Global Warming and Game Theory

Global Warming 

According to NASA, over the last hundred years, the global average surface temperature increased from 0.6 to 0.9 degrees Celsius. In the last fifty years, the rate of temperature increase has doubled. What causes global warming? The warming of earths surface is caused by an increase in greenhouse gasses in the atmosphere, caused by human activity. Heat enters earth as solar radiation in the form of light from the sun. Earth's atmosphere deflects some of this light, however about 70 percent of that light enters the Earth and is absorbed, warming the land and the oceans. Earth emits its own heat in the form on infrared rays. Some of this heat escapes, while some is absorbed by greenhouse gas and then re-emitted back to earth. As more and more greenhouse gasses are emitted into earths atmosphere, more and more heat is trapped, resulting in a steady rise in overall temperature levels.

Greenhouse Effect

$CO_2$ is naturally in the earth's atmosphere, however due to human activity and greenhouse gas emissions (mainly through the burning of oil, natural gas, and coal) humans have altered the earths carbon cycle. Before the Industrial Revolution, levels of $CO_2$ in the atmosphere averaged around 270 ppm (parts per million). By 1960 the level was up to 313 ppm, and by 2013 the level reached a high of 400ppm. From 1990-2013 $CO_2$ emissions have increased by 7 percent. The main cause of these emissions from energy and transportation. The surest way to reduce $CO_2$ emissions is to reduce the consumption of fossil fuels - however with a constantly increasing world population and an expanding economy, energy use is ever increasing, increasing emissions.

Curb Emissions? A Prisoner's Dilemma Representation

If we know curbing $CO_2$ emissions would be in our best interest for our future, why are we not doing so? The refusal to put a cap on $CO_2$ emissions can be represented through the prisoners dilemma. Let's take the two countries with the highest $CO_2$ emissions, China and the United States. The best outcome would be for both countries to cut emissions, invest in green technology, install filters in factories, and produce less overall. If one country, let's say China decides to reduce emissions, But the United States does not, China loses money in transitioning to greener technology, loses money as its production level falls, and the US is still polluting, so they do not reap the benefit of having a global change in $CO_2$ emissions. On the other hand, if China decides to not reduce emissions, but the United States does, China will reap the benefit of fewer emissions in the global atmosphere and they will not spend money on green technology themselves. If both countries decide to not reduce emissions, they will both face the consequence of global warming in the long term, but in the short term their profits will remain the same. This is represented in the payoff matrix below.

 

As we can see, the dominate strategy for both players is to do nothing and not reduce $CO_2$ emissions. If both countries follow their dominant strategy, the best outcome will never be achieved, and everyone will remain worse off.  

Assisted Tit for Tat Strategy

The choice of a country to/to not curb emissions is not limited to one time period. For example if in 2017 the US decides to not curb emissions, in 2018 it would have another opportunity to decide whether to curb/not curb emissions. Games that occur multiple times over multiple periods are known as repeated/iterated games. In repeated games players can use the decision of the other players from the previous round to determine their strategy for the current round. 

A tit-for-tat strategy in game theory is a strategy in which a players strategy is to start off cooperating. The next round, the player operating under the tit-for-tat strategy will do what ever the other player did the previous round.

Let's say U.S citizens pressure their government to curb emissions, so the U.S decides to cut emissions in the first round. Let P represent the initial round of the game. The table below demonstrates the tit-for-tat strategy.

As you can see, in the second round China would be faced with the decision to continut polluting or cut emissions. If China knows the US is under pressure from its citizens to curb emissions, China will decide to continue poluting as that decision will yield the highest utility.

Hence, the tit-for-tat strategy does not bring about the best solution for both countries in this situation. Clemons and Shimmelbusch, in their paper The Environmental Prisoner's Dilemma, propose an assisted