Suppose we have an economy with N people. There is a public good which is only...

 

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Suppose we have an economy with N people. There is a public good which is only provided if at least one person gives up c € (0, 1) dollars to pay for it. Consider the following game. Each person simultaneously decides whether or not to contribute c. If at least one person contributes c, each contributor gets 1 c and each non-contributor gets 1. If no one contributes, everyone gets a payoff of zero. (a) What are the pure strategy equilibria of this game? (b) Find the unique symmetric mixed strategy equilibrium of this game. (By "symmetric," I mean an equilibrium in which everyone uses the same mixed strategy.) What is the probability that at least one person contributes in this equilibrium? Show that as N → ∞, the probability that someone contributes converges to something strictly between zero and one. (That is, there is uncertainty about the outcome in the limit.) Suppose we have an economy with N people. There is a public good which is only provided if at least one person gives up c € (0, 1) dollars to pay for it. Consider the following game. Each person simultaneously decides whether or not to contribute c. If at least one person contributes c, each contributor gets 1 c and each non-contributor gets 1. If no one contributes, everyone gets a payoff of zero. (a) What are the pure strategy equilibria of this game? (b) Find the unique symmetric mixed strategy equilibrium of this game. (By "symmetric," I mean an equilibrium in which everyone uses the same mixed strategy.) What is the probability that at least one person contributes in this equilibrium? Show that as N → ∞, the probability that someone contributes converges to something strictly between zero and one. (That is, there is uncertainty about the outcome in the limit.)

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