The rock-off fairness fallacy (SOI cross-post)

I have a new blog! Dan and I are doing a joint blog over at thesubjectsofinterest.blogspot.com

Here's my latest post from over there.

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I've noticed that there is an increasing trend for people to resolve disputes or allocate resources using the game of Rock Paper Scissors. The procedure is sometimes called a 'rock-off', as in "let's rock-off for the last slice of pizza"

This procedure is fine in a two person game (assuming no one cheats), but I often see people happily submitting to three-way rock-offs. In these arrangements two people rock-off, then the third person plays the winner, and the winner of that second rock-off is declared the overall winner.

But this procedure is inherently unfair!

Imagine that John, Fred, and Mary are rocking-off for a slice of pizza. John and Fred play first. Mary plays the winner.

For John to win overall he has to win the first encounter against Fred and then a subsequent encounter against Mary. John has a .5 probability of winning the first time and .5 probability of winning the second time. .5 x .5 = .25, so he has a 25% chance of winning the pizza.

The same applies for Fred. He has to win first against John and then against Mary. Both times he has a .5 probability of winning, .5 x .5 = .25, so he too has a 25% chance of winning the pizza.

But Mary, she gets it easy. No matter what happens she only has to play once. Regardless of whether she's playing John or Fred she has a .5 probability of winning that rock off. So her chance of winning the pizza is 50%.

Mary has double the chance of winning overall because she only plays the winner. Great if you're Mary, bad if you're John or Fred.

Better to just draw straws.