Key prediction: Collapse is ubiquitous and occurs at peak power.

The phenomenon of collapse has occurred across diverse world cultures throughout history, and has affected polities of all sizes. Here we describe cases where a strong state, which was able to exert significant power, has experienced collapse or unrest. Such states have strong leadership relative to their subjects, evidenced by a) the successful imposition of will, or b) the ability to expend resource in prolonged offensive wars, extensive building programs, etc. These examples contrast slow decline in which the ultimate collapse does not occur at a period of strength. In the following examples and many more, collapse or unrest followed displays of strength relative to contemporary states and/or the same state at different times.

The 5th century BC saw the peak of the Achaemenid Persian Empire, which was then the largest empire in history. The early reign of the emperor Xerxes, following that of his father Darius, was beset by internal rebellion in Babylonia and Egypt. Darius' reign involved extensive military activity, and while Xerxes was eventually able to return to the offensive, he had to start his reign regaining dominion over the territory he inherited [30].

Alexander the Great's empire is a prime example of fragmentation of a powerful state at its peak in 323BC. The division can be thought of as a collapse, as it resulted in violent upheaval [31]. The successor states or Diadochi engaged in extensive warfare over the following decades, and many subdivided further [32].

The Roman Empire underwent several cycles of expansion and instability in the first and second centuries AD. For example, the reign of the Roman emperor Domitian was lambasted by writers of his time as being tyrannical and autocratic [33]. Contemporary detractors claimed that he ignored tradition, executed senators who opposed him and openly asserted primacy over the senate [34]. After his assassination, the new emperor Nerva had a short and impotent tenure, due to revolts by the military. The indulgence and autocracy of the Roman emperor Commodus [35] is often described as the beginning of the decline of the Roman empire [4], leading to a protracted civil war. Yet the Nerva-Antonine dynasty that preceded Commodus was amongst the most successful periods in Roman history. The remnants of the Roman empire, centred on Byzantium, exhibited periodic instability well into the Middle Ages [36].

Beginning in the 6th century after the adoption of Islam, Arab armies expanded their domain westwards towards Morocco, east into Persia and as far north as France. The empire collapsed in the mid-8th century, and the successor states suffered further internal conflict and dissolution in the following centuries [3].

Contemporaneous to European medieval civilisation, the Khmer civilisation of Indochina achieved regional hegemony and its capital Angkor may have been the largest city in the world [37]. It experienced multiple damaging episodes of civil war, notably prior to the accession of Suryavarman I [38]. Angkor provides an excellent example of a state whose power rested on appropriation from vassals; indeed, much of Jayavarman II's power depended on a rice surplus [39]. Its ultimate collapse is associated with climate [12],

Finally, the Mongol empire was the largest contiguous land empire in history. In the wake of their initial conquests the empire divided, at times violently, into smaller, culturally heterogeneous polities [2].

Prediction: Cascading civil unrest.

It is not clear a-priori whether states should end rapidly, slowly, or require external influences to fail. A brief examination of history leads to the conclusion that civil unrest can very quickly accelerate, leading to at least the potential for rapid state collapse. In history and in our model, uncoordinated defections often produce their own momentum. As more factions choose to leave a state, the perceived legitimacy of that state is eroded encouraging further defections.

This is seen in many of the examples discussed, but we draw attention to the Seleucids and the Byzantines. The furthest vassals of the 1st century BC Seleucid empire seceded early, leading to a geographical wave of defections [31], [32]. In the middle ages, the Byzantine emperors were frequently beset by a single rebellion or usurpation that preceded rebellions around the empire [40]–[43].

Assumption: Inequality before unrest.

Our model assumes that perceived inequality can influence the decision to defect. This is backed up by unrest frequently following both centralisation and increasing inequality. Once again, examples can be found across documented history.

From the 14th century BC, an excellent example of unrest stemming from centralised power is the reign of Amenhotep IV (Akhenaton) of Egypt. He introduced massive centralisation in the form of a new religion. The strength of his regime is clear from the foundation of a new capital and decoration of the empire with sun temples and other paraphernalia. His reign was followed by unrest and internal disorganisation [44].

Many examples of inequality leading to unrest can be taken from the late Roman republic in the first century BC. The Social (or Marsic) War [5], as well as three Servile Wars were fought by Rome during or immediately after periods of expansion abroad. The stated grievance in these cases was explicitly inequality; the slaves wished to be elevated from their abject position in society, whilst the Socii demanded an end to their second-class status within the Republic. Caesar's civil war [45] also took place immediately after a period of offensive foreign campaigns, notably the annexation of Gaul. Caesar explicitly states that inequitable distribution of wealth and power was a cause.

