponding numbers of points to go. For example, if you are ahead 12–10 in a 15 point match you have 3 points to go and your opponent has 5, so your probability of winning according to the table is 67%; his is 33%.
The procedure for evaluating a cube decision is as follows: for each possible cube action (take or pass, double or don't double), determine the probability of the likely results as best as possible based on your assessment of the position. Then multiply these probabilities by the match equity figures from the table, add them up, and the decision with the higher total is the winner. Let's look at a simple example. Suppose you are ahead 12–9 in a 15 point match, and find yourself redoubled to 4. There is no gammon danger, and your assessment of the position is that you will win the game 35% of the time. Should you take? If you drop it will be 12–11, which from the table makes your equity 60%. Suppose you take. 35% of the time you win outright; the other 65% you are down 13–12, with 41% equity. Consequently, your equity in the take is 1 × .35 + .41 × .65 = .616. Therefore, the take is correct.

Position 1. 19 point match Black 16, White 13 Black owns the cube at 2 

Position 2. 17 point match Black 11, White 14 Black owns the cube at 2 
It might seem difficult to make these calculations in your head at the table, but it must be done. The match equity table can be learned easily, and even a rough approximation in the calcultions will often prevent gross blunders. The importance of doing this is shown in the following example (Position 3) which came up in the finals of the jackpot tournament in Monte Carlo.

Position 3. 13 point match Black 1, White 6 Black owns the cube at 4 
More articles by Kit Woolsey  
More articles on match equity  
Return to: Backgammon Galore 