A popular bull case for Bitcoin is “the bubble theory of money” where:

1. The asset rises in price due to speculation and forms a bubble
2. The bubble doesn’t fully “pop” as once the rate of new buyers slows down, the volatility decreases and the price stabilizes
3. When the price stabilizes, people will use it as money

There’s a bit more to it (Vijay Boyapati wrote about it exhaustively here), but the gist is sufficient for this piece.

JP Koning wrote a critique of this theory, expressing doubt that a “Keynesian Beauty Contest” would ever stabilize. He’s not convinced that the price of Bitcoin holds in step two of the bubble theory. It’s an excellent piece and you should read it, but after reading it, I was still unsure whether a Keynesian Beauty Contest would ever stabilize.

So this post is about that. What is a Keynesian Beauty Contest? When does it stabilize and when does it not? And how does it play a role in and out of crypto.

## Keynesian Beauty Contests

A Keynesian Beauty Contest describes a game where the players are incentivized to take actions based on predicting the actions of other players (who are also trying to predict the actions of other players). Keynes introduced the concept in a 1936 paper called The General Theory of Employment, Interest and Money:

…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 practice the fourth, fifth and higher degrees.”

Player behaviors are sometimes described as playing at n-degree where:

• zero-degree players pick at random
• first-degree players pick who they think is most beautiful
• second-degree players pick who they think other people think is most beautiful
• third-degree players pick who they think other people think other people think is most beautiful
• and so on and so forth

The concept of “Keynesian Beauty Contests” come up most frequently around markets, particularly speculative ones, because its actors are trying to predict what other market participants are thinking. First-degree traders buy what they think is undervalued. Second-degree traders buy what they think other people think is undervalued. And so on.

This makes for a market susceptible to speculative bubbles because there’s no “fundamental value” that the market can anchor on. As I mentioned in “The Narrative Bubble Loop”:

an environment where narratives fuel speculative bubbles have the following three properties:

1. Lack of reliable or relevant historical data to form valuations
2. Conditions that attract retail investors, oftentimes poor regulation
3. Relative strength of narratives to grab attention in an opportunity rich investment environment.

This maps neatly with what we’ve observed in crypto. Because we lack a proven valuation model for cryptoassets, narratives drive investment decisions.

Without reliable or relevant historical data to form valuations, market participants don’t align on a valuation method for the assets. As a result, they play a “Keynesian Beauty Contest” and try to predict what each other are thinking1.

This leads to high volatility and fragile market structures, as we’ve seen with the spectacular run-up and correction of all crypto markets. So do Keynesian Beauty Contests ever stabilize?

## Nash Equilibriums

Keynesian Beauty Contests are often studied in experimental economics with the following number guessing game2:

• Three or greater players play this game
• To win the game, guess a number between 0 and 100 that is closest to 66% of the average number guessed

The largest number that a self-interested player would choose is 66 (the winning number if everybody picked 100). Thus, they should expect everybody to pick a number of 66 or lower. Since they might expect others will think this way, the next highest number they should pick is 66% of 66, or 43. And so on and so forth. This experiment has been conducted in many settings and the answers tend to be around 20.

If the same players play multiple times, eventually the number reaches an equilibrium of zero, as players have no reason to believe other players will pick numbers larger than it.

In a 2004 study by Kocher and Sutter titled “Time is Money”, longer periods between rounds of the guessing game led to a more rapid trend towards the equilibrium

Here is a case where a Keynesian Beauty Contest stabilizes. This is known as a “Nash Equilibrium” where players do not change their behavior while knowing the equilibrium behavior of everybody else. The current state is most optimal for all.

This game shows us that Keynesian Beauty Contests stabilize when there is a Nash Equilibrium3.

You might argue that stocks also form a Nash equilibrium around their “fundamental value.” Koning cites $AMZN (Amazon) as a stock that is no longer volatile. Why has Amazon stabilized, and will bitcoin do the same? When Amazon shares debuted back in 1997, earnings were non-existent. […] I’d argue that the stabilization of Amazon hasn’t been driven by a larger market cap and/or growing trading volumes. Under the hood, something fundamental has changed. The company’s business has matured and earnings have become much more stable and predictable. And so has its stock price, which is just a reflection of these fundamentals. In other words, market consensus on a “fundamental value” creates a Nash equilibrium. This requires a valuation methodology4 (e.g. multiple of earnings, discounted cash flow). ## Does the price of Bitcoin have a Nash equilibrium? Unlike the number guessing game, the price of Bitcoin does not have a Nash equilibrium. There is no valuation methodology that everybody agrees on (and there likely won’t be one). At any given point in time, some people can think it is rat poison and others can think it will be the global reserve currency. And even if the vast majority of people think it will be the global reserve currency, it’s impossible to assign an equilibrium value to it. The best we can do is anchor the value to known assets like gold or money supply. If enough people agree that the price should be equal to (or some multiple of) an anchor, you reach a sort of equilibrium. For instance, if a lot of people agree that Bitcoin should be worth as much as gold, then each Bitcoin should be worth$380K. However, this only holds for as long as everybody thinks that everybody else believes in this anchor.

This isn’t quite a Nash equilibrium. Recall a Nash equilibrium has the following requirements:

• everybody knows the equilibrium behavior of everybody else
• everybody does not change their behavior

These requirements are not met because Gold is just one of many possible equilibrium behaviors5 and self-interested market participants will constantly try to manipulate the equilibrium in their favor. Equilibrium exists at any given point in time (the spot price) but nobody knows what the equilibrium behavior of everybody else is and everybody is willing to change their behavior given new information of the equilibrium behavior of others.

