How Bad Can A Nuclear War Get?

In “How Likely is World War III?”, Stephen suggested the chance of an extinction-level war occurring sometime this century is just under 1%. This was a simple, rough estimate, made in the following steps:

  1. Assume that wars, i.e. conflicts that cause at least 1000 battle deaths, continue to break out at their historical average rate of one about every two years.
  2. Assume that the distribution of battle deaths in wars follows a power law.
  3. Use parameters for the power law distribution estimated by Bear Braumoeller in Only the Dead to calculate the chance that any given war escalates to 8 billion battle deaths
  4. Work out the likelihood of such a war given the expected number of wars between now and 2100.

Not everybody was convinced. Arden Koehler of 80,000 Hours, for example, slammed it as “[overstating] the risk because it doesn’t consider that wars would be unlikely to continue once 90% or more of the population has been killed.” While our friendship may never recover, I (Stephen) have to admit that some skepticism is justified. An extinction-level war would be 30-to-100 times larger than World War II, the most severe war humanity has experienced so far. Is it reasonable to just assume the number goes up? Would the same escalatory dynamics that shape smaller wars apply at this scale?

Forecasting the likelihood of enormous wars is difficult. Stephen’s extrapolatory approach creates estimates that are sensitive to the data included and the kind of distribution fit, particularly in the tails. But such efforts are important despite their defects. Estimates of the likelihood of major conflict are an important consideration for cause prioritization. And out-of-sample conflicts may account for most of the x-risk accounted for by global conflict. So in this post, we interrogate two of the assumptions made in “How Likely is World War III?”:

  1. Does the distribution of battle deaths follow a power law?
  2. What do we know about the extreme tails of this distribution?

Our findings are:

  • That Battle deaths per war are plausibly distributed according to a power law, but few analyses have compared the power law fit to the fit of other distributions. Plus, it’s hard to say what the tails of the distribution look like beyond the wars we’ve experienced so far.
  • To become more confident in the power law fit, and learn more about the tails, we have to consider theory: what drives war, and how might these factors change as wars get bigger?
  • Perhaps some factors limit the size of the war, such as increasing logistical complexity. One candidate for such a factor is technology. But while it seems plausible that in the past, humanity’s war-making capacity was not sufficient to threaten extinction, this is no longer the case.
  • This suggests that wars could get very, very bad: we shouldn’t rule out the possibility that war could cause human extinction.

Battle deaths and power laws

Fitting power laws

One way to gauge the probability of out-of-sample events is to find a probability distribution, a mathematical function which gives estimates for how likely different events are, which describes the available data. If we can find a well-fitting distribution, then we can use it to predict the likelihood of events larger than anything we’ve observed, but within the range of the function describing the distribution.

Several researchers have proposed that the number of battle deaths per war is distributed according to a power law. Power law distributions are described by the formula . This means that the probability of event x is proportional to the value of x raised to the power . These distributions look like this:

Figure 1. Power law distributions of various scaling parameters

α, the scaling parameter, determines the shape of the distribution. As the figure above shows, the smaller the α, the longer the tails of the distribution. There’s a simple way to interpret this parameter: if one event is twice as large as another, it is  times less likely.

Long tails are an important feature of power laws. Events many orders of magnitude larger than the mean or median observation are unlikely, but not impossible. However, we need to be careful when fitting power laws. Other long-tailed distributions may fit the same data, but imply different probabilities for out-of-sample events. Since the tails of power laws are so long, improperly fitting this distribution to our data can lead us to overestimate the likelihood of out-of-sample events.

Power laws and conflict data

Rani conducted an informal literature review to assess the strength of the evidence for a power law distribution of war deaths. You can find a summary of her review, including the goodness-of-fit tests, here.

She found six papers and one book in which a power law distribution is fitted to data on battle deaths per war. In each case, a power law was found to be a plausible fit for the conflict data. Estimates for the of conflict death data range from 1.35 to 1.74, with a mean of 1.60. Anything in this range would be considered a relatively low value of  (most empirical power laws have values between 2 and 3). In other words, if these estimates are accurate, the tail of the battle death distributions is scarily long.

