Salviati: Simplicio, do you think the Bitcoin markets are efficient?

Simplicio: If you'd asked me two years ago, I would have said yes. I know hindsight is 20/20, but even at the time, I think the fact that relatively few people were trading it would have risen to prominence in my analysis.

Salviati: And what about today?

Simplicio: Today, it seems like there's no shortage of trading volume. The hedge funds of the world have heard of Bitcoin, and had their quants do their fancy analyses on it, and they actively trade it.

Salviati: Well, I'm certainly not a quant, but I think I've spotted a systematic market inefficiency. Would you like to hear it?

Simplicio: Nah, I'm good.

Salviati: Did you hear what I said? I think I've spotted an exploitable pattern of price movements in a $10 Billion market. If I'm right, it could make us a lot of money.

Simplicio: Sure, but you won't convince me that whatever pattern you're thinking of is a "reliable" one.

Salviati: Come on, you don't even know what my argument is.

Simplicio: But I know how your argument is going to be structured. First you're going to identify some property of Bitcoin prices in past data. Then you'll explain some causal model you have which supposedly accounts for why prices have had that property in the past. Then you'll say that your model will continue to account for that same property in future Bitcoin prices.

Salviati: Yeah, so? What's wrong with that?

Simplicio: The problem is that you are not a trained quant, and therefore, your brain is not capable of bringing a worthwhile property of Bitcoin prices to your attention.

Salviati: Dude, I just want to let you know because this happens often and no one else is ever going to say anything: you're being a dick.

Simplicio: Look, quants are good at their job. To a first approximation, quants are like perfect Bayesian reasoners who maintain a probability distribution over the "reliability" of *every single property* of Bitcoin prices that you and I are capable of formulating. So this argument you're going to make to me, a quant has already made to another quant, and the other quant has incorporated it into his hedge fund's trading algorithms.

Salviati: Fine, but so what if quants have already figured out my argument for themselves? We can make money on it too.

Simplicio: No, we can't. I told you I'm pretty confident that the market is

efficient, i.e.

anti-inductive, meaning the quants of the world haven't left behind any reliable patterns that an armchair investor like you can detect and profit from.

Salviati: Would you just shut up and let me say my argument?

Simplicio: Whatever, knock yourself out.

Salviati: Ok, here goes. Everyone knows Bitcoin prices are volatile, right?

Simplicio: Yeah, highly volatile. But at any given moment, you don't know if the volatility is going to move the price up or down next. From your state of knowledge,

it looks like a random walk. If today's Bitcoin price is $1000, then tomorrow's price is as likely to be $900 as it is to be $1100.

Salviati: I agree that the Random Walk Hypothesis provides a good model of prices in efficient markets, and that the size of a each step in a random walk provides a good model of price volatility in efficient markets.

Simplicio: See, I told you you wouldn't convince me.

Salviati: Ah, but my empirical observation of Bitcoin prices is inconsistent with the Random Walk hypothesis. So the only thing I'm led to conclude is that the Bitcoin market is not efficient.

Simplicio: What do you mean "inconsistent"?

Salviati: I mean Bitcoin's past prices don't look much like a random walk. They look more like a random walk *on a log scale*. If today's price is $1000, then tomorrow's price is equally likely to be $900 or $1111. So if I buy $1000 of Bitcoin today, I expect to have 0.5($900) + 0.5($1111) = $1005.50 tomorrow.

Simplicio: How do you know that? Did you write a script to loop through Bitcoin's daily closing price on Mt. Gox and simulate the behavior of a Bayesian reasoner with a variable-step-size random-walk prior and a second Bayesian reasoner with a variable-step-size log-random-walk prior, and thus calculate a much higher

Bayesian Score for the log-random-walk model?

Salviati: Yeah, I did.

[This is a fictional dialogue. The truth is, I was too lazy to do that. Can someone please do that? I would much appreciate it. --Liron.]

Salviati: So, have I convinced you that the market is anti-inductive now?