If you accept this logic, Bitcoin does not have a Nash equilibrium and therefore does not have a reason to stabilize.

But doesn’t this logic also suggest that no liquid assets have a real Nash equilibrium? Tesla has a “fundamental value” yet there are fanatics that think its highly undervalued and fanatics that think its highly overvalued. They aren’t all happily organizing around one way to value the company. Unlike the number guessing game that has a clear equilibrium, real world assets have multiple equilibriums of varying strength.

A useful way to think of this is in the context of Ben Graham’s popular adage, “in the short run, the market is a voting machine but in the long run, it is a weighing machine.”

The transition from voting to weighing implies a shift towards equilibrium. But that equilibrium is not always a single equilibrium, but rather all equilibriums adjusted for their popularity and strength (or, “Nashiness”–the relative dominance of the equilibrium6).

## Plausible path to Bitcoin stability

Bitcoin’s stability depends on the design of its “weighing machine7,” which takes each equilibrium behavior (e.g. anchoring to Gold, believing it’s worth zero) adjusted for popularity and strength, and returns a net weight (price).

Stability would come from relative stasis in the design of the “weighing machine.” For example reductions in:

• volume of entries and exits from the market (fiat inflows, outflows)
• variance in the composition of equilibriums (i.e. narratives)
• changes in strength for individual equilibriums

Maximum stabilization would come when the entire world is using Bitcoin and everybody agrees on one way to value it (e.g. MV = PQ)8. Indeed this is along the lines of what proponents of the “bubble theory of bitcoin” argue. Once we reach “hyperbitcoinization”, the asset will stabilize around some very large number.

It’s easy to see how today’s price is a Keynesian Beauty Contest with no Nash equilibriums. It’s easy to see how a future where everybody uses Bitcoin could stabilize. It’s much less clear how we get from today to that stable future9.

I expect “Narrative Bubble Loops” over and over until we find a narrative that resonates sufficiently. Whether that dominant narrative is Bitcoin is the reserve currency of the world, Bitcoin is fundamentally worth nothing10, or something in the middle remains to be seen.

Disclosure: I hold Bitcoin.

Thanks to Rae for conversations that contributed to this post

1. This is why mass movements form that compete for true believers to spread narratives. It has a material effect on a speculative bubble’s “Keynesian Beauty Contest.”

2. This game and many experiments like this were created by Rosemarie Nagel. I found them in this presentation

3. It’s worth noting that in Nagel’s experiments, she observed deviations from the equilibrium valuation of zero. They attributed this to either (1) confusion (not perceiving the Nash Equilibrium) or (2) sophistication (correctly predicting some players would not pick the equilibrium value). These experiments were also conducted in controlled environments and do not account for “cheating”. If the players were allowed to collude outside of the game, they could try and manipulate the game so that one of them wins. For example, two players could increase the winning number by having one of them pick 100 and the other a number higher than the expected equilibrium value.

4. The process of reaching consensus on a valuation methodology can be influenced by owners of the thing to be valued. You could argue that \$AMZN’s success is due to Bezos’ ability to convince the market to evaluate his company on his terms, rather than what was previously the consensus valuation methodology (i.e. net profits).

5. Why not money supply? Or chuck-e-cheese tokens? Or the market cap of Visa? Or 50% of the total market cap of crypto? Or some multiple of NVT? There are reasonable arguments for each of these (some more reasonable than others).

6. What an awful word. If it gets popular, I apologize for using it.

7. The lack of one clear Nash equilibrium leads to variance in player behavior. This is important for systems that rely on Keynesian Beauty Contest mechanisms like TCR voting. These systems rely on “the truth” being the Nash equilibrium. If the truth is not the single Nash equilibrium in all cases the TCR wants to handle, then the system needs another check to correct for errors (where the truth is not the Nash equilibrium). Also interesting, Vitalik’s 2014 essay on SchellingCoin that explores the truth as a Schelling Point (a Nash equilibrium that emerges due to properties that make it stick out from other options).

8. Some of you may wonder how this applies to “productive” cryptoassets like work tokens. If you value something like Augur’s REP token, for instance, you would have to apply a tremendous multiple to current network rewards to achieve the current REP valuation. Or you might apply a multiple to the total gas costs for a network like Ethereum. Interestingly, there are cases where the total gas costs might go down while usage goes up (for example, if most activity moves to layer 2). These cases highlight how nascent our thinking is on valuing cryptoassets. As industry evolves, I suspect we will slowly start to align on the design of the “weighing machine” for the various types of cryptoassets, but these machines will have to change as the networks themselves change.

9. After publishing this piece, I received several messages with similar feedback that using the same model to evaluate equities and Bitcoin is flawed. I agree and that’s one of the main points of this piece. The Keynseian Beauty Contest framework is neutral–it’s not positive or negative, and can be applied dispassionately to anything. The lack of cashflows make it harder for the current market to determine a Nash equilibrium. But one can argue–and many do!–that Bitcoin’s strong properties of money will lead it to a Nash equilibrium that is very high. Still, the point remains that there is not yet a natural Nash equilibrium for Bitcoin.

10. While the Lindy effect is hard to prove statistically, there is merit to the argument that the longer that Bitcoin is not worth zero, the harder it is to form an equilibrium of “Bitcoin is worth zero.”