Let’s Make a Deal

The Problem

In the misnomered “Monty Hall” problem, the rules are set out as follows:

1. You as the contestant are faced with the choice of three doors, behind exactly one of which is money or something else of significant value. You choose a door.
2. The host, who knows where the money is, then opens a door that does not hide anything of value.
3. You are then given the option to keep the current choice or switch. What do you do?

The simple answer is: Switch. On your first choice, you have a 1/3 chance of selecting correctly. The host is obliged by the rules not to reveal the money. Therefore, since you had a 2/3 chance of selecting incorrectly at first, you should switch.

Let’s contrast this with another TV show, “Deal or No Deal”. We’ll ignore the banker, the lesser valued cases, and the drama, and make it simple:

1. You as the contestant are faced with the choice of thirty cases, inside exactly one of which is a good deal of money. You choose a case.
2. You, who do not know where the money is, choose other cases to open. You continue so long as the cases are empty, until you are faced with two cases (the one you have and the last one you don’t have).
3. You are then given the option to keep the current choice or switch. What do you do?

In this case, it doesn’t matter. On your first choice, you only had a 1/30 chance of selecting correctly, so you might be tempted to think you have a 29/30 chance of winning if you switch, especially if you’re familiar with the “Monty Hall” problem. But you don’t. You have a 2/30 chance at the outset of getting to the point of being able to choose between the last two cases, and a 28/30 chance of losing before then.

The key difference is that, in the “Monty Hall” problem, the host knows where the money is and makes your choice for you. You have a 100% chance of making it to the final choice. There are only two initial possibilities: You choose correctly and have a chance to switch, or you choose incorrectly and have a chance to switch. Since the odds of the first event happening are 33%, the odds of the second are 67%.

In “Deal or No Deal”, there are three initial possibilities: You choose correctly and have a chance to switch, you choose incorrectly and have a chance to switch, or you choose incorrectly and lose along the way. The last possibility has by far the highest likelihood of occurring.

So what’s the point?

The so-called “Monty Hall” problem is often held up as a case where mathematical intuition fails us. On the actual show, people didn’t always switch. People nattered about switching. They were cajoled by the audience, and by Monty Hall himself. See how stupid people are? When we lay out the mathematics, it’s obvious that people should switch, and yet… they didn’t, not reliably.

Here’s the reality: That’s not really how “Let’s Make a Deal” worked. Sure, sometimes Monty would play it out that way, but other times he’d make people stick to their original choice. Sometimes he’d dangle money to buy contestants off.

If “Let’s Make a Deal” really worked reliably as described above, the show would be giving away money more often than not. That assumes that the show’s producers want that. The reality, as the contestants and audience know, is that the producers want to give away just enough money to keep people watching the show, thinking that they too might someday get some of that money.

In “Deal or No Deal”, the trick is that the banker generally calculates the average winnings based on the remaining cases, and then offers slightly less than that. Both shows are designed to minimize the show’s actual payout while maximizing the contestants’ expectations of winning.

In both cases, the contestants ought to be aware that the hosts are not on their side. “Deal or No Deal” deliberately messes with this notion by having a bad guy (the banker), leaving the host to be a good guy, but there’s no such pretense in “Let’s Make a Deal”.

If the show really worked consistently like the problem, then contestants would quickly figure out the trick, and Monty Hall would be a damned fool.

Ja, und?

This isn’t a trivial matter. “Monty Hall is just used to frame the problem,” goes the defense. “People are familiar with the show, so it helps people understand the problem.”

The question is: Does it help people understand the problem, or does it confuse people? Would people have a different intuition about the problem if it were framed differently? Consider some possibilities:

1. A friend has hidden a token in one of three boxes. If you find the token, he’ll let you have the larger piece of cake.
2. A game show host has hidden a million dollars in one of three boxes. If you find it, he’ll let you have it. Otherwise, he keeps it.
3. A game show host has hidden a token in one of three boxes. If you find it, you’ll be the next contestant. Otherwise, the person to your left will be.

The rules for finding the hidden item are the same as above: The hider knows where it is, and won’t reveal it.

The probability in each of these cases is the same: You have a 2/3 chance if you switch. The question I’m curious about: Are intuitions different based on the scenario? In the first case, the hider is a friend, and only has a minor reason for trickery. In the second case, the hider has a significant impetus for trickery. In the third case, the hider has no reason for trickery at all.

I don’t know whether this changes mathematical intuition or not. I’m not sure if there’s ever been research on this. My point, though, is that we tend to take it as a given that people have poor mathematical intuition in this situation when the real issue may well be that people’s judgment is clouded by real world knowledge contained in the “framing”.