The Uppsala Conflict Data Programme (ucdp) provides free, high-quality data on the durations of armed conflicts. This is their extensive conflict termination dataset covering all armed conflicts (interstate and intrastate) where there have been flare-ups of violence (called episodes in the data) after the second world war.

The following plot is a survival curve of the duration of conflict episodes started in the pc age2² It is often convenient to limit analysis to “very recent history”. I define that as the launch of the first ibm pc in 1981. It also happens to coincide with the eradiction of smallpox in 1980, meaning we can split history into the smallpox age and the pc age.. The crosses represent those that are still ongoing.

For predicting the probability of peace in the near future, we need the hazard rate. Extracting that from this data is conceptually simple – it’s the same idea as when we discussed the Supreme Court – but it turned out to be algorithmically tricky, for two reasons:

• The conflict episode data is quantised to the nearest next year. If an episode ends 3.2 years in, it will be reported in the data as having gone on for 4 years.
• A large number of episodes end early. Almost a third end after the first year, then almost a fifth in the second year.

These two properties in combination make for a weird shape of the distribution in the area around 0–2 years: conflicts that ended after 0.2 years and conflicts that ended after 0.9 years are both reported as having ended after exactly one year. The strange steppy shape of the distribution throws a typical maximum likelihood fit for a loop. After much trying of various ways of dealing with it, I figured out how to do it properly.

This gives us a theoretical distribution that fits the data well. The duration \(t\) of violent episodes is distributed as roughly

\[t \sim{} \mathrm{Weibull}(\lambda=3.4,\, k=0.5)\]

We can superimpose this on the survival function from before to get a visual confirmation of the fit. Keep in mind that we’re not saying the duration of an episode is literally determined by the Weibull distribution, only that the curve of the Weibull distribution happens to lie very close to the actual durations, so we can use the Weibull curve as an approximation of the underlying process, whatever it is.

The Weibull distribution has two parameters: the scale \(\lambda\), and the shape \(k\). The interpretation of the shape parameter \(k\) is quite easy:

• \(k=1\) means the hazard rate is constant, which implies an exponential survival distribution: at any given time, a fixed percentage of the population experiences the event of interest. (In this case, that would be the end of the conflict episode.)
• \(k>1\) means the hazard rate is increasing, which implies a survival distribution that is calm at first but then quickly drops off. There is a specific region of ages at which the event of interest happens.
• \(k<1\) means the hazard rate is decreasing, which implies a survival distribution where the majority of the population experiences the event early, and then some go on for a very long time without experiencing the event.

For episode durations, we have \(k = 0.5 < 1\), which means most episodes end early, but then some go on for a very long time. If we hold \(k\) fixed at 0.5, we also get a natural interpretation of the scale parameter \(\lambda\): the average duration of a episode is \(2\lambda\). Thus, the average duration of a conflict episode in the pc age is 7 years. This might seem surprising, because the average observed episode duration in the ucdp data is something like 3.8 years! But that’s because many of those conflicts have not yet terminated, so the observed duration is biased downward.

There’s another way this is surprising: the majority of conflicts end within two years, so how can the average length be seven years? That’s thanks to the heavy-tailed nature of conflict lengths. There are a few that go on for a very long time and pull up the average length by a fair bit.

We have more information we can use. The Sudan conflict is a civil war, whereas the Ukraine conflict is a war between two states. That could have a large effect on the probability of peace; when I read the news, it seems like civil wars are more difficult to stop.

Of the 411 conflict episodes in the pc age, there are only 33 interstate wars. But we can still split the data up into two categories and fit the survival distribution with civil war as a predictor.

This confirms our intuition that civil wars last longer. It might even be the case that interstate wars have a completely different type of hazard rate than civil wars, but we’ll treat them the same for now.

The theoretical Weibull distribution we fitted to the data has the same shape as before (\(k\) = 0.5), but the scale now depends on whether it’s an interstate war or a civil war.

Interstate war	Civil war
\(\lambda\) = 1.3	\(\lambda\) = 3.6

As it happens, most of the average length of episodes in the data is driven by civil wars. The mean length of an interstate war in the pc age is 2.5 years, whereas for a civil war it is over 7 years. Updating our duration-agnostic forecast split up into conflict type, we have

Interstate war	Civil war
22 %	10 %

As we have seen, the peace probability depends a lot on conflict duration, so these duration-agnostic forecasts aren’t great, but they are a good way to interpret the scale of the Weibull distribution: a randomly selected interstate war is more than twice as likely to end in the next year as a civil war.

