Undoubtedly some backgammon players with exceptional memories will eventually learn a complete chart of take points for tournament doubles by heart. But most of us have only limited memories. Can we find some other method of determining how good our chances need be to justify taking the cube at a particular score?
Yes, if we have memorized or can estimate fairly closely our match equities at various scores. By a match equity, I mean simply the probability of winning the match with that score. We can use match equities to calculate take points. We treat the match equity obtainable from passing as our benchmark, and consider the match equity from taking and losing as our risk which we measure against our gain from taking and winning.
Suppose we lead 9–7 in an 11-point match when our opponent offers an initial double. By passing, we keep an 9–8 lead. This gives us roughly 60% chances to win the match, so we use 60% as our basis of comparison.
By taking and losing, we tie the match a 9–9, giving us 50% chances. But by taking and winning , we win the match outright, 11–7; i.e., obtain 100% chances.
We see immediately that we risk 10% match equity to gain 40%. We are receiving 4-to-1 odds by taking, better than the 3-to-1 odds we get in money backgammon. This allos us to take a little bit loosely, giving us a 20% take point.
Taking another example, suppose we lead 5–3 in a 9-point match and have already doubled our opponent. Now he sends the cube back to us at 4. Shall we take, or shall we surrender 2 points?
By passing, we tie the match at 5–5 and obtain 50% chances. By taking and losing, we trail 7–5 and become 2-to-1 underdogs, with a chance to win the match. By taking and winning, we win the match outright — 100% winning chances. Thus we risk 17% to gain 50%, almost 3-to-1 odds. We should be only slightly more conservative than in money backgammon, with a take point just above 25%.
The same procedure works in helping us determine gammon risks that are worth taking. Using our same two examples, suppose we have already taken the cube at 2 leading 9–7 in an 11-point match. We take as our benchmark the match equity obtainable from losing a plain game. This would tie the score at 9–9 and give use 50% chances in the match.
Losing a gammonn would lose the match, while winning the game would win the match. Our risk of 50% equity from getting gammoned equals our gain of 50% equity from winning the game. Thus we see we must take our opponent’s gammon threats very seriously indeed. It is just as important to save the gammon as it is to try to win the game; unlike money backgammon, where winning the game is exactly twice as important as saving the gammon.
With the cube at 4 with a 5–3 leading in a 9-point match, our equity from losing a plain game is our 33% chance to win the match from a 7–5 deficit. Getting gammoned loses the match, sacrificing all of this 33% equity. But winning the game wins the match, picking up 67% equity. In this situation. slightly more than in money backgammon perhaps, we must try to win the game, even at a cost of a possible gammon.
How can we estimate match equities themselves? Prior to either side reaching match point, we can use the number of points each player needs as rough odds. After the Crawford game, we can take 1⁄2 to the power of the number of games the trailer has to win as an estimate of his equity. We can use this also going into the Crawford game, if we add an extra .05 when winning a gammon in the Crawford game reduces the number of games the trailer must win.