A small scale but important example was the reign of Richard I of England in the 12th–13th century AD, leading to the signing of the Magna Carta. To fund conflict in France and the Levant during the reign of Richard I, the crown made increasing demands on vassals. Following this, John I was forced to make concessions to the nobility [46]. This can be thought of as an example of factions resisting an attempt to appropriate resource and power.

As a final example, the 1905 [47] and February 1917 [48] revolutions in Russia are often attributed to the inequitable distribution of wealth and power between the ruling classes and the majority of the population [49] – the term ‘autocrat’ was even part of the official title of the Tsars.

Assumption: All else being equal, power will agglomerate.

Our model makes the assumption that power tends to accumulate over time. Although there are counter-examples, the tendency towards conglomeration is relatively ubiquitous. Specific examples include the concentration of land holdings into the latifundia of ancient Rome [4], the extensive provincial landholdings of the late Sassanid empire [50], and the distinction between land-holding nobility and the rest of society in Feudal Europe [51]. Mercantile quasi-states such as the Hanseatic League [52], Italian merchant republics [53] and European colonial enterprises [54] exhibit this behaviour, amongst many others.

Assumption: Defection can be pragmatic.

We have assumed some degree of rationality about the choice to revolt. Whilst we are very careful not to assume that people are rational, we must assume that there is some link between the resource expected from defecting, and the choice to defect.

As an extreme but clear cut and common example, one instigator of internal conflict is the inability of a faction to hold political power or influence by peaceful means. This occurs when a faction perceives that it receives an inadequate degree of power, or that it's power is being eroded. The perception of inequity may encourage the use of violence in order to redress the apparent imbalance. Excellent examples of this appear in the late Republican period of Rome [55]; of particular note is the career and attempted rebellion of Catiline [56]. The repeated, and often unsuccessful, peasant revolts of medieval Europe also exhibit many of these features [57].

Features of the basic model.

We can take the continuous time limit of our model, which removes intrinsic noise due to discretisation. We can also take the continuous faction limit, which leads to a Partial Differential Equation model. These models (Section S1 in File S1) are not readily solvable but do allow us to understand why our model behaves as it does.

During cooperation, the power of each faction departs exponentially from the baseline. Defection cascades occur because:

1. A defection always makes cooperators worse off;
2. Later defections have a greater impact than early defections, making a cascade more likely as more factions defect;
3. Failed defection cascades erode the power of weaker non-defecting factions most, helping future cascades to succeed.

The dynamics follow 4 distinct phases that repeat in a cycle:

1. Cooperation: Power becomes concentrated in the leading factions. Weaker factions may defect in an uncoordinated manner.
2. Collapse: Defectors coordinate into a cascade when the cumulative power distribution is everywhere above a threshold.
3. Defection: Defection continues until power becomes sufficiently diffuse to permit cooperation. The strongest factions may cooperate first in failed state formation attempts.
4. Recovery: A cooperation cascade occurs in much the same way as the defection cascade, when the cumulative power distribution is everywhere below a (complex) threshold.

Additionally, we obtain a bound on the duration of cooperation and defection periods by allowing all non-leading factions to behave identically. In this case we can obtain closed-form expressions for the duration of cooperation and defection phases. The initial conditions can be very important in determining how close the bound is, from which we conjecture that this model has no general analytic solution, although bounds can be found and special cases solved.

Unequal resource distribution.

Resource is distributed unevenly in practice, which we model by replacing the raw resource with for faction . means that initially powerful factions have less resource. Figure 3e shows that a resource-weak leader can either persist or be usurped. Periodicity and collapse events persist, and further, Section S2.1 and Figure S1 in File S1 shows that a resource-weak leader results in reduced average conflict.

Uncertain outcomes.

The political power process is contingent on events outside of complete control of faction leaders. We model this by adding noise (normal, with mean ; see Section S2.1 in File S1) to the obtained power change before normalisation. Figure 3f shows that small levels of noise do not effect the qualitative behaviour. Moderate levels lead to a leader turnover and uncertain state lifetimes, whilst high levels prevent both coordination of both state formation and collapse (Figure S2 in File S1).

Biased decision making and random choices.

People are not naive resource optimisers. Decisions may be biased, hard to change, poorly calculated, or made with respect to longer term goals and otherwise unobserved features.

Complex decisions can be allowed for by introducing a random function for the decision threshold of each faction. We include ‘persistence’ via a bias towards the previous action, and random fluctuations in whether to favour defection or cooperation via a Gaussian Process [59] (Section S2.3 in File S1). This is determined by two parameters: the magnitude of the fluctuations and their correlation over time . To interpret , factions are effectively making ‘new random decisions’ every time units. If , then decisions appear ‘noisy’, and if is very large, each faction will appear (randomly) biased.