The refined table of peace probabilities conditional on duration is more useful for forecasting.

Duration	Interstate war	Civil war
1 year	39 %	26 %
2 years	27 %	17 %
3 years	23 %	14 %
4 years	20 %	12 %
5 years	18 %	11 %
5–10 years	15 %	9 %
10–20 years	11 %	7 %
20–40 years	8 %	5 %
>40 years	6 %	3 %

Now we get an even more accurate forecast of peace in the two wars we opened the article with.

Question	By duration	By type	Change
Ukraine	13 %	20 %	+0.51 log-odds
Sudan	15 %	14 %	−0.08 log-odds

For the first time, we have a forecast that is lower for Sudan than Ukraine. That’s because this is the first time we have accounted for it being a civil war, and civil wars generally last longer.

In the conflict episode termination dataset, the ucdp have coded intensity as a binary variable: low intensity is 25–1000 annual battle deaths, and more than that counts as a full-scale war. This is a dichotomisation, meaning it throws a third of the data in the rubbish bin. Fortunately, the ucdp publishes separate data sets for battle deaths, and both data sets are indexed by conflict id and year, meaning we can inner join them to get a complete data set of both durations and battle deaths.

We cannot include the raw sum of battle deaths as a predictor, because then it would leak the duration of the conflict. What we can do is count the average number of yearly battle deaths as the intensity of the conflict, and include that as a predictor. However, the distribution of battle deaths is extremely heavy-tailed.

This chart goes out so far to the right because there are data points there.

Almost all conflicts have a high battle death rate; the median is nearly 100 deaths per year. Yet there are a handful of conflicts that are staggeringly violent at more than 10,000 battle deaths per year. The worst one has over 100,000 battle deaths per year. This handful of really deadly conflicts in the pc age are

• Ethiopian Tigray war (2020–2022): 103,000 battle deaths/year.
• Russia–Ukraine (2022–): 80,800 battle deaths/year.
• Ethiopia–Eritrea (1998–2000): 32,700 battle deaths/year.
• Syrian civil war (2011–2024): 20,500 battle deaths/year.
• Israel–Palestine (2018–ongoing): 12,200 battle deaths/year.

This is insanity.

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32,000 battle deaths per year is almost 100 people dying every day. And that’s just among combatants, not counting civilians. I don’t even want to look up civilian casualties.

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What is going on in the world?

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Deep breaths.

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I sometimes have to remind myself why I do this. The examples in this article are gruesome, but my hope is that (a) people realise that these things really happen in real life, and (b) at least one reader will make the world a better place using the tools I share.

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If we take the logarithm of the battle death rate we get a more tractable distribution.

Here we have taken the logarithm base ten, so that the number indicates how many zeroes there are in the number. For example, 4 should be read as 10,000. From this plot, it is clear that the median is somewhere around 100 annual battle deaths. The standard deviation is nearly an order of magnitude.

When we include this variable in the model, the variable that indicated whether the fight was over territory or government drops out. It seems that it mostly served as a proxy for the level of violence in the conflict. The shape \(k\) goes up to 0.7, and to get the scale, we need an estimation of the order of magnitude of the intensity.

Log-intensity	1	2	3	4	5
Interstate war	\(\lambda\) = 0.4	1.7	8.7	43	214
Civil war	\(\lambda\) = 0.8	3.7	19	92	460

For interpretability reasons, we can convert this to a probability of peace when ignorant of their duration

Log-intensity	1	2	3	4	5
Interstate war	74 %	32 %	8 %	2 %	<1 %
Civil war	52 %	17 %	4 %	1 %	<1 %

This would indicate probabilities of <1 % for the Ukraine conflict (interstate ware with annual battle death rate \(10^5\)), and 4 % for the Sudan conflict (civil war with annual battle death rate \(10^3\)). However, we also need to account for their duration.

