Permutation-fair dice are 
fair dice whose rolled values
are independent random variables,
with no possible ties between different dice, such that every ordering of the dice
from largest to smallest has probability 
. Here 
is a positive integer
and 
is the factorial (Ford et al. 2023, p. 81).
Every nonempty subset of 
dice also has all its 
orderings equally likely, since each such ordering extends
to 
of the full orderings. Each die therefore has probability 
of each rank within that subset.
Thus permutation-fair dice are Go First Dice, but
the converse need not hold (Ford et al. 2023, p. 81).
An example consists of three dice with equally likely face labels
 |
There are 48 equally likely outcomes, and each of the six orderings of 
, 
, and 
occurs eight times. Thus the dice need not have the same number
of faces (Ford et al. 2023, p. 81).
