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Non transitive dice

Suppose you were given three dice with different numbers than usual dice. One is red, one green, and one blue. Your friend asks you to play a betting game. You pick a die and your friend then picks a die. Now you both roll the dice and whoever has the highest number wins. It seems like a very simple game. Also, you pick first so intuition is likely to say that you have the advantage as you can pick the "best" die. However, this is highly misleading. On closer inspection you find that the dice have the following faces:

Red: 1, 4, 4, 4, 4, 4
Green: 2, 2, 2, 5, 5, 5
Blue: 3, 3, 3, 3, 3, 6

These seem to be balanced. After all, the average roll on each die is 3.5. However, somehow your friend keeps on winning. Consider the red and green dice. If the red die rolls 1 and the green die rolls 2, the green die wins. This has probability 1/12. Likewise, if the red die rolls 1 and the green die rolls 5 (a probability of 1/12), the green die wins. Similarly, if the red die rolls 4 and the green die rolls 5 (a probability of 5/12), the green die wins. These are the only ways in which the green die can win. The total probability that the green die wins is 7/12. Thus, in the long run, the green die beats the red die. Likewise, the probability that the blue die beats the red die can be shown to be 7/12.

Now, if in the long run the green die beats the red die and the blue die beats the green die, you might think that the red die is the "weak" die and the blue die is the "strong" die. Thus you would expent the blue die to, in the long run, beat the red die. However, the red die beats the blue die with a probability of 25/36, or almost 70%. This is an even more resounding win than before. The dice can be likened to the game of rock-paper-scissors. This is how your friend wins so often. Since you pick first, your friend keeps on picking the die that beats yours.

Now suppose you have these dice and are winning against another friend. He becomes very suspicious after a while so you show him the secret. You then pick first but this time raise the stakes. Now both players roll the dice twice and add up the scores. Your friend is probably initially happy as he thinks he has an increased chance of winning. However, he is in for another shock. Suppose you pick the red die. He picks the green die, expecting to win, but now each person rolls twice. In this case red beats green with a probability of 85/144. Likewise, green beats blue with a probability of 85/144. Also, blue beats red with a probability of 671/1296. You are advised to be cautious here and rarely, if ever, pick the blue die. This is as the winning probability is roughly 52% so the game could go either way.

Now back to single die rolls. The worst case probability of winning is 7/12. With six-sided dice, this is optimal. With dice with arbitrarily many sides, the optimal probability approaches φ - 1, which is roughly 0.618. You can also have four dice, in which case the optimal winning probability is 2/3. This is achieved by Efron dice, named after Bradley Efron who invented them. This time, the dice are as follows:

Red: 3, 3, 3, 3, 3, 3
Green: 2, 2, 2, 2, 6, 6
Yellow: 1, 1, 1, 5, 5, 5
Blue: 0, 0, 4, 4, 4, 4

Now Red beats Green beats Yellow beats Blue beats Red, each with a probability of 2/3. Unfortunately you cannot reverse the chain by rolling twice. Incidentally, Green beats Blue with a probability of 5/9 but Red and Yellow dice win exactly half of the time against each other.

Another, more interesting set of dice is Van Deventer dice, named after the Dutch puzzle inventor Oskar van Deventer. It has 7 dice and has the remarkable property that you can invite two friends and win against both at the same time. Here are the dice:

Red: 7, 7, 10, 10, 16, 16
Orange: 5, 5, 13, 13, 15, 15
Yellow: 3, 3, 9, 9, 21, 21
Green: 1, 1, 12, 12, 20, 20
Blue: 6, 6, 8, 8, 19, 19
Indigo: 4, 4, 11, 11, 18, 18
Violet: 2, 2, 14, 14, 17, 17

In this case the chains are very complex. Red beats Orange beats Yellow beats Green beats Blue beats Indigo beats Violet beats Red. Also, Red beats Yellow beats Blue beats Violet beats Orange beats Green beats Indigo beats Red. In addition, there is a third chain. Red beats Blue beats Orange beats Indigo beats Yellow beats Violet bears Green beats Red. In every single case, the probability of winning is 5/9. This system ensures that no matter which two dice your friends choose, you can always find one that beats both.

However, the probability of beating both opponents is still only 39%. Since you have the advantage against both opponents, this is higher than the chance of losing against both opponents, which is roughly 28%. Suppose you each bet £1 on the result. The person with the lowest number loses the money. The person with the middle number gets his money back. The person who has the highest number wins £2, giving an overall profit of £1. Every time you play, you will gain roughly 11p on average.


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