Complexity_ A Guided Tour - Melanie Mitchell [104]
Interestingly, Axelrod followed up his tournaments by a set of experiments in which he used a genetic algorithm to evolve strategies for the Prisoner’s Dilemma. The fitness of an evolving strategy is its score after playing many repeated games with the other evolving strategies in the population. The genetic algorithm evolved strategies with the same or similar behavior as TIT FOR TAT.
EXTENSIONS OF THE PRISONER’S DILEMMA
Axelrod’s studies of the Prisoner’s Dilemma made a big splash starting in the 1980s, particularly in the social sciences. People have studied all kinds of variations on the game—different payoff matrices, different number of players, multiplayer games in which players can decide whom to play with, and so on. Two of the most interesting variations experimented with adding social norms and spatial structure, respectively.
Adding Social Norms
Axelrod experimented with adding norms to the Prisoner’s Dilemma, where norms correspond to social censure (in the form of negative points) for defecting when others catch the defector in the act. In Axelrod’s multiplayer game, every time a player defects, there is some probability that some other players will witness that defection. In addition to a strategy for playing a version of the Prisoner’s Dilemma, each player also has a strategy for deciding whether to punish (subtract points from) a defector if the punisher witnesses the defection.
In particular, each player’s strategies consist of two numbers: a probability of defecting (boldness) and a probability of punishing a defection that the player witnesses (vengefulness). In the initial population of players, these probability values are assigned at random to each individual.
At each generation, the population plays a round of the game: each player in the population plays a single game against all other players, and each time a player defects, there is some probability that the defection is witnessed by other population members. Each witness will punish the defector with a probability defined by the witness’s vengefulness value.
At the end of each round, an evolutionary process takes place: a new population of players is created from parent strategies that are selected based on fitness (number of points earned). The parents create offspring that are mutated copies of themselves: each child can have slightly different boldness and vengefulness numbers than its parent. If the population starts out with most players’ vengefulness set to zero (e.g., no social norms), then defectors will come to dominate the population. Axelrod initially expected to find that norms would facilitate the evolution of cooperation in the population—that is, vengefulness would evolve to counter boldness.
However, it turned out that norms alone were not enough for cooperation to emerge reliably. In a second experiment, Axelrod added metanorms, in which there were punishers to punish the nonpunishers, if you know what I mean. Sort of like people in the supermarket who give me disapproving looks when I don’t discipline my children for chasing each other up the aisles and colliding with innocent shoppers. In my case the metanorm usually works. Axelrod also found that metanorms did the trick—if punishers of nonpunishers were around, the nonpunishers evolved to be more likely to punish, and the punished defectors evolved to be more likely to cooperate. In Axelrod’s words, “Meta-norms can promote and sustain cooperation in a population.”
Adding Spatial Structure
The second extension that I find particularly interesting is the work done by mathematical biologist Martin Nowak and collaborators on adding spatial structure to the Prisoner’s Dilemma. In Axelrod’s original simulations, there was no notion of space—it was equally likely for any player to encounter any other player, with no sense of distance