Summary: An Optimization Problem With a Surprisingly
D. Drinen, K. G. Kennedy, and W. M. Priestley
Suppose you and n of your friends play the following game. A random number
from the uniform distribution on [0, 1] will be generated. This number is called
the target. Each of you will independently guess what the target number will
be and the person whose guess is closest will be declared the winner. In order
to investigate an optimal strategy for this game, we need to assume something
about your friends' guesses.
Consider first the case where you have complete knowledge of all your friends'
guesses before you make yours. In that case the optimal strategy is trivial:
simply order their guesses and then find the largest gap between successive
guesses. If that gap is at least twice as large as that between 0 and the smallest
guess and also that between the largest guess and 1, then position yourself
halfway between those two guesses. If not, then position yourself as close as the
rules allow to the left of the smallest guess or the right of the largest guess, as
appropriate. Your friends would likely not find this game to be worth playing
and even you would probably not find it interesting.
Let us now assume that you do not have exact knowledge of your friends'