Quantitative determination of the level of cooperation in the presence   of punishment in three public good experiments
  Strong reciprocity is a fundamental human characteristic associated with our extraordinary sociality and cooperation. Laboratory experiments on social dilemma games and many field studies have quantified well-defined levels of cooperation and propensity to punish/reward. The level of cooperation is observed to be strongly dependent on the availability of punishments and/or rewards. Here, we suggest that the propensity for altruistic punishment and reward is an emergent property that has co-evolved with cooperation by providing an efficient feedback mechanism through both biological and cultural interactions. By favoring high survival probability and large individual gains, the propensity for altruistic punishment and rewards reconciles self- and group interests. We show that a simple cost/benefit analysis at the level of a single agent, who anticipates the action of her fellows, determines an optimal level of altruistic punishment, which explains quantitatively experimental results on the third-party punishment game, the ultimatum game and altruistic punishment games. We also report numerical simulations of an evolutionary agent-based model of repeated agent interactions with feedback-by-punishments, which confirms that the propensity to punish is a robust emergent property selected by the evolutionary rules of the model. 
D. Darcet and D. Sornette