Problem 1. In a city full of sunshine there are a number of people. They can either drive their cars (to work or elsewhere), which can cause pollution and traffic jams, or they can ride bicycles (which is not problematic, given that in that city it almost never rains). The problem is that riding a bicycle can be dangerous (and unhealthy) if there are many cars on the road. The more cars there are on the road, the more costly it is to ride the bicycle, however, if there are no cars, then for each person, the utility of riding a bicycle is higher than their utility of driving. You may also assume that the more cars on the road also negatively affect the utility of driving (due to traffic jams). Assume that there is no private information (everyone's utility parameters are known to all people) and model this situation as a game. Specifically, look for equilibria (pure strategies) and comment on their Pareto efficiency.
Problem 2. Now suppose that each person in the city can either be of a lazy kind (in which case the person has a relatively low utility from riding a bicycle) or of an active kind (in which case the person has a relatively high utility from riding a bicycle). Each person knows for themselves whether they are lazy or active, and knows that every other person is lazy with probability 1 and active with probability 1. Modify your game from Problem 1 to account for this private information, and find some equilibrium of this new game (remember that as in the class, each person will now take into account their private information when making their choice). Is the equilibrium efficient? In your model, does there exist a Pareto efficient equilibrium outcome?
'Hint: Consider first the case where there are only 2 persons in the city, and then argue as formally as you can regarding the case where there are more than 2 persons.