The Paradox of Cryptocurrency Valuations

This is a rather long post, so please bear with me. I will first describe an apparent mathematical paradox, and then relate it to the price of cryptocurrencies.

Paradox

Let us perform a Gedankenexperiment (a thought experiment).
I promise you that this will be extremely relevant to cryptocurrencies, so don’t just skip this part.
Imagine that you are presented with two sealed envelopes, and told that each envelope contains a valid check (cheque) made out in your name.
You are also told that the value of one of the checks is double that of the other check. (but not which envelope contains the larger amount).
The envelopes are labelled ‘A’ and ‘B’, and you can only take one of them.
Great! You excitedly decide to open envelope ‘A’. You receive a lovely surprise that it contains a check for $100. You can buy a nice pair of Nikes with that :)
Now, you are given the option to swap and take envelope ‘B’ instead, but you will need to give up the first envelope and its $100 prize. Should you swap to the second envelope?

Think about this for a moment. At first glance, it seems like there wouldn’t be any good reason to swap, since your prize might either double or halve.
However, this is what is known as an asymmetrical bet. In fact, you have an even chance of gaining $100 or losing $50. In other words, you will end up with either $50 or $200 in your pocket, so your overall expected prize will actually be (200/2+50/2) = $125.
So, mathematically, you should certainly swap to the other envelope, since then your expected prize is 25% more than your current prize.

This is all well and good, but there is an inherent paradox just around the corner...

You now know that when you opened envelope ‘A’ and found $100, you should indeed decide to swap to envelop ‘B’ instead.
But regardless of the value stored in ‘A’, you should always swap to ‘B’. So you don’t even have to actually open ‘A’. Why not just choose ‘B’ in the first place?
But hold on… if you were to open envelope ‘B’ first, then logic insists that you should then swap to envelope ‘A’.
How can both envelopes be worth 25% more than the other one?

I am interested to hear your answers to the apparent paradox, but here are my two explanations of how to resolve it:
Answer 1. Resolve by Dismissing
Arguably, the initial setup of the problem is *not possible* in the real world. It is not possible to have two checks written down that has one double the value of other one, and with no preconceived information on the values of the checks.
(This is akin to the old saw about ‘what happens if an irresistible force meets an immovable object?’. These two things cannot co-exist in the same universe, so the question is nonsensical to pose, let alone answer.)
Why is the original description of the check/envelope scenario invalid? Because in the real world there are always ‘a priori’ facts known about checks. For instance, would a check for $100 be possible? Sure. $1000? Sure. $1,000,000,000? Well I guess, but perhaps not so likely, right?
What about $1,000,000,000,000,000,000? You would have to say that a check for this amount is invalid or useless, and so the relative chance of the check being worth an amount N is not completely flat over the set of all positive numbers N.
At some high dollar value, the likelihood of the ‘other check’ being a ‘doubled’ check is slightly less than 50%, and the chance of a ‘halved’ check is slightly more than 50%.
In this way the paradox can be resolved, since the initial setup of the situation is not possible in the real world.

Answer 2. Resolve by Infinity
An alternative solution to the paradox (and one that is more relevant to my upcoming segue into cryptocurrency) is the following:
Given the original statement of the situation, each envelope does indeed contain 25% more than the other envelope. There is only one manner in which this is possible, and that is if the expected return (value) of both envelopes is infinite. (or $0, but we can ignore this case)
Infinity plus 25% is still infinity. In a real mathematical sense, this is what we implied when we formulated the original problem statement. If there is truly no upper limit on the value of the checks, then the expected return is indeed infinite.
You can keep switching from envelope ‘A’ to envelope ‘B’ and back again in your mind (without even opening them) and each time your expected return will actually increase by 25%! It was infinitely large after the first choice, and 25% larger when you swapped to the other one. And again 25% larger when you swapped back to the original, etc! (ah! The joys of infinity!)

CryptoCurrency Valuations

Now, let us draw some comparisons between this paradox and the market valuations of cryptocurrencies. I shall use the term ‘bitcoin’ here, but feel free to substitute your favorite coin in your mind whenever I write ‘bitcoin’.

Let us run a new thought experiment:
Crypto_Gedankenexperiment
Imagine that it is 2021, and you are just hearing about an invention named ‘bitcoin’. Some people make outrageous claims about hyperbitcoinization and how it will change the future of money.
You are reliably informed that there is no logic by which to value the tokens of this new invention. In 12 months from now, one bitcoin is equally likely to be half its current value, or double its current value.
You are very kindly gifted a single bitcoin. Should you keep the bitcoin for a year, or sell it immediately for its current price?

As you might expect, this is also an asymmetrical bet. If you sell the bitcoin, you get some amount of dollars, $B. However if you keep the bitcoin, you have equal chances of doubling or halving your cash, which is a good thing!
At the end of a year, you will (on average) end up with 1.25 times $B. You have just made 25% by HODLing.
Given the situation as described, you should definitely hold onto your bitcoin.

But wait, this doesn’t just apply to you… it applies to everyone in the world. If you are going to make 25% just by HODLing, then everyone else can do the same. In an informed marketplace, this surely must push today’s price for a bitcoin up by nearly 25%, right? (minus a bit due to risks, inertia, interest rates, available capital, etc)
But if the price $B today rises, that doesn’t affect the computation! (In the same way that you don’t actually need to open envelope ‘A’ in the original paradox to know that you should still swap to ‘B’.) You still make 25% just by waiting, even though today’s price is higher than you previously thought!
Man! I sure want some of that action!

The parallels with the original Gedankenexperiment above are clear. If we really have no range expectations for the price of a bitcoin, and all values are equally likely, then the actual mathematical expected value for one bitcoin is infinite.
This is the only way in which the value 1 year from now is 25% higher than today, even if everyone knows it and factors it into their current buying decisions.

Bitcoin Price Probability Distribution

Of course, in the real world, we don’t (quite) ascribe equal probabilities to all possible dollar values for a bitcoin. At some top end, we say… well a $Billion per coin is unrealistic.
But recent history is littered with false conclusions about the extent of the ‘unknowingness’ of the true price for bitcoin.
I would argue that the distribution curve for the bitcoin price is indeed flatter (more constant) than anyone ever expected, and it is flattening more every year. A year ago, when BTC cost $5,000, the pronounced downward (i.e. less probable) curve of the expected price distribution started at, say, $10,000. Today, the curve is nearly flat out across to the $50,000 range and beyond.

Summary

• If we have no a priori information about the price probability distribution for an item, then the expected value of that item is infinite.
• If the value of an item can either double or halve over time with equal probability, then in the long term its value increases without bound.

Disclaimer

Note that none of the information above should be construed as investment advice. Do your own research and come to your own conclusions.