Crazy Dice

Can you choose a winner?

Here we have 4 unusual dice — one has a 3 on all six faces, one has four 4s and two blank faces,
one has three 5s and three 1s, and the last has two 6s and four 2s.

Suppose you are competing with a friend. Each of you can choose a die, and the winner is the person who most often rolls the largest value. Being a fair-minded individual, your friend kindly allows you to choose first. Which die should you choose?

Simulate the game using the app below and see if that helps you make your choice.

Scores

333333 | 0-0 | 0-0 | 0-0 |

4444-- | 0-0 | 0-0 | |

555111 | 0-0 | ||

662222 |

(Scores are written a-b, where a is the number of wins for the dice from the label to the left, and b is the the number of wins for the die from the label below.)

Is your friend actually being as fair as it seems?

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.