There is a theory in criminology that says that if a broken window in a building is left unrepaired, that building will soon have all its other windows broken. The rationale behind it is that signs of urban disorder and untended public spaces elicit antisocial behaviour. The idea is reminiscent of the concept of “criticality” in physics, whereby small, seemingly insignificant changes, can trigger large-scale sequences of events with unexpected consequences.

With this analogy begins a recent work by D’Orsogna & Perc^{1}, in which they have compiled and explained several statistical physics models that have been adapted to the singularities of criminal behaviour. In the next lines, we’ll take a look at some of them and see if we can use mathematics to prevent crime.

**Mapping crime**

A simple system to study crime is residential burglary since the targets don’t move and one can focus solely on the actions of the burglars. In a model developed by Short et al., the criminals are considered to act in an area that is close to their residence, but not too close. Target houses are chosen based on two things: their “*attractiveness*” (i.e. ease of access, perceived wealth) and the burglar’s past experiences, by which a successful rob can encourage new robberies at the same residence or those located in the vicinity.

Depending on the values given to the parameters, the model predicts that three stable situations can emerge: 1) crime can be evenly spread throughout the area; 2) crime hotspots can transiently emerge and persist for some time before disappearing and diffusing elsewhere, or 3) persistent crime hotspots emerge surrounded by areas with low criminality. An interesting observation is that when there is a large number of criminals, the most likely outcome is either the first model, in which there are no crime hotspots or the third one, where hotspots are persistent.

Police intervention has also been studied in the context of this model by setting to zero the crime rates at hotspots for a given time frame. One of the conclusions is that only some crime hotspots can be permanently eradicated by police intervention, while the most active spots are only displaced.

**Criminal games**

Another interesting approach to study crime is to apply Evolutionary Game Theory models. These models are devised as games in which two players are confronted and have to decide simultaneously if they want to cooperate or defect. Depending on both decisions, each player then receives a pay-off, and by setting different values to the pay-offs, different games emerge.

Probably the most famous game is the ‘Prisoner’s dilemma’. Although its story can be told in many ways, it was initially presented as two criminals that are arrested at the same time and imprisoned. They are both guilty, but there is not enough evidence to fully convict them for all their crimes, so the prosecutors try to negotiate: if both of them accuse each other, they get two years in prison each; if one accuses the other but the other remains silent, the informer gets out and the accused gets three years in prison; finally, if both remain silent, each one gets only one year in prison.

Although it might not seem obvious at first glance, the only rational strategy for each player in the ‘Prisoner’s dilemma’ is to accuse the other. Why? Imagine you are one of the prisoners: if the other prisoner accuses you, you should also accuse him because that way you will serve two years instead of three. On the other hand, if the other prisoner remains silent, you should accuse him because then you go free instead of spending a year in prison. So, despite the fact that if they cooperate they can get a better deal for both, the rational strategy is to defect. The idea is the same that underlies the famous concept of the “Tragedy of the commons”, by which a selfish, yet rational, behaviour, ends up being against one’s interest.

But the “Prisoner’s dilemma” is just one of many games. By changing the pay-offs, other situations emerge in which the rational strategy might be different than defection. Higher layers of complexity can be added by including different types of players with different strategies and pay-offs. Each round two randomly players are chosen to interact and their pay-offs are calculated.

An interesting evolutionary game to model crime was constructed considering four types of players, each one with a different strategy:

1) Paladins: model citizens that never commit crimes and are always willing to collaborate with the authorities.

2) Villains: the bad guys, they commit crimes and never collaborate with the authorities.

3) Informants: an intermediate figure, they commit crimes but collaborate with the authorities.

4) Apathetics: who neither commit crimes nor collaborate with the authorities.

There is a population of players and each round a criminal (either a villain or an informant) and a potential victim (that could be anyone else) are randomly selected. After every interaction, the criminal gets a positive pay-off and the victim loses the same amount. If the victim is either an apathetic or a villain, the crime goes unreported and the round ends. On the other hand, if the victim is either an informant or a paladin, an investigation begins. Chances that an investigation succeeds depend on the total number of paladins and informants within the total population. In the case of a conviction, the victim gets back its loss and the criminal gets a punishment. If the crime goes unpunished, the criminal retains its pay-off and the victim has a further loss, that accounts for a possible retaliation. Importantly, at the end of each round, the player with the lowest pay-off changes its strategy, which means that the number of paladins, villains, informants and apathetics within a population is dynamic.

By changing the initial number of each type of players or the different pay-offs, one can model different kind of situations and their outcomes. One of the conclusions of the model is that informants are crucial to the emergence of a crime-free society. In fact, a transition from a crime-dominated society to a crime-free society can be achieved by imposing an optimal number of informants, an observation that has also been observed in behavioural experiments with humans.

**Can we solve a crime like we solve an equation?**

The short answer is that no, we can’t. There are seldom simple solutions to complex problems. However, models such as those commented here can offer different and interesting angles to help in the search of solutions. And if we want to win the fight against crime, we are going to need all the help we can get.

## References

- D’Orsogna, MR & Perc M. (2015) Statistical physics of crime: A review.
*Phys Life Rev.*doi: 10.1016/j.plrev.2015.02.002 ↩