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.

If you were a quant, you would know that random walks on a log scale (geometric Brownian motion) are what people normally use for asset prices. It's what's beneath Black-Scholes, for example. An additive random walk can go negative, which prices can't, but a log random walk is always positive.

(Also note that the fact that the EV is higher tomorrow than today isn't that meaningful, because of time discounting- if the EV tomorrow is the same as the EV today in nominal terms, you should sell and buy something that's expected to go up. How does the expected future growth rate compare to other opportunities?)

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.

Do we really ? My own view is quite the opposite - a kinda reverse bell curve, with two possible outcomes :

1. Bitcoin dies, either because the crypto behind it is broken (due to mathematical progress or Moore's law) or because it gets replaced by other, "second generation" cryptocurrencies, or because states successfully fight it, or any other reason - and then it'll have a very low value, maybe even less than $1 for a BTC.

2. Bitcoin survives, and then, because it's inherently deflationary (fixed monetary mass for an always growing amount of real world wealth) there is no limit to how high the value of single BTC can grow.

But maybe it depends what exactly "long-term" is ?

Simplicio: There are zillions of exotic markets out there. I think you fixate on Bitcoins because they are fun and shiny. Why not instead try to outguess local real estate markets? You are much more likely to be successful.

So a word of warning if you are thinking of playing the long-game here. The source of your observation is that bitcoin is (or has been) apparently undervalued.

But an other explanation is that its a bubble. While spot trading in bitcoin is quite liquid, the derivatives markets in bitcoin are fairly untested, relatively new, and fairly low volume. Right now, if I believe bitcoin is expected to climb, I can easily pile in, but shorting bitcoin takes more effort.

As the new bitcoin derivatives markets grow, its will give counterparties with negative opinions of bitcoin more voice, which may let the air out of the bubble a bit.

Edit: Also, we expect prices to take a multiplicative random walk, the meat here is whether or not the increase in expected value beats inflation/beats other asset markets you can easily invest in.

This bitcoin conversation has run for almost a week now, and given the site I'd expect the level of reasoning to be quite high, yet when I hit "^Ftax" or "^Fgovern" or "^Fpolitic" almost nothing shows up, which causes me a measure of confusion, because these (much more than "magnitudes") are key nodes in my causal reasoning about the future value of bitcoins.

From my perspective, the plausible socio-political implications of bitcoin are large enough, and different enough from what I see commonly discussed, that it causes me to question the quality of my own thinking and seek education.

In 1789 Benajmin Franklin wrote a letter wherein he said:

Our new Constitution is now established, and has an appearance that promises permanency; but in this world nothing can be said to be certain, except death and taxes.

It could be that I'm wrong in my reasoning, but it appears to me that bitcoin allows tax evasion and black markets to function on such a breathtaking scale that if bitcoin persists and expands into common use then I anticipate, like tomatoes in winter, the withering of formal governmental power in its current form (based as it is on tax collection and the ability to regulate the market via indirect oversight) and perhaps even the withering of the public good of reasonably just protection services provided by democratically elected law makers.

Sharply put, it seems moderately plausible to me that either extant governments smash the bitcoin infrastructure, or bitcoin financially strangles modern nation states.

In more detail: If bitcoin turns out to be ineradicable so long as people have access to the untamed Internet (and this seems like an open but fundamentally empirically determinable question) it suggests to me that human communities may collectively face a choice between cutting their wires and jamming their airwaves or else lose the ability to form reasonable transparent organizations with elected officials who manage the local violence monopoly by paying law enforcers better wages than are available to criminals.

Or perhaps I'm underestimating the extent of the revamping that would be necessary? Still, it is hard to see how the IRS, SEC, ATF, or Fed could maintain their status quo operations if bitcoins become the de facto world currency. Traffic in drugs and slaves are relatively limited now, but I'm not sure it would stay so in a bitcoin dominated economy. Ransom payments become significantly more feasible with bitcoin, and the kidnapping market seems likely to grow if bitcoins persist.

