Hidden patterns of reciprocity

Reciprocity can help the evolution of cooperation. To model both types of reciprocity, we need the concept of strategy. In the case of direct reciprocity there are four second-order action rules (Simple Tit-for-tat, Contrite Tit-for-tat, Pavlov, and Grim Trigger), which are able to promote cooperation. In the case of indirect reciprocity the key component of cooperation is the assessment rule. There are, again, four elementary second-order assessment rules (Image Scoring, Simple Standing, Stern Judging, and Shunning). The eight concepts can be formalized in an ontologically thin way we need only an action predicate and a value function, two agent concepts, and the constant of goodness. The formalism helps us to discover that the action and assessment rules can be paired, and that they show the same patterns. The logic of these patterns can be interpreted with the concept of punishment that has an inherent paradoxical nature.