I don't know anything more about intransitive dice than I did when math-fun last discussed this, in a discussion Bill Gosper kicked off in Nov. 2001. But at the time I did hunt down Marting Gardner's piece on Efron's dice, which included the quote:
It has been proved, Efron writes, that 2/3 is the greatest possible advantage that can be achieved with four dice. For three sets of numbers the maximum advantage is .618, but this cannot be obtained with dice beacause the sets must have more than six numbers. If more than four sets are used (numbers to be randomly selected within each set), the possible advantage approaches a limit of 3/4 as the number of sets increases.
So that at least answers your "I believe I saw..." question. The original Efron dice, 004444/111555/222266/333333, do attain that 2/3 bound for four dice, by the way. I did a little literature searching last time. Aside from the Dec 1970 SciAm column, there's a May 1976 Math magazine column by Richard Tenney and Caxton Foster, and there's a paper of Zalman Usiskin from 1976: "Max-min probabilities in the voting paradox." (Ann. Math. Statist. 35, 1964, pp 857--862.) This paper proves the bound of 3/4 when the number of independent random variables goes to infinity. Perhaps a reasonable way to find anything written on the subject recently is to search for articles which refer to the above... --Michael Kleber On 4/5/06, dasimov@earthlink.net <dasimov@earthlink.net> wrote:
Does anyone know if there is a recent summary of all that's known about intransitive dice?
Searching on MathSciNet doesn't seem to get me very far.
Suppose you have a cyclically ordered sequence of n dice, each of which has s sides, and each of which beats the (cyclically) next one with the same probability p > 1/2.
I believe I saw some reference that claims p <= 3/4 under all circumstances; maybe another that sharpened this to p <= sqrt(5 - 1)/2 = .618....
--Dan
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