Problem
Suppose that the St Clair river, which flows along the border between Ontario and Michigan, becomes polluted and needs to be cleaned. The American and Canadian governments need to make a decision on whether to clean the river or not. They must make this decision simultaneously. If both countries invest in cleaning the river, they each get a payoff of 730. If one country invests in cleaning the river, but the other one doesn't, the country that spends the money on the cleanup gets a payoff of 136, while the other country gets to enjoy the benefits of the clean river without having to spend any money, and therefore gets a payoff of 1139. If neither country invests in cleaning the river, they must both deal with the consequences of the pollution, and they each get a payoff of -595.
If Canada receives information that the U.S. will clean the river, and reacts optimally based on this information, how much will be the payoff for Canada?
If Canada instead receives information that the U.S. is not planning to clean the river, and Canada makes an optimal choice based on this information, how much will be the payoff for Canada?
i. Suppose that the U.S. and Canada are again making decisions simultaneously. Find the Nash Equilibrium in mixed strategies.
ii. In the mixed strategy Nash Equilibrium, what is the probability that Canada invests in cleaning the river? (Write the probability as a number between 0 and 1, rounded to the second decimal point, e.g. 0.37.)
iii. In the mixed strategy Nash Equilibrium, what is the probability that the U.S. invests in cleaning the river?
iv. In the mixed strategy Nash Equilibrium, what is the probability that both countries end up cleaning the river?
v. In the mixed strategy Nash Equilibrium, what is the probability that the river remains polluted (i.e. that neither country ends up investing in cleaning it)?