No matter which die you choose, there is another that will beat it, on average, two out of every three rolls. For example, consider the 33333 die and the 662222 die. Since the first die is always 3, the second die loses when it rolls a 2, i.e. 4 of the 6 possibilities or two-thirds of the time. What about the other pairs?

Construct a cycle that connects each die to the die that beats it with greatest likelihood. This is called a Condorcet cycle, and is just like a game of rock/paper/scissors. Interestingly, as the example below shows, it is also relevant to a preferential voting result where there is no winning candidate.

Suppose in an election there are 3 candidates, call them A, B and C, and 300 voters. After counting the ballots it is found that:

• 100 voters ranked A > B > C
• 100 voters ranked B > C > A
• 100 voters ranked C > A > B

So who won?

Well, each of A, B and C received 100 first preference votes, so that doesn't decide anything.

Since two-thirds of voters preferred A over B, and two-thirds of voters preferred C over A, it seems C should be declared the winner. But two-thirds of voters preferred B over C! This cycle is the Condorcet paradox.

This particular presentation of the paradox is from the wonderful book, The Heart of Mathematics: An invitation to effective thinking, by Edward Burger and Michael Starbird.