Let's Make a Deal: Game theory is confuisng

Discover magazine used to have an installment called Fuzzy Math, in which the author described some sort of mathematical probability which was contrary to what seemed intuitive.

Fuzzy math

From the game show Let's Make a Deal, this article describes a contestent given a choice of three doors, one has a prize behind it. The contestant chooses a door, the host then opens one of the other two doors to show that nothing is behind it. The contestant can then either choose to stick with the first choice or change to the remaining door.



Through "fuzzy math'' the author reasons that it is always smarter to choose the other door not originally picked, even though you would think it doesn't matter, the chance of winning should be 50-50.

Honestly, I'm not even sure if this is really fuzzy math or just game theory at it's finest. It seems to me that you can use reasoning to make the optimal choice go either way. The example in class of mr. row and mrs. column choosing either to invest or to save obviously had a situation that was ideal for both players. But in this Fuzzy Math, things seem really, uh , fuzzy to me.

What do you think about the suggestions of this article?
Do you think game theory gets fuzzy?

This is actually not game theory. It is a game about statistics. The way it works is that after you pick one door, Hall will open another door showing nothing is behind it. The key part is that Hall will only open a door with nothing behind it which means his pick is not random. If his pick was random then there would be a 50-50 chance that the prize will behind the last two doors because there would be some chance that he would pick a door with the prize behind it and the game would be over.

Here is another way of thinking about it. If there are three door all with equal chance the prize is behind it, then there is a 1/3 chance you picked the door with the prize behind it on your first pick and a 2/3 chance one of the two other doors have the prize behind it. Since Hall is only going to pick the door without the prize, then the chance you did not pick the correct door is still 2/3 expect now there is only one door in the group rather than two. That is why it is always better to switch.

That example is a good example of what happens when you someones does actions that are not random, the normal statistic math we are use to doesn't apply.