Statistical reliability analysis for a most dangerous occupation: Roman emperor

Abstract

Popular culture associates the lives of Roman emperors with luxury, cruelty, and debauchery, sometimes rightfully so. One missing attribute in this list is, surprisingly, that this mighty office was most dangerous for its holder. Of the 69 rulers of the unified Roman Empire, from Augustus (d. 14 CE) to Theodosius (d. 395 CE), 62% suffered violent death. This has been known for a while, if not quantitatively at least qualitatively. What is not known, however, and has never been examined is the time-to-violent-death of Roman emperors. This work adopts the statistical tools of survival data analysis to an unlikely population, Roman emperors, and it examines a particular event in their rule, not unlike the focus of reliability engineering, but instead of their time-to-failure, their time-to-violent-death. We investigate the temporal signature of this seemingly haphazardous stochastic process that is the violent death of a Roman emperor, and we examine whether there is some structure underlying the randomness in this process or not. Nonparametric and parametric results show that: (i) emperors faced a significantly high risk of violent death in the first year of their rule, which is reminiscent of infant mortality in reliability engineering; (ii) their risk of violent death further increased after 12 years, which is reminiscent of wear-out period in reliability engineering; (iii) their failure rate displayed a bathtub-like curve, similar to that of a host of mechanical engineering items and electronic components. Results also showed that the stochastic process underlying the violent deaths of emperors is remarkably well captured by a (mixture) Weibull distribution. We discuss the interpretation and possible reasons for this uncanny result, and we propose a number of fruitful venues for future work to help better understand the deeper etiology of the spectacle of regicide of Roman emperors.

Full text

ARTICLE
Statistical reliability analysis for a most dangerous
occupation: Roman emperor
Joseph Homer Saleh1*
ABSTRACT Popular culture associates the lives of Roman emperors with luxury, cruelty,
and debauchery, sometimes rightfully so. One missing attribute in this list is, surprisingly, that
this mighty office was most dangerous for its holder. Of the 69 rulers of the unified Roman
Empire, from Augustus (d. 14 CE) to Theodosius (d. 395 CE), 62% suffered violent death.
This has been known for a while, if not quantitatively at least qualitatively. What is not
known, however, and has never been examined is the time-to-violent-death of Roman
emperors. This work adopts the statistical tools of survival data analysis to an unlikely
population, Roman emperors, and it examines a particular event in their rule, not unlike the
focus of reliability engineering, but instead of their time-to-failure, their time-to-violent-death.
We investigate the temporal signature of this seemingly haphazardous stochastic process
that is the violent death of a Roman emperor, and we examine whether there is some
structure underlying the randomness in this process or not. Nonparametric and parametric
results show that: (i) emperors faced a significantly high risk of violent death in the first year
of their rule, which is reminiscent of infant mortality in reliability engineering; (ii) their risk of
violent death further increased after 12 years, which is reminiscent of wear-out period in
reliability engineering; (iii) their failure rate displayed a bathtub-like curve, similar to that of a
host of mechanical engineering items and electronic components. Results also showed that
the stochastic process underlying the violent deaths of emperors is remarkably well captured
by a (mixture) Weibull distribution. We discuss the interpretation and possible reasons for
this uncanny result, and we propose a number of fruitful venues for future work to help better
understand the deeper etiology of the spectacle of regicide of Roman emperors.
https://doi.org/10.1057/s41599-019-0366-y OPEN
1Georgia Institute of Technology, Atlanta, USA. *email: jsaleh@gatech.edu
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Introduction
What did a Roman emperor have in common with a
gladiator? The latter had better odds of surviving a
fight than the former had of avoiding a violent death.
Popular culture traditionally associates the lives of Roman
emperors with luxury, cruelty, and debauchery, sometimes
rightfully so. One missing attribute in this list is, surprisingly, that
this mighty office was most dangerous for its holder, as the sta-
tistics will show. Consider the following: of the 69 rulers of the
unified Roman Empire, from Augustus (d. 14 CE) to Theodosius
(d. 395 CE), 43 emperors suffered violent death, that is 62%,
either by assassination, the most common mode of death, suicide,
or during combat with a foreign enemy of Rome
1
(Fig. 1). To put
this statistic in context and better appreciate its magnitude,
compare it with what is considered nowadays a seriously dan-
gerous activity: Himalaya mountaineering. Climbers that summit
above 8000 m in the Himalayas have a risk of death of about 4%,
a relatively consistent figure for the past 50 years (Amalberti et al.,
2005). Roman emperors had an order of magnitude greater risk of
violent death than these intrepid climbers.