Not that payment for protection services would completely disappear forever... Presumably we would switch to tax collectors (either hired by the existing but revamped governments or perhaps the henchmen of whatever violence monopolies replace them) who force people to transfer digital cash in amounts assessed based on visible or statistically inferrable indicators of wealth, or be jailed. Tax evasion in such a world seems likely to take the form of pretending to be poor, which seems to have weird implications for the personal status of the super rich? Facebook-based estimates of taxes owed would be amusing, but less ironic forms of surveillance could work as well. If the super rich were the ones hiring the tax collectors, that could reduce the number of sociologically confusing discrepancies, but it starts sounding somewhat feudal...

The Treasury Department must have people thinking about this? Or maybe the private individuals composing the Treasury Department have non-trivial personal stakes in bitcoin and no civic virtue? Or maybe the problem is international in scope and there's an element of realpolitik where some nation states expect to weather the "bitcoin winter" better than others? Or maybe... I don't know... There are a lot of things that could be going on...

In this family of scenarios, it seems like there would be large changes to many parts of the economy, many of which I expect would take a lot of people, including me, by surprise. Maybe drone-based mass surveillance and law enforcement could patch the gaps by enforcing laws so thoroughly at the physical layer than the digital layer can remain anarchic for a while longer? It seems like the kind of "everything is changing, faster and faster" thing that I might expect to be sprung on people in the lead up to various (somewhat disturbing) versions of an Kurzweil-style "smooth singularity"... Kurzweil did predict runaway deflation after all, and 2014 is the sort of year you'd read about something like this happening in a Stross novel.

So, anyway...

I see people trading in bitcoins. I don't see the government moving to destroy the bit coin markets, or talking about the bitcoin market as though bitcoins were a social scourge that fundamentally disrupts the business model of status quo governments. But I also I don't see people preparing for a profound restructuring of the political economy and everything effected by the current political economy. Thus, I am confused. I don't see other people even talking about these sorts of implications, as though they are important open questions. Thus, I am doubly confused.

My best guess as to my confusion's cause is twofold. I probably lack an adequate understanding of the big picture pragmatics of political economy, also I suspect that the really smart money in the bitcoin market is staying mostly silent so as to harvest money more efficiently. For myself, the political/moral dimension of the bitcoin market has frightened me away from it thus far... whether there is a "bitcoin winter" or a successful smashing of the bitcoin infrastructure, both outcomes seem to suggest that personal and/or political action might be prudent.

The value of information seems high. If anyone could respond here or via PM to help correct my confusion, I would very much appreciate the education!

They look more like a random walk on a log scale.

Not surprising - the risk free rate (http://en.wikipedia.org/wiki/Risk-free_rate) is exponential, and any efficient asset has to do at least as well in expectation. So expected exponential growth in asset value is exactly the behaviour you'd expect.

Or put more prosaically: if I invest money at x% interest in a bank, I have exponential growth. Therefore any investment that would tempt me away from a bank account, must offer at least exponential growth.

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.

Can you elaborate on why you think this is true?

Several people already pointed out that what is most frequently used as a model for prices is precisely the log random walk. This should have been obvious since no asset can have a negative price, and it is known that the long-term probability of the one-dimensional random walk reaching any specified point (such as zero) is 1 (same for 2D, not so in 3+ D) - this part of how casinos make money.

It is very easy to spot why the random walk model doesn't make sense for prices, just take the sentence that says "If today's Bitcoin price is $1000, then tomorrow's price is as likely to be $900 as it is to be $1100." Now if the normal random walk model were true, then you could also have said that the price is as likely to deviate by +X as it is by -X for any X. Now take X=1000: is it as likely to cost $0 as it is to cost $2000?

The fact that there are a stock has a random walk behavior in the log-scale does not imply that its expected future price is larger than its current price. Imagine that when a bitcoin is worth 1000$ today it has equal chance tomorrow of going up 100$ or going down 100$. Then if it goes to 900$, it has an equal chance of going up or down 90$ the next day, and similarly, if the price becomes 1100$, it has an equal chance of going up or down 110$. Then the log-price would have variations on the log scale, but at each step the expected value stays the same.

Please place more trust in the competence of mainstream institutions next time.

Salviati's claim at the end reminds me of Einstein asserting that mass has more inertia than you think, but you'll only notice that when its velocity is really high.