Inequality between identity groups and social unrest

Abstract

Economic, social and political inequality between different identity groups is an important contributor to violent conflicts within societies. To deepen our understanding of the underlying social dynamics, we develop a mathematical model describing cooperation and conflict in a society composed of multiple factions engaged in economic and political interactions. Our model predicts that growing economic and political inequality tends to lead to the collapse of cooperation between factions that were initially seeking to cooperate. Certain mechanisms can delay this process, including the decoupling of political and economic power through rule of law and allegiance to the state or dominant faction. Counterintuitively, anti-conformity (a social norm for independent action) can also stabilize society, by preventing initial defections from cooperation from cascading through society. However, the availability of certain material resources that can be acquired by the state without cooperation with other factions has the opposite effect. We test several of these predictions using a multivariate statistical analysis of data covering 75 countries worldwide. Using social unrest as a proxy for the breakdown of cooperation in society, we find support for many of the predictions from our theory.

1. Introduction

Economic, social and political inequality is an inherent feature of social complexity and has profound effects on human societies [1–5]. Despite declines in the first part of the twentieth century, economic inequality has increased among Western countries since the 1970s [6], which may be responsible for economic inefficiencies [7–9], bad governance [10,11] and crime where inequality is conspicuous [12–15]. Recent quantitative studies also link economic inequality with political instability and conflict [5,16–22] for which the underlying mechanisms are poorly understood [23].

Earlier work focused on inequality between individuals and/or between households, which is sometimes referred to as vertical inequality. More recently, studies, especially those concerned with civil conflicts, have focused on horizontal inequality, i.e. economic, social and political inequality between different identity groups (e.g. ethnic, regional, religious, cultural) [24–28]. Current efforts to measure horizontal inequality [9,29–33] have not yet led to a consensus measure like those established for vertical inequality, such as income Gini coefficient [4] and income share held by the top 1% [34].

Both forms of inequality have been shown to have harmful social effects. Vertical inequality has a negative effect on economic efficiency [9], the production of public goods [10] and government quality [11]. Horizontal inequality promotes civil conflict between identity groups [24,26,35–39], increases ethnic voting [27,40], reduces public good provision [10] and destabilizes democracy [19]. Though economic grievance is often believed to be a prime motivator [41], political exclusion provides leaders with the incentive to change the status quo, e.g. by exploiting the group’s beliefs concerning its religion, history, sacred values, etc. [42]. At the same time, there may be cultural and institutional mechanisms which stabilize unequal societies. A cultural norm such as ‘high confidence in the state’ may increase tolerance of economic inequality [43,44]. Moreover, inclusive social institutions—such as checks and balances and rule of law—decouple economic and political power, meaning elites cannot convert political power into economic gain so easily [45–47].

Social and cultural processes underling the dynamics of cooperation and conflict in heterogeneous societies are complex. Given this complexity, mathematical modelling may provide some additional insights on these dynamics. While cooperation and conflict in heterogeneous societies are richly researched, horizontal inequality is not yet well modelled mathematically, so we do not understand the logic of crucial parameters and patterns across time and heterogeneous interactions. Our goals here are, first, to develop a mathematical framework for modelling the joint dynamics of cooperation, power and inequality in heterogeneous societies, and, second, to test our model’s key predictions empirically.

We build on earlier mathematical models of conflicts between different parts of the society. In particular, a number of studies [48–56] modelled contests for power between two or three factions in the society (e.g. the elite, middle class and commoners or the authoritarian government and the military or two political groups) the winner of which sets a preferred type of the economy and political arrangements (e.g. democratic or despotic). Esteban & Ray [57] studied a conflict between a number of different factions over the control of the distribution of goods produced by the society; the conflict was modelled as a Tullock contest [58,59]. Esteban & Ray [57] showed how the equilibrium contributions to conflict depend on the indices of inequality, fractionalization and polarization [60] in the society. Some models from cooperative game theory studied coalition formation [61]. In these models, the power of individual factions was constant and determined endogenously, while economic factors were disregarded. Lawson & Oak [62] introduced a novel approach focusing on the non-equilibrium dynamics of resources and power in a society with an arbitrary number of factions engaged in the redistribution of a fixed amount of resource. Previous work has not considered, however, the possibility of cooperation between factions in the production of collective goods and the associated collective action problem [63]. By contrast, our models will focus on the non-equilibrium dynamics of interactions between cooperation in the production of collective goods and conflict over their division. We will show that these processes exhibit an inherent tendency for cycling in power, inequality and economic production.

Moreover, our models will explicitly consider mechanisms for state stability, largely ignored in previous work, including the effects of social norms (conformity and allegiance to the state), decoupling between economic and political power (rule of law), heterogeneity in resources between factions and the availability of certain material resources that can be acquired by the state without cooperation with other factions. We note that some of these factors, e.g. normative commitments and conformity, are not frequently emphasized in the literature on conflicts and horizontal inequality. Nevertheless, they are firmly established in economics, psychology, sociology and cultural evolution research as important factors of human decision-making [55,64–69]. Therefore, we include them in our model.

Our models describe the processes of societal evolution at the meso-scale [70], which is intermediate between the macro-scale processes shaping the structure and demography of the society [71–74] and micro-scale processes governing the behaviour of individuals [75–80]. Our starting point is the assumption that a successful functioning of the society (both economic and political) requires mutually beneficial cooperation between its different factions. Cooperation is sustainable only if it is beneficial to all parties. Subsequently, as factions seeking to cooperate are jockeying for power, inequality among them grows and cooperation becomes no longer beneficial. Our focus is on the breakup of cooperation between factions which is a necessary condition for internal conflict and instability. Following previous work, we postulate that the horizontal inequality may enhance group grievances breaking cooperation between factions which in turn may facilitate mobilization for conflict. Rather than model conflict explicitly, we focus instead on some of its necessary conditions. Our models reveal some unexpected combined effects of interacting factors that had previously been studied separately. After formulating our model, we analyse its dynamics, describe the effects of different parameters and make several testable predictions. Finally, using a multivariate statistical analysis covering 76 countries worldwide, we find evidence consistent with some of the model’s key predictions.

The paper makes both theoretical and empirical contributions. Theoretically, we develop a novel mathematical model describing the dynamics of economic cooperation and political competition in heterogeneous societies in the presence of evolving horizontal inequality. We also propose novel causal mechanisms for why horizontal inequality can lead to the breakdown of cooperation. Our work goes beyond the study of political instability. Quantitative analyses find that horizontal inequality has many effects beyond its effect on civil wars. For example, Baldwin & Huber [10] find that horizontal inequality reduces the provision of public goods. We provide causal mechanisms for why horizontal inequality leads to the breakdown of cooperation, which, in turn, reduces public good provision.

