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3. The key insight of the prisoner's dilemma-that cooperation is sustained by reciprocity in repeated interactions extends to environments where multiple players are enforcing a norm. An example of such a model is developed by Panchanathan and Boyd (2004): ". .. we consider a large population subdivided into randomly formed social groups of size n. Social life consists of two stages. First, individuals decide whether or not to contribute to a one-shot collective action game at a net personal cost C' in order to create a benefit B shared equally amongst the n - 1 other group members, where B > C. Second, individuals engage in a multi-period 'mutual aid game'. . . In each period of the mutual aid game, one randomly selected individual from each group is 'needy'. Each of his n - 1 neighbours can help him an amount b at a personal cost c, where b > c > 0. Each individual's behavioural history is known to all group members. This assumption is essential because it is known that indirect reciprocity cannot evolve when information quality is poor. The mutual aid game repeats with probability w and terminates with probability 1 - w, thus lasting for 1/(1 - w) periods on average." The "shunner" strategy is of primary interest:"Shunners contribute to the collective action and then try to help those needy individuals who have good reputations during the mutual aid game, but mistakenly fail owing to errors with probability e. . . Shunners never help needy recipients who are in bad standing." (a) The authors argue that (shunner, shunner, ..., shunner) is an equilibrium iff (",1) (1 ) (6- c)(1 - we) > C. However, they only consider deviations to two other strategies, and not to all possible strategies in this game. Under what conditions is (shunner, shunner, . .., shunner) an equilibrium of the game when you consider deviations to all possible strategies of this game? You should find two conditions: the same one that the authors found, and also _$ 2 1_ . To simplify the math, feel free to set e = 0. to the (b) When is contribution to the public good sustained as part of a Nash equilibrium? What property must any strategy that sustains contributions to the public good have