Match Equity

Evaluating Match Equity Kit Woolsey, 1982

 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.

 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
Position 1.
19 point match
Black 16, White 13
Black owns the cube at 2
In Position 1, Black is on roll, ahead 16–13 in a 19-point match, and owns a 2-cube. Should he redouble? If he does, should White take? Let's examine White's problem first. If he passes he is behind 18–13, with equity of 10%. If he takes he will, of course, send it back for the match. For White to win Black has to roll a non-double and White needs double 2 or better, with probability 5/6 × 5/36 = .116. Therefore, White has a take. Now should Black double? As we have just concluded, if he doubles, the match will be on the line and his equity will be .884. If he does not double he has an 18–13 lead with probability .884, and a 15–15 lead with probability .116. So using the figures from the table his match equity would be .884 × .90 + .116 × .60 = .865. Therefore, it is correct for Black to double.

 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
Position 2.
17 point match
Black 11, White 14
Black owns the cube at 2
In Position 2, Black is on roll trailing 14–11 in a 17-point match and he owns a 2-cube. Should he redouble on the come? Most good players would automatically redouble, figuring that things would be pretty grim if they missed, so they might as well gamble. Let's run it through the analysis. First, we must make some assessments about the position. I estimate that Black is about even money to get off the gammon if he doesn't hit, and that if he hits and has to play the game to a conclusion he will win 80% of the time (if you disagree substantially with these estimates, your final conclusions may also be different). Suppose Black doubles. He must hit, probability 11/36, then he must win, probability .8, and this gives him a 15–14 lead for an equtiy of .59. Consequently, equity in doubling is 11/36 × .8 × .59 = .144. Suppose Black holds the cube. Now a hit (11/36) is a win (since White must then drop the redouble) for a 13–14 deficit and an equity of .40, and a miss (25/36) leaves him a 50-50 shot at trailling 11–16 with an equity of .10. So Black's total equity in holding the cube is 11/36 × .40 + 25/36 × 1/2 × .10 = .157. Therefore it is correct for Black to hold the cube, and those experts who doubled would be sacrificing an additional 1.3% of their already slim chances.

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.
 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
Position 3.
13 point match
Black 1, White 6
Black owns the cube at 4
A dull race, each side having 50 pips to go. What made it interesting was the I was Black, on roll, behind 6–1 in a 13-point match, and I owned a 4-cube. Needless to say, there was a lot of money at stake. Should I make it 8? If the game is played to completion I will win about 60% of the time. If I hold the cube the figure goes up to about 70% due to the cube leverage. Being behind 10–1 my equity would be about 7%. If I double, 40% of the time I lose, 60% of the time I have a 9–6 lead for equity of .72, so my equity in doubling is .72 × .60 = .432. If I hold the cube, 30% of the time I am behind 10–1 with equity of .07, 70% of the time I am behind 6–5 with equity .43, so my equity in holding the cube is .07 × .30 + .43 × .70 = .322. So even if some of the estimates are off by quite a bit doubling is still clear-cut. In practice I made the mistake of waiting a roll (and most good players I talked to afterward agreed with this action!), I outrolled my opponent on the next roll and doubled him out, but then went on to lose the match. There is no way to know what would have happened, of course, but I sure would like to be able to go back and send that 8-cube over!

 More articles by Kit Woolsey More articles on match equity Return to:  Backgammon Galore