On Mar 26, 11:14*pm, bob wrote:
On Mar 25, 9:11*pm, wrote:

Almost surely, there _are_ situations like that in backgammon. *Can't
you mimic the above scenario in backgammon by postulating 18 monster
rolls for each side?
But, suppose there are such situations in backgammon, why does it then
follow that the value of a cube, as well as its position, can affect
theoretical money play? *This seems to be a hole in your argument. *So
let us assume that your scenario is exactly replicated in backgammon
where there's a stalemate but each side has 18 monster winning rolls,
and no gammons are possible. *Please explain why this scenario leads
to the conclusion that the scenario with a 2 cube is essentially
different than the scenario with a 1024 cube.

* *I doubt that there is a siutation like that in backgammon. Note
that you would need the 18 non-monster rolls to lead to repeating
positions. There has been one proposed but it only can arise as the
result of an illegal checker play.

*Here is why such a situation will affect theoretical money play:

*Theoretical money play means making the move that maximizes equity,
assuming perfect play from the opponent. Equity means the expected
value of the position. Take my coin flipping example again. If both
players use the always double/take strategy then the player on turn
will win $2 with probability 1/2, will lose $4 with probability 1/4,
will win $8 with probability 1/8, will lose $16 with probability
1/16 ... *The expected value is 2(1/2) + (-4)(1/4) + (8)(1/8) + (-16)
(1/16) + ... which does not converge. The expected value is not even
defined.

Bob Koca

Yes, indeed. I realised that. However, you appeared to claim that
this no-equity position implies that the value of the cube needs to be
taken into account when enumerating positions. You don't show this.
In your example, whether the cube is on 2 or 4 or whatever, it's still
the same no-equity verdict.

Paul Epstein