The odds of survival for a Roman emperor were roughly
equivalent to playing the Russian roulette with a six-chambered
revolver, in which the participant places not one but four bullets,
spins the cylinder to randomize the outcome, and pulls the trigger
with the muzzle against his head.
This sinister comparison conjures the notion of random vari-
able, a central protagonist in our story. That the likelihood of
violent death for an emperor was high was well-known, if not
quantitatively at least qualitatively as far back as Edward Gibbon’s
publication of the first volume of his Decline and Fall of the
Roman Empire (1776). In discussing the highly energetic emperor
Aurelian (ruled from 270 to 275 CE), nicknamed restitutor orbis,
restorer of the world or the unity of the Roman empire in the
troubled third century (Watson, 1999), one can almost feel
Gibbon’s disappointment when reflecting on the emperor’s
murder
2
:
Such was the unhappy condition of the Roman emperors,
that, whatever might be their conduct, their fate was
commonly the same. A life of pleasure or virtue, of severity
or mildness, of indolence or glory, alike led to an untimely
grave; and almost every reign is closed by the same disgusting
repetition of treason and murder [emphasis added].
He then added, “the Roman senators heard, without surprise,
that another emperor had been assassinated in his camp”.
It is worth taking a little historical detour at this point, back to
the time before this track record of treason and murder Gibbon
refers to was even started, to revisit a popular landmark quotation
and understand it in a new light, Julius Caesar’s“Iacta alea est”…
Of dice and men. And Roman emperors. As Caesar con-
templated his decision to cross the shallow Rubicon river with his
legion, a capital offense for himself and his soldiers, he is said to
have exclaimed, according to Suetonius, “iacta alea est”(Sueto-
nius, 1998). The expression is commonly translated as “the die is
cast”, or more resolutely, “let the die be cast”, and it is taken to
signify the passing of the point of no return in an irrevocable
course of action. For anyone familiar with the game of dice, which
Roman soldiers and several emperors were fond of, the emphasis
of the exclamation is less on the passing of the point of no return,
and more on the aleatory nature of the possible outcome. What is
the likelihood that the outcome will be, say, odd? Or a particular
number given a four-sided or six-sided dice? What outcomes did
Caesar contemplate? There is obviously the fate of his under-
taking: will the Roman Republic endure, or is this the last nail in
its coffin? What will his own fate be? Will he be successful, or will
he meet a violent death? In crossing the Rubicon, Caesar sig-
nificantly narrowed down the scope of possible outcomes he
could face.
Beyond this original throw of dice, Caesar provided a
metaphor, not just for himself, but for every subsequent emperor:
upon confirmation by the senate, or being chosen by the legions
or the praetorian guards, every emperor could have exclaimed,
“iacta alea est”and was in effect throwing a dice for his life. In
retrospect, we now know that collectively the rulers of the unified
Roman Empire had a 62% chance of dying a violent death. This is
roughly the equivalent of associating four outcomes, for example
{1, 2, 3, 4} with this gruesome end, and rolling a six-sided fair
dice. Only those who rolled a 5 or 6 got to die of a natural death.
This is a seemingly more benign but equally potent metaphor
than the previous Russian roulette.
What is not known, however, and has never been examined to
date is another random variable associated with these rulers, their
time-to-failure or time-to-violent-death. A brief discussion of
reliability engineering is in order to better understand this idea
and the focus of this work.
Reliability engineering and Roman emperors: from time-to-
failure to time-to-violent-death
Reliability is a popular concept that has been celebrated for years
as a commendable attribute of a person or an equipment.
Although many words in the English language have been coined
by or attributed to Shakespeare, it seems that we owe the word
reliability to another English author, the poet Samuel Coleridge
(1772–1834). In praising a friend, Coleridge wrote (Coleridge,
1983; Saleh and Marais, 2006):
He inflicts none of those small pains and discomforts which
irregular men scatter about them and which in the
aggregate so often become formidable obstacles both to
happiness and utility; while on the contrary he bestows all
the pleasures, and inspires all that ease of mind on those
around him or connected with him, with perfect con-
sistency, and (if such a word might be framed) absolute
reliability.
