# Monty Hall, meet Game Theory

### Game Theory

For years I have been moderately obsessed with Game Theory, for me it is more of a window into the nature of the universe. Things are not optimized in societies, they never will be, because what is best for one person may not be for another. The classic example in Game Theory is the Prisoner’s Dilemma, where if the parties cooperate (in this case they don’t betray one another), they both do not lose, but do not necessarily win. However, if one party betrays the other and one party does not, the betrayer “wins.” This means without cooperation it’s always best to betray (since you have a better chance to win), but be mindful if both of you betray both will lose.

Prisoner’s Dilemma Prisoner B stays silent (cooperates) Prisoner B betrays (defects)
Prisoner A stays silent (cooperates) Each serves 2 year Prisoner A: 20 years
Prisoner B: goes free
Prisoner A betrays (defects) Prisoner A: goes free
Prisoner B: 20 years
Each serves 10 years

Hence, the outcome will usually lead to both parties betraying one another, because both parties will attempt to maximize their winnings. In turn, both will end up losing. If you are interested, or do not know Game Theory I highly recommend Game Theory 101: The Complete Textbook for a basic knowledge and Game Theory: Analysis of Conflict for a more in-depth understanding.

### Monty Hall

Let’s make a deal was the game show hosted and created Monty Hall. The Monty Hall Problem was actually not created by Monty Hall at all, but it was loosely based on the idea of Let’s make a deal. The problem states:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice? – Wikipedia

I will not go into the details, but it turns out you have 2/3 chance of winning the game if you switch (don’t believe me?).

### Monty Hall, Meet Game Theory

As it turns out, Game Theory has a solution to this problem, take the best choice. In other words, just always switch after they open the door. This is not very interesting, so we will make the following assumption:

Where the car (or any prize) is hidden by the game host.

This assumption allows you to throw away the dominating strategy to always switch if offered. Instead, we can base switching off two things, (1) the game host and (2) the game host. If we further assume that the game hosts poker face is poor or we have an excellent eye, this can create a situation where it is not best to switch. The conundrum, is that we often do not know that the host has a bad poker face and that he is not tricking us, so we have to tread carefully here. It turns out, that in this case it is better to go with the dominating strategy.

However, if for example you were playing with a friend you should be able to figure out their “tell.” That being said, I did this (I called it a trick, since he was a statistics guy), my friend gave me the option to switch every round. Here was my score after ~750 games with 3 cards over the course of about a month:

Correct Choice: 556
Incorrect Choice: 194
My Accuracy: 74.13%

My strategy was simple: look for a tell, if I see it go with it, if I do not see a tell defer to the dominating strategy (aka switching). Now if I went with the switching every round my accuracy should have been 66.67%, this means I did 7.46% better. This does not necessarily prove the theory, however I think it is strong evidence given the number of rounds.

### Problems with this theory

Every person has a number of different tells and each person is different. Some people will have trouble determining if someone is presenting them with a tell, others might have an excellent poker face. For example, my friend eventually caught on and attempted to start tricking me, which actually worked to my benefit since he was always giving me a tell of some kind. This actually devolved the game into a common Game Theory problem, a zero-sum game and very similar to poker, since at this point he was frustrated and did not want me to win.

I found it interesting that as time went on and as he became an adversary, it became easier not only to be correct, but it devolved the game into a game I understood and had a clear strategy. Further, this means that the game strategy changed over the course of the game and potentially at some point if my friend stopped caring all together I would have to go back to the dominating strategy.

### Summery

If you are confident that enough that you know when you are receiving a tell while playing the a Monty Hall type game, it is likely in your best interest to switch. However, in general the dominating strategy (switching) is still better.

## One thought on “Monty Hall, meet Game Theory”

1. You could reinterpret this story in Bayesian terms. If you’re playing a typical game, your prior odds are 1/3 when you first pick a door. Then, the host eliminates one door. So, now, updating on this new evidence, your odds are 1/3 with this door, or 2/3 when switching.

But let’s say you then observe a sly smile! And you figure about 75% of the time that sly smile indicates the host is trying to fool you. So, now you can update your odds on that evidence, giving you 61% probability that the car is behind the first door.

I like this a lot better. Consider: What if you think there’s a 57% chance that what you saw was a tell? Well, with a naive strategy of always-stay-on-tell, you’re acting irrationally. Switching has a 60% win rate!

Now, if I’m the host, how can I minimize your win rate? You’ve taken out the easy route — I have to be the one to hide the prize given your rules. Here’s what I’d do. Take drugs to disrupt short term memory formation, hide the prize, and leave a note to my future self about which door to eliminate in which situation! If I’m worried that I can figure out where the car is based on my notes, I can just blind myself — “only read the next page if Austin picks door number 3.”

In such a situation, it doesn’t matter if I have a tell. I have no evidence to reveal!