4. A new bridge has just been built in a town, which has a population of N > 1. Every day, everyone in this town chooses his / her own probability of using the bridge, denoted by 9,7, where 3' = 1, 2, - - -, N. For each individual, the payoff of using the bridge is given by where a E (0, 1). That is, using the bridge provides convenience to everyone. But the more people trying to cross the bridge, the worse the trafc ow on the bridge, and thus the lower the marginal benet of using the bridge. If an individual does not use the bridge, the payoff is c > 0, which is constant. (a) [10 points] Consider the daily simultaneous decisions of those living in this town on 9,. Find the daily choice 9* in a symmetric Nash Equilibrium. (b) [10 points] Consider a mayor, who is benevolent and cares about everyone equally. What is the socially optimal level of bridge usage, 9'}? (c) [5 points] Compare 9* and 1:}. What is your policy recommendation to the mayor in order to correct the inefcient equilibrium outcome- That is, would you recommend a tax on using the bridge or a coupon (is, a positive transfer to an individual for using the bridge)? Explain the intuition