Since then, reliability engineering has developed into an
important discipline that pervades many aspects of our modern,
technologically intensive world (Hoyland and Rausand, 2009).
The foundational idea in reliability engineering is that the time-
to-failure, T
f
, of an item is stochastic in nature, it is a random
variable. Roughly speaking, reliability, S(t), is defined as the
probability that an item is still operational at time t; it has not
failed and is till performing its function up to this time:
StðÞPr Tf>tðÞ ð1Þ
S(t) is also known as the survival or survivor function. Reliability
engineering is then concerned with quantifying this probability
Fig. 1 Deaths and failure modes of the rulers of the unified Roman Empire.
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over the life of an item. This is sufficient for our purposes.
Another useful concept in reliability engineering is that of failure
rate. It is less important for our story but is worth mentioning.
While reliability is the complement of the cumulative distribution
function (CDF) of the random variable time-to-failure (T
f
), the
failure rate λ(t) is its conditional probability density function
defined as
λtðÞPr t<TftþdtjTf>tðÞ
dt¼ 1
StðÞ
dStðÞ
dtð2Þ
In words, it is the likelihood per unit time that a failure will
occur between tand t+dtgiven that it has not yet occurred by
the time t.
What does this have to do with Roman emperors? It is not that
they were unlike Coleridge’s friend, it is that we can treat their
time-to-violent death as a random variable, and examine it with
the same tools of reliability engineering, or more broadly with the
statistical tools of life data analysis—the double entendre (life) is
not intended but unavoidable here.
Life data analysis, also known as survival analysis, refers to the
analysis and probabilistic modeling of the time-to-event as a
random variable. The event can be broadly defined, and several
academic disciplines have adopted this statistical tool for their
particular interests. For example, beyond reliability engineering
where the interest is in the time to failure of an item, survival
analysis is also broadly used in medical research and public
health, where the interest is in, for example, the time to devel-
opment of a disease, the time to remission after treatment, or the
time to death after prognostic. More details can be added to
further qualify these events. Survival analysis is also increasingly
used in quantitative political science and sociology to examine,
for example, the time to break-down of cease fires, the time to
landing a first job after graduation, or the time to having one’s
first child.
The next section extends the application of these statistical
tools to an unlikely population, Roman emperors, and it exam-
ines a particular event in their rule, not unlike the focus of
reliability engineering, but instead of their-time-to-failure, their
time-to-violent-death.
Data and methods
The data for this work was obtained for De Imperatoribus
Romanis, a peer-reviewed online encyclopedia of Roman
emperors [DIR] (De Imperatoribus Romanis DIR, 2019). The
entry for each emperor was written by a leading scholar in the
field. The Imperial Index provides the list of the rulers of the
united Roman empire, along with the many usurpers who
unsuccessfully claimed the mantle. The data is fairly standard and
generally accepted, and consequently it was considered beyond
the scope of the present work to subject it to further quality
control. Only legitimate emperors confirmed by the Roman
senate are here considered
3
.
For example, after the troubled period following the murder of
both Commodus
4
(d. 192 CE), and Pertinax (d. 193 CE), a
notorious event in the history of Rome took place: the praetorian
guards effectively sold the empire to the highest bidder, Didius
Julianus. This shameful event came to be known as the “auction
of the empire”, and it was neither the first time nor the last that
the praetorians acted as emperor-makers (the first such instance
was the selection of Claudius following the murder of Gaius
Little-Boots or Caligula, as he was nicknamed by the soldiers of
his famed father, Germanicus). Even though Didius Julianus was
considered a usurper by his successor, Septimius Severus and
others, the fact that he was proclaimed by the senate makes him a
“legitimate”emperor for our purposes. As a side note, Didius
Julianus is noted for two memorable feats: the manner in which
he became emperor, and for holding one of the shortest reigns for
an emperor, 66 days, before he was executed.
