Michael Soper wrote:
> I've ready several of the 'proofs' about -2:-2 matches and I remain
> unconvinced. The proofs I've read all seem to imply that nothing can be
> gained by not doubling, I think that this is incorrect.
> Clearly there is no point to doubling if one has no market losers, since
> the cube can be passed on the next roll. So, in order to prove doubling
> is correct it must be proven that there is no situation where the equity
> is higher without doubling. I believe such situations exist.
> For example positions can be reached using Jellyfish that have X on roll
> with 40+% gammons, and 30%+ losses without, I believe, either side having
> a previous market losing sequence. These positions are most easily
> reached with X rolling a 44 or 55 to make a 4 point board and O dancing.
> This is X's best sequence and O still has a take.
> In these positions X appears to be better off playing for a gammon, in
> which case he doesn't double. O has no market losers until he gets a
> shot so he definitely shouldn't double. When O is later able to run off
> a gammon without ever getting a shot, I claim X then has the first
> optimal double to cash the game.
> Anyway I have yet to see a proof that this type of position is impossible
> and without that the proofs seem incomplete. If I missed this from
> someone's proofs I apologize.
Of course such a position can be reached. However you are incorrect when
you say that X is better off playing for a gammon in such a position.
The key is that, although X may not have lost his market yet, from such a
position there certainly will be market losers (i.e. sequences after
which X will be better than 70% to win). Granted these may be positions
which X has a high gammon winning probability, but that won gammon is not
certain. Since O will of course make sure that he doesn't lose his
market (O being the perfect player he is), there are no scenarios where X
gains from not doubling -- if X loses the game he will never lose only 1
point. Consequently, by not doubling now X costs himself in the
variations where he wins the game but doesn't win a gammon -- if he ever
gets to a position where he doubles and O correctly drops then X will have
wished he had doubled earlier. As I have shown X has no corresponding
gain from not doubling now, so faililng to double does cost equity.
It is true that there are positions where, at this score, it would be
correct for X to play for a gammon, since his equity in playing on is
higher than if he doubles and O correctly passes. However, for such a
position to have been reached it would mean that X had at some previous
point failed to double when O still had a take but X had market losers,
such as the position you describe. As I have shown, not doubling in this
position is an error by X. Therefore, a position where it is correct to
play for a gammon cannot be reached without somebody having made a cube
Hope this explains it.