Think about that for a little bit while I discuss some recent research*. They tested to see how pigeons responded to this problem, with food as a reward. They put the pigeons in a box and had them peck one of three lit buttons. Just like in the game show, one of the buttons was associated with a food reward and the others had nothing. After the pigeons pecked a button, the buttons would go dark and then two buttons would light up, with the button that stayed dark being one that had no food and that was not chosen first. The pigeon then chose which lit button to peck. It either pecked the same one it had before, or switched.
They found that
... birds adjusted their probability of switching and staying to approximate the optimal strategy.
They then claim that
Replication of the procedure with human participants showed that humans failed to adopt optimal strategies, even with extensive training.
But I read the paper closely, and found that this statement is not true. There were two important differences. First, the human subjects had far fewer trials, 200 versus over 2500 for the birds. The birds did not choose optimally on the first day; it took days and days of 100 trials each before they learned the right strategy.
Second, and most importantly, the humans did not receive any actual reward for choosing right. They were given course credit for showing up and participating, and told to try to win as many points as possible, but there was zero incentive for actually doing so aside from seeing the word 'Win' on the computer screen. I imagine that the humans (undergraduate psychology students) just wanted to get out of there as fast as possible, and kept hitting the same button until the experiment was over.
This illustrates a very important point about learning and efficient behavior. All organisms, including people, are fairly bad at reasoning and predicting but fairly good at learning through trial and error. After many rounds of trial and error, behavior starts to approach the optimal, efficient, and rational behavior that economists and statisticians and game theory experts calculate. But you have to give them repeated trials, and the feedback has to be immediate and relevant. Learning the efficient behavior pattern will only happen when the subject is immediately given a real reward for doing the right thing.
Now for the solution. You should always switch doors. Your chances of winning are one in three if you stay and two in three if you switch. This is because there was a two in three chance that you picked a door with nothing. If your door has nothing, then the host must open the other door with nothing and you will always win by switching. Pigeons who switched were twice as likely to get food as pigeons who did not switch, and they learned this fact after a few days and adjusted their behavior.
* Are birds smarter than mathematicians? Pigeons (Columba livia) perform optimally on a version of the Monty Hall Dilemma
Walter Herbranson & Julia Schroeder
Journal of Comparative Psychology, February 2010, Pages 1-13
I could not find any free copy of the article online