*posted: January 1, 2022*

tl;dr: The most interesting thing is what happens after you do so...

As mentioned in my post The problem with PhDs, I have several relatives with doctorates in my family, either a PhD, ScD, JD, or MD, which are all the apex of college degree attainment for various disciplines and hence categorically equivalent. The events described below actually did happen decades ago at a family holiday gathering, although I may be a bit hazy on some of the details. I intend no embarrassment to anyone, which is why I won’t provide any significant identifying details, not even gender.

A PhD relative, myself, and a few other family members were seated around the kitchen table, in transition between one activity, probably a meal, and another, probably a board game. We may have been discussing game theory, a topic of interest to me as someone with passions for computers, logic, and mathematics, although I’ve never taken a formal course in that subject. I have programmed games. My relative almost certainly had not studied game theory, as his/her doctorate was in a non-STEM field. An impartial observer might therefore postulate that I knew a little more about game theory than my relative. But PhDs, recall from my prior post, like to think they know everything. So I challenged my relative to a simple game.

The game is an incredibly simple one-player game that was well known at the time, since it was often featured on the popular television game show Let’s Make a Deal, originally hosted by Monty Hall. Here’s how Monty Hall would describe the game:

*“Welcome, contestant, to Let’s Make a Deal! Behind me you see three closed doors, numbered “1”, “2”, and “3”. Behind one of those doors is a fabulous prize; behind the other two doors is nothing. You don’t know which door contains the prize, but I do. I’m going to first ask you to select a door. I will open one of the doors you did not select, and I will give you the opportunity to change your mind. You will then receive what is behind the door you select. Are you ready? Good! Please select a door.”*

The game theory question: what strategy should the contestant employ to maximize the chances of receiving the prize? Think of your own answer before reading further.

There are only two decisions to be made by the contestant: the original door selection, and whether to change that selection after a door is opened. It should be pretty clear to most that the original door selection doesn’t matter: the contestant has no clue where the prize is, so there’s a ⅓ chance that whatever door the contestant first chooses contains the prize. There’s only one decision the contestant makes that matters: whether or not to change his/her selection after an unselected door is opened.

Here’s how most people, including my relative, misanalyze the game. When the first selection is made, there is a ⅓ chance of choosing the prize. Then, after another door is opened, there are just two unopened doors. Now there is a ½ chance that the prize is behind either unopened door. So changing one’s selection does not improve one’s odds; in fact, it might be a trick by the host, who is trying to get the contestant to make a mistake. Best to stay with one’s original selection and not switch.

Unfortunately for my relative, as I tried to patiently explain, the best strategy is to switch one’s selection. Here’s how I tried then, and try now, to explain this simple game.

The host has actually given the contestant a piece of valuable information by opening a door that does not contain a prize. Originally, there is a ⅓ chance that the prize is behind the contestant’s selected door, and a ⅔ chance that it is behind one of the other two doors. The host knows where the prize is, and opens one of the unselected doors that does not contain a prize. That ⅔ chance of the prize being behind one of two unselected doors now consolidates onto the one unselected door. The contestant should switch, to improve the odds of getting the prize from ⅓ to ⅔.

That explanation didn’t work with my PhD relative. So I broke it down further by trying to lay out the entire set of possible scenarios for the game. Let’s say the contestant originally chooses door “1”. If the contestant chooses another number, just consistently switch the numbers in the analysis which follows. There’s a ⅓ chance the prize is behind door 1, and a ⅔ chance the prize is behind doors 2 or 3:

- If the prize is behind door 3: the host opens door 2, which is empty, so there’s now (from the contestant’s perspective) a ⅔ chance the prize is behind door 3: the contestant should switch
- If the prize is behind door 2: the host opens door 3, which is empty, so there’s now (from the contestant’s perspective) a ⅔ chance the prize is behind door 2: the contestant should switch
- If the prize is behind door 1: the host opens either door 2 or 3 (it doesn’t matter): this is the only scenario of the three in which the contestant loses by switching

So in two out of the three scenarios, the contestant wins by switching; in only one out of three scenarios does the contestant lose. Switching doesn’t guarantee the contestant wins, but it does double the odds, from ⅓ to ⅔.

I still couldn’t get my PhD relative to accept any explanation of mine for how this simple game works. That’s when I learned how difficult it can be to correct a PhD. PhD’s aren’t used to being corrected; they are used to being the smartest person in the room. I’m sure my relative thought of himself/herself as smarter than me, and smarter than a television personality like Monty Hall, and able to outsmart a silly TV game show. Maybe my relative was reacting negatively to the suggestion that he/she should “change your mind”: PhD’s are used to being absolutely correct from the start. Maybe my relative thought that the suggestion of switching was a trick being pulled by someone who, lacking intelligence, was resorting to subterfuge to get him/her to make the wrong decision.

I don’t fully know what my relative was thinking. After refusing to accept any of my explanations, he/she went off to ponder the game further, and didn’t participate in the family’s next activity. He/she never did say that my analysis was correct. As I’ve seen on many other occasions, it is hard for some people, especially really smart people with a PhD, to admit they are wrong.

*Related post: The problem with PhDs*

*Related post: Mistakes weren’t made*