Figure 3g and Figure S3 in File S1 show that a range of has no qualitative effect, whilst moderate values lead to unpredictable leader turnover, make both the magnitude of a collapse event and the duration of a stable period uncertain. Figure 3h-i demonstrate that small to moderate levels of noise (Figure S4 in File S1) don't effect the dynamics, and further, when decisions are more correlated in time (Figure S5 in File S1) then state formation is more stable, even in the presence of high decision noise. This happens because power has time to equilibrate around the random choice of decision boundary; i.e. factions adapt to the stubbornness of other factions, and adjust their own bargaining positions accordingly.

Spatial structure.

Some political scenarios are best described with a spatial model. For example, factions may be local leaders of villages, or semi-autonomous regions of a larger state. We replace by (Section S2.4 in File S1), i.e. both the resource penalty for defection, and the political gains from doing so, decay with distance from the capital (leading faction). The average , i.e. is unchanged, and distance is calculated on a ring (so factions and are neighbours).

The spatial model (Figure 3j) allows for a variety of different scenarios. The state grows from the capital (Figure S6 in File S1) and collapses as in the non-spatial model. Collapse may be from the outside in, or the inside out. There may be a well defined maximum spatial extent (hatched parameter region of Figure 3j).

Modified intrinsic noise.

We chose to define Model 1 as an iterated game, which has consequences for the way that noise enters the system. Although the basic model is deterministic, the discretisation of time can produce ‘chaotic’ dynamics (e.g. large in Figure 3b) as small variations in the value of political power have large effects. Is this ‘intrinsic noise’ important for the dynamics?

To address this issue we constructed a modified version of the model in which only a single faction makes a decision at a time (using the Gillespie Algorithm [60]) with an average timestep of (Section S2.5 in File S1). Figure S7 in File S1 compares this model with the basic model and shows that there is no qualitative change. Additionally, the continuous time version of the model (Section S1 in File S1) matches the qualitative data. We view these issues as ‘modelling degrees of freedom’ and only consider model behaviours that are present for all choices.

Non-uniform defection penalty.

If the penalty for defection decreases with the number of defectors, both defection during cooperation and cooperation during defection are harder. This makes the phenomenon of collapse more likely to occur, as we show numerically (Section S2.6 and Figure S8 in File S1). Leader replacement is also easier under this model, if there is at least some noise.

Non-linear relationship between power and resource.

Power and resource are simply related in our model. However, we find that a family of non-linear functions do not effect the qualitative dynamics (Section S2.7 in File S1). Since we can map resource levels to a decision boundary in the power distribution, we conjecture that most ‘reasonable’ increasing functions will demonstrate collapse.

Proposition 1.

For Model 1a (the continuous time model) with two players, there exists a defection strategy defined by a power lower bound for a given upper bound with , which when both players use it the resource obtained is for the stronger and weaker players respectively. (For proof, see Methods).

The existence of this longer term strategy leads to an interesting extension. We now allow and to use two potential strategies in a long-term meta-game, which both dominate the short term strategies. Passive players use the strategy from Proposition 1, and aggressive players cooperate only as the dominant faction. We consider the payoff averaged over many cycles, assuming that during each state formation the player with the initially higher power is chosen randomly. Therefore a passive player always ends up with lower power than an aggressive player, two aggressive players obtain the ‘greedy’ payoff , and two passive players share leadership over time. The average payoff matrix is given in Table 2.

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Table 2. Long term payoff for strategies taking into account future payoff in the two player version of the game, averaged over many iterations.

https://doi.org/10.1371/journal.pone.0096523.t002

As , this is a form of the prisoners' dilemma. We note an analogy of the ‘passive’ strategy with democratic parties sharing power over time, as opposed to corrupt or autocratic systems in which this is impossible.

The case of three or more factions can be understood analogously. Optimally, weaker factions will act together to remove power from the leader, and such coordination follows naturally from the ‘passive’ strategy. Consider that factions cannot solve for the optimal thresholds but can choose them empirically. Cooperation first occurs when all factions have equal power (to within ). Each defection now occurs at the factions' chosen threshold. As in the simple model, defections reduce cooperators' resource leading to defection cascades. Aggressive factions still do not cooperate unless they are the leader and will not take part in the state. We therefore conjecture that any number of short-term resource maximisers, and/or ‘passive’ long-term strategists, can form states that experience dramatic collapse events.

Proof of Proposition 1.

Power during cooperation follows:starting at . The time taken to reach is . The resource obtained per unit time in the cooperation state is . Therefore the total resource obtained is:During the defection phase, resource is obtained at rate and power accrued at rate , so the defection duration and therefore . The average rate of resource acquisition is(3)This can be solved for giving a maximum resource when the resource rate . Although there is no explicit form, the maxima exists at positive and is non-trivial (i.e. not a boundary). During this time, the leading faction obtains a higher payoff during the cooperation phase , with a higher average rate .