This is one of the most insightful fairy tales. The most famous rendition is by Hans Christian Andersen, but, according to Wikipedia, there are various earlier versions around. Here is a very short version. Swindlers sell the king what they claim to be a magnificent set of new clothes with the feature that only competent or clever people can see them. When a series of officials and then the king himself inspect the new clothes, they see nothing but pretend otherwise. Eventually, the king leads a procession through the town and still everyone claims to see the magnificent clothes until a child finally cries out that the emperor is naked. At this point, everyone realizes that they have been fooled. In this blog post, I want to provide a first decision-theoretic and then game-theoretic underpinning for this story (with a big thank you to PB). I will use psychological game theory, even if that’s not strictly necessary, and one can also see it a bit as rational herding.

Let me first look at this problem from just one person’s point of view. This person should probably realize that there are two possibilities: that what the swindlers say is true or that this is all a hoax. The swindlers probably didn’t introduce themselves as swindlers, only we the reader already know that there are swindlers. But a reasonable person should arguably attach at least some positive probability to the possibility that this is a hoax, in which case even clever people would see no clothes. Call this (subjective) probability of a hoax some number between zero and one, with the (subjective) probability of the swindlers being truthful. Next, let me allow for the possibility that this person we are looking at does not fully know whether or not they are stupid. Let also between zero and one, be this (subjective) probability of being stupid, with the (subjective) probability of being not stupid, call it clever. Suppose that this person, furthermore, if they believe the swindlers to be truthful, believes that they are fully truthful. This means that there are only stupid and clever people (nothing in between) and stupid people, with probability one, do not see the clothes, while clever people see the clothes with probability one. If the whole thing is a hoax, then, of course, everyone, clever or stupid, would see no clothes.

Now suppose, as seems to be the relevant case at hand, that this person doesn’t see any clothes. If this person is rational, despite perhaps being stupid, what should this person now believe about their own stupidity and about whether or not the whole affair is a hoax? We are interested in the probability that this person is stupid conditional on this person seeing no clothes, which is given by

Why? The numerator is equal to because the probability of being stupid and seeing no clothes is the same as that of being stupid. After all, irrespective of whether this is a hoax or true, a stupid person would not see any clothes. The denominator is equal to because in the case of the whole thing being a hoax, probability all people (also the clever ones) would see no clothes, and only in the case of it being true, probability only stupid people, probability would see no clothes.

Anyway, what does this mean? Well, it means that this person when seeing no clothes updates their belief about their own stupidity to something higher than before, as long as (otherwise they are ex-ante sure that this is a hoax, in which case they learn nothing about their own stupidity from seeing no clothes) and as long as (otherwise they know they are clever and no evidence can change that). Perhaps importantly, if this person is fairly sure that the whole thing is no hoax, that is when is close to zero, then the fact that they see no clothes would essentially tell them with almost certainty that they are stupid.

We can similarly compute this person’s updated belief about the whole affair being a hoax. The probability of that is given by

This person will, therefore, also think that the probability that this is a hoax has gone up after they see no clothes provided (otherwise they are sure that this is all true, and no amount of evidence can change that) and provided (otherwise they are sure they are stupid and so they learn nothing about whether or not this is a hoax from seeing no clothes).

So, this is what this rational, yet possibly stupid, person learns from the fact that they see no clothes. But why do they not honestly declare that they have not seen any clothes? To answer that properly, we need to think about the other people involved in this interaction and this takes us to game theory.

The, or at least a way to rationalize this person’s behavior is by assuming that they care about what others think of them. Their utility from their own (and everybody else’s) actions depends (at least to a sufficient extent) on the induced posterior beliefs that the other people have about this person’s stupidity that arise from these actions. If this is what we assume, we are in fact in the more recently developed realm of psychological game theory (See this 2022 article by Battigalli and Dufwenberg for a recent survey). In standard game theory, utilities (also called payoffs) only depend on states and actions, not on beliefs. Having said that, there is an easy, and not entirely implausible, way to use standard game theory here as well. Instead of caring about what others think of them, people (especially those officials perhaps) might just care about keeping their job, and they keep their job only if they are deemed clever. But I do think that psychological game theory is, if nothing else, at least a nice shortcut to modeling situations such as this one.

Anyway, let us assume that everybody involved in this interaction cares mostly about what others think of them: the subjective probability that others attach to them being stupid negatively enters the utility function. For instance, a concrete model could have a utility function for a person of minus the average subjective probability that others attach to this person being stupid.

When you think about it this way, you realize that how certain a person is about their own stupidity is not so very relevant to their own choices. Even if you know that you are clever, and, therefore, when seeing no clothes, know that this is all a hoax, you might say that you see clothes. It all depends on whether the others know that you are clever. If others are unsure whether you are clever or not, then you will say you see clothes so that the others don’t think that you are stupid, even if you know that you are not.

So, when you see no clothes, and are asked to make a public statement about what you saw, the relevant parameters are not your own assessments and but everybody else’s assessments about whether or not this is a hoax and about your possible stupidity.  As we have seen above that the subjective probability assessment of you being stupid would go up if you were truthful, and as you dislike this, you will not be truthful and say that you have seen clothes.

This is also why it is not enough for one person (who privately thinks themselves very clever) to declare that they have seen no clothes for all to suddenly realize that this is a hoax. For this to happen, it takes a person, who everybody sees as not possibly being stupid, to declare that the emperor is naked. I am not sure why a child satisfies this criterion, but I seem to remember that there was an argument in the story – maybe it was not about stupidity, but about being an adulterer or -ess, or something else that could not possibly apply to a child, but could apply to all adults. 

I now want to finally turn to the social dilemma that this behavior induces. In the story, there is a series of people who observe the state of the king’s clothing and they successively (somewhat publicly) declare that they see clothes. So far, I have, essentially, only argued why the first person might declare that they see clothes when they do not. But if the second person understands that the first person would always declare to have seen clothes, irrespective of what they have actually seen – and a bit of introspection might lead to this conclusion – they have learned nothing from the first person’s statement. So, the problem for the second person is just the same as that for the first person. As long as nobody has declared to have seen no clothes, the problem is the same for everyone (ok, this is true under the assumption that we are all somewhat alike). The tragedy of this phenomenon is that, if it hadn’t been for the child, nobody would have learned that the swindlers were exactly that, swindlers. Everybody would have privately updated the chance of this being a hoax to a slightly higher probability, yet all together they would have had enough information to find out the truth. If everybody (against their incentives) had truthfully declared what they saw, all would have quickly found out that nobody had seen any clothes, and as long as they all believe that there is at least some proportion of non-stupid people in this town (that is that ), they would have realized that the chance of this being a hoax, conditional on all this information, is pretty close to one. The behavior of the people in the story of the emperor’s new clothes, thus, prevents learning, somewhat like in the case of rational herding (as in a previous post) where learning stops after people start to discard their own information and imitate others instead.