An Optimization Problem with a Surprisingly Simple Solution

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’ guesses. Most of the papers on similar games (see, for example, [1], [7], [2], and [3]) make the assumption that your friends will behave game-theoretically. That is, they will make guesses based on what they think your guess will be, knowing that your guess also depends on what they are likely to guess. With this approach it seems necessary also to assume the number of players is small, for if n is large such strategical thinking is probably not as feasible and the computations may become much more difficult. Here we are primarily interested in large n, so we take a different viewpoint: we assume that you have probabilistic knowledge about your friends’ guesses. That is, you do not know what the guesses will be, but you know that they will all come from a particular probability distribution. We will consider several distributions in the sections ahead, but for now let us focus on the mathematics of a relatively simple case. If your friends are all computers programmed to guess a uniform random number between 0 and 1, then the optimal strategy can be found. Before we do it, though, take a moment to guess what the answer is. If you