Intransitive dice

A set of <a href="/facts/Dice/aURlPtbp">dice</a> is <a href="/facts/Intransitivity/eRDjvMDo">intransitive</a> (or nontransitive) if it contains X>2 dice, X1, X2, and X3... with the property that X1 rolls higher than X2 more than half the time, and X2 rolls higher than X3 etc... more than half the time, but where it is not true that X1 rolls higher than Xn more than half the time. In other words, a set of dice is intransitive if the <a href="/facts/Binary_relation/r9BwGgaK">binary relation</a> – X rolls a higher number than Y more than half the time – on its elements is not <a href="/facts/Transitive_relation/9r90y50r">transitive</a>. More simply, X1 normally beats X2, X2 normally beats X3, but X1 does not normally beat Xn.
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).

Intransitive dice open-in-new

Intransitive dice