Pick the Smallest Number (That Nobody Else Picks)

It's the start of another school year and I have a game I play with all of my new AS classes. Everyone gets a piece of scrap paper and you have to pick an integer from 1 to infinity. The person with the small number that nobody else picks is the winner.

Once everyone has had 10 seconds or so to write down a number you ask, "Who had number 1?" If only one person does then they win, if not you ask for 2s and so on. Typically with a class of 15 - 30 you will get either 1 or 2 winning. It rarely gets beyond that.

After people have seen that picking the number 17 or so was not going to win you play the game again. Typically everyone reacts to the first game by picking smaller numbers and you will have loads of people picking 1 or 2. This time some number slightly higher will win.

Play this an infinite number of times (which I am yet to do with a class) and the guesses should even out to what we call a Nash Equilibrium. I think it would be fun to run a computer simulation of this to find the optimal guess for playing against n people.

After everyone has caught on I suggest we play again, but with me in the game. However I tell the class that I am going to pick the number 2 and hold it up for them to see. What is the correct strategy to play here? What would you pick?

Well 1 undercuts me, and usually you will get perhaps a quarter of the class try it. 2 as a choice definitely loses. But unless someone takes me down and picks it then all the higher numbers will lose. You need some selfless individual to make it possible for anyone else in the class to win.

I have probably played this with 10 classes over my time as a teacher (from 11 year olds up to 18 year old Further Mathematicians) and the pattern in all of these games is the same. Someone always picks 2 (except, for the very first time today I got away with it and managed to win).

What mathematics can we do on this game? Well we can solve some of the smaller cases. First have a think what the strategy is in a two person game.

Well, if you always pick 1 you either win or draw, so there is never any reason for either person to pick anything but 1. The game devolves.

How about in a 3 person game? Firstly let's simplify it by pointing out it isn't worth picking a number bigger than 3 at any point. For each of the three options that you could pick we can make a table of whether you win, draw or lose:

If you pick 1 then you have 4 options of winning and 1 of drawing. This is the best option in isolation. But as option 1 becomes more popular you will start losing or drawing with more frequency and so you switch to 2 or 3. But as more people do that 1 becomes more popular again; we have a cycle emerging. However, unlike the similar line of logic with Rock Paper Scissors not all of the options are equal. The correct choice is to vary your pick, with option 1 half of the time and the other two options a quarter of the time each.

For those of you familiar with the film "A Beautiful Mind", this line of thinking in game theory is called a Nash Equilibrium, named after John Nash, the protagonist of the film. Nash and his wife died in a car crash last year (2015) aged 86.