We contribute to the empirical literature on horizontal inequality and political inequality in at least two ways. First, while previous work has shown that horizontal inequality is associated with civil wars [24,26,35–39,81], coups d’etat [82] and democratic breakdowns [83], there is little work on its effect on other forms of violence. Since the model describes the mechanisms through which horizontal inequality leads to the breakdown of cooperation, the empirical analysis focuses on small-scale forms of unrest, such as riots and anti-government demonstrations, rather than civil wars or coups. The latter require more organization and do not necessarily follow directly from the breakdown of cooperation. Small-scale conflicts, on the other hand, are more spontaneous. We thus contribute to the limited quantitative literature on the effect of horizontal inequality on small-scale violence. Crucially, different forms of violence follow different processes and can have different determinants. In fact, while the previous literature has shown that vertical inequality fosters small-scale violence, most authors find that it does not encourage civil wars [20,84,85]. Second, we provide empirical evidence that certain shared cultural values can reduce the chances of instability. For example, we find that values of conformity are associated with a higher likelihood of violence. To our knowledge, we are the first to report this finding.

2. Model

We consider a society composed of n factions (i.e. political, ethnic, religious or regional groups) in discrete time. Each faction is characterized by its relative political power f_i (0 ≤ f_i ≤ 1 for all i; $\sum _{i=1}^n{f}_i=1$). Following the general approach of Hurwic [86] to the evolution of social institutions, we assume that factions are engaged in an economic game about material resources and a separate political game about power. Specifically, at each time step the factions first cooperate or not (x_i = 1 or 0) in an economic nonlinear collective goods game played according to the current state of a dynamic set of rules. Then they participate in a contest for the power to change the rules of the economic game to be played at the next time step, in terms of how the collective goods are divided among the factions.

In interpreting our model, we will use the terminology of the structural-demographic theory of revolutions [71,72,87]. Accordingly, cooperating factions will be viewed as a coalition of the ‘elites’ led by the ‘dominant faction’, i.e. the faction with the largest power. Defecting factions are viewed as ‘counter-elites’. In contrast to the structural-demographic theory, which focuses on the demographic sizes of different factions in the society, our emphasis will be on their political power, economic resources and cooperation among them in the production of collective goods.

2.1. Economic game

Let $R_i^0$ be the resource owned by faction i (endowment), E the resource owned by the society (e.g. some natural resources or some other wealth) and available for distribution among the elites, and P(X) the additional resource produced by $X=\sum x_i$ cooperating factions. We will use an S-shaped production function which captures the law of diminishing return in a simple form:

$$P(X)=b\frac{X^{\kappa }}{X^{\kappa }+X_0^{\kappa }},$$ 2.1

where b is the maximum possible benefit of the collective action, X₀ is a half-effort parameter, and κ is a steepness parameter (X₀, κ > 0) [88]. The larger X₀, the more cooperating factions X are needed to produce the goods; at X = X₀, the cooperating factions secure half of the maximum possible amount. As κ → ∞, function P(X) approaches a step function. It is convenient to use a scaled half-effort parameter x₀ = X₀/n, whose value is always between 0 and 1.

The material payoff of faction i is defined as

$$R_i(x_i)=R_i^0+[v_i(E+P(X))-c]x_i,$$ 2.2a

where c is the cost of a faction’s effort in the collective action and v_i is the share of the collective goods E + P(X) going to faction i which depends on its power:

$$v_i=\frac{\hspace{0.17em}{f}_i}{\sum _C{f}_j},$$ 2.2b

where the sum is over the set C of cooperating factions. That is, the factions face a conflict over the division of jointly owned (E) and produced (P) resources. In this model of ‘club goods’ [89], only the coalition of elites (i.e. cooperating factions with x_i = 1) share the goods dividing them according to their power, whereas the counter-elites (defecting factions with x_i = 0) just keep their endowment $R_i^0$. Factions should cooperate only if their power and corresponding share of collective goods are sufficiently large [63,88,90]. Note that each time a faction moves from the elite to the counter-elite, production P is reduced which punishes the remaining cooperating elites more than the counter-elites (who have little left to lose when they defect). Also note that in models of collective action, the most important factor for agents’ decision-making is the relationship between benefit and cost. In our model, the faction’s benefit and cost depend on its power and effort in the collective action, respectively.

2.2. Political game

All factions in the society are engaged in a political contest the outcome of which modifies political power f_i to

$$f_i^{\prime }=\frac{y_i}{\sum _j{y}_j},$$ 2.3a

where y_i is the effective effort of faction i in the political game (the prime means the value at the next iteration). Recurrence equation (2.3a) is a form of the Tullock contest [59]. In general, the effectiveness of a faction in the political contest should increase with its economic resources R_i and political power f_i, with more politically powerful factions being able to use the available resource more efficiently than less politically powerful factions in shaping the rules of the economic game to their own advantage [91]. We capture these intuitions by defining y_i as

$$y_i=R_i(1-\epsilon +\epsilon f_i),$$ 2.3b

where the incumbency effect parameter ɛ controls the strength of dependence of y_i on power f_i (0 ≤ ɛ ≤ 1). If ɛ = 0, then y_i = R_i and only the amount of the faction’s material resource R_i matters; if ɛ = 1, then y_i = R_if_i, so that the material resource and power combine multiplicatively in defining y_i. Parameter ɛ captures the efficiency of ‘rule of law’ mechanisms in the society acting towards maintaining a level playing field and preventing politically powerful factions from bending the rules of competition in their favour. High values of ɛ are indicative of low degree of ‘rule of law’. We note that equations (2.3a,b) defining the dynamics of power are related to the replicator equation [92,93] which is widely used in modelling biological, cultural, and social processes (see the electronic supplementary material).

2.3. Utility function

Following earlier work (e.g. [94–98]), we define the utility function as the sum of a material and normative components. For a non-dominant faction,

$$u_i(x_i)=R_i(x_i)+[{\eta }_0{x}_s+{\eta }_1(2\tilde{x}-1)]x_i,$$ 2.4a

where η₀ is the normative value of allegiance to the state (embodied in the most powerful faction which makes effort x_s) and η₁ is the value of conformity with the majority of peer factions (among which the average effort is $\tilde{x}$). Parameter η₀ can also be viewed as a measure of the legitimacy felt towards the most dominant faction or the ruling regime. For the dominant faction,

$$u_s(x_s)=R_s(x_s)+{\eta }_0{x}_s,$$ 2.4b

that is, the state faction has a normative value η₀ of contributing to production.

