Suppose someone challenges you to this game. Feeling confident about his judgment and ability to outguess you, he is willing to lay you $101 to $100 per play. We’ll assume you too feel your challenger has the best of it in terms of judgment. Nevertheless, by employing game theory, you can gladly accept the proposition with the assurance that you have the best of it. All you have to do is flip a coin to decide whether to put out one or two fingers.
If the coin comes up say, heads, you put out one finger; if it comes up tails, you put out two fingers. What has this procedure done? It has completely destroyed your opponent’s ability to outguess you. The chances of your putting out one or two fingers are 50-50. The chances of a coin coming up heads or tails are 50-50.
However, instead of your thinking about whether to put out one or two fingers, the coin is making the decisions for you, and most importantly it is randomizing the decisions. Your opponent might be able to outguess you, but you are forcing him to outguess an inanimate object, which is impossible. One might as well try to guess whether a roulette ball is going to land on the red or the black.