The Drake equation is a probabilistic argument used to estimate the number of active, communicative extraterrestrial civilizations in the Milky Way galaxy. The idea is to express the number of such civilizations as a product of quantities in a manner that’s true by definition, but also such that one can talk somewhat more intelligently about estimating the individual factors than one can talk about directly estimating the number of active, communicative extraterrestrial civilizations. XKCD has poked fun at the Drake equation in at least two comics. Viewed as an exercise aimed at obtaining precise actionable estimates, the Drake equation is probably futile. But viewed as a way to start thinking about the problem, it is arguably useful. The main reason it’s bad for estimation is that the multiplicative nature of the model means that the huge uncertainty in measurement for each of the factors is also multiplicative, leading to a gigantic uncertainty in the overall estimation.

Here’s my Drake-like attempt:

$latex \text{Utility of a particular form of open borders advocacy} = Wxyz$

Here:

- $latex W$ is the naive estimate of the gains from complete open borders (using, for instance, the double world GDP ballpark).
- $latex x$ is a fudge factor to represent the idea that “things rarely turn out as well as we expect them to.” If we set $latex x = 0.1$, for instance, that’s tantamount to saying that, due to all the numerous problems that our naive models fail to account for, the actual gains from open borders would be only 10% of the advertised gains. The product so far, namely $latex Wx$, describes what we
*really*expect the gains from open borders to be. - $latex y$ is the fraction to which the world can realistically move in the direction of open borders. The product $latex Wxy$ is total expected gain from however far one can realistically move in the open borders direction.
- $latex z$ is the extent to which a particular effort at advocacy or discussions moves the world toward open borders, as a fraction of what is realistically possible. For instance, setting $latex z = 10^{-4}$ for Open Borders the website would mean that the creation of the website, and work on the website, has moved the world 1/10,000 of the way it feasibly could in the direction of open borders.

The restrictionist or pessimist might well view $latex x$ as a negative number, making open borders advocacy a great disservice to humanity. For our purposes, however, we’ll consider estimates where the values are positive, yet sufficiently small as to account for considerable uncertainty. Let’s say that, for the *Open Borders* website, the numbers look as follows, with the numbers in US dollars (note that of the four numbers, $latex z$ is the only one that requires particular knowledge of the *Open Borders* website):

$latex W = \$ 50 \text{ trillion}, x = 0.01, y = 0.001, z = 0.0001$

The 50 trillion figure can be calculated as just one year’s gain based on the double world GDP estimates. Note that there are some complications when considering potential delays in opening borders, as well as discount rates for the future and economic growth in the future. But since the starting numbers are anyway very rough guesses, there’s not much point in trying to do a very elaborate estimation exercise to calculate $latex W$ (for what it’s worth, I did some estimates based on assumptions about discount rates and economic growth, and I got a figure of about twice that much in expected value even if open borders are delayed by several years and the gains are slow to arrive and temporary). Note also that the fudge factor $latex x$ of 0.01 is essentially taking a very pessimistic view of the estimation exercise, by claiming that 99% of the claimed gains will not in fact materialize, or will be canceled by other losses.

With these numbers, the value of the website comes out to 50,000 US dollars. That’s not huge, but it’s about the same order of magnitude as the cost of time spent on the website (about 1500-2000 hours). With these numbers, therefore, the site just about breaks even in terms of social value generated versus time spent.

Here’s an optimistic version of the numbers:

$latex W = \$ 50 \text{ trillion}, x = 0.1, y = 0.1, z = 0.0001$

With this view, the naive estimate overstates the gains, but only ten-fold, it’s also possible for the world to realistically move 10% of the way toward open borders, and *Open Borders* the website has moved the world 1/10,000 of the way toward the theoretically possible limit. With these numbers, the expected value of *Open Borders* comes to about $50 million.

Obviously, the above estimation exercises are very naive, and there’s a sense in which this might feel like Pascal’s mugging. The key point that emerges here, though, is that the position *yes, open borders would have gains, but the gains from what’s realistically possible in that direction are too small to be worthwhile* isn’t a very tenable position. Open borders is a radical proposal — for better or worse. To arrive at such a position, you’d need to have $latex x,y,z$ all *very small* — but still positive. If you’re coming that close to zero, then you might as well offer some good reason why you don’t go all the way to zero — or beyond, to the negative territory. If the restrictionist position were right, then, it would entail showing at least one of these (or more precisely, an odd number of these, but never mind that):

- $latex x$ is zero or negative: Economists have badly estimated not just the magnitude, but rather, the
*sign*of the effect of open borders. The best attempts in the direction of demonstrating that the expected sign is negative is the killing the goose that lays the golden eggs argument. And while I think there’s considerable plausibility to that argument, and it may well point toward certain keyhole solutions being desirable, I am not convinced that these come anywhere near toggling the expected sign of the gains from open borders. - $latex y$ is zero or negative: It’s impossible to move in the direction of open borders at all.
- $latex z$ is zero or negative: Open borders advocacy (or at any rate, the specific advocacy effort being considered) hurts the move toward open borders more than it helps. Tyler Cowen took this sort of approach in his recent blog post that generated considerable response (including from Nathan and John).

An alternative position is that we just don’t know enough to even estimate the signs of the quantities, and that more research is needed. I certainly agree about the need for more research, and I think a strong case could be made for an agenda that focuses extensively on research before clearly coming down on one side or another, while favoring continued experimentation with liberalization and keyhole solutions at the margin. But what’s not justifiable is the absolute certainty that many people seem to have that the *status quo* is approximately optimal, or that radical liberalization of movement simply isn’t a paradigm worth investigating because the gains are too small.

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