Suppose that it people use M3 to commute to London by car. Each drivers chooses how many times to travel a year, s. The total number of trips of all drivers except i in a year is xi = ∑j≠isj. Let the total annual benefit that an individual i derives from getting to London be v (so = 2√7. Assume that there is a cost of travelling of c per trip (petrol, car depreciation etc) (this is not an annual cost, but a cost per trip). Finally, assume also that there is a congestion cost per trip equal to ax i where a is a positive parameter (that is, the more drivers other than i use the M3, the more it costs driver i to travel).
(a) Formulate this situation as a normal-form game. In paticular, write what is the utility/payoff fuction for commuter i.
(b) Find the best response functions and characterize the Nash Equilibrium. (Hint: you may obtain an equation that is difficult to solve. In this case do not solve it, but in any case explain).
(c) Characterize the efficient number of trips (maximizing the sum of all utilities).
(d) Now suppose that the authorities introduce a fee/toll for using the M3, equal to t per trip. Formulate this game and characterize the equilibrium (decentralized) outcome with such a toll. Explain how the authorities should choose the toll to bring the equilibrium number of trips to the efficient level.