Figure 2. Clauset (“Trends and fluctuations in the severity of interstate wars”)’s power law model of war severity. Note that the axes are logarithmic and do not start at 0.

How scary exactly? In Only the Dead, political scientist Bear Braumoeller uses his estimated parameters to infer the probability of enormous wars. His distribution gives a 1 in 200 chance of a given war escalating to be twice as bad as World War II and a 3 in 10,000 chance of it causing 8 billion deaths (i.e. human extinction).

That’s bad news. If wars continue to break out at their historical-average rate of one every two years, Braumoeller’s estimates imply a ~18% chance of war twice as bad as World War II and a ~1.2% chance of an extinction war occurring between now and 2100.

Breaking the law

Before we all start digging bunkers, though, it’s worth highlighting a few caveats.

First, and most importantly, only two papers in the review also check whether other distributions might fit the same data. Clauset, Shalizi, and Newman (2009) consider four other distributions, while Rafael González-Val (2015) also considers a lognormal fit. Both papers find that alternative distributions also fit the Correlates of War data well. In fact, when Clauset, Shalizi, and Newman compare the fit of the different distributions, they find no reason to prefer the power law.

Second, there are potentially important data limitations. Wars are relatively rare, especially large wars that kill an appreciable fraction of the world, and death counts are uncertain for many wars. The small sample size makes our estimates of the underlying distributions more uncertain. It also means that our parameter estimates are sensitive, and could change in important ways if uncertain death counts are revised. Unfortunately, our estimates of the probability of extreme tail events are super sensitive to changes in these parameters.

Third, most of the papers we considered used the Correlates of War dataset. Analyses using other datasets may produce different parameter estimates. The CoW dataset also excludes many countries from its dataset in the pre-WWII period. If, for some reason, the excluded countries were more likely than average to fight especially large or small wars, then the estimates using these data may be biased.

Fourth, analyses that group together wars over time implicitly ignore potential changes in the distribution over time. Perhaps the probability distribution for deaths in a 21st-century war is different from the distributions for 20th or 19th-century wars. Whether the frequency or severity of war has been changing over time is a complicated question. Economic growth, globalization, technological change, and political and social institutions are dynamic and all plausibly influence the conduct of war.

What does the tail of the distribution look like?

To summarize the previous section: battle deaths are plausibly distributed according to a power law. But there is some evidence that other distributions also fit the same data. We also have to contend with the fact that we have a small sample size and the distribution could be changing over time.

Suppose, though, that we could establish with confidence that battle deaths are power-law distributed. Extrapolating would still pose a problem. Power laws with  have an infinite mean. But the range of the battle death distribution can’t stretch to infinity. It’s upper bound must be, at most, the global population. And it could even be lower than this, with the distribution bending downwards at some point between the largest war we’ve observed and the logical limit. We can’t tell if or where the distribution is capped based on data alone. We need to consider “physical motivating or theoretical factors.”

Cioffi-Revilla and Midlarsky (2004), for example, argue that enormous wars are less likely than they should be according to a power law. They venture two explanations for this observation. First, they suggest that as a larger proportion of the population is drawn into the war, public pressure on the government to end the war grows. This “democratization” of war limits the extent to which they can scale. Second, the international system is “finite”. As wars escalate they become more complex. They suggest that the diplomatic, strategic, and logistical dynamics of enormous wars may be too difficult to manage.

While we don’t find either of those explanations fully convincing, at least Cioffi-Revilla and Midlarsky are trying to sketch the tail of the distribution. This is a rich vein for future research. We’d be very interested in seeing analyses that try to connect the observed distribution of outcomes to insights from the IR literature on models of escalation. These would give more insight into the shape of the tail.

Here we want to touch on a narrower, and perhaps distinctly EA, claim. Are there technological reasons to think the distribution of outcomes is bounded at a severity level below extinction? In other words, do we have the weaponry to kill ourselves in war?



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