[Z.e.r.o]

The number of matches you need is n!
where n! = 1*2*3*4*...*(n-1)*(n)
and is called factorial and can be used to determine the numbers of permutations of n tokens.

In a 3 players match where anyone is playing a role only once is 3! (1*2*3 = 6)

Empirical solution:
123
213
312
321
132
231

If I'm not right you can spank me with a crowbar