2-away, 2-away Cube Decisions

One way to strive to understand something is try to explain it to someone else who may not know the concept—or better still, does know it well, and can tell you if you are right. With that in mind, I will try to explain the doubling situation at the 2-away, 2-away match score (-2, -2). My goal as always is to learn the concept myself by trying to put it into words, so questions and comments will be much appreciated.

Terminology

(-2, -2) = e.g. 5 to 5 in a match of 7, or 9 to 9 in a match of 11. Each needs 2 points to win.

Take point = the winning probability you must have in order to accept a cube. In a single game, not part of a match, it is 25%. In a match it could be anywhere from 15% to 40% depending on the score. [See end note]

Market loser = any one roll of your dice that decreases your opponent's winning probability below his take point

Cubing decisions between equal players (A, B) in the  -2, -2 situation must begin by asking: What happens if A offers the cube to B, and B drops?

When you start the game at -2, -2, each has a 50% chance of winning the match. But if B drops, he loses one point and the score at the start of the next game is -1, -2.  In this case (with two equal players, meaning each has a 50% chance of winning a specific game), A's chance of winning the match has dropped to 30%, and B's has increased to 70%. [This statistical detail is explained in an end note; but we know without any math that one has to go up and the other down.]

Thus the take point in -2, -2 has to be 30%. If you conclude that you have a 30% or bigger chance of winning when you are offered the cube you should take it, but if A missed his market and you have less than 30% chance to win,  you should drop it.

Thus if you have the slightest lead with any chance at all that you will lose your market on the next roll, you should double... ie suppose A has a 60% chance of winning, not normally enough to double, but A sees a couple of market loser rolls, then A should double and B should take... ie at present B has a 40% chance of winning, but if B drops he has then only 30% chance starting with the next game, so it is a logical take to move forward at the 40%.

Suppose A does not double when at 60% and then does roll a market loser and is now at 75% chance of winning.
B will then drop a double since this is below 30%, and A will start the next game at 70% chance of winning, when
A could have had 75 or maybe even 80% with good market losers had he doubled at the right time.

The main message is that knowing this analysis, you should either give a double with the slightest lead, or accept one when just marginally behind, both of which will generally take place early in the game. All of this is predicat- ed on the assumption that both players can analyze their probabilities of winning with some level of accuracy. The usual guidelines of race, board, and threats and some guess at market losers should do the job most of the time.

That, I believe, is the main argument, but no one ever gets away that easily in the explanation. Suppose you are the 60% ahead, but then do not double, and on the next roll you do badly. Not only did you not lose your market (ending up 75 or 80% favorite) but now you are behind and do not even have a double, and your opponent is 60% ahead. So are you happy that you did not double? According to the expert thinking on this topic, the right an- swer is No, because now your opponent will double you and you should take. The situation is the same. It is just you who are now playing with 40% chance of winning instead of 30. The key to the analysis is that both players know the game well enough that they will not lose their market by doubling too late, coupled with a case where we know the take point very well.


With all that said, the -2, -2 is not strictly a double-take within the first couple rolls. Some good rolls do not have market losers, plus folks might make mistakes. One such mistake is your opponent does indeed get some good rolls in a row and did not double and now  has a strong lead, ie greater than 70%, say 80%, leaving you with just
20% in this game. If he then doubles, you are happy to drop and raise your chances back to 30%. Thus if you think your opponent might make a mistake, then you might hang on a bit and see how things evolve.

Note this does not mean that if you missed your market you do not double. You still double and take 70-30 on the next games rather than risk losing the present one with turn of the luck—unless, of course, you really missed it and can play on for a gammon, but if no gammon is in sight, you double and start again. You did, however,  make a mistake. You could be winning the match with this one game had you doubled at the right time.

Note on -2,-1 with equal players and 20% gammon chance.
Trailer will win the first game 50% of the time, and of these 50% he will get a gammon 20% of the time, meaning he wins the match in one game 0.2 x 0.5 = 0.1 = 10% of the time.  Of the 40% that he wins a single game, he will then have to play a second game of which he will win 50%, or 0.5 x 0.4 = 0.2 meaning he wins the match in two games  20% of the time... thus he wins either way a total of 30% and opponent wins 70%.  Put another way, match equity at -2, -1 is 30, 70.

Note on why the take point 25% when no match scoring is involved.
To say you have a 25% chance of winning means you win 1 out of 4 games starting with the identical board each time. When confronted with a cube on this board, if you drop all 4 games, the score at the end will be (0, 4), meaning you are down 4 pts, or playing for 1$ per point you lost $4.

If you accept the cube you will lose 3 at 2pt each and win one at 2 pt, so the final score will be 2, 6, again if playing for $1 per point you have lost $4. So in games where the match score does not matter, your take point is 25% chance of winning... or something near that. In other words, to accept the cube, you do not have to think you can win this game, you only have to think you could win 1 game like this if you played four of them starting from this identical set up... with your opponent on roll.

To decide what your actual numerical chances of winning are, however, is another matter. Expert players can do it to within a few percent. After playing some years myself, I would be lucky to get this right to within 15 or 20%, except when there are just a few checkers left and I can evaluate the actual rolls that are left. There are several general guidelines on when to double and when to take, and these must be, on average, relying on the right numerical values, but they do not themselves provide specific percentages.

One way to develop this skill is play against a computer or phone app that keeps the winning chances posted as you play, and then watching this as games evolve, try to develop a feeling for various configurations and their values.


Please add comments, corrections, suggestions.

— David