Simplicio: Well, you've empirically demonstrated that the log Random Walk Hypothesis was a good model for predicting Bitcoin prices in the past. But that's just a historical pattern. My original point was that you're not qualified to evaluate which historical patterns are *reliable* patterns. The Bitcoin markets are full of pattern-annihilating forces, and you're not qualified to evaluate which past-data-fitting models are eligible for future-data-fitting.

Salviati: Ok, I'm not saying you have to believe that the future accuracy of log-Random-Walk will probably be higher than the future accuracy of linear Random Walk. I'm just saying you should perform a Bayesian update in the direction of that conclusion.

Simplicio: Ok, but the only reason the update has nonzero strength is because I assigned an a-priori chance of 10% to the set of possible worlds wherein Bitcoin markets were inefficient, and that set of possible worlds gives a higher probability that a model like your log-Random-Walk model would fit the price data well. So I update my beliefs to promote the hypothesis that Bitcoin is inefficient, and in particular that it is inefficient in a log-Random-Walk way.

Salviati: Thanks. And hey, guess what: I think I've traced the source of the log-Random-Walk regularity.

Simplicio: I'm surprised you waited this long to mention that.

Salviati: I figured that if I mentioned it earlier, you'd snap back about how efficient markets sever the causal connection between would-be price-regularity-causing dynamics, and actual prices.

Simplicio: Fair enough.

Salviati: Anyway, the reason Bitcoin prices follow a log-Random-Walk is because they reflect the long-term Expected Value of Bitcoin's actual utility.

Simplicio: Bitcoin has no real utility.

Salviati: It does. It's liquid in novel, qualitatively different ways. It's kind of anonymous. It's a more stable unit of account than the official currencies of some countries.

Simplicio: Come on, how much utility is all that really worth in expectation?

Salviati: I don't know. The Bitcoin economy could be anywhere from hundreds of millions of dollars, to trillions of dollars. Our belief about the long-term future value of a single BTC is spread out across a range whose 90% confidence interval is something like [$10, $100,000] for 1BTC.

Simplicio: Are you saying it's spread out over the interval [$10, $100,000] in a uniform distribution?

Salviati: Nope, it's closer to a bell curve centered at $1000 on a log scale. It gives equal probability of ~10% both to the $10-100 range and to the $10,000-100,000 range.

Simplicio: How do you know that everyone's beliefs are shaped like that?

Salviati: Because everyone has a causal model in their head with a node for "order of magnitude of Bitcoin's value", and that node varies in the characteristically linear fashion of a Bayes net.

Simplicio: I don't feel confident in that explanation.

Salviati: Then take whatever explanation you give yourself to explain the effectiveness of

Fermi estimates. Those output a bell curve on a log scale too, and seems like estimating Bitcoin's future value should have a lot of methodology in common with doing back-of-the-envelope calculations about the blast radius of a nuclear bomb.

Simplicio: Alright.

Salviati: So the causality of Bitcoin prices roughly looks like this:

[Beliefs about order of magnitude of Bitcoin's future value] --> [Beliefs about Bitcoin's future price] --> [Trading decisions]

Simplicio: Okay, I see how the first node can fluctuate a lot in reaction to daily news events, and that would have a disproportionately high effect on the last node. But how can an efficient market avoid that kind of log-scale fluctuation? Efficient markets always reflect a consensus estimate of an asset's price, and it's rational to arrive at an estimate that fluctuates on a log scale!

Salviati: Actually, I think a truly efficient market shouldn't just skip around across orders of magnitudes, just because expectations of future prices do. I think truly efficient markets show some degree of "drag", which should be invisible in typical cases like publicly-traded stocks, but become noticeable in cases of order-of-magnitude value-uncertainty like Bitcoin.

Simplicio: So you think you're the only one smart enough to notice that it's worth trading Bitcoin so as to create drag on Bitcoin's log-scale random walk?

Salviati: Yeah, I think maybe I am.

**Salviati is claiming that his empirical observations show a lack of drag on Bitcoin price shifts, which would be actionable evidence of inefficiency. Discuss.**