The two main components of our utility function are material payoffs and the effects of social influences. These are the main forces driving human behaviour as studied by the two most commonly used mathematical theories in social sciences: game theory [93,99–101] and social influence theory [102–107]. Our approach unifies them in a single framework leading to more realistic and comprehensive models [76,96–98,108–110]. We separate social influences into peer influences and the authority/state influence because these two types of influence have different nature and power [75,97,98,111–115].

2.4. Strategy update

We will assume that each faction updates its action in the economic game, that is, chooses x_i value, randomly and independently with probability ν. Assuming bounded rationality [116], we postulate that each updating faction uses myopic best response to maximize its utility u_i. To capture errors in decision-making, which are unavoidable in any realistic situation, we use the Quantal Response Equilibrium approach with logit error [117]. The corresponding mathematical model has a non-negative precision parameter λ (see the electronic supplementary material). If λ = 0, the factions cooperate or defect with equal probabilities; if λ → ∞, the factions always use the best strategy. We will also assume that factions pay a fixed cost δ for changing their action (i.e.moving between the elite and counter-elite).

2.5. Intuitions

The behaviour of the two main components of our model, the economic game and the political game, is well understood when each is acting in isolation. In particular, if the factions’ power f is fixed, the society will split into high-power factions that cooperate and low-power factions that defect [118]. Similarly, the existing theory of the replicator equation, which describes selection of the ‘strongest’ among many competitors, tells us that if the factions’ resources R are fixed, then over time all power will concentrate on just one faction. What is not well understood, however, is how these components work in tandem, i.e. condensation of power versus the fractionation of cooperation. Below we study what happens when both resources and power change dynamically as a result of bounded rational decision-making processes involved in the economic and political games.

3. Theoretical results

3.1. Basic model

Consider first a basic model with equal initial power (f_i = 1/n for all i), no normative values (η₀ = η₁ = 0), no variation in endowment (R_i = R₀ for all i), and no natural resource to divide among the elites (E = 0). The electronic supplementary material provides some analytical results for this model. We focus here on the results of agent-based simulations of the more interesting cases. We will assume that all factions cooperate initially and, to reduce the effects of stochasticity, will postulate infinite precision initially (λ = ∞).

3.1.1. Three dynamic regimes

Numerical simulations show that in the basic model there are only three possible dynamics: complete loss of cooperation, stable hierarchy, and continuous turnover. Under complete loss of cooperation and production, all factions have similar power. Under stable hierarchy, one faction persists on top of the hierarchy with some fluctuations in the power and identities of the elites and counter-elites. Under continuous turnover of dominant factions, cycles of cooperation and defection are coupled with cycles in power and inequality. The regimes of stability and turnover are illustrated in figures 1a,c, 2a,c and 1b,d, 2b,d, respectively. The top graph in each set shows which factions cooperate and which do not at each time, and how coordinated these actions are. The solid segments of black and white strips correspond to periods of apparent stability in the society [62]. During these periods, inequality nevertheless keeps growing (as shown in the two other parts of each set).

Figure 1.

Examples of social dynamics in the basic model: with no turnover (a,c) or with turnover (b,d) of the most dominant factions. In each set of graphs, top graph: factions cooperating at time t are shown as black pixels, defecting factions as white pixels; middle graph: different colours show power f_i of individual factions; bottom graph: dynamics of Gini index based on the resources R_i. (The Gini index based on power exhibits very similar dynamics.) The red stars mark the first collapse of cooperation at time T (when X hits 1 for the first time). The green stars mark the end of the transient period τ (explicitly defined in subsection ‘Cycling’). Other parameters: x₀ = 0.75, κ = 2, δ = 0.25, γ = η₀ = η₁ = r = 0. (a) n = 2, b = 10, ɛ = 0.03. (b) n = 2, b = 5, ɛ = 0.01. (c) n = 4, b = 15, ɛ = 0.03. (d) n = 4, b = 10, ɛ = 0.03.

Figure 2.

Same as in figure 1 but with n = 6 and 8. (a) n = 6, b = 15, ɛ = 0.05. (b) n = 6, b = 15, ɛ = 0.03. (c) n = 8, b = 20, ɛ = 0.01. (d) n = 8, b = 20, ɛ = 0.03.

3.1.2. Effects of parameters

The first regime is expected when the benefit b of cooperation is small, collective goods are difficult to produce (half-effort parameter x₀ is large), and the number of factions n is large. The second regime is expected when the benefit b of cooperation is large, collective goods are easy to produce (half-effort parameter x₀ is large), and the number of factions n is small. The third regime is expected at intermediate values of b, n and x₀.

Figure 3 illustrates the effect of changing parameters on long-term average behaviour of the model. Figure 3a shows the average number of contributing factions (which is zero in the lower left because there is no cooperation). Also the proportion of contributing factions (and, thus, the total level of production) is reducing with the number of factions n. Figure 3b shows that the average power of the most dominant factions increases with b and decreases with n. The incumbency parameter ɛ does not affect the average level of cooperation or the power of the most dominant faction. Figure 3c characterizes the turnover of the most dominant factions observed during the simulation run. We measured the turnover using the Simpson index, which is the number of dominant factions during the simulation run appropriately weighted by the time they were in power (see the electronic supplementary material). This number is equal to 1 in the top right part of the graph where the dynamic regime (ii) out of three is observed. Significant turnover of dominant factions (i.e. regime (iii)) is observed along the diagonal of figure 3c. Even in the absence of turnover of the dominant factions, there can be significant fluctuations in the power of factions and production levels. The effects of parameters on the period of fluctuations in the power of the most dominant faction are illustrated in figure 3d (see electronic supplementary material for how period was estimated). Note that increasing incumbency parameter ɛ decreases the period of fluctuations in power in regime (ii) and increases the turnover of dominant factions in regime (iii).

Figure 3.

