Equity in Backgammon - Part 2
d) Cube Leverage
Lets assume that you lead 1-0 in a match to 3. If the doubling cube is never put into play, your match-winning chances are as follows:
i) 10% of the time you will win a gammon and win the match
ii) 40% of the time you will win a single game, and be 75% to win the match
iii) 40% of the time you will lose a single game, and be 50% to win the match
iv) 10% of the time you will lose a gammon game, and be 30% to win the match
10% + (40% * 75%) + (40% * 50%) + (10% * 30%) is 63%. However, from the data that was actually accumulated it shows that the leading side wins only about 60%. Why?
If you recall, we showed that the trailer at this current score can double far earlier than usual. We did not discuss gammons as well.
Let’s look over the winning chances, including the cube, which each side needs to win the game. 75% win chances need to be reached by the leader. Gammons for his side are irrelevant if he doubles. If we abide by our 20% gammons rule, when the trailing player reaches 65.8% wins, he will be winning 13.2% gammons. Winning 65.8% with 13.2% gammons is the same as to having 50% match-winning chances, the same as if the opponent drops a double.
If we take a basic model, then each player starts with 50% chances to win the game and to win with the cube the leader has to go from 50% to 75%. But the trailing player has to go only from 50% to 65.8%. If for simplicity sake we round this off to 65%, we discover that the trailing player has to go 15% only while the leader has to go 25%. In this case, the leader should win only 3 games for every 5 won by the trailing player!
You can not accomplish much simply by knowing match equities. Match equities are stepping stones for further concepts. We have already discussed simple examples, of doubling at scores of 1-0 in 3-point matches. More complex examples will be given in later discussions.
II. Position Equity
The equity of a position is its mathematical expectation, such as this basic example: if you are 60% to win and 40% to lose, with no gammons possible, your equity if 0.60 – 0.40, or 0.20 points.
How about considering gammons and backgammons? These are reflected easily. Lets review a position with the following odds:
Backgammon win: 2%
Gammon win: 30%
Simple win: 38%
Simple loss: 20%
Gammon loss: 9%
Backgammon loss: 1%
The equity for this position will be calculated as follows:
.02 * 3 + .30 * 2 + .38 * 1 - .20 * 1 - .09 * 2 - .01 * 3
Or 0.63 points.
Please click here to read part 3 of our Equity in Backgammon article or go to our
Backgammon Articles page.
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