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Improvers Corner Number Two

Posted in Backgammon articles
November 13, 2006, 05:41:00 PM
Written by dorbel
Views: 31133
Rating: (Rates: 8)

Improvers Corner Number Two

 

Last week we looked at a mid-game positional double that was a marginal take/pass for money but a clear pass at 3-away, 3-away. It’s a position worth filing away as a reference position, but Sarah asked, “Why is the match decision different to the money decision?”
Just about everybody knows that to take a money cube (ignoring gammons for simplicity) you need a minimum cubeless winning chance of 25%. Owning the cube one can usually take with a little less than that, but 25% cubeless is the reference point. How is that derived? We need to be able to do a risk/gain analysis, a technique that is useful in many gambling situations. It works like this. When the cube is offered, we can pass, paying one point. That is the base line. We can also take and then of course we risk paying an extra point, because a loss will cost us two points (again ignoring gammons for simplicity) rather than one. To counter balance that risk, there is of course the chance that we can win two points and if we do that, we will be three points better off than if we had passed. Instead of losing one we win two. All clear so far? Thus we have risked one point to try and gain three points, so if we can win the game one time in four, we break even and if we do even slightly better then we will gain. There is of course a formula for this and to determine what winning chances we need we divide the Risk by the Risk plus the Gain. R/R+G in money backgammon is 1/1+3 = ¼ = 25%. In every money game that will remain the same because the parameters are constant.
In match play the parameters are never constant, except in the early stages of long matches. In the 3 point match the cube decisions are different to money at every score! Why is this? Because not every point in a match is equally valuable. Sounds crazy doesn’t it, but the reason is this. At every score, we can state with some accuracy the chances of either player winning the match. At 0-0 of course both have a 50% chance. At 1-0 it is 60/40. At 2-0 Crawford it is 75/25 and at 2-1 Crawford it is 70/30. Some Match Equity tables give very slightly different figures, but for practical purposes these nice round numbers will do. You can see that winning the first point gains you 10%, but winning the second point gains you an extra15%. The second point is more valuable than the first because it gets you to the Crawford game, a very strong position to be in.
Now let’s apply our risk/gain analysis to the question of taking a cube in the first game of the match. If we pass to trail 0-1, our match equity (ME) drops from 50% to 40%. We lose 10% and that is the base line for our calculation. If we take and lose to trail 0-2, our ME will drop to 25%, so we are risking an extra15% of our ME, i.e. 40-25. If we take and win to lead 2-0, then our ME will be 75% and we will have gained 35%, i.e 75-40. Risk = 15%, Gain = 35%.
Risk/Risk + Gain = 15/15+35 = 15/50 = 30%. We need 30% cubeless winning chances to take the first cube of a three point match, rather than the “normal” 25%. Because we have rounded off all the figures to make the calculation easy, the actual figure is more probably 31%. From this we can see that you need to pass doubles that would normally be takes in the 25-30% region and it is obvious that you need to double earlier as well, because the chances of losing your market (going past the point at which your opponent has a correct take) are greater.
I have a feeling that this article is going to raise more follow up questions, but that’s ok, that way we will learn something. Bring them on! Until next week, Enjoy the Game!

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