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Jul 21, 2000, 3:00:00 AM7/21/00

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I'm interested in the following position:

Money game cube action?

+24-23-22-21-20-19-------18-17-16-15-14-13-+

| X X O O O O | | O O X |

| X O O O | | X |

| | | |

| | X | |

| | | |

| |BAR| |

| | | |

| | O | |

| | | |

| O X X X | | O |

| O O X X X X | | X X O |

+-1--2--3--4--5--6--------7--8--9-10-11-12-+

Pipcount X: 169 O: 169

CubeValue: Any

Once one thinks about how the entering numbers play, I think it becomes

clear that this is both double/take with the cube centered and

redouble/take if the side on roll has the cube. (I don't think it would

be a take without the substantial value of holding the cube.) Since one

fails to enter 1/4 of the time, the variance of this position (related

to the expected square of the final payoff) is infinite, since the

contribution to the expected square of the payoff from entering after

one throw is positive and equals the contribution from the possibility

of entering after the nth throw, for any n.

If this position can be reached by _optimal_ play, then the variance of

backgammon is infinite. This should not be a surprise, since that is

what happens in the continuous limit (doubling in at 80%). It has been

claimed that there are positions where the equity doesn't exist. It is

actually pretty easy to construct positions like the one above

(redouble/take with both sides on the bar against a 3-point board), but

this one seems more promising than most to arise from actual play.

So, can you construct a sequence of die rolls and plays that you are

confident are optimal (or at least that JF or Snowie would play) which

leads to this position? Less satisfying but still of interest would be a

game ending in this such that one side has played optimally, and the

other side might have erred. If this position can't be reached by

optimal play, or even by plausible play by strong players, then it is

not nearly as interesting an example.

What would it mean if variance doesn't exist? In long money sessions,

the results would not approach a classic bell-curve. One can try to

apply variance reduction techniques when the variance is infinite; I'll

say more about this later.

Douglas Zare

Jul 25, 2000, 3:00:00 AM7/25/00

to

Douglas Zare <za...@math.columbia.edu> writes:

> I'm interested in the following position:

>

> Money game cube action?

> +24-23-22-21-20-19-------18-17-16-15-14-13-+

> | X X O O O O | | O O X |

> | X O O O | | X |

> | | X | |

> | | | |

> | |BAR| |

> | | | |

> | | O | |

> | O X X X | | O |

> | O O X X X X | | X X O |

> +-1--2--3--4--5--6--------7--8--9-10-11-12-+

> CubeValue: Any> I'm interested in the following position:

>

> Money game cube action?

> +24-23-22-21-20-19-------18-17-16-15-14-13-+

> | X X O O O O | | O O X |

> | X O O O | | X |

> | | X | |

> | | | |

> | |BAR| |

> | | | |

> | | O | |

> | O X X X | | O |

> | O O X X X X | | X X O |

> +-1--2--3--4--5--6--------7--8--9-10-11-12-+

>

> If this position can be reached by _optimal_ play, then the variance of

> backgammon is infinite. This should not be a surprise, since that is

> what happens in the continuous limit (doubling in at 80%). It has been

> claimed that there are positions where the equity doesn't exist. It is

> actually pretty easy to construct positions like the one above

> (redouble/take with both sides on the bar against a 3-point board), but

> this one seems more promising than most to arise from actual play.

This is very interesting. Unfortunately Deja News won't let us at the

discussions on r.g.b. (from late '97) about undefined equity

positions, but essentially I believe (given only some assumptions

about the results of the games once either player enters) that you can

prove that the correct cube action for both the positions discussed

then and your position above is redouble/take. Such arguments do not

rely on the equity or volatility; rather, I can prove that any

behaviour other than redouble/take is suboptimal, or equivalently,

that it is better to double n+1 times and then stop doubling than it

is to double n times and stop, for all n. From there I conclude that

either the optimal strategy is redouble/take, or that no optimal

strategy is defined; it comes down to a matter of definition.