Another point with the data that should be mentioned is
censoring. Censoring in a statistical sense occurs when life data for
the analysis of a set of items is “incomplete”. This situation occurs
frequently in medical research and reliability engineering, and it
can happen because some individuals under study are lost to
follow-up, or some items are removed prior to the observation of
failure or the event of interest. In our case, about 38% of the
individuals are (right-)censored by virtue of having died of illness
or old age. But two emperors, Diocletian and Maximianus, are
censored through another improbable mechanism: abdication.
Diocletian rose through the military ranks to become emperor in
284 CE, and he put an end to the imperial crisis that plagued
much of the third century, since the murder of the young Alex-
ander Severus (d. 235 CE). Diocletian also undertook many
reforms, administrative and military, which gave the Roman
empire another lease on life for centuries to come, at least in the
East. Then, in 305 CE he abdicated and devoted himself to gar-
dening for the rest of his life at his palace in Split on the beautiful
Croatian coast
5
(d. 316 CE). For the purpose of this study
therefore, Diocletian, and his co-emperor Maximianus, who also
abdicated, perhaps unwillingly, contribute censored life data to
this analysis. The fact that Maximianus was killed or committed
suicide 5 years after he had abdicated does not qualify him to be
counted among the emperors who met a violent death since he
was no longer one.
The quality and limitation of the sources has to be acknowl-
edged, even for such major events as the death or assassination of
an emperor. The sources for the ancient history of interest here
are unevenly distributed. They are patchy at times, sometimes
contradictory, and often proceed with innuendos and inferences.
For example, Suetonius relates several rumors that Caligula, or an
attendant, may have poisoned or smothered the emperor Tiberius
(d. 37 CE) after the latter had fallen ill. The extent of hatred this
emperor inspired across all social classes in Rome may have
contributed to these rumors. But the likelihood that Tiberius’s
death was the result of foul play is slim, and the general consensus
tilts away from this possibility. However, this lack of certainty in
things related to ancient history has to be contended with,
especially in the cases of dubious deaths of emperors when
competing narratives are available. A similar example occurs with
Claudius (d. 54 CE), but the general consensus in this case is that
he was indeed poisoned by his wife Agrippina to expedite the
ascension of her son, Nero. For the purpose of this work, Clau-
dius is therefore considered to have met a violent death, whereas
Tiberius is not.
Two more cases are worth noting, both for their strangeness
and for the parallel they offer with the deaths of Tiberius and
Claudius. Numerianus was acclaimed Augustus in 283 CE upon
his father’s death. He was married to the daughter of the prefect
of the praetorian guard, Flavius Aper. Upon returning from Syria,
the young emperor fell ill, and his entourage, including Aper, let it
be known that the emperor had an eye inflammation, and
therefore had to travel in a closed litter. For several days, no one
checked on the emperor until the soldiers waiting on him noticed
the smell of decay. They opened the litter only to find the
decomposing body of the emperor. An imperial commander
accused Aper of this sordid deed and cut him down. The soldiers
then proclaimed this commander, Valerius Diocles, emperor, and
he would become Diocletian, one of the most capable emperors,
militarily and administratively. So what are we to make of this
story? Numerianus dies under mysterious circumstances. But was
it Aper’s ambition to eliminate his son-in-law and assume the
mantle in his stead? Was Diocles more cleverly devious and
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eliminated in one stone both the emperor and the Praetorian
Prefect to become emperor himself? Or was Numerianus simply
unlucky and succumbed to his illness, after which events just
unfolded haphazardly? There can only be speculation at this
point, and these issues cannot be resolved. For the purpose of this
work, Numerianus’s death is treated like that of Tiberius not
Claudius, that is, it is not considered the result of foul play. A
similar situation occurred some 40 years earlier, with the suspi-
cious death of the child emperor Gordian III (d. 244 CE). The
machinations of his newly appointed Praetorian Prefect Julius
Philippus, later the emperor Philip the Arab, are suspected in this
deed. These again are speculation, and for the purpose of this
work, Gordian III death is treated like that of Tiberius. The sta-
tistical results that follow therefore err on the side of caution and
are most likely conservative.
Aside from these vexing little problems with the data, the rest
of the statistical analysis in this work is straightforward: it starts
with the nonparametric Kaplan–Meier estimator for handling
censored data, then proceeds with the maximum-likelihood
method for estimating the parameters of a particular probability
distribution function
6
. Emperors who met a violent death within
the first year of their rule are assigned a 0.5-year reign, since the
exact number of days is not always accurately known. After the
first year of rule, the time is rounded up to the year when death
occurred. The data is provided in the appendix in Table A1.