Long-term average values of (a) the number of contributing factions, (b) power of the dominant faction, (c) an effective number of different dominant factions, and (d) the period of fluctuations in power. For parameter values corresponding to the grey graphs, there are no contributing factions. For parameter values corresponding to the black graphs, fluctuations in power are insignificant. Baseline parameters: c = 1, x₀ = 0.75, δ = 0.25, κ = 2, λ = ∞, ν = 0.5, γ = 0.0, η₀ = 0, η₁ = 0, r = 0. The averages and confidence intervals over 200 runs each of length 4000 time steps.

3.1.3. Cycling

To shed more light on social volatility and the dynamics of cycling in cooperation and inequality we focused on the first cycle, i.e. starting with complete cooperation at t = 0 and ending with its ‘collapse’ at time T defined as the time when X becomes ≤1. We also compute the duration of the initial cooperative phase τ during which inequality grows but cooperation remains high and stable (see figure 1 and the electronic supplementary material). The parameter τ was defined as the time to reach for the first time the long-term average value of the Gini index. Considering the model dynamics on longer time-scales over multiple cycles is less informative because in most realistic situations the nature and the number of factions as well as various forces captured by the model parameters are likely to change with each new cycle. Figure 4 illustrates the effects of parameters n, b and ɛ plus one additional parameter on τ (figure 4a) and T (figure 4b,c,d). Increasing incumbency ɛ, decreasing the number of factions n, and decreasing the benefit b accelerate the collapse of cooperation.

Figure 4.

The duration of the transient phase τ (a) and the time to the collapse of cooperation T (b–d) as affected by the number of factions n, benefit b, incumbency ε, and one extra parameter. The additional parameter is: (a) initial power inequality γ; (b) allegiance to the state η₀; (c) conformity with peers η₁; (d) extra resource r. In the shaded graphs in (d), the effect of r on T is opposite to that in the white graphs. Baseline parameters: c = 1, x₀ = 0.75, δ = 0.25, κ = 2, λ = ∞, ν = 0.5, γ = 0.0, η₀ = 0, η₁ = 0, r = 0. The averages and confidence intervals over 200 runs each of length 4000 time steps. (a) γ = 0. (b) η₀ = 0. (c) η₁ = 0. (d) r = 0.

3.2. Additional factors

Next we consider several additional factors. To isolate their effects, we add only one new factor at a time. The red lines in figure 4 show the values for the basic model with the additional parameter set to 0; the green and blue lines correspond to two different positive values of the corresponding parameter.

3.2.1. Effects of initial variation in power

In figure 4a, the additional parameter 0 ≤ γ ≤ 1 characterizes the initial variation in power. With γ = 0, each faction has an equal power 1/n initially; with γ = 1, the initial distribution of power is drawn from a ‘broken stick’ distribution (see the electronic supplementary material). As expected, increasing incumbency ɛ and/or initial power inequality γ shorten the initial cooperating phase τ. The effects of γ on T are insignificant (not shown).

3.2.2. Effects of the strength of allegiance to the state and conformity

Figure 4b shows that increasing the normative value η₀ of allegiance to the state increases the stability of the system and can often prevent its collapse (on the time scale studied, e.g. with n = 8). However, conformity with the majority of peer factions η₁ accelerates the collapse of cooperation (figure 4c). This happens because once a majority of low-power factions are defecting, the other factions are ‘pulled’ to defect as well. This effect is only present when the number of factions is not too small.

3.2.3. Effects of collectively owned resource

Finally, we consider the effects of a collectively owned resource E available to sharing among the elites which is independent of the extent of their cooperation. We write its amount as E = r × b, where r is a new parameter measuring the amount of this benefit relative to the maximum amount nb that can be produced by collective action. Figure 4d shows that r has a nonlinear effect on T. If cooperation can potentially bring large benefits (e.g. with b = 15 or 20 and n = 6 or 8), increasing r decreases T. That is, high natural resources disincentives cooperation—a phenomenon called the ‘resource curse’ [119]. If the potential benefits of cooperation are not high enough, however, increasing r increases T because some factions are motivated to cooperate with the state to receive their share of resources.

3.2.4. Effects of the inequality in endowments

Introducing inequality in endowments R₀ results in a possibility of new dynamics: stable cooperation of high-endowment factions with low-endowment factions not contributing at all or exhibiting cyclic or stochastic dynamics (see figure S1 in the electronic supplementary material for an example). Introducing errors in the decision-making process (that is, decreasing precision parameter λ) increases stochasticity of the system without affecting the results qualitatively (see the electronic supplementary material).

3.2.5. Theoretical predictions

Our modelling results thus lead to a number of predictions:

• 1. Horizontal inequality should increase when cooperation between groups is high (generating large amounts of resources).
• 2. By coupling economic and political power, a strong incumbency effect ɛ (weak rule of law) should increase horizontal inequality.
• 3. Horizontal inequality should increase social unrest and accelerate the collapse of cooperation (shorter T).
• 4. High normative value η₀, i.e. support for the state’s institutions, will tend to stabilize cooperation (longer T).
• 5. High normative value η₁, i.e. conformity, should, counterintuitively, destabilize cooperation (shorter T).
• 6. Increasing the number of factions should decrease cooperation and increase turnover of dominant factions and/or oscillations in power and production.
• 7. High initial inequality in power should decrease stability.
• 8. Variation in faction-specific endowments, R₀, should increase resilience of the society to high inequality.
• 9. The presence of additional resources r not dependent on cooperation (e.g. natural resources, state rents) should decrease cooperation.

4. Empirical analysis

4.1. Data and methodological approach

This section looks at the empirical evidence in favour of hypotheses 2–6. We focus on hypothesis 3, on the effect of economic horizontal inequality on social unrest. The unit of analysis is the country-year and the main sample covers more than 1800 observations on 75 countries between 1991 and 2016.

The model describes the process through which horizontal inequality leads to the breakdown of cooperation between factions. It, therefore, corresponds more closely to mild forms of violence that are spontaneous and require minimal organization. We thus focus on small-scale forms of conflict. Our dependent variables capture six forms of social unrest: riots, strikes, assassinations, anti-government demonstrations, guerrillas and revolutions. With the partial exception of guerrillas and revolutions, all of these represent relatively small-scale forms of conflict.

The main dependent variable (Unrest Index) is an index that gives how many of these six forms of social unrest a country has experienced within a given year. For example, if country A has experienced a riot and a strike in year 2005, but not an assassination, a demonstration, a guerrilla or a revolution, it receives a score of 2. This variable ranges between 0 and 6.

