A dynamic competition model of regime change

Journal of the Operational Research Society (2015) 66, 1939–1947 © 2015 Operational Research Society Ltd. All rights reserved. 0160-5682/15 www.palgrave-journals.com/jors/ A dynamic competition model of regime change Richard Syms* and Laszlo Solymar EEE Department, Imperial College London, London, UK A dynamic competition model for an oppressive government opposed by rebels is proposed, based on coupled differential equations with constant coefficients. Depending on their values, the model allows scenarios representing a stable, oppressive government and violent regime change. With constant coefficients, there can be no limit cycles. However, cycles emerge if rebels and governments switch characteristics after a revolution, if resources change hands and rebel motivations switch from grievance to greed. This mechanism is proposed as an explanation for the establishment of a new repressive regime after the overthrow of a similar regime. Journal of the Operational Research Society (2015) 66(11), 1939–1947. doi:10.1057/jors.2015.28 Published online 22 April 2015 Keywords: conflict analysis; simulation; system dynamics The online version of this article is available Open Access 1. Introduction It is an unfortunate truth that many governments are oppressive. Throughout recorded history, powerful minorities have exploited the wealth of larger communities by force (Lundahl, 1997). The penalty for objection is typically severe, with legal niceties, such as ‘treason’, and ‘crime against the state’ used to exaggerate the offence and euphemisms such as ‘execution’ providing cover for acts of murder designed to ensure retention of power. This behaviour is widespread, and many modern regimes would still commit almost any crime to retain power (Rummel, 1994). Economic inequality, suppression of rights and marginalisation of religious or ethnic groups often result in continual unrest. However, oppressive governments are depressingly stable. Collapse can follow from the economic failure of a kleptocratic regime (Gasiorowski, 1995), or the increasingly erratic behaviour of a tyrant (Gladd, 2002). The end is often at the hand of small, determined groups, who seize power following the decay of the regime. However, these acts have mainly not improved the lot of the majority, with successful revolutionaries often developing an equally oppressive rule (Weede and Muller, 1997). This cycle is driven by greed (Collier and Hoeffler, 2004), and simply involves the replacement of one kleptocracy with another (Grossman, 1999). Recent world events have attracted considerable interest, leading to the development of mathematical models designed to analyse the progress of insurgencies (Blank et al, 2008; Kaplan et al, 2010; Atkinson et al, 2012; Toft and Zhukov, 2012; MacKay, 2014). Similar approaches have been developed for domestic conflicts, including civil war (Garrison, 2008; Zhukov, 2013), guerrilla war (Dietchman, *Correspondence: Richard Syms, EEE Department, Imperial College London, Exhibition Road, London SW7 2AZ, UK. E-mail: 1962; Intriligator and Brito, 1988) and protest, coercion and revolution (Tsebelis and Sprague, 1989). The majority use a differential formulation and are inspired by combat modelling or population biology. However, with the main exception of a three-party model by Feichtinger and Forst (1996), the outcome is usually a choice between either party winning or a stalemate. Here, we focus on the often-ignored cyclic outcome of revolution. We avoid considering individual motivations, because these have been considered elsewhere (Tullock, 1971). Instead, we direct our attention to the dynamics of the conflict itself. Such assumptions are highly restrictive, but do at least allow the development of a model that can highlight a truism: cycles of repression are common, because it is easier to take over an existing kleptocracy than to establish a new one. We choose a simple differential model with few coefficients, concentrating on aspects of the struggle that might prevent or allow regime change: popular support, resources and weapons. We do not attempt to simulate a particular event, but merely develop a model that appears to display the correct behaviour. At the least, this should describe three scenarios: stable points (representing an established, oppressive government), abrupt changes in stability (regime change), and limit cycles (a return to oppression). The model is introduced in Section 2, and phase plane analysis is presented in Section 3. The conditions representing stable and unstable regimes are discussed in Section 4, and cyclic regime change is described in Section 5. The assumptions of the model and possible extensions are discussed in Section 6 and conclusions are drawn in Section 7. 2. Dynamic model A common feature of struggles for liberation is how small a fraction of the population is involved. The majority is inactive, and unless there is a fully blown civil war its number is 1940 Journal of the Operational Research Society Vol. 66, No. 11 relatively stable. We therefore ignore this overall population and assume that the competition is between two sub-groups, government (G) and rebels (R). Both are drawn from the population. Since the government has access to resources, its forces are assumed to be mercenaries. A reasonable description for their growth is one whose rate is proportional to current strength, since this will dwindle as resources are exhausted. In contrast, rebel forces grow spontaneously due to anger with the regime. There may be many triggers for anti-government feeling; here we simply assume a background of dissent, and model rebel recruitment as a constant rate. To limit the span of the competition, we also assume that recruitment follows the logistic law. In the absence of interaction, the time dependence of the forces may then be described as: dG ¼g2 Gð1 - GÞ dt dR ¼r1 ð1 - RÞ dt ð1Þ Here g2 and r1 are constants representing the effectiveness of government and rebel recruitment and the terms 1 − G and 1 − R set the carrying capacity of each side to unity. In reality, values of GC (limited by government budgets) and RC (limited by the pool of potential activists) might be expected, with RC≫GC. However, models with non-unity carrying capacity can be placed in the form above by appropriate scaling of variables. Note that these assumptions do not imply that G and R represent fractions of the total population. Instead, they are fractions of a maximum likely strength that is in each case less than the total. These equations have the well-known solutions for initial conditions G = G0 and R = R0 at t = 0 of G¼ G0 fG0 + ð1 - G0 Þ expð - g2 t Þg R ¼1 + ðR0 - 1Þ expð - r1 t Þ G¼ G0 ðg3 G0 - r3 R0 Þ fg3 G0 - r3 R0 exp ½ - ðg3 G0 - r3 R0 Þt g R¼ R0 ðr3 R0 - g3 G0 Þ fr3 R0 - g3 G0 exp ½ - ðr3 R0 - g3 G0 Þt g ð4Þ If g3G0 > r3R0, the upper exponential will decay to zero, while the lower one will grow. As a result, G will tend to G0 − r3R0/ (...truncated)