Go First Dice are a nonempty finite set of fair dice whose rolled values are independent
random variables. No two different dice can tie, and every die in any nonempty
subset is equally likely to occupy each rank when the
rolls are ordered from largest to smallest. For a subset
of 
dice, each die has probability 
of each rank. They allow players to
determine an order of play without rerolling ties or favoring a player (Ford et
al. 2023, pp. 75-77).
If all 
dice have 
equally likely faces, where 
and 
are positive integers, each nonempty subset
of 
dice has 
equally likely outcomes. Equal probabilities
for the 
ranks require 
to divide 
.
In particular, every prime not exceeding 
must divide 
. For five dice, 
must therefore be a multiple of 30 (Ford et al. 2023,
p. 77).
Permutation-fair dice satisfy the stronger requirement that every ordering of the dice be equally likely. Equal rank probabilities for every subset do not imply this stronger property. The five 60-sided Go First Dice of Ford et al. (2023, pp. 76, 81) have the required rank probabilities, but their orderings are not all equally likely.
