The Dice Game You Can Never Win
Imagine a set of 4 dice in front of us, you can pick any dice and then I pick a dice. We then roll these many times, the winner is the person who rolls a higher number more times. No matter what dice you pick I guarantee that I will win. How?
Let’s take a step back. Take a standard dice:
If we both roll this dice many times you will win just as often as I will, this makes sense as we are both rolling identical dice.
Let’s now take a look at the following four dice:
This situation is a bit trickier, what is the best dice to pick? We can play them off against each other to see.
The colored green squares correspond to the dice mentioned on the left side of the image winning while the red colored squares correspond to the dice mentioned on the top side of the image winning.
You can skip these images if you want to just see the results.
Summarizing these:
- Dice A beats Dice B
- Dice B beats Dice C
- Dice C beats Dice D
Let’s start the game, you go first and pick a dice. What dice should you pick?
The only choice seems to be Dice A, if you pick any other dice I can simply pick the one that beats it from the list above.
Ok great you’ve picked Dice A. I pick Dice D and I win! How?
Let’s plot the outcomes to see this, green squares correspond to Dice A winning and on red squares Dice D wins:
What’s going on here ?
Intuitively we expect that if A beats B, B beats C and C beats D then A should beat D?
The technical term for this expectation is called transitiveness.
The key to understand the result that Dice D beats Dice A is that these set of dice are not-transitive!
What other systems are not transitive ?
Consider the Rock, Paper, Scissors game.
In this game Paper beats Rock, Rock beats Scissors and Scissors beats Paper. This game is also non-transitive.
We really don’t have great intuitions for non-transitive games. Evolution may have played a role in this, if food pile A is larger than food pile B and food pile B is larger than food pile C, it’s reasonable to conclude that pile A is larger than pile C.
We just don’t stumble upon many situations where non-transitiveness occurs.
