It is not just that I made many small errors in my first post. Somehow, I got sidetracked into developing my own match-equity table from scratch! So, here is the redo, with conventional references to METs and an expanded conclusion that you may want to skip ahead to.
CUBELESS EQUITY IN A CASH GAME
In a cash game, this one is easy. Black cashes, regardless of the relative strengths of the players. Look at the odds:
- Black wins on his first roll when he rolls doubles two or better.
- If Black rolls 1-1 or a non-double, White can win by rolling any double on his turn.
- Otherwise, Black wins.
- White’s cubeless chance of victory is the compound probability that Black does not roll double twos or better (31/36) and that White rolls any double (1/6). The product is 31/216, or about 14.35%.
- Black’s cubeless chances are (100% - 14.35%), or 85.65%.
Of course, if Black cashes now, his chance of winning is 100%.
EQUAL PLAYERS IN A MATCH
In a match between equally skilled players, Problem 91 is a no-double/take. Here is the breakdown:
If
Black doubles, and White takes, then Black will win the match with a victory in this game. If he gets a turn, White has a mandatory redouble. Thus, both sides will be playing for the match. Black will win 85.65% of the time.
If
Black doubles, and White passes, then the score will be 6-0, with the Crawford game upcoming. Assuming there are no gammons or backgammons, White will have to win the next four games in order to win the match. His chance of doing that is 50% raised to the fourth power, or 6.25%. Even if you give him one gammon victory, White still must win three games in a row. His match winning odds are then 12.5%. And with a lucky backgammon, White might win the match in just two games. A
match-equity table (MET), such as the one found at
http://www.bkgm.com/articles/GOL/Oct03/met.htm, provides a statistical average of all these possibilities, giving White an overall 9.1% chance to win the match. To find Black’s
match winning chance (MWC), we can look in the match-equity table, or we can just subtract. The difference between 100% and White’s MWC is 90.9%.
If
Black does not double, the computation is more complex. The game will be played to its conclusion, with Black winning some 85.65% of the time. A win will take Black to the Crawford game, leading 6-0 on the score sheet. From there, as we determined above, Black will go on to win the match about 90.9% of the time. On the other hand, Black will lose this game 14.35% of the time, taking him to a match score of 4-2. In the parlance of match-equity tables, Black will then be 3 games away from victory, while White is 5 games away. At the so-called
3-away/5-away score, the match equity table gives Black a 64.6% chance of victory. By combining all these numbers, we find that Black’s probability of winning the match when he does not double is 85.65% * 90.9% + 14.35% * 64.6%, which is about 87.13%.
So, here is what we now know:
- If Black doubles, his MWC is 85.65% when White accepts and 90.9% when White passes. Clearly, White should take if doubled.
- If Black does not double, his MWC is 87.13%. Because White's take will reduce it to 85.65%, Black should not double.
A MATCH BETWEEN UNEQUAL PLAYERS
The situation changes when the players are of greatly differing skill levels. If the weaker player has the Black checkers in Problem 91, he should double in a shot. Even when his opponent takes, he loses only about 1.48% of his match equity. But what a deal! For this price, he buys an 85.65% chance to
win the match based on dice alone against his world-class opponent. Playing White, the stronger player may not wish to risk the entire match on a few shakes of the dice. He may reasonably choose to pass the double he would take when playing against another expert.
Conversely, the stronger player as Black should not double. Just look at the numbers. He is not supposed to double even against a normal opponent. Why would an expert want to amp up the luck factor against a novice or intermediate? If he does, he had better have a rock-solid read on his opponent, and
know that he will pass. If the expert’s double is accepted, then we will strip him of his expert rating. As White, of course, the weaker player should gobble up the cube, and throw it back at the first opportunity.
MY SOLUTIONS
Part (a):
Double, pass
Part (b):
No double, take
Last edited by Taper_Mike; 01-14-2011 at 06:01 AM.