The model:
A population (unstructured) of n individuals with the option to help or not.
Random pairs of players are chosen. One is the donor, and can help the recipient at a cost c to himself, and the recipient receives a benefit b (b > c). If the donor doesn't help, both receive zero payoff.
Each player has an image score, s, which is known to every other player. If a donor cooperates, his image score increases by one unit, otherwise it decreases by one unit.
Test 1: donors decide to help according to the image score of the recipient.
A strategy has the number k, indicating that the individual will provide help if and only if the potential recipient's image score is at least k.
They ran simulations in which n = 100 and k values were distributed so that -5 indicates pure cooperate and +6 indicates always defect. At the beginning of the run, all individuals have image score 0, and these are capped and +/- 5.
In each generation, 125 pairs are chosen (each player has an average of 2.5 interactions) and then at the end of each, players produce offspring proportional to their payoff.
This run ended with k = 0 fixated in the population. They claim that as long as the fixation k value is ≤ 0, then cooperation has won.
Mutation. They then ran the same simulations with a 0.001 probability that an offspring would have a random strategy. A fixation of k = 0 allowed less discerning cooperators such as k = -4 or -5 to drift in. This in turn allowed them to be exploited by defectors with k = +4 or +5, which could then invade. This was countered by the re-emergence of more discerning cooperators.
Incomplete information. It may be more reasonable to assume that an interaction between two individuals is observed only by a subset of the population. Only these "onlookers" and the recipient are able to update the donor's image score. They ran simulations where each interaction was observed only by 10 randomly chosen players, and varied the overall population size. When the group of onlookers is large relative to the population, then cooperation evolved, but not as much when the population was large relative to the small group of onlookers.
Two types of strategies. "And" strategies involve cooperation if the image score of the recipient is larger than some value and the image score of the donor is less than a certain value. The idea being that if an individual already has a high image score, one doesn't need to keep improving it.
"Or" strategies result in cooperation if the image score of the recipient is larger than some value or the image score of the donor is less than a certain value. The idea is that if an individual has a low image score, he may want to improve it regardless of the other person's image score.
In both cases, highly cooperative populations evolve. However, if strategies only consider their own image and do not account for the image of the recipient, cooperation does not emerge.
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