Apparent strength conceals instability in a model for the collapse of historical states

Abstract

An explanation for the political processes leading to the sudden collapse of empires and states would be useful for understanding both historical and contemporary political events. We examine political disintegration across eras, cultures and geographical scale to form a simple hypothesis that can be expressed verbally yet formulated mathematically. Factions within a state make choices described by game-theory about whether to accept the political status quo, or to attempt to better their circumstances through costly rebellion. In lieu of precise data we verify our model using sensitivity analysis. We find that a small amount of dissatisfaction is typically harmless to the state, but can trigger sudden collapse when there is a sufficient buildup of political inequality. Contrary to intuition, a state is predicted to be least stable when its leadership is at the height of its political power and thus most able to exert its influence through external warfare, lavish expense or autocratic decree.

Figure 1. Qualitative model.

Factions (a) decide whether to cooperate or defect. Then (b) raw resource is collected, which (c) is either reduced (for defectors) or redistributed according to power (for cooperators). Power grows (d) proportional to resource, with a defection bonus, and (e) is normalised so that the total power remains constant. This effectively reduces power for some and increases it for others, potentially changing their behaviour next round.

Figure 2. Characteristic behaviour of our model.

a) Defection behaviour with state formation and collapse. Defection is shown in grey, cooperation in white, and the leading faction in black (which always cooperates). b) The power of factions over time. c) The resource of factions over time. The power and resource of the non-leader factions converge, with the result that periodic coordination and defection periods occur. (Parameters: , , and .)

Figure 3. Effect of parameters/model extensions on the qualitative dynamics.

The plots are shaded to show whether model qualitatively behaves as Figure 2. The model either matches (solid), deviates (dense hatching) or fails (thin shading). The qualitative fit is based on quantitative scores (see Methods). Firstly, ‘State formation’ () is high when states are large and collapse rapidly to few factions. Secondly, ‘Periodicity’ () is high if there is periodic predictability to decisions. Thirdly, ‘State size’ () is high if state formation and collapse affect all factions. Finally, ‘Capital stability’ () is high if the leading faction does not change from the initial leader (relevant only for plots e–h). The qualitative model is matched if , , and . It deviates if , or . Otherwise the qualitative model fails. Also shown (where possible) is the parameter value from Figure 2 (vertical line).