First, for interstate wars:

Log-intensity	1	2	3	4	5
1 year	77 %	40 %	15 %	5 %	2 %
2 years	67 %	33 %	12 %	4 %	1 %
3 years	62 %	30 %	11 %	4 %	1 %
4 years	59 %	27 %	10 %	3 %	1 %
5 years	56 %	26 %	9 %	3 %	1 %
5–10 years	53 %	23 %	8 %	3 %	<1 %
10–20 years	47 %	20 %	7 %	2 %	<1 %
20–40 years	41 %	17 %	6 %	2 %	<1 %
>40 years	33 %	14 %	4 %	1 %	<1 %

and then for civil wars:

Log-intensity	1	2	3	4	5
1 year	59 %	26 %	9 %	3 %	1 %
2 years	49 %	21 %	7 %	2 %	<1 %
3 years	45 %	18 %	6 %	2 %	<1 %
4 years	42 %	17 %	6 %	2 %	<1 %
5 years	40 %	16 %	5 %	2 %	<1 %
5–10 years	37 %	14 %	5 %	2 %	<1 %
10–20 years	32 %	12 %	4 %	1 %	<1 %
20–40 years	27 %	10 %	3 %	1 %	<1 %
>40 years	22 %	8 %	2 %	<1 %	<1 %

Of course, we don’t actually have data to fill all these fine-grained buckets. We have made assumptions about how the data we do have are related to each other, and used that to interpolate and extrapolate to where data is missing. With this, our forecast for the two conflicts we opened with should be

Question	By contention	By raw intensity
Ukraine	21 %	1 %
Sudan	12 %	6 %

Well … I’m not sure what I think about that. Our model leads us to predict 1 % for every extremely high-intensity (\(10^5\)) conflict. That’s obviously the wrong prediction: those conflicts do end far sooner than in the 100 years implied by the 1 % success rate. I would personally maybe not distinguish between conflict intensities above \(10^3\), and use the probabilities for \(10^3\) conflicts for those too. We can encode that intuition into the model by breaking the intensity up into three bands:

• Low intensity: fewer than 50 annual battle deaths. (30 % of the data.)
• High intensity: more than 200 annual battle deaths. (31 % of the data.)

In between, we find the median, baseline intensity. The shape of the Weibull distribution fitted to these predictors is 0.6.

Intensity	Low	Median	High
Interstate war	\(\lambda\) = 0.5	1.4	6.2
Civil war	\(\lambda\) = 1.0	3.1	13

Translated to duration-agnostic probabilities of peace for interpretation:

Intensity	Low	Median	High
Interstate war	55 %	29 %	9 %
Civil war	36 %	16 %	4 %

The duration-dependent probabilities for interstate wars would then be

Intensity	Low	Median	High
1 year	64 %	42 %	20 %
2 years	51 %	32 %	14 %
3 years	45 %	27 %	12 %
4 years	41 %	25 %	11 %
5 years	38 %	23 %	10 %
5–10 years	34 %	20 %	9 %
10–20 years	28 %	16 %	7 %
20–40 years	22 %	12 %	5 %
>40 years	17 %	10 %	4 %

For civil wars they are

Intensity	Low	Median	High
1 year	49 %	28 %	13 %
2 years	37 %	21 %	9 %
3 years	32 %	18 %	8 %
4 years	29 %	16 %	7 %
5 years	27 %	15 %	7 %
5–10 years	24 %	13 %	6 %
10–20 years	19 %	10 %	4 %
20–40 years	15 %	8 %	3 %
>40 years	12 %	6 %	2 %

Converted to a forecast, we get.

Question	By contention	By intensity
Ukraine	21 %	11 %
Sudan	12 %	8 %

These probabilities still need to be adjusted because the resolution criteria for the forecasting competition are looser than in the ucdp data, so I would add maybe one log-odds to both probabilities, and end up with a final forecast of

Question	By contention	Forecast	Change
Ukraine	21 %	25 %	+0.23 log-odds
Sudan	12 %	19 %	+0.54 log-odds

That feels reasonable. It is lower than my naïve forecasts of 38 % and 43 % respectively, but that also reflects my main takeaway from this analysis: conflicts last much longer than I think. Since so many of them resolve so quickly, it is easy to accidentally think that those that are currently ongoing will resolve soon, too. But by looking at those that are ongoing now, we are filtering for those that are long-running, thanks to the heavy tails of the distribution of their duration.

Although it is of no help for answering the forecasting question we have, we can also analyse the cost of conflict episodes. We have battle death estimations of 362 of the episodes in the pc age.

Half of conflict episodes end before 300 battle deaths. Of those that go on, half again end before 3,000 battle deaths. Of those that go on, half again end before 30,000 battle deaths. Of those that go on, we don’t yet have numbers to accurately estimate when half of them end, but it wouldn’t be weird to assume that they end before 300,000 battle deaths.

But that also means half of those continue further, even after 300,000 deaths in battle. That’s 0.3 million people whose lives are no longer. Whose family misses them. Whose children will either never be, or never see their parent again.