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Craig Connell wrote:
> Could you explain again why it is proper to double in a volatile
> situation. What I don't understand is that if the situation is very
> volatile, that probably means the next few rolls will drastically
> effect the outcome of the game. Given that, your winning chances
> cannot be extremely high (if your opponent is lucky on the next few
> rolls your gonna lose). Assuming the match is tied at say 3 away, 3
> away (my market window table says the perfect double is at 72% at that
> point) why is it proper to double?
If you think about it, what really happens when you turn the cube is that
you are doubling the stakes you are playing for. Given that, it is
potentially correct to double any time you are the favorite. The only
reason not to double when you are the favorite is if it is likely that
after the next exchange (you roll, he rolls) if things go well for you he
will still have a take.
Let's suppose you are playing a game of backgammon, you are a small
favorite, and you are on roll. Suddenly God comes down and says: I'm
changing the rules this game -- you may double now if you wish, but if
you don't double now I'm removing the doubling cube from the game. What
should you do? Clearly you should double -- you are doubling the stakes
when you are the favorite and this is your last opportunity to do so.
Back to reality. Let's look at the classic position where you have two
men left, one on the five point and one on the two point, and your
opponent has one on the ace point. You probability of winning is 19/36.
Since this is the last roll of the game it is essentially like the
situation described above where you must double now or never, so it is
clearly correct to double now. Note that the volatility could not be any
higher -- everything is riding on the next roll.
Getting back to more complex types of positions, let's suppose you have
some position where you are 65% to win. However this 65% comes out as
follows: After the next exchange of rolls, half the time you are 85% and
the other half you are 45%. Very volatile, that's for sure. Should you
double? Absolutely! If things go well it is very costly for you not to
have doubled -- your opponent will now pass if you double, and since you
are 85% to win this is a very correct pass -- you will have lost
considerable equity. If things go badly you are 45% to win -- so you
have doubled the stakes as a small underdog, not a big deal.
Consequently, the cost of not doubling and being wrong is greater than
the cost of doubling and being wrong.
The above example is artificial, of course -- I doubt if one can come up
with a position which meets these criteria. It is possible to get close,
however, with normal positions. Take the following position, for example:
13 14 15 16 17 18 19 20 21 22 23 24
+------------------------------------------+
| | | X X X X |
| | | X X X X |
| | | X X X |
| | | X |
| | | |
| | | |
| | | |
| | | O |
| | | O O O O O O |
| O O X | | O O O O O O |
+------------------------------------------+
12 11 10 9 8 7 6 5 4 3 2 1
O is on roll. X has two men off. O has 20 numbers out of 36 which hit.
If O hits he is a huge favorite, perhaps 85-90% to win. If O misses it
is a close race with O appearing to be a small underdog. Thus this
position roughly meets the criteria I gave for my artificial example.
Even though O is nowhere near X's drop point, the position is so volatile
that it is correct for O to double.
In theory it would be nice to be able to double right at the opponent's
minimum take point. In practice, backgammon just isn't that way. It is
often too volatile. If there is a significant chance that you may shoot
way beyond your opponent's minimum take point by a lot on the next
exchange of rolls, then it is important to double now even if your
advantage isn't too great. The cost of losing your market by a lot is
very expensive. That is why very volatile positions (that is, positions
where the equity is likely to swing a lot on the next exchange of rolls)
call for early doubles.
Kit
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