The measures of riots, strikes, assassinations, anti-government demonstrations, guerrillas and revolutions are taken from Banks’ dataset [120] (see the electronic supplementary material, for definitions). These variables count the number of riots, etc., that a country has experienced within a given year. To compute the Unrest Index, we recode the variables as six dummy variables that take the value one if a country experiences at least one riot, etc., in a year. We then add them.

We also run models on these six dummy variables. For example, in one of the models, we estimate the effect of horizontal inequality on the probability that a country experiences at least one riot within a given year. We recode the variables as dummy variables—rather than use the original continuous variables—to make sure that the results are not driven by outliers that have experienced a large number of riots, etc. We show in electronic supplementary material, table S5, that the results are largely unchanged when we employ the continuous variables instead.

Our key independent variable is economic horizontal inequality. It is a Gini coefficient that captures inequality between ethnic groups based on luminosity data and ethnic group definitions from Ethnologue [9]. In the electronic supplementary material, we show that the results are largely robust to the use of two alternative measures of horizontal inequality (electronic supplementary material, tables S6 and S7). The model describes the consequences of inequality between factions. Of course, ethnicity is only one of multiple possible ways in which factions can be defined. For practical reasons, we had to focus the empirical analysis on a single type of faction. Therefore, the sample only covers countries in which ethnicity is politically relevant according to the Ethnic Power Relations dataset [121]. In the electronic supplementary material, we show that the results are largely unchanged when we include all countries, regardless of whether or not ethnicity is politically relevant (electronic supplementary material, table S4).

We use rule of law [122] as an empirical proxy for incumbency because one function it serves is to ensure politically powerful elites must obey the same rules as everybody else [123], thus opportunities to convert political power into economic power are reduced. The two cultural norms—support for institutions and conformity—are composite variables of relevant World and European Values Survey (WEVS) items (see electronic supplementary material). Support for institutions is comprised of survey items asking if people have confidence in the government, political parties, civil service, armed forces, press and police. Conformity is the reverse of the well-established ‘autonomy index’ [124], where low conformity nations value traits like independence and determination and high conformity nations value obedience and religious faith. We use principal component analysis to compress these multiple items into one-dimensional measures for both ‘support for institutions’ and ‘conformity’. Support for institutions captures the beliefs that underlie the norm for ‘conformity with the state’ in our model. Likewise, a tendency to conform (and not act autonomously) captures the belief system that underlie the model’s ‘conformity with peers’ parameter. Finally, we measure the number of factions using the country’s number of politically relevant ethnic groups taken from the Ethnic Power Relations dataset [121]. Table 1 lists the proxies used for the key model parameters.

Table 1.

Proxy measures for model parameters.

model variable or parameter	its proxy in data
Gini index for payoffs	horizontal inequality [9]
time to collapse of cooperation T	social unrest [120]
incumbency ɛ	rule of law [122]
allegiance to the state η0	support for institutions [12,13]
conformity with peers η1	conformity [12,13]

The analysis controls for variables usually included in studies of small-scale conflicts: dummy variables for former British and French colonies, polity score and polity score squared, ethnic fractionalization [121], GDP per capita logged and the growth rate [125]. We include the Polity score squared because the previous literature finds that partial democracies (i.e. countries with intermediate polity scores) are more likely to experience political violence [126].

A limitation with cross-cultural studies like ours is Galton’s problem: countries with shared cultural histories will not be fully independent in terms of other measures [127–129]. Whereas language phylogenies have often been used as a proxy for cultural proximity [128,129], this approach would not be appropriate for our framework because our units of cultural groups are ethnic groups, which often have different languages and other cultural attributes.

Instead, we focus on geographical proximity. Countries that are neighbours, for example, are likely to have many of the same ethnic groups. Moreover, the previous literature has shown that political unrest often diffuses across neighbours [130]. To account for spatial diffusion, we construct a variable, Spatial DV, which gives the average social unrest level of the country’s neighbours within a given year. This variable is constructed for each dependent variable (Unrest Index, Riot, etc.). For example, if a country has three neighbours, the Spatial DV in the models in which the dependent variable is Unrest Index is the average Unrest Index score of these three countries. In the models using Riot, it is the proportion of them that have experienced at least one riot during that year. Neighbours are defined as countries that share a land border or that are separated by a stretch of water less than 400 miles (data taken from the Correlates of War Project [131]). We present the data used to measure the important quantities in detail in the electronic supplementary material. Electronic supplementary material, table S3, shows basic descriptive statistics.

All models are estimated in STATA. The models in which the dependent variable is Unrest Index are estimated using ordinary least squares (command reg in STATA). The models with the riot, strike, assassination, demonstrations, guerrilla and revolution dummy variables are estimated using Probit models (command probit in STATA). All models include lagged dependent variables. Therefore, we employ dynamic models, in that they capture the association between the explanatory variables and changes in social unrest. In order to address temporal autocorrelation, in all models standard errors are clustered by country (using the command cluster) [132]. This enables us to account for the fact that observations from the same country at different points in time are not independent from one another. We present other tests/diagnostics of temporal autocorrelation in the electronic supplementary material (tables S10 and S11).

Unfortunately, the data do not enable us to test the cycling relationship implied by the model. Ideally, we would have an exogenous time-variant instrument for horizontal inequality that we could use in two stage estimations. However, we are not aware of any instrument for horizontal inequality that is exogenous to social unrest. Some authors have used the ratio of land suitable for sugar production and wheat production as an instrument for vertical inequality [133]. Countries with more land suitable for sugar production have tended to be more unequal historically than those that produce wheat. There are at least three problems with using the sugar-to-wheat land ratio in this study: (i) it is an instrument for vertical inequality, not horizontal inequality; (ii) it fails the exogenous requirement—land endowment affects social unrest through mechanisms other than horizontal inequality; and (iii) it is not time-variant, so it would not enable us to test the cycling relationship implied by the model.

In fact, there is virtually no variation in horizontal inequality within the relatively short period under study (25 years), which makes it difficult to test the cycling relationship. There are two reasons why measures of horizontal inequality show little within-country variation. First, the data are imperfect. The main measure is based on inequality in luminosity between regions inhabited by different ethnic groups, which does not change much in time.

