When you were a kid and had a gumball to share but 2 of your friends wanted the gumball, what would you do?
When I was a kid, and now that I have two kids of my own, a common way to resolve disputes like this is to say “Pick a number between 1 and 10 and whoever is closer gets the gumball”.
This works with any number of people and with any number of rewards because you can say the N closest people out of M people total get a reward. Ties can be broken by repeating the process.
It SEEMS like a purely random choice too, doesn’t it?
Well, if you want to win, the right move is to always guess 5 or 6, actually. It makes sense if you think about how 5.5 is right in the middle, so 5 and 6 are the numbers least far from all the other numbers.
If you are only playing against one other person, and both the real number and the guess are really random (uniform independent random numbers), guessing 5 will let you win 55% of the time, and tie 14% of the time. That means you will lose only 31% of the time, or less than one out of three times!
There is a way to make this game fair again though, and that is by letting the numbers wrap around between 1 and 10 as if they were on a circle instead of a number line.
So, if the right answer is 1, and you guess 10, without wrap around, the distance would be 9. With wrap around, the distance would be 1.
You calculate wrap around distance by saying “if the difference is greater than 5, then the difference is actually 10 minus that number”.
Playing that way, it puts all guesses on equal footing. Every number acts like a 5, because every number is 5 or less away from every other number.
In that scenario, guessing a 5 will cause you to win 41% of the time, tie 18% of the time, and lose 41% of the time.
The game is fair again, as is shown experimentally below.
The code that generated this output is at:
Another way kids have to solve these sorts of situations is through “inka binka” and similar, where there is a song and a deterministic process to pick a winner.
My 5 year old already realized the same position wins every time, which i was pretty proud of hehe.
This sort of stuff is pretty neat though and makes me think people out there must be studying childhood computational sociology 😛