Results and discussion
In this section, we investigate the temporal signature of this
seemingly haphazardous stochastic process that is the violent
death of a Roman emperor. We also examine whether there is
some structure underlying the randomness of this process or not,
and we discuss parallels with results in reliability engineering.
Non-parametric and parametric analyses, and interpretation.
The data in Table A1 is treated with the Kaplan–Meier estimator,
and the results are provided in Fig. 2. The longest rule was that of
Augustus, who established the principate, as the early empire was
called. He ruled for 45 years, and since he died peacefully of old
age, he contributes censored data to the analysis. The results in
Fig. 2are shown up to this last time-to-violent death.
Figure 2reads as follows: at the 3-year mark for example, an
emperor had 64% chance of not having met a violent death; at the
7-year mark, those chances drop to 50%, a mere coin toss. The
likelihood of a violent death is the complement of these figures,
36% and 50%, respectively.
What does this mean? Three salient features in this figure are
important to note:
i. 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;
ii. 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…
iii. …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.
A Weibull plot for the previous nonparametric results is
provided in Fig. 3. The plot displays ln{−ln[S(t)]} as a function of
ln(t). If the data points obtained are aligned, it can be concluded
that the data effectively arises from a Weibull distribution, that is,
the underlying parametric distribution giving rise to this violent-
death process is indeed a Weibull. The Weibull survivor function
is widely used in reliability engineering and survival analysis
because it is a highly flexible parametric model, and it is given by
StðÞ¼exp t
θ

β
 ð3Þ
βis termed the shape parameter, and θthe (temporal) scale
parameter or characteristic life. A shape parameter β<1 is
indicative of or reflects the prevalence of infant mortality in the
items under study, whereas β> 1 indicates the prevalence of wear-
out failures (decreasing versus increasing failure rates, respec-
tively). Equation (3) is equivalent to a linear Weibull plot (Eq.
(4)):
StðÞ¼exp t
θ

β

,ln ln StðÞ½fg¼βln tðÞβln θðÞ ð4Þ
It is interesting that a stochastic process as unconventional and
haphazardous as the violent death of a Roman emperor—over a
Fig. 2 Survivor (reliability) function of Roman emperors as a function of
their time in office.
Fig. 3 Weibull plot of the survivor function of Roman emperors, and linear
least square fit(R2=0.962).
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long four-century period and across a vastly changed world—has
a systematic underlying structure, and is remarkably well
captured by a Weibull distribution. The fact that this result is
completely otiose does not diminish its uncanniness!
Why this underlying structure? In statistical theory, the
Weibull function is an extreme value distribution, which captures
the minimum value of a large collection of random observations.
To clarify this point, consider for example a system with a large
number nof components placed in series to fulfill a specific
function. The failure of any one component results in the failure
of the system, its function is no longer provided. The time to
failure of the system is therefore the minimum time to failure of
any one of its component. Statistical extreme value theory tells us
that regardless of the underlying failure distribution of the
components, when nis very large, the time to failure of the
system approaches a Weibull distribution. Notice in this example
the difference between component level and system level
considerations, and how the result at the aggregate system level
is independent of the failure distribution of any one component.
Extreme value theory is also applicable in another related context:
consider a single monolithic item. There are no components in
this item. But assume that there are ndifferent competing failure
processes of this item, whichever one occurs first breaks the item.
When nis very large, this will also result in a Weibull distribution
of the time to failure of the item regardless of the distribution of
each failure process.
The extension of these observations to the violent death of
Roman emperors has to be done cautiously. But they offer
nonetheless a fruitful venue for exploration. The fact that the time
signature of the stochastic process of interest here is remarkably
well captured by a Weibull distribution suggests that it is perhaps
indeed the result of a very large number of underlying processes
conspiring to violently eliminate the emperor. The fact that there
were many pathways to the violent death of an emperor, with
large numbers of individuals and motivations for undertaking the
grisly task, makes the Weibull, an extreme value distribution,
theoretically plausible in this case.