I hadn't previously considered positions where the equity was defined

but the variance was not, but I do believe yours is one (again, given

the assumptions about the equity once either player enters). So it

appears that positions exist where both equity and variance are

defined (non-contact positions are trivial examples of this); and

others also exist where equity is defined but variance is not

(including yours), and also those where neither equity nor variance

exist (e.g. the one Paul Tanenbaum posted in '97).

To generalise a bit, what can we say about the higher order moments

beyond equity and variance? (Equity is, loosely speaking, the first

moment; variance is the second; the third would be related to the

result of the game to the third power, which is, er, the cube cubed.

Some of these moments are raw (measured about zero) and some are

central (measured about the mean), but we can gloss over that for

now.) Obviously in all non-contact positions, all moments must be

defined, because the finite pip count must decrease with every move

made and that places an upper bound on the cube value. But given

sufficient contact, we could find a positive probability that the cube

will reach any given level, and so interesting things could happen at

higher moments.

I once proposed a model for the probability distribution function of

the cube value/game result based on some empirical data; the article

(tentatively) survives at:

http://www.dejanews.com/=dnc/getdoc.xp?AN=491955947

If we assume this model holds, then we believe that once the cube has

been accepted at n (n>=2), it will be redoubled and accepted at 2n

with probability 0.152. This probability is low enough that variance

(i.e. the second order moment) is defined in general, because the

probability of successive terms of the series decreases by a factor of

0.152 per term while the squares of the results only increase by 2^2,

and so the series converges. But at the third moment, the results

increase by a factor of 2^3 per term, which exceeds the decay from the

probability; the series diverges and the third (and higher) moments

are undefined.

> So, can you construct a sequence of die rolls and plays that you are

> confident are optimal (or at least that JF or Snowie would play) which

> leads to this position?

I don't know. What about the general question? Prove or disprove:

for all positions P that may be legally reached, there exists a

sequence of dice rolls D such that an optimal play of D from the start

position leads to P.

Cheers,

Gary.

--

Gary Wong, Department of Computer Science, University of Arizona

ga...@cs.arizona.edu http://www.cs.arizona.edu/~gary/

Jul 25, 2000, 3:00:00 AM7/25/00

to

Gary Wong <ga...@cs.arizona.edu> writes:

> I don't know. What about the general question? Prove or disprove:

> for all positions P that may be legally reached, there exists a

> sequence of dice rolls D such that an optimal play of D from the start

> position leads to P.

> I don't know. What about the general question? Prove or disprove:

> for all positions P that may be legally reached, there exists a

> sequence of dice rolls D such that an optimal play of D from the start

> position leads to P.

I'm sure that's not true. Here's an example of a position (with X to

move) which is legally reachable, but where O's last move cannot have

been optimal:

+24-23-22-21-20-19-------18-17-16-15-14-13-+

| O X X X X | | X |

| O | | |

| O | | |

| (15) | | |

| |BAR| |

| (10) | | |

| X | | |

| X | | |

| X | | |

+-1--2--3--4--5--6--------7--8--9-10-11-12-+

O's last roll was either 5-2 or 6-2, played without hitting; in either

case, hitting would have been better.

David desJardins

Jul 25, 2000, 3:00:00 AM7/25/00

to

My previous posting was wrong (the previous roll could have been

double-2).

double-2).

Here's a better example of a position (X to move) which is reachable

with legal play, but where O must have made a suboptimal move:

+24-23-22-21-20-19-------18-17-16-15-14-13-+

| O | | |

| O | | |

| O | | |

| (14) | | |

| |BAR| |

| | | |

| | | |

| X X X X X X | | |

| X X X X X X | | O X X X |

+-1--2--3--4--5--6--------7--8--9-10-11-12-+

O's last move didn't move the man on the 7 point, when it obviously

would have been better to do so.

David desJardins

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