On Mar 27, 1:13*am, wrote:
On Mar 27, 12:43*pm, bob wrote:





On Mar 26, 9:13*pm, wrote:

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- Hide quoted text -

- Show quoted text -

* Torben wrote in this thread how a set of equations could be written
and solved to find the equity of any
backgammon position. The solutions only make sense as equities though
if the equities are all defined.
In my coin example one can give an equation and solve it but that does
not mean it gives the expected value of the game. If double take is
correct then the player on turn either wins 2 points or gives his
opponent the exact same situation but with the cube doubled. It is
very tempting though wrong to think that E(X) = (1/2)(2) + (1/2)
(-2E(X))
whose solution gives E(X) = 1/2.

Bob Koca- Hide quoted text -

- Show quoted text -

Ok but you did say that your paradox makes the potential number of
positions infinite. *I still don't see how. *You have never explained,
why, even assuming your paradoxical scenarios exist, a 32 cube should
be regarded differently to a 64 cube. *Both lead to the same
conclusion -- equity undefined.

You said this:

If you are talking about
money backgammon though then the cube position and value makes the
NUMBER OF POSITIONS INFINITE. Now one might say that its position is
all that matters since if you know the correct theoretical play
holding a 2 cube then you also know the correct theoretical play
holding a 4 or any higher value cube. There is a problem though in
that the equations might not have a solution....

(caps added)

Yet you never explain why your paradox would lead to an infinity of
positions.

Paul- Hide quoted text -


I clearly said that considering the position and value gives an
infinite number of positions. Ignoring the value does give a finite
number of positions. One must be careful though since equations can be
made but there is no guarantee that the solutions actually give
equities as one might expect.

Bob Koca