Introduction
Game-theoretic cooperation with stylized migration.
Migration: Individuals prefer better neighborhoods.
Success-driven migration: agents consider different alternative locations within a certain migration range, reflecting on the effort they are willing/able to invest on identifying better neighborhoods.
How favorable a neighborhood is expected to be is predicted via test interactions with individual in that area ("neighborhood testing"). These investments are usually small compared with potential gains/losses after moving. Agents move to their neighborhood that promises to be the best.
The authors claim that migration has received little attention in game theory because it has been found that mobility can hurt cooperation by supporting defector invasion, but that this only applies when individuals move randomly. They show that success-driven migration can support the survival and spread of cooperation, as well as its spontaneous outbreak from a world of defectors with various sources of noise.
The Model
N agents on an L x L lattice with periodic boundaries. Each cell is either empty or contains one agent. Asynchronous update in a random order. The selected individual plays the PD game with all 4 neighbors and compares payoffs with those neighbors. After, the strategy of the best neighbor (if better) is copied with probability 1 - r. With prob r, strategy is random reset.
Noise 1: An individual spontaneously chooses to cooperate with probability q or defect with probability 1 - q until the next strategy change.
Migration: Before the imitation step, an individual explores the expected payoffs for the empty sites in a the Moore neighborhood of radius M. If the fictitious payoff is higher than in the current location, the agent moves to the site with the highest payoff, and in the even of a tie, chooses the closest one.
Results
Without noise, cooperators do all right in all conditions, as in the Nowak models, and do best with both imitation and migration. With noise, defectors dominate for both imitation-only and migration-only conditions, but dynamic zones of cooperation can persevere if there is both imitation and migration (fig 1).
Figure 2 shows a "defector's paradise," and how success-driven migration allows cooperation to be sustained.
Noise 3: Noise 1 + Noise 2
Without migration, cooperators come to dominate in all conditions in the "defector's paradise" condition. With success-driven migration, however, cooperators do very well, and completely defeat defectors in the Noise 2 condition.
Noise 3 and success-driven migration can even produce the spontaneous emergence of cooperation from a defector only population. Even though the probability of a defector turning into a cooperator randomly is very low, and much lower than for a cooperator turning into a defector, it can take a long time, but a large enough cluster of cooperation can emerge and then come to dominate the population (fig 4).
Discussion
Our results help to explain why cooperation can be frequent even if individuals would behave selfishly in the vast majority of interactions. Although one may think that migration would weaken social ties and cooperation, there is another side of it that helps to establish cooperation in the first place, without the need to modify the payoff structure. We suggest that, besides the ability for strategic interactions and learning, the ability to move has played a crucial role for the evolution of large-scale cooperation and social behavior.
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