$7.$ Consider the following payoffs from two countries $($Canada and the U$.$S$.)$ negotiating over reducing an air pollutant that crosses the boundaries of the two countries. Each has two strategies available to it: reduce the pollutant or not. The costs of reducing the pollutant are $\backslash (\backslash$$ $8\backslash )$ million for Canada and $\backslash (\backslash$$ $16\backslash )$ million for the U$.$S$.$ If a country does not reduce the pollutant, its costs are $\backslash (\backslash$$ $0\backslash ).$ The benefits of reducing the pollutant are $\backslash (\backslash$$ $20\backslash )$ million if both countries reduce the pollutant, $\backslash (\backslash$$ $10\backslash )$ million if only one country reduces the pollutant, and $\backslash (\backslash$$ $0\backslash )$ if no country reduces. As was the case in the international environmental agreements game studied in class, each country receives the benefits.

$($a$).$ Fill in the payoff matrix for this one$-$shot game. As in class, the payoff for each country $($in millions$)$ should be given in the table as $($Payoff for U$.$S$.,$ Payoff for Canada$).2$ pts

Canada

Reduce Pollutant

Don't Reduce

$($b$)$ Is there a Nash equilibrium in this one$-$shot game? Be sure to identify any dominant strategy for either country and explain your answer. $3$ pts

$($c$)$ Using the game above, describe a tax mechanism and solve for the tax level that would generate a Nash Equilibrium in which both countries reduce the pollutant. $\backslash (\backslash$mathbf$\{3\}\backslash )$ pts