The Survival Function of Roman Emperors
How long could a Roman emperor expect to survive after taking power?
That’s a challenging question to answer because the majority of Roman emperors met violent ends. However, new research suggests such results weren’t as unpredictable or as random as you might first think.
In modern engineering, the concept of reliability describes the probability that an item will still be operational at some point of time in the future.
Usually, reliability is applied to things like electrical circuits and mechanical devices, but Joseph Saleh has applied the concept to politics, and more specifically, to mathematically describe the survival function of Roman emperors.
“It’s interesting that a seemingly random process as unconventional and perilous as the violent death of a Roman emperor–over a four-century period and across a vastly changed world–appears to have a systematic structure remarkably well captured by a statistical model widely used in engineering. Although they may appear as random events when taken singularly, these results indicate that there may have been underlying processes governing the length of each rule until death.”
The following chart from Saleh’s paper shows the mixture Weibull survivor function he was able to map to the available empirical data for how long ancient Rome’s emperors lived after they assumed the purple.
Looking at the nonparametric estimation of the average remaining lifespans of Roman emperors, shown as the heavy black line in the chart above, Saleh offers several observations:
- Emperors faced a significantly high risk of violent death in the first year of their rule. This risk remained high but progressively dropped over the next 7 years. This is reminiscent of infant mortality in reliability engineering, a phase during which weak components fail early on after they have been put into service, often because of design or manufacturing defects for example. Roman emperors therefore experienced a form of infant mortality;
- The reliability or survivor function stabilizes by the 8th year of rule. The emperors could lower their guard a bit if they made it to 8 years…
- … but not for long: the risk of violent death picks up again after 12 years of rule. This suggests that new mechanisms or processes drove another round of murder. This is reminiscent of wear-out period in reliability engineering, a phase during which the failure rate of components, especially mechanical items, increases because of fatigue, corrosion, or wear-out. Roman emperors therefore also experienced wear-out mortality.
We’ve built the following tool to estimate the likely survival potential of a generic Roman emperor from Saleh’s math. If you’re accessing this article on a site that republishes our RSS news feed, please click through to our site to access a working version.
In the tool, we’ve arbitrarily capped the maximum number of years a Roman emperor might survive to 45 years, which corresponds to the Emperor Augustus’ reign, the longest on record.
Saleh’s paper also provides a chart showing the hazard function, or failure rate, for Roman emperors, which reveals a unique pattern.
This pattern is the familiar “bathtub curve” that characterizes how many real world components behave in reliability analysis. Saleh provides an interesting interpretation of how this pattern applies to the lives of Roman emperors:
- The decreasing failure rate early on, the signature of infant mortality, reflects as noted previously a prevalence of weak emperors who were incapable at the onset of their rule to handle the demands of their environment and circumstances. The fact that the failure rate was decreasing though suggests a competition between antagonistic processes, on the one hand those that sought to violently eliminate emperors (elimination), and on the other hand those that reflected the emperors learning curve to better protect themselves and perhaps eliminate their opponents (preservation). Examples abound in Roman history of this competition. Up to the first 12 years of one’s rule, the preservation processes steadily improved their performance, and the situation can be casually summarized as “whatever didn’t kill them [the Roman emperors] made them stronger” or less likely to meet a violent death;
- The increasing failure rate after 12 years of rule, the signature of wear-out failures, reflects as noted previously an uptake in failures through degradation with time, fatigue, or increased harshness in their circumstances. A growing mismatch between capabilities and demands under changing (geo-)political circumstances. This can be due to a number of reasons discussed previously. The fact that the failure rate was increasing after this 12-year mark suggests again a competition between the same antagonistic processes noted in (i), and this time the preservation ones were on the losing end of this competition. This result can be causally summarized as “whatever didn’t kill them made them weaker” after a 12-year rule.
The existence of the pattern means that the probability of how long a Roman emperor might last is the result of both random chance and deterministic factors, rather than just chance alone, as perhaps best imagined by the “roll of the dice” Julius Caesar figuratively cast before crossing the Rubicon on his way to taking power as Rome’s age of emperors began.
Part of what makes Saleh’s analysis so intriguing is that the same concept can be applied to other nations, or forms of government, that have developed in the centuries since the fall of the Roman empire. It will be interesting to discover what patterns they might share with the Roman emperors.
Image Credit: iam_os
Saleh, J.H. Statistical reliability analysis for a most dangerous occupation: Roman emperor. Palgrave Communications 5, 155 (2019). doi: 10.1057/s41599-019-0366-y.
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