Nontransitive dice are dice for which the comparison "more likely to roll a larger number" contains a directed graph
cycle. Write 
when the rolled values of 
and 
are independent
random variables satisfying 
. When ties are impossible, this is equivalent
to 
.
It also describes the advantage when tied rolls are independently repeated. There
can be dice 
,
,
and 
with 
,
,
and 
,
so this relation need not be transitive
(Gardner 1974, pp. 121-122).
For example, take three six-sided fair dice whose face labels are
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
Each die has three equally likely values. Counting the nine equally likely pairs of values gives 
.
Thus each die is more likely to beat one rival and lose to another, despite each
die being individually fair. This is the dice form of Gardner's three-packet card
example (Gardner 1974, p. 122). Efron's dice
provide a four-die construction.