Mixture Weibull distributions. A closer inspection of Fig. 3
shows two distinctive slopes for the data points, before and after
ln(t)≈2.5, which corresponds to the onset of the wear-out failures
seen in Fig. 2. A mixture Weibull distribution is therefore fitted to
the data, and the maximum-likelihood estimates of its parameters
are provided as follows:
b
StðÞ¼0:876 exp t
12:835

0:618
hi
þ0:124 exp t
14:833

13:387
hi
ð5Þ
Equation (5) provides an analytical confirmation of the
previous observations, that Roman emperors experienced both
infant mortality (β=0.618) and wear-out mortality (β=13.387)
in the form of violent death. This parametric result is shown in
Fig. 4.
The emperors who experienced infant mortality were not
unlike engineering components that suffer early failures after they
are put to use: weak by design or fundamentally incapable of
meeting the demands of their environment and circumstances.
Examples from each century abound, for example Galba (d.69
CE), Pertinax (d. 193 CE), Macrinus (d. 218 CE), and Severus II
(d. 307 CE). These were times of upheaval, and in the first two
cases, these turned out to be times of transition to new dynasties
(the Flavian, and the Severan, respectively). Emperors’infant
mortality can be seen, in part, as both causes and consequences of
times of crisis and instability
7
.
The emperors who experienced wear-out mortality met their
end through different failure mechanisms. Consider first that
some engineering components experience an uptake in failures
(wear-out failures) after they have been in service for a long time.
They may have been sturdy at first and benefitted from clement
operational environments to start with. But through degradation,
fatigue, or increased harshness in their operational environment,
they begin to experience wear-out failures. The emperors who
survived the first 8 years of their rule, as seen in Fig. 2, had a grace
period of about 4 years. Violent death came to them afterward
(wear-out mortality) because, for instance, their old enemies had
regrouped or new ones emerged, because they had alienated an
increasing number of parties, or because new weaknesses in the
imperial rule appeared or grew. These new murderous processes
clearly had a different temporal signature than those driving the
emperors’infant mortality, as seen in Fig. 3and in the different
characteristic life parameters of each Weibull distribution in Eq.
(5). For example, the death of Domitian after a 15-year rule (d.96
CE), or Commodus after a 12-year rule (d. 192 CE), or Gallienus
after a 15-year rule (d. 268 CE) are illustrative of wear-out
mortality
8
.
The failure rate (Eq. (2)) of the parametric fit (Eq. (4)) is given
in Fig. 5. The result shows a remarkable bathtub-like curve, a
model widely used, and empirically confirmed in reliability
engineering for a host of mechanical and electronic components.
Roman emperors, like these engineering items, therefore
experienced a bathtub-like failure rate.
The results in Fig. 5lends themselves to an interesting
interpretation:
i. The decreasing failure rate early on, the signature of infant
mortality, reflects as noted previously a prevalence of weak
Fig. 4 Mixture Weibull survivor (reliability) function of Roman emperors,
and the nonparametric results.
Fig. 5 Failure rate of Roman emperors (parametric fit of the time-to-violent-
death).
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emperors who were incapable at the onset of their rule to
the 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;
ii. 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.
Beyond these specific details, what does it mean to find a
coherent structure within a stochastic process of historical nature
as the one here examined? Roughly speaking, the result implies
the existence of systemic factors and some level of determinism,
in an average sense or expected value, superimposed on the
underlying randomness of the phenomenon here examined. In
other words, the process is not completely aleatory; it has some
deterministic factors overlaid on its randomness. Conan Doyle, in
Sherlock Holmes: The Sign of Four, expressed this general idea
rather accurately when he wrote:
While the individual man is an insoluble puzzle, in the
aggregate he becomes a mathematical certainty. You can, for
example, never foretell what any one man will do, but you
can say with precision what an average number will be up to.
The results in this section suggest a similar idea underlies the
violent death of Roman emperors.
Etiology and suggestion for future work. The previous subsec-
tions investigated the temporal signature of the phenomenon here
examined, the violent death of emperors, a spectacle of brutality
and violence not unlike the gladiatorial games, except it stretched
over four centuries and affected the entire Roman world (Millar,
1977).
