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Energy Policy: Trying to Avoid a Nash Equilibrium

http://ftalphaville.ft.com/blog/2011/09/09/673476/a-greek-t-bill-oddity/

The article above displays how even the most complicated political debates can be broken down by an economist into the simplest of terms. The superpowers of the world have debated for decades how to impose regulations which will improve the social welfare of the world through cutting down on their carbon dioxide emissions–yet year after year countries seem to throw compromise to the wind as they continue to pump out higher and higher emissions than in previous years. The reason? The situation described is the extremely common case that an economist would describe as a Nash Equilibrium that does not benefit the two parties.

Think of the Prisoner’s Dilemma, everyone’s favorite introduction-to-game-theory anecdote. If the two prisioners were to cooperate, they could surely maximize the net social welfare between the two of them. However, the pursuance of each individual’s dominant strategy results in a Nash Equilibrium that is far more worse off relevant to the state in which the two could have cooperated.

The battle over carbon emissions is in reality no different. While I claim no ability to accurately project what the relative utilities of a respective polluting nation would be when measured in tandem with another nation, let us for a moment consider a basic example with a number of assumptions for the sake of simplification: there are two countries in the world, each of the same size and substance. Both countries have the same marginal costs to emitting carbon dioxide (please disregard your vantage points regarding the harms of global warming or pollution and view this as a given utility cost function) and the same marginal benefits (added production, among other things). Seeing as how firms do pollute to some extent in a free market, let us also accept the assumption that at some level of pollution the marginal benefits exceed the marginal costs, and it is thus utility maximizing to pollute. For arguments sake let us also assume a convex cost function (marginal costs are increasing) and a concave benefit function (marginal benefits are decreasing). Finally, let us assume that when one of the two countries pollutes the costs are felt by both countries, that if a country does pollute it will do so until marginal revenue is equal to marginal cost at pollution level X, thus maximizing its own utility (disregard for now the possibility of partial pollution), and that countries make their pollution decisions at the same time (thus if a country pollutes it will base its marginal cost only its own level of pollution X).

Based on these assumptions, one can easily perceive the game that has been created. For each country, there exist two options: pollute or not pollute. Thus with two countries, our game has 4 outcomes:
1) C1 and C2 both do not pollute: If both countries were to not pollute, both would have a net utility of zero as there would be no costs or gains.
2) C1 and C2 both pollute: If both countries were to pollute, the total amount of pollution would cause both of them to suffer–some amount of pollution X led to a maximized utility, but doubling that amount of pollution with a convex cost function means the costs incurred from X to 2X > the costs from 0 to X, and thus the net utility for each country is negative.
3) C1 pollutes, C2 does not: C2 will have negative utility as it will have no benefits but only costs from the pollution of C1, while C2 will be at its utility maximizing pollution level and will thus have positive utility.
4) C2 pollutes, C1 does not: same scenario as 3 but vice-versa–C1 will have negative utility and C2 will have positive utility.

It is easy to see here how for C1 and C2, polluting is the dominant strategy; no matter what the other country’s strategy, both country’s are better off polluting (either from 0 to positive or from some negative to a lower negative). What the article points out is that while some countries are attempting to take the non-pollution strategy, other country’s are “free-riding” and continuing to follow the dominant strategy–after all, this is the best strategy for that country.

The article postulates that in order to reach a social welfare maximizing solution, rather than allow “free-riding” countries to continue to reap the benefits of following their dominant strategy, environmentally conscience countries should also follow their own dominant strategy. This would force both countries to the original Nash Equilibrium which is harmful to everyone. At this point, rather than continuing to pollute at their individually profit-maximizing level, a polluting country may be willing to negotiate and impose a negotiated regulation that forces both countries into the social-welfare maximizing strategy of not polluting for both countries. More realistically, this would force the two countries to act as oligopolies and reach a profit maximizing solution for both countries at some lower level of pollution (as opposed to one country acting as a monopolist). Thus this is exactly what the article proposes when it states that such a strategy could increase pollution in the short term (the time spent at the original Nash Equilibrium), but in the long run could produce a pair of strategies which improves social welfare.

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