Puzzle: Suppose there are 4 dice: Blue, Green, Red, and White. These dice have different numbers than usual printed on their faces. After observing a long sequence of experiments rolling pairs of these dice, you conclude that
Answer: We cannot conclude P>50%. In fact, P can be as small as 1/3 (i.e. 33.33%) and as large as 100%! A set of dice that yield a Blue>Green>Red>White>Blue "probability cycle" (where "A>B" means "rolling A will produce a higher roll than B with probability 2/3") is given in the top row of this table.
blue | green | red | white |
---|---|---|---|
004444 | 333333 | 222266 | 111555 |
446666 | 555555 | 222266 | 111333 |
However, if the numbers painted on each die's 6 faces instead are those given in the bottom row of the table, then Blue>Green>Red>White and the Blue die will beat the White die 100% of the time.
The fact that 1/3 and 100% both are best possible bounds is obvious for 100%,
and not-so-obvious for 1/3 – but proven by
S.Trybula: On the paradox of N real variables, Zastos. Matem. (Applied) 8 (1965) 143-154
and also independently by Z.Usiskin: Max-min probabilities in voting paradoxes,
Annals of Math'l Statistics 35,2 (1964) 857-862.
Usiskin indeed showed that with 2K "dice" (i.e. independent probability distributions
over the reals), the maximin intransitivity probability P(2K)
was bounded by
The exact maximum possible intransitivity probabilities P(n) for n=2, n=3, and n=4 dice, namely P(2)=1/2, P(3)=(√5-1)/2, and P(4)=2/3, all meet the following upper bound claimed by [Ilya I. Bogdanov: Intransitive Roulette, Mathematical education, ser. 3, no. 14 (2010) 240-255; this journal and this paper both are in Russian].
Since my Russian is poor (actually, nonexistent) I am not sure whether Bogdanov is claiming the "≤" is tight for all n.
Other references:
Colin R. Blyth: Some Probability Paradoxes in Choice from Among Random Alternatives,
J.American Statistical Association 67, 338 (June 1972) 366-373
R.P. Savage Jr: The Paradox of Nontransitive Dice,
American Mathematical Monthly 101,5 (May 1994) 429-436.
H.Steinhaus & S.Trybula: On a paradox in applied probability,
Bull. Acad. Polo. Sei. 7 (1959) 67-69.