^{ }and Duéñez-Guzmán (henceforth U&DG). The main point of the paper is that it is possible to maintain cooperation in an evolving population if there is a sub-population of corrupt police.

Yes, in this paper, corrupt policers are a force for good. In a simple evolutionary model, you will have a population of cooperators and defectors, and the defectors always win out. The question is if we can enforce cooperation by having "policers" who punish defectors, incurring a personal cost to do so. This paper concludes that it

*is*possible, but only if we allow policers to be "corrupt". A corrupt policer simply means that they punish defectors while simultaneously defecting themselves. They're hypocrites, in other words.

I'm not an expert in this field, but a couple years ago I discussed another evolutionary game theory study, where completely different conclusions were reached. As I surmised, this is because different models were used, and here I briefly explain the difference:

- In the other model, the population plays a game called the Iterated Prisoner's Dilemma. Individuals pair up, and play the prisoner's dilemma with their partners repeatedly, while responding to their partner's strategy. The players in that model used "mixed" strategies, meaning that their decisions were determined by continuously varying probabilities.
- In the U&DG model, the population plays the "Corruption Game." In this game, there are four strategies: cooperate, defect, honest police, and corrupt police. Here, the players use "pure" strategies, meaning that they just pick one of four discrete options. This vastly simplifies the problem.

[image modified from paper]

The upper left gray square is just the prisoner's dilemma. But each player also has the option of being a policer, which means they will spend one point to punish a defector. In this example, the defector gets penalized 10 points if they're a policer, 20 points otherwise.

It's only an example because you can replace these numbers with arbitrary parameters that obey certain conditions. Which is what they do in the paper. The paper finds that the situation depends on the arbitrary constants, specifically on how powerful policers are relative to non-policers.* There are three regimes, helpfully illustrated by these three triangles:

The triangle on the right shows what happens if policers are not so powerful. There is only one stable equilibrium (d) in which everyone defects. The triangle on the left shows that if policers are very powerful, then a second equilibrium (k) will appear, describing a population of entirely corrupt policers. In the middle triangle, there is an equilibrium (x) consisting of a mix of cooperators and corrupt policers. The x point has better outcomes for society as a whole, despite the corruption.

U&DG have a lovely model, although it's not very optimistic, and rather simplistic. It may apply to many situations but I'm not sure how well it describes the situation with police officers, who are publicly funded, after all. If we give police officers less power, that won't make them disappear, because we can just increase wages until enough people want to be officers. Also, I would have thought that corrupt police officers get along with each other.

Next time I'll discuss another paper, which appears to overturn these conclusions by demonstrating another equilibrium in which everyone is an honest policer.

Duéñez-Guzmán EA, Sadedin S (2012) Evolving Righteousness in a Corrupt World. PLoS ONE 7(9): e44432. doi:10.1371/journal.pone.0044432

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*Specifically, the constant that matters is q+d, as compared to p and s in this payoff grid. Policers are powerful if q+d < s, and weak if q+d > p.