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The Last Rolls - Part 2

by Douglas Zare
25 January 2004

Douglas Zare

In Part 1, we considered the theory of the bearoff near pure n-roll versus n-roll positions. In this column, we consider actual examples of close cube decisions where one side is close to an n-roll position

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Subject: Re: The Last Rolls - Part 2

From: DBrthrtn Member has a photo available

Date: 26 Jan 2004 19:14 EST

D2D is a No Double by 0.020 and a Beaver by 0.007 - I'd call this a "near-miss Kauder paradox bearoff" (nearly a correct double/correct beaver). Can a true Kauder paradox exist in a bearoff? The only ones I've ever seen were contact positions when activating gammons (Jacoby rule) was a factor, but in a bearoff that would be a moot point. These days I suppose one could do an exhaustive search of a bearoff database to find the answer. I suspect there aren't any, but perhaps the "near-misses" might make another interesting set of reference positions. More great stuff, thanks Doug.

Subject: Re: The Last Rolls - Part 2

From: zare

Date: 26 Jan 2004 20:50 EST

The Kauder paradox only happens because of the Jacoby Rule. When there are no gammons,

Opponent's <= Centered,

where these represent your equity on the same cube level. With the cube centered, you can only hurt yourself by declaring that you will not double from the center.

In order to double, it must be that

Centered <= 2 Opponent's.

These two inequalities combine to say

Opponent's <= 2 Opponent's.

Subtracting Opponent's from both sides,

0 <= Opponent's.

That's the opposite of what is needed for your opponent to have a correct beaver. A priori, it could be that all of the equities are 0 so that the double and beaver would be optional, but I believe that doesn't happen in actual backgammon. Position D2d is just close.

Speaking of paradoxes, the reason that Red is strictly better off in D2d with the cube in the center than owned (on 1) by White is because White can avoid redoubling in Jacoby paradox positions after Red rolls 3-1 or 2-1. For example, after Red rolls 2-1, leaving 52 vs. 5, White should double (improving from 0.056 to 0.111) but is better off not redoubling (0.187 is better than 0.111).

Douglas Zare

Subject: Re: The Last Rolls - Part 2

From: DBrthrtn Member has a photo available

Date: 31 Jan 2004 18:08 EST

Thanks Doug that is clear now. Also, interesting about how D2d would sometimes steer itself into a Jacoby paradox.

I wonder what the closest to zero all the equities can be in a bearoff position (no double, double/take, double/beaver)? Maybe the way to ask is "what's the bearoff position with the minimum difference in equity between a no double and a double/beaver". For D2d that difference is 0.027. Probably the only way to find out is by brute force searching of a database (gotta get one - oh well, one of these days....). Maybe D2d is the one....

Subject: Re: The Last Rolls - Part 2

From: zare

Date: 02 Feb 2004 16:07 EST

I haven't made an exhaustive search, but 332 vs 52 is much closer. No Double is worth 0.000686 and Double/Beaver is worth -0.002401. My guess is that this is the closest.

Another close double, close beaver is 6211 vs. 641: ND 0.014455, DB -0.001155.

To find these, I asked Trice's Bearoff Quizmaster to show me positions among the first 10,000 such that double/take is worth between -0.010 and 0.010. There are many other such positions, but in most borderline beavers, No Double is worth much more.

Douglas Zare

Subject: RE: The Last Rolls - Part 2

From: DBrthrtn Member has a photo available

Date: 02 Feb 2004 18:31 EST

Thanks again Doug.

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