Second, and perhaps more importantly, inequality is highly sticky within countries over time [19,134]. Apart from the dynamics described by our model, most other explanations for changes in inequality go back to major events such as major wars, the way different countries have been colonized, the political institutions that were in place, for example, in the middle age, resource endowment, etc. Short of major events, inequality (vertical and horizontal) does not change much in the short run. Houle [83], for example, constructs a measure of horizontal inequality using survey data covering more than 20 years. This measure also changes very little across survey-waves (which are conducted in different years) within the same countries, even though each survey-wave relies on different respondents. Even major events rarely have an immediate effect on inequality. For example, transition from autocracy to democracy has been found to only affect inequality in the very long run, notably because most of its effect operates through educational opportunities, which only translate into income changes once individuals have completed their education [135].

Therefore, we believe that the dynamic relationship implied by the model operates in the long run, while our data only enable us to look at the correlation between horizontal inequality and social unrest in the short run. This empirical analysis thus only presents correlational support for the predictions and mechanisms implied by the theoretical model. We cannot be sure that our results are not driven by some form of endogeneity between social unrest and the explanatory variables of interest, particularly horizontal inequality. At the same time and as we have noted, horizontal inequality is highly sticky within countries within the short time period of our data. This suggests that most of the correlation between horizontal inequality and social unrest is driven by horizontal inequality, rather than the other way around.

In fact, most previous studies on horizontal inequality and social unrest use an estimation strategy similar to ours and many assume that horizontal inequality is unchanged over several decades [38,81–83]. Thus, although our empirical approach has limitations, it is consistent with the literature.

Not only does the lack of within-country variation make it difficult to use techniques such as instrumental variable estimation, but it also prevents us from including country fixed effects. Moreover, regardless of within-country variation, country fixed effects would not be optimal in small time panels like ours [136] and the inclusion of both country fixed effects and lagged dependent variables creates bias, known as the Nickell bias [137].

4.2. Empirical results

Our main results are shown in table 2. All models report robust standard errors clustered by country. Consistent with hypothesis 3, we find that horizontal inequality is positively correlated with Unrest Index, riots, revolutions and anti-government demonstrations. The coefficients on assassinations, strikes and guerrillas are of the (positive) predicted sign, but are not statistically significant.

Table 2.

Determinants of social unrest. All models are run with STATA. Model 1 is run using ordinary least squares (with the command reg) and models 2–7 using Probit estimations (with the command probit). All independent variables are lagged. Robust standard errors clustered by country in parentheses.

	dependent variables
	index	riot	assass.	strike	guerrilla	revol.	demonst.
	(1)	(2)	(3)	(4)	(5)	(6)	(7)
lagged DV	0.492	0.964	1.005	0.946	1.731	1.797	0.735
	(0.032)***	(0.092)***	(0.100)***	(0.135)***	(0.126)***	(0.147)***	(0.079)***
spatial DV	0.203	0.972	0.593	1.115	0.390	−0.209	0.813
	(0.040)***	(0.153)***	(0.244)**	(0.256)***	(0.305)	(0.218)	(0.149)***
horizontal inequality	0.441	0.507	0.232	0.363	0.478	0.614	0.551
	(0.148)***	(0.188)***	(0.234)	(0.256)	(0.302)	(0.251)**	(0.195)***
rule of law	−0.187	−0.216	−0.298	−0.022	−0.203	−0.139	−0.176
	(0.063)***	(0.098)**	(0.096)***	(0.117)	(0.099)**	(0.092)	(0.079)**
conformity	0.229	0.049	0.256	0.263	0.418	0.338	0.252
	(0.109)**	(0.141)	(0.151)*	(0.205)	(0.177)**	(0.181)*	(0.131)*
respect for institutions	−0.014	−0.010	−0.151	−0.030	0.216	0.170	−0.118
	(0.093)	(0.144)	(0.134)	(0.216)	(0.160)	(0.134)	(0.115)
number of groups	0.018	0.015	0.018	0.012	0.018	0.004	0.034
	(0.005)***	(0.009)*	(0.009)*	(0.008)	(0.007)***	(0.009)	(0.005)***
former British colony	.069	0.266	0.229	0.324	−0.075	−0.274	0.125
	(0.126)	(0.153)*	(0.150)	(0.177)*	(0.192)	(0.194)	(0.159)
former French colony	.012	0.023	0.067	0.104	0.179	−0.083	−0.122
	(0.081)	(0.092)	(0.220)	(0.173)	(0.182)	(0.156)	(0.113)
polity score	.017	0.011	0.035	0.044	0.020	0.004	0.011
	(0.007)**	(0.011)	(0.011)***	(0.013)***	(0.012)*	(0.011)	(0.009)
polity score squared	−0.002	−0.002	−0.002	−0.0001	−0.004	−0.002	−0.003
	(0.002)	(0.002)	(0.002)	(0.003)	(0.002)	(0.003)	(0.002)
ethnic fractionalization	−0.082	0.010	−0.317	−0.502	0.012	−0.118	−0.209
	(0.155)	(0.235)	(0.241)	(0.326)	(0.281)	(0.253)	(0.208)
GDP per capita (logged)	0.189	0.231	0.049	−0.015	0.239	−0.101	0.330
	(0.049)***	(0.071)***	(0.077)	(0.089)	(0.074)***	(0.076)	(0.063)***
growth	−0.005	−0.003	−0.011	−0.008	−0.0002	−0.010	−0.011
	(0.003)	(0.005)	(0.006)*	(0.008)	(0.008)	(0.005)**	(0.005)**
N	1889	1889	1889	1889	1889	1889	1889
log-Lik.	−2699.511	−862.001	−580.783	−547.316	−517.109	−451.489	−1035.998

*p < 0.1; **p < 0.05; ***p < 0.01.

Figure 5 illustrates the magnitude of the associations for each dependent variable. The first panel gives the predicted Unrest Index score across horizontal inequality values. The six other panels give the predicted probability that a given country experiences at least one riot, etc., across all horizontal inequality levels. All other explanatory variables are kept at their median. For example, the predicted Unrest Index score of a country with a horizontal inequality value at the 10th percentile of the distribution (0.165) is 0.97, while the predicted score of an identical country with a horizontal inequality value at the 90th percentile (0.888) is 1.289. Importantly, these predicted values give differences within a single year. In the long run, we would thus expect countries with high horizontal inequality levels to be cumulatively more unstable than the latter.

Figure 5.

Effect of horizontal inequality on the predicted probability that a country experiences social unrest in a given year. Shaded areas represent 95% confidence intervals.

