Abstract
We consider indirect reciprocity with optional interactions and private information. A game is offered between two players and accepted unless it is known that the other person is a defector. Whenever a defector manages to exploit a cooperator, his or her reputation is revealed to others in the population with some probability. Therefore, people have different private information about the reputation of others, which is a setting that is difficult to analyze in the theory of indirect reciprocity. Since a defector loses a fraction of his social ties each time he exploits a cooperator, he is less efficient at exploiting cooperators in subsequent rounds. We analytically calculate the critical benefit-to-cost ratio above which cooperation is successful in various settings. We demonstrate quantitative agreement with simulation results of a corresponding Wright–Fisher process with optional interactions and private information. We also deduce a simple necessary condition for the critical benefit-to-cost ratio.
Highlights
The evolution of human cooperation is an intensely researched topic in the biological and economic sciences [1,2,3,4]
Various subtopics have been researched, including the effects of image scoring and good standing strategies [23], the reputation dynamics that lead to evolution of indirect reciprocity [24,25], involuntary defection [26], games among more than two players [27,28], the ability of cheaters to disrupt stable strategies [29], costly information transfer [30], trinary reputation models [31], mixing of social norms [32], and others
The defector participates in m rounds and is offered to play a game m times, each time with a random cooperator
Summary
The evolution of human cooperation is an intensely researched topic in the biological and economic sciences [1,2,3,4]. Various subtopics have been researched, including the effects of image scoring and good standing strategies [23], the reputation dynamics that lead to evolution of indirect reciprocity [24,25], involuntary defection [26], games among more than two players [27,28], the ability of cheaters to disrupt stable strategies [29], costly information transfer [30], trinary (instead of binary) reputation models [31], mixing of social norms [32], and others. (Alternately, one could consider that two defectors do not play a game together; since a game between two defectors has no effect on payoffs and results in no information transfer, such a distinction is inconsequential for our model.) Each time a defector plays a game with a cooperator, his or her identity is revealed to each of the NC cooperators in the population with probability p. What is the critical value of b/c above which cooperators are more abundant than defectors when their numbers, NC and ND , are averaged over many successive generations of the mutation–selection dynamics? What is the critical value of b/c above which, with no mutation, defection fixes in the population with probability less than 1/N when starting with a single defector? Can these questions be answered analytically?
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