Game with Infinite Mirrors

In 1930's John Maynard Keynes was thinking about operation of stock markets. It was not long after Black Tuesday after all and understanding why the stock prices may not reflect the true value of the underlying enterprise and why they can swing rapidly was of utmost interest.

He came up with the simile that is today known as "Keynesian beauty contest":

Professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one’s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees.

In other words, the dumbest participants will pick the six faces they personally liked the best. Smarter participants will think about what kind of faces people in general tend to prefer and pick those. Then there are those who would try to guess which faces would people consider to be generally likeable. And so on.

It should be said that that there were actual experiments performed and yes, it turns out that surveys did yield different results when subjects were asked to choose what they personally liked and when they were asked about what most people would prefer.

But if you are like me you probably have this nagging feeling that we may be hitting on something more general here and that the beauty contest model may apply even outside of the investment sphere.

Is that so?

Well, let's think about pricing in general. Imagine you want to buy a banana. How do you know what's the fair price to pay? The answer that immediately comes to mind is that the fair price is the price at which the supply and the demand match. But hold your horses! You know nothing about banana supply or banana demand. You have never heard of the terrible banana plague that decimated the plantations south of Malacaya. You've never heard of the growing hunger for bananas in the quickly expanding Chinese economy. So, how do you know what the fair price is?

An alternative answer could be that you base your estimate on the utility of a banana. On how owning one would benefit you personally. You would discard all external factors. But then: How come that you value banana less than a truffle? Quite a few people prefer bananas to truffles, yet they are willing to pay only a fraction of truffle's price for bananas.

To make it short, I believe that people engage in a kind of Keynesian beauty contest when estimating prices of goods. They check what other people consider to be a fair price and base their estimate on that. They may go to a supermarket, check the banana price and accept that. Then they may chat with their friends and find out that they consider the price they've paid in the supermarket outrageous. And so on.

The stakes when buying a banana are low, so there's probably no reasoning being done at third, let alone fourth, level. Still, the mechanism of reasoning at the meta-level, inherent to the beauty contest model, is still recognizable here.

Now, let me explain why I introduced the banana example: In the stock trading case the subject of the estimate is pretty abstract. What's the value of a company? It's not just all the assets it owns. It's also the people who work there. It's ingenuity of its director. It's their business plan. It's the contracts it has signed. It's the current state of the market. It's whether they have competition and whether the competition is going to obliterate them. Surely, this reasoning on ever higher and higher levels of abstraction stems from this complex nature of stock valuation.

But no. The banana example shows that the beauty contest aspect is retained even with such a mundane and tangible object as a banana.

The system is fed inputs from the real world, like facts about the company or facts about bananas. The inputs are then evaluated on different levels: What do I think about these facts? What would an average person think about these facts? What would an average person think about average attitude to these facts?

And you may ask a question: Would it be possible to split the "real world inputs" part of the evaluation process from "infinite mirror" part of it? Would it be possible to think about this meta-reasoning process as such while not being distracted by the ugly real world facts? If so, we may be able to learn something about the nature of the process.

Let's do the following thought experiment: We'll do the old Keynesian beauty contest (hundred photos of faces, people asked to choose six prettiest faces) but this time all the photos will be blank.

It this case you have no "real world facts" to base your reasoning on. The hundred photos are exactly the same. The best you can do is to pick six photos at radom and hope that other contestants will accidentally prefer the same photos as you.

So far, no surprises here. The newspaper evaluates the results and announces the winning photos. Then it runs the contest for the second time.

The contestants still have no "real world inputs". The photos are still perfectly blank. No faces, no nothing. However, they do have the results from the previous round. A lot of people will, because of the lack of better option, choose the photos that won in the previous round. And lo and behold, the same set of photos have won twice in a row! When the contest is run for the third time everbody bets on those photos. The system have reached an equilibrium.

Think about it in game theoretic sense: it's a game with Nash equilibrium at every single point. Any set of six photos can become the winning set forever.

But do we encounter such games in the real world?

Well, we can get pretty close.

Consider a group of strangers stranded on unihabited island. They have no shared language. What one calls "tree" the others call "derevo", "arbol", "Baum" or "shajara".

If they want to communicate, if they want to refer to a tree, they have to choose. None of these words is better than others but they still have to choose. And once the choice is made, there is no going back.

This is idea is not unknown in linguistics. It's called "arbitrariness of the sign" and was first formulated at the beginning of XX. century by Ferdinand de Saussure. It says that there's no inherent connection between the word and what it means. "Shajara" is as good a word to signify tree as "fa" or "strom" or, for what it's worth, "gruxyid".

De Saussure is of course quick to point out that "arbitrary" doesn't mean that the word can be replaced at will:

The word arbitrary also calls for comment. The term should not imply that the choice of the signifier is left entirely to the speaker (we shall see below that the individual does not have the power to change a sign in any way once it has become established in the linguistic community) ; I mean that it is unmotivated, i.e. arbitrary in that it actually has no natural connection with the signified.

Now, with the mechanism of Keynesian beauty contest (calling it "infinite mirrors" would be a better fit IMO, but whatever) we can explain the arbitrariness of the sign in much more precise, game-theoreric way. We can even make a mathematical model of the process which was not possible with the vague Saussurian definition.

Anyway, let's get back to the beauty contest. It may seem that all this blog post is a pointless philosophic exercise with no pratical consequences. But I don't think that's true. For example, it makes it possible to answer some interesting questions. Consider this one: Why doesn't an anarchist who doesn't believe in money throw his money into a gutter? Why does he keeps it for himself or, possibly, gives it to someone else? In other words, why does he still value money in spite of his convictions?

And the infinite mirror model answers: Because your belief is irrelevant when valuating stuff. Instead, you try to estimate the value assigned to an object by everyone else.

The model can also predict some weird, yet believable, scenarios: Imagine that everyone is sick and tired of word "tree" and would prefer to use word "gruxyid" instead. However, given that everybody still uses word "tree" everyone thinks that everyone else prefers "tree" and thus the word "gruxyid", which everyone thinks is superior, never gets used.

By the way, the last years of communist era in Ostblok may have been like that: Nobody really believed in the system any more but it still went on because, shrug, people assumed it will go on.

All the above begs a question: Given that the same mechanism can be identified in fields as disparate as economics, linguistics and political science can it be that we are looking as some kind of generic social phenomenon that transcends individual social sciences?

Maybe. Or maybe not. In any case, it's an interesting topic to investigate.

August 12th, 2017