The electronic supplementary material reports multiple robustness tests of the association between horizontal inequality and social unrest. Specifically, the results are largely robust when we: (i) extend the sample to countries in which ethnicity is not politically relevant (electronic supplementary material, table S4); (ii) use a different operationalization of the dependent variables (electronic supplementary material, table S5); (iii) use alternative measures of horizontal inequality (electronic supplementary material, tables S6 and S7); (iv) omit countries with horizontal inequality values at either extremes (electronic supplementary material, tables S8 and S9); and (v) employ alternative strategies to address temporal autocorrelation (electronic supplementary material, tables S10 and S11).

On balance, we also find evidence consistent with hypotheses 2, 4 and 6: countries with robust rule of law (hypothesis 2) are more stable, while those with higher conformity values (hypothesis 4) and more factions/ethnic groups (hypothesis 6) are more unstable. To our knowledge, we are the first to document a positive association between conformity values and political unrest. The association between the number of ethnic groups and unrest is also interesting given that ethnic and religious fractionalization has been found to bear no relationship with political violence [84]. The evidence in favour of hypothesis 5 is much weaker as we fail to find any significant correlation between respect for institutions and unrest. Although never statistically significant, for some forms of unrest—such as assassinations—respect for institutions is indeed related with less instability. However, it is positively correlated with other forms of instability, such as guerrillas.

As expected, political instability among neighbours is usually associated with political instability at home. There are several mechanisms that could drive this correlation. As discussed above, political instability may diffuse across neighbours. Moreover, neighbours may share a common culture, which makes them likely to experience similar forms of social unrest. Finally, neighbours may share similar characteristics, such as economic structures or dependence on international superpowers, which renders them vulnerable to the same political and economic shocks. Although the variable Spatial DV is too broad to distinguish different mechanisms, it serves as a control for spatial autocorrelation and shared cultural histories, enabling estimation of the association between horizontal inequality and political unrest.

4.3. Discussion

Recent discussions, supported by data analysis, have identified economic, social, and political inequality between identity groups as a possible threat to the stability of society [5,16–19,21,24,35–39,81–83]. Here we have modelled cooperation and conflict in a society composed of multiple factions engaged in economic and political interactions. We explicitly assumed that politically powerful factions are attempting to shape the rules of economic interactions to their own advantage. Our model showed how inequality arising among factions that otherwise wish to cooperate led to conflict, at least until new elites emerged in the model’s dynamic cycle. Our model captures the effects of rule of law, social norms of support for the state and conformity with peers. It also makes predictions regarding the effects of the number of factions in the society, initial inequality in power, heterogeneous endowments and resources, and the presence of additional resources available for redistribution among the elites independently of the success of collective action.

Our work represents the first attempt to mathematically model the effects of horizontal inequality on social dynamics. Our models include a number of factors which are firmly established as crucial drivers of human decision-making but nevertheless have not received due attention in the literature on horizontal inequality. More generally, we have expanded the theoretical tool kit of evolutionary game theory by developing a novel approach in which agents are engaged in two separate games: an economic game about material resources and a political game about the power to set the rules of the economic game. While earlier work used static or statistical models, our inherently dynamic approach shows how the growth in inequality and breakdown in cooperation develop in time. Our model is thus able to predict relevant time scales rather than just the direction (positive/negative) of various effects. An interesting feature of the joint dynamics of cooperation and inequality revealed by our model is their inherently cyclic behaviour (as illustrated in figures 1 and 2).

The model predicts that growing inequality tends to lead to the collapse of cooperation between factions, which could then trigger conflict. This process can be impeded by several mechanisms. One is the decoupling of political and economic power through rule of law, which prevents a ‘power grab’ leading to dangerous levels of inequality. Our model indicates, that a social norm for allegiance to the state also delays factional conflict, whereas a social norm for conformity with other factions hastens the onset of conflict by facilitating a cascade of defections. Division of the society into multiple factions decreases its stability. One additional mechanism is heterogeneity in endowments between the factions, which makes societies more stable to the negative effects of inequality by giving incentives to powerful factions to cooperate with the state. The availability of certain material resources that can be acquired by the state without cooperation with other factions makes societies less stable (resource curse).

We tested some of the predictions of the theoretical model using country-year data from 76 countries between 1991 and 2016. Our main finding is that, consistent with our model, horizontal inequality is associated with social unrest, which we employ as a proxy for the breakdown of cooperation between factions. With the exception of the predicted association between respect for institutions and political instability, the evidence is consistent with the core predictions made by the model.

Although our results are very encouraging, there are a number of data-related considerations for future work. First, and perhaps most importantly, the current horizontal inequality data do not enable us to access the direction of the causality in the relationship between horizontal inequality and political instability. As noted above, we believe that, since inequality (including horizontal inequality) is highly sticky within countries, the relationship captured in our analysis is mainly driven by the effect of horizontal inequality on instability within the short period of time we study. However, having horizontal inequality data for longer periods of time would enable one to better capture the nature of the relationship. To be clear, this limitation is shared by almost all previous large-N quantitative studies of horizontal inequality and political violence, although in our case it prevents us from testing the cycling relationships predicted by the model. Developing horizontal inequality data spanning longer time periods should thus be a priority for future work.

Second, we do not have ideal measures for all variables included in the model. Most notably we use rule of law as a proxy for the decoupling of economic and political power. However, the rule of law is probably too broad because it serves a number of different functions. For example, the rule of law may promote a dynamic private sector economy, which in turn would decrease political instability. This relationship is not directly captured by the model.

Our modelling work adds to the general knowledge on the relationship between horizontal inequality and instability in a number of ways. First, it provides a general framework for describing the dynamics of this relationship. It also characterizes the importance of economic, cultural and social factors, including legal checks and balances, cultural allegiance to the state, and conformity which are factors not commonly emphasized in the literature on horizontal inequality. Our model makes quantitative predictions about the effects of parameters on various characteristics including the time-scales involved.

On the empirical side, we tested several hypotheses emerging from interpreting our modelling results. We make two main empirical contributions. First, we test the association between horizontal inequality and low-scale conflicts, rather than complex conflicts such as civil wars or coups. Our theoretical model is more closely related to low-scale conflicts because it focuses on the breakdown of cooperation between factions. The second contribution is that we assess novel culture-based hypotheses, notably regarding the role of social norms like conformity and institutional confidence.

