Game variations and descriptions of intransitive die and their behaviour
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A set of dice is intransitive (or nontransitive) if it contains three dice, A, B, and C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but where it is not true that A rolls higher than C more than half the time. In other words, a set of dice is intransitive if the binary relation – X rolls a higher number than Y more than half the time – on its elements is not transitive. More simply, A normally beats B, B normally beats C, but A does not normally beat C.
It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. This is different in that instead of only "A does not normally beat C" it is now "C normally beats A". Using such a set of dice, one can invent games which are biased in ways that people unused to intransitive dice might not expect (see Example).[1][2][3][4]
^Weisstein, Eric W. "Efron's Dice". Wolfram MathWorld. Retrieved 12 January 2021.
^Bogomolny, Alexander. "Non-transitive Dice". Cut the Knot. Archived from the original on 2016-01-12.
^Savage, Richard P. (May 1994). "The Paradox of Nontransitive Dice". The American Mathematical Monthly. 101 (5): 429–436. doi:10.2307/2974903. JSTOR 2974903.
^Rump, Christopher M. (June 2001). "Strategies for Rolling the Efron Dice". Mathematics Magazine. 74 (3): 212–216. doi:10.2307/2690722. JSTOR 2690722. Retrieved 12 January 2021.
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