What has not been explored is the etiology or causal basis of
this phenomenon, or why emperors repeatedly met violent deaths
in the first place, not just temporally how. The immediate causes
of violent deaths of Roman emperors are frequently discussed in
the literature. They can be found for example in Scarre’s(1995)
“Chronicle of Roman Emperors”, and a short summary is
provided in Retief and Cilliers’(2005)“Causes of death among
the Caesars (27 BC–AD 476)”. The entries include statements
such as “murdered by the sword/dagger […]”,“poisoned by
[name of individual]”,or“decapitated by the soldiers”. These
explanations are of little interest, and they do not reflect the
complex nature of causality in this context. The causal basis of the
phenomenon here examined intersects a number of fundamental
issues in Roman history, the development and pathologies of the
Roman monarchy for example, the problem of imperial
succession, the role of the praetorian guard, the loyalties of the
legions, and the geographic extent and resources of the empire, to
mention a few. These issues and the complex nature of their
relations with the phenomenon here examined are left as a
fruitful venue for Roman historians to examine. It is worth noting
that the the spectacle of regicide of Roman emperors is related a
reciprocal way, as both a causal factor and a consequence, to the
decline and fall of the Roman empire. As such, it deserves careful
attention in future work.
Conclusion
On his deathbed, Augustus called for a mirror, examined himself,
and had his hair combed. Then, as Suetonius recounts
9
:
he called in his friends and asked whether it seemed to
them that he had played the comedy of life well. He added:
since well I have played my part, all clap your hand, and
from the stage dismiss me with applause.
Marcus Aurelius closed his Meditations on a similar, albeit more
somber note, of life as a theater and actors sometimes getting
dismissed from the stage after fewer than the whole five acts
(Meditations, tr.1989)“It does not matter”, he adds stoically,
“whetherhebeholdstheworldalongerorshortertime”.Thisisa
fitting metaphor for all Roman emperors, and it serves to frame the
scope of this work, namely for how long the emperors “beheld the
[Roman] world”before they were dismissed from the stage, vio-
lently if that was the case. Edward Gibbon offered a similar view for
the entirety of the history of decline and fall of the Roman empire:
By a philosophic observer, the system of Roman govern-
ment might have been mistaken for a splendid theater,
filled with players of every character and degree, who
repeated the language and imitated the passion of their
original model.
This work began in jest by comparing Roman emperors with
gladiators, and it noted that the odds of survival of the former
were worse than those of the latter. There is perhaps more to this
comparison than meets the eye. There was a particular appeal to
gladiatorial games in the Roman world (Fagan, 2011). Whatever
its reasons
10
, it is undeniable that these games offered a spectacle
of extreme brutality, like an unscripted theatrical play with vio-
lence as the main protagonist, and gladiators the creative agents
of its delivery. Roman emperors performed in similar games,
except instead of delivering their role in single afternoon, they
took several years, sometimes only a few months to complete it
before they were dismissed from the stage. They also faced more
diverse hazards, and stealthier adversaries than those encountered
by the gladiators in the arena. Incidentally, the emperor Com-
modus would blur the line of this analogy and go down into the
arena and fight gladiators (as well as wild beats).
In examining their time-to-violent-death, this work found that
of Roman emperors experienced infant mortality as well as wear-
out failures. Their failure rate displayed a bathtub curve, similar to
that of a host of mechanical engineering items and electronic
components. More interestingly, it was found that a stochastic
process as unconventional and haphazardous as the violent death
of a Roman emperor has a definite underlying structure, and is
remarkably well captured by a Weibull distribution. The inter-
pretation and possible reasons for this result were discussed. Some
fruitful venues for future work were proposed to help understand
the deeper etiology of the violent deaths of Roman emperors.
In seeking to uncover the causal basis of the spectacle of
imperial regicide, one important causal factor should not be
overlooked. It was briefly hinted at earlier in the section “Data
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and methods”, when Diocletian was being urged to reclaim the
purple a few years after he had abdicated, he replied that he
would never trade the peace and happiness of his garden “for the
storms of a never-satisfied greed”. Since Augustus and Marcus
Aurelius conceived of life as a play, we leave it to a Greek play-
wright, Euripides, as quoted by Plutarch in the Life of Sulla, to
present this final causal factor in the phenomenon here examined:
[…] passing afterwards through a long course of civil
bloodshed and incurable divisions proved Euripides to have
been truly wise and thoroughly acquainted with the causes
of disorders in the body politic, when he forewarned all
men to beware of Ambition, as of all the higher Powers [it
is] the most destructive and pernicious to her votaries
11
.