It is well established in the cross-national quantitative literature that ethnic fractionalization is statistically unrelated to the probability of political violence, regardless of whether one looks at large-scale or small-scale forms of violence [82,84,85]. This result is confirmed in our analysis: ethnic fractionalization is statistically insignificant in three out of seven models, and when it is significant, it is associated with less unrest. This finding is often seen as a puzzle since the case study literature shows that most civil wars are fought along ethnic or religious lines. Our results suggest that it is not ethnic fractionalization per se that matters but whether ethnicity is politically salient. Horizontal inequality could be one of the factors that increase the salience of ethnicity. In the light of this literature, our results on the number of ethnic groups is also interesting. They suggest that it is the total number of groups, not ethnic fractionalization, that is associated with political instability.

Social norms and institutions are a ubiquitous component of our social life and decision-making [66,138,139]. While there is a growing number of theoretical studies in economics and cultural evolution that account for psychological or sociological factors [96,98,140,141], these efforts have not led so far to falsifiable predictions in studies of social conflict that distinguish between material and non-material forces [41]. Our model offers such predictions. Moreover, historians have argued that some societies become more successful and/or stable than others due to their social norms and institutions [142]. Our theoretical and empirical results support these conclusions.

Our model was intended to describe states that rely on large-scale cooperation between its segments operating under largely stable economic and political rules. Adapting the model to other types of societies, e.g. those relying on a forced transfer of resources and goods up a social hierarchy, is an interesting direction for future work. Horizontal inequality was a factor in some prehistoric economic organization and chiefdoms [143]. In contrast to our model, cycles of breakdown in complexity of early societies [144–149] are typically driven by deaths of the rulers, exogenous events, or certain demographic processes [74,150] that we did not consider here. Some data suggest that conflict among feudal elites—landlords, clerics, kings, and officeholders—gave rise to capitalism itself [150]. Our modelling framework might be useful for describing these processes.

More generally, data show that while inequality within human civilizations has generally increased from the Stone Age to today [5], significant reductions in inequality have repeatedly occurred, each typically preceded by violent events [4,5,151]. As high inequality becomes unsustainable, it becomes reduced again by wars, social strife and/or revolutions [4].

We did not model the (violent) conflict between the factions explicitly but only a necessary condition for it—the breakdown of cooperation between factions. Nevertheless, in our model the factions switching to opposition reduced the maximum amount of benefits potentially available to the ruling elites punishing them as a result. More generally, the presence of counter-elites can not only decrease the overall benefit but also increase the costs of production.

Our models can be generalized in a number of ways. For example, one can add a normative ‘bonus’ to a faction’s effective effort y_i for being in the opposition [62] or impose a material penalty on defecting factions as a result of some kind of ‘punishment’ administered by the state. One can allow for political concessions from the state [47,48,91], effects of foresight [152,153], and/or social learning [154] in decision-making. One can make utility of conformity dependent on ‘affinity’ between factions which in turn can depend on the history of past decisions. We modelled bounded rationality of our agents in a simple way using myopic best response subject to errors. Recent approaches using forward-looking agents or the ‘theory of mind’ (reviewed in [153]) suggest added complexity to incorporate into future revisions of our model. We used an S-shaped production function aimed to capture the law of diminishing returns in economics in a simple form. From game theoretic work, we know that the shape of production function can affect the resulting dynamics and equilibria. We treated parameters η₀ and η₁ as constant. Support for the state and the level of conformity could be a function of inequality within society, however. In our model, we have treated each faction as a single unit which makes or not a fixed contribution to the production of collective goods. In reality, factions are composed of individuals and can have certain structure. Factions can decide how much effort/resources to invest into competing for power, and it may not be optimal to invest all resources, especially when the chance to increase the share of the collective good in the next stage is slim. The presence of vertical inequality within factions can have a significant effect to their propensity to remain loyal to the state or to rebel [27,28,155]. The effects of these additional important forces and factors remain to be explored. The methods of experimental economics have proved to be useful in uncovering patterns and processes of human decision-making both at the individual level [156,157] and at the group level [158,159]. Our model is relatively simple and thus might be amenable to experimental implementation. It would be interesting to test its predictions in an experimental set-up. Overall, sharpening the focus of theoretical work will be hardly possible without additional empirical data to validate and parametrize the models.

There are also limitations to this study in articulating a model simulation with real-world statistics aggregated at the scale of nations, which do not match simulation results in a one-to-one fashion. To address this, future studies could test the model results more directly, either at the scale of experimental psychology under controlled conditions, or potentially using detailed historical data. This could correct for the approximate connections we make between rule of law as an empirical proxy for incumbency, for example. Future models will also address questions of endogeneity and reverse causality.

Our findings have important implications in light of the economic consequences of the COVID-19 pandemic. Research shows that the coronavirus has increased inequality, particularly inequality between ethnic groups worldwide. Low income households have experienced larger income reduction and the effect has been more pronounced among certain ethnic groups, often those that were already disadvantaged before the pandemic [160–162]. Moreover, according to the World Bank, more than 80 million people attained extreme poverty in 2020 because of the pandemic [163]. At the same time, according to various news reports, many of the world’s richest people have become richer both because of the pandemic and because of the policies adopted by governments to respond to it [164–166]. Within the USA, the proportion of African–Americans among a county population predicted higher death and infection rates [167], which is likely due to socio-economic inequality (high population households, underlying health conditions, tendency to work service jobs). These disparities in pandemic effect have fuelled tensions between groups. This suggests that mitigating the effect of COVID-19 on horizontal inequality may be crucial if we are to limit its destabilizing effects.

Data accessibility

Data and replication codes are available at https://houlec.com/research/ and https://datadryad.org/stash/share/ME4jCUQ01MjZgxcY5Pp00qF30nbg90N8tmFn1YBcLdc.

Authors' contributions

C.H.: conceptualization, data curation, investigation, methodology, validation, visualization, writing—original draft, writing—review and editing; D.J.R.: conceptualization, data curation, investigation, methodology, resources, writing—original draft, writing—review and editing; R.A.B.: conceptualization, investigation, resources, writing—original draft, writing—review and editing; S.G.: conceptualization, formal analysis, funding acquisition, resources, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests.

Funding

S.G. was supported by the US Army Research Office grant nos. W911NF-14-1-0637 and W911NF-17-1-0150, the Office of Naval Research grant no. W911NF-18-1-0138, the Air Force Office of Scientific Research grant no. FA9550-21-1-0217, the National Institute for Mathematical and Biological Synthesis through NSF award no. EF-0830858, and by the University of Tennessee, Knoxville. D.J.R. and R.A.B. received funding from the College of Arts and Sciences and the Office of Research and Engagement at the University of Tennessee.