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Jacoby Rule - a challenge

Posted By: Ray Kershaw
Date: Wednesday, 9 March 2011, at 4:53 p.m.

In Response To: Reasons for the Jacoby Rule (Stein Kulseth)

Stein asks: How big an error is it even possible to make by not doubling a position that would not be a double without Jacoby?

I think the theoretical answer is 1/3 of a point. This assumes no beavers and that the outcome does not depend on the location of the cube.

Suppose I have a 1/3 prob of winning a gammon and a 2/3 prob of losing a single. Then without Jacoby, my expectation is zero whether or not I double. With Jacoby, my expectation is -1/3 if I do not double and zero if I do double.

So if my prob of winning a gammon were just under 1/3 and of losing a single were just over 1/3, then I should not double without Jacoby and should double with Jacoby.

All this is Kauder Paradox territory where (without beavers) I can double with a gammon win prob of 1/4. But there I should not double without Jacoby. Stein's constraint raises 1/4 to 1/3.

The answer is 1/2 a point if I have a 1/4 prob of winning a backgammon and a 3/4 prob of losing a single.

Perhaps someone can construct a position which gets close to 1/2 a point ?

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