Nontransitive dice are really messed up:

Let's limit ourselves to normal six-sided dice. We have three dice with the following numbers on their faces:

A: 2,2,4,4,9,9

B: 1,1,6,6,8,8

C: 3,3,5,5,7,7

Each die is 'fair' in that it is symmetrical and evenly weighted - in the long run it will land on each face equally often.

Now we're going to have a competition - we each choose one die and then roll them - whoever rolls the highest number will win one dollar from the other player. Then we can choose again - we're going to repeat this one hundred times before the game is over.

I generously let you always make first choice, so that you can choose the die that you think stands the best chance of winning - you can change your choice each time or stick with the same die if you prefer.

The amazing fact is that this is a sucker game - whichever die you choose I can choose another one that will (in the long run) win me money.

A beats B 5/9 of the time.

B beats C 5/9 of the time.

C beats A 5/9 of the time.

So in our 100-roll match, I would expect to win about 56 of the rolls and only lose 44 meaning I would expect to take about $12 from you.