Four Boxes

The following article has nothing to do with MMORPGs. I had read about a game theory problem online and wanted to replicate the results with a larger sample (although I felt it was more of a psychological problem than a game theory problem). Respondents were shown the following:

Sam is going to hide $100 in 1 of 4 boxes, as seen below. After Sam has hidden the money, Pat has one chance to guess where the money is hidden. If Pat guesses correctly, Pat will get the money, otherwise Sam gets the money.

Each respondent was randomly assigned into two conditions:

1) If you were Sam, where would you hide the money?
2) If you were Pat, where would you look for the money?

Respondents were also asked to briefly explain their reasoning behind their choice using the abbreviations A1, B, A2, and A3 in their explanations.

Before going on to see the results and the reasoning that respondents gave for their choice, you may want to think for a moment as to where you would have hidden or looked for the money.


Below are samples of the explanation of choices from respondents. Responses were interesting in that many explanations were very convincing in a particular selection, but in their overall persuasiveness began to reveal the intractable "what if" black hole that could easily be used to argue for equal distribution of choices among the four boxes.

Hide (A1):

When making guesses, people very rarely pick the first option. Thus, hide it in box 1. [m, 19]

The first box seems the least likely to be picked by Pat. B is obvious, but too obvious so it may be picked. The 4th A is the farthest from B so instinctively it may be picked. The 3rd A is between these two boxes and draws more attention than the First A. Therefore, the 1st A seems the most likely not to be picked. [m, 35]

Find (A1):

It wouldn't be B, as that's just too obvious (or is it?) I guessed box 1 as no-one ever seems to pick the first answer for anything, so it would be that one (knowing my luck) [f, 21]

The only possible basis for anything other than random choice would be an inference about Sam's psychology. How would he choose, given that he can see the labels as well as I can. He might decide not to put it under the 'B' box simply because that's too obvious. On the other hand, he might put it precisely there because, if I chose a different box, that would be just that much more funny. Lacking any other information, I decided: 1) 2 is too obvious. 2) 3 is also in the middle, so too obvious. 3) 4 is the most distant from 2, so too obvious. 4) Therefore, 1 is the least distinguished. So, I picked 1 - even though I really have no confidence in any of these inferences at all. [m, 42]

Hide (B):

Surely, Pat will not guess the only box that is labeled differently, box 2. sometimes the most obvious place is the most conspicuous because it's too obvious for suspect. [m, 20]

Here is where having a psychologist for a husband messes with my answering this. I would say Sam would hide the money in B, because it's too obvious a location. If Pat thinks about it at all, Pat will probably pick one of the As. [f, 28]

Find (B):

If I was Sam, I'd put the money in B under the assumption that Pat would think B was the most obvious, and therefore the least likely, choice. Of course, this is under the assumptions that Pat is a relatively bright person, but not one whose studied interdependency problems (i.e. game theorist, CIA operative, etc), and that Sam is relatively greedy. [m, 29]

They want you to think its not in B but it really is. They try to trick you. [m, 15]

Hide (A2):

Box 2 differentiates itself and draws attention. Boxes 1 and 4, on either end, may naturally draw attention as well. Box 3 seems least suspicious even if each box has a statistically equal chance 1:4 of being chosen. [m, 23]

2 is too obvious, which makes some avoid it. But in some cases might make you pick it, just because it's SO obvious. 1 would be the next to worst choice based on the location of the 'obvious' 2. 4 is less obvious so has potential, but it's on the end, so again, it's too obvious. So, I would put it in box 3. [f, 44]

Find (A2):

Box 1 and 4 are on the end and are more obvious choices for Pat. Box 2 is different and also a natural choice. Therefore Sam would put the money in box 3 as it is a less obvious choice out of the four, being the same letter and not on the end of the row. [f, 18]

Since there doesn't seem to be any 'logical' solution, a random logical solution is that box 3 has a letter that is wrong if you wanted to spell ABBA. Since the other letters are correct for ABBA, box 3 is the one most unlike the others and therefore most likely to attract the attention of Sam. :) [m, 27]

Hide (A3):

Box 2 was a tempting choice, but the likelihood someone else would guess that box because it's different than the others would probably be high. I chose the last box because if it were me guessing, box 4 would be my last guess. If I knew more about Pat and the way he thinks, I might have chosen differently. [f, 38]

We read left to right. I consider the first box an obvious choice in addition to B, simply because it's different. I suggest far right as being the least likely because it is last, and because it's one of three like it. [m, 24]

Find (A3):

Most people will assume (often unconsciously) that the money is hidden between the end boxes, 'covered' by them. Therefore they will go for a middle box, and as B stands out as different to the other three A boxes, will most likely go for it as the most attractive option. As Sam has evidently run this game before, and it is not a random placement of the money, then choosing not-B, one of the end boxes, and the one farthest away from B appears to be the sensible choice. [M, 30]

B seems a little too obvious (which is why the money should probably be ther after all). Either of the A's bordering B just seem too convenient. [m, 24]

I hope you're semi-convinced that there is good reason to hide it in any of the four boxes, and it is this overall persuasiveness that leads to the "what if" black-hole scenario, and several respondents argued that the choices would be equally distributed among the four boxes:

Completely random guess. I don't know Sam so I can't bring any reasoning to bear on the question, and even then the game of second guessing the competition again makes it completely random. [m, 38]

There is a 25% chance to hit the correct box, regardless the name of the box. Sam can hide the money in any of the boxes without restrictions, so its only luck to choose the correct box. [m, 35]

Random choice. There is an equal chance for the money to be in any of the boxes. (At least that's what I >think<) [m, 46]


What is striking is that this is not the case. The distribution of choices was far from random. 40% of respondents would hide it in Box A2, and almost 60% of respondents would look in that box as their guess.

One possible psychological reasoning is that people have been shown to like the third option in a multiple choice format. This is why the correct answer on multiple-choice exams is much more likely to be "C", and also why students are most likely to guess the answer "C" on multiple choice exams. To show that this is not simply the "third option" effect. I switched the B box to the third position and continued running the survey.

The results showed that whatever was going on, it wasn't the "third option" effect, and that despite all the convoluted reasoning, there's something very compelling about that middle A box. While it now becomes easy to claim that there was good reason to expect that this is because "xyz", it's also true that it's easy to generate explanations post-facto and that it would have been possible to generate post-facto explanations for any of the four choices.

I'm leaning towards the sense that there is something illogical (psychological) rather than logical about this overall trend. I don't think Box A2 is any more or less logical a choice as any other box. But I would love to hear your comments.