# The influence of influence in Prisoner’s Dilemma

Cooperation wins out over betrayal when successful prisoners recruit followers

Despite the risk of sleeping with the fishes, prisoners will always choose to be rats — at least, idealized prisoners always do in a classic mathematical scenario meant to explore cooperation and betrayal. But tweaking the scenario to allow successful prisoners to expand their circle of influence causes cooperation to blossom, new research shows.

This new variation of the Prisoner’s Dilemma, perhaps the most famous scenario studied in a branch of mathematics called game theory, helps illuminate when and how the players in the game will choose cooperation over betrayal. Previous research has shown that the web of connections among prisoners can lead to cooperation, but this study is the first that allows that web of connections to evolve over time, simulating the ability of successful prisoners to, essentially, win friends and influence people.

Cooperation always grew to become the dominant strategy whenever the sphere of influence of any one player was limited to a medium-sized “Goldilocks Zone” of about 50 players. If these social networks were too small or too large, the game would become overrun by betrayal, the researchers report online and in an upcoming issue of Europhysics Letters.

“The question that we wanted to answer is how can such cooperative networks evolve” starting from a simple, randomized network, says Matjaž Perc, a physicist at the University of Maribor in Slovenia. “We found that by adding this simple process for evolving the network connections, we arrived at, not the same graph, but a very similar graph” to those that promoted cooperation in previous research.

Countless police shows and action movies have depicted the basic Prisoner’s Dilemma: Two crooks are caught for a minor crime, and police officers lean on each crook to rat out the other one for a more serious offense. If prisoner A betrays prisoner B, A will go free and B will go to jail for five years, and vice versa. If both prisoners rat, they’ll both be convicted of the greater crime but will serve only three years because they helped the police. However, if the prisoners cooperate and both remain silent, they’ll both avoid the serious charge and serve just one year each for the lesser crime.

Mathematicians have known since 1950 that these fictional prisoners will always betray each other, assuming they can’t communicate with each other and they act based on their rational self-interest. That’s because, regardless of which strategy the other player chooses, a player always gets a shorter prison sentence by ratting — freedom instead of one year if the other player is silent, and three years instead of five if the other player chooses betrayal. If the scenario involves many prisoners who all know each other and whose fates are all intertwined, the result will be the same.

In real life, many of the prisoners would not know each other, of course. To simulate this, a fertile area of game theory research has explored the Prisoner’s Dilemma when the prisoners are connected only to other prisoners that they “know,” creating social networks. Each player competes only with players that he is connected to.

In 2005, physicists in Portugal showed that these network connections can tilt the balance of the game in favor of cooperation. In the new work, Perc and his colleagues added the ability of these network connections to evolve depending on the success of each player. As the game is repeated, a player that performs well — that is, gets the least jail time — can influence one of his acquaintances to adopt his  strategy, whether cooperation or betrayal. That player also gets to add one more prisoner to his circle of acquaintances, widening his circle of influence.

This clustering effect is what enables cooperation to survive.

“It was shown that if you make cooperators and defectors to compete, the defector will defeat the cooperators. The only way a cooperator can survive is by forming clusters of cooperators,” comments Marcelo Kuperman, a physicist at the National Commission for Atomic Energy in Bariloche, Argentina.