A fundamental engine of the spectacle of regicide was not
structural in nature, nor within the legions or the flawed system of
government of the empire, but intrinsic to the actors themselves:
they were votaries of Ambition on the stage of one of the most
consequential adventure in human history that was the Roman
empire. The individuals should not be neglected in future work. It
was the mutual interactions between the motivations and ambi-
tions of individuals on the one hand, Vespasian, Septimus Severus,
and Diocletian for example, and the social, political, and military
factors on the other hand that led to spectacle of regicide of Roman
emperors. The whole was subject to historical contingencies and
some level of randomness in the timing and alignment of factors.
This observation harkens back to Caesar’s“iacta alea est”;this
work has shown that the die he cast, and every emperor after him,
was loaded and not completely fair or aleatory.
Data availability
The dataset used in this work is publically available in De
Imperatoribus Romanis, a peer-reviewed online encyclopedia of
Roman emperors [DIR]: http://www.roman-emperors.org/.
Received: 20 June 2019; Accepted: 11 November 2019;
Notes
1 When calculated on a century basis, the statistics show the same order of magnitude.
For example, in the first and fourth century CE, roughly 58% of the emperors
suffered a violent death, and in the third century, 77%, a reflection of the convulsion
and crisis of the third century. Only during the second century does the rate fall to
the low 30%. This was the period of the “five good emperors”, as Gibbon calls them,
from Nerva (d. 98) to Marcus Aurelius (d. 180).
2 The city of Orléans in France is named after him, and consequently, New Orleans.
And by extension, with a bit of a stretch, one can associate with him Jazz!
3 This introduces a subtle bias in the analysis which has to be acknowledged. By
focusing on legitimate emperors only, we discard the many contenders for the
imperial power who failed to secure senate confirmation, and were thus killed usually
early on after their uprising. Senate confirmation, however, sometimes came after-
the-fact as a recognition of a fait-accompli that a would-be usurper and or his legions
made it to Rome and obtained the senatorial recognition of the emperorship. Suffice
to remember that Vespasian for example, Septimius Severus, and Julian to mention a
few were, for a brief time, usurpers before obtaining senate confirmation. The
implications for our purposes is that this introduces a survivor bias in our analysis,
and that the results in section “Results and discussion”for the survivor function are
likely conservative and underestimate the spectacle of regicide of Roman emperors,
especially in the first few years of their reign. We are grateful for an anonymous
reviewer who brought this up to our attention.
4 Portrayed skillfully by Joaquim Phoenix in the movie Gladiator. Fans are still waiting
for the second installment of this movie.
5 In a famous statement upon being urged to reclaim the purple, Diocletian is said to
have replied, “if you could show the cabbage I planted with my own hands to your
emperor, he definitely wouldn’t dare suggest I replace the peace and happiness of this
place with the storms of a never-satisfied greed”[Epitome de Caesaribus]. It is
however not known what kind of cabbages he grew.
6 The data are assumed to be independent and identically distributed (iid), as generally
done in survival data analysis. This assumption, however, may be challenged for some
emperors (e.g., father–son pairs), and this constitutes a potential limitation of the
analysis.
7 To paraphrase Thomas Paine, “these are times that try [an empire’s] soul”.
8 Each of these cases has an interesting underlying pathway to violent death. These
narratives, however, are beyond the scope of this work.
9 It is enjoyable to imagine and hope that Suetonius’work be turned into a TV series
someday. If this happens, what music would go with Augustus’death scene? Rex
Tremendae from Mozart’s requiem would be a good start as the camera pans outside
his dwelling (he was on his way to Rome and had just passed Naples when his illness
took him). He was one, a rex tremendae, and was on the verge of being deified. Then,
within his dimly lit chamber, the music switches to the more intimate prelude of
Bach’s Cello Suite No. 2 in D minor. The music fades, labored breathing is heard,
then, cue to the actor portraying Augustus…
10 One only needs to consider the current appeal of the mixed martial arts (MMA)
bloody fights, and then add to that swords, spears, and tridents for example to
understand the appeal of gladiatorial games.
11 A devoted follower or a zealous acolyte.
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Competing interests
The author declares no competing interests.
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