There are three dice, with different numbers on each side.
Two players each choose one of the three dice, and roll once; high roll wins. You get to / have the misfortune to go first. Which of the three dice should you choose?
Follow-up; is it possible to make a perfect set of dice like these, so that
* P(A defeats B) = P(B defeats C) = P(C defeats A) = P > 1/2
* all 18 numbers are natural numbers
* no ties can happen; and
* the average of the numbers on the six sides of each die is the same?
If there are multiple such solutions, how high can P be?
If it is not possible, why not? Perhaps it could be done with dice with a number of sides other than six?