> I can not understand why a rational actor would ever pick a number larger than the number of participants. If there are two of you and you're both always pic

> I can not understand why a rational actor would ever pick a number larger than the number of participants. If there are two of you and you're both always picking 1 or 2, you're going to both lose half the time when you collide. You'll win 1/4 of the time. If you instead pick from 1-3 you're only going to collide 1/3 of the time, and you'll win 1/3 of the time. 1-4, 1/4 collision, 3/8 win rate. (I think I've got the maths right, based on picking numbers uniformly.) If you accept this is logical but decide to actually only ever pick 1 or 2, your logical opponent will surely decide the same and you'll both be worse off.