# Epstein's riot model

Previously discussed Granovetter threshold model is just one of the numerous simple collective action model. This time we continue the same topic by considering another, yet a bit more complex, riot model, which was proposed by Epstein in [1]. This model is rather interesting in a sense that it is not static as original Granovetter model is. It has interesting temporal dynamics builtin. In a recent paper by British mathematicians [2] this model was applied to explain the patterns observed in 2013 London riots.

## Model

Briefly the model could be described as follows. People randomly move on the lattice. Each of them feels a personal hardship, a grievance towards government. If there are now policemen around the people will express their attitude publicly. Yet they will keep quiet if policemen are around. Unless a critical mass of protesters gathers, then outburst of violence is unavoidable. The most unlucky will go to jail, while the remaining protesters will have to keep their opinions to themselves.

From the brief description above it should be more or less clear what do policemen do. They are randomly moving around the lattice and looking for protesters (their vision radius is given by \( v \)). If they encounter protesters, they arrest random protester. Policeman moves to the cell containing random protester and sends him to jail for a certain number of turns (randomly picked from the interval \( [1,J_{max}] \)). Policemen themselves do not experience grievance. Their number is random, yet influenced by the density, \( \rho_{cop} \), parameter.

On the lattice we show policemen in blue. While number of jailed agents is shown as a black curve in the plot.

The behavior of ordinary agents (the population) is a bit more complex. They also move around the lattice randomly. As they move around they observe the number of protesters, \( A \), and the number of policemen, \( C \), in their vision radius (also given by \( v \)). According to these numbers they perceive level of risk:

\begin{equation} R_i= r_i \theta(C-A+0.1) , \end{equation}

here \( \theta(x) \) is a Heaviside theta function, while \( r_i \) is a measure of agent's risk aversion (in the model it is a random number in interval \( [0,1] \)). The \( 0.1 \) term is included purely for the sake of simplicity, so that \( \theta(x) \) yields only \( 0 \) or \( 1 \). Perception of risk influences the agent's decision to protest:

\begin{equation} H_i (1-L) - R_i > T , \end{equation}

here \( H_i \) is the hardship experienced by the agent (in model it is a random value from interval \( [0,1] \)), \( L \) is the perceived legitimacy of the current regime (parameter of the model), while \( T \) is a action threshold (parameter of the model). If this condition is not satisfied, then agent hides his discontent.

In the lattice we use green color to represent ordinary agents, while protesting agents are shown in red. Furthermore red curve in the plot shows a temporal evolution of the number of protesters.

We have not yet discussed a meaning of \( \rho_{civ} \). This parameters sets the initial desired density of the ordinary agents on the lattice.

## Interactive applet

## References

- J. M. Epstein. Modeling civil violence: An agent-based computational approach. Proceedings of the National Academy of Sciences 99: 7243-7250 (2002). doi: 10.1073/pnas.092080199.
- T. P. Davies, H. M. Fry, A. G. Wilson, S. R. Bishop. A mathematical model of the London riots and their policing. Scientific Reports 3: 1303 (2013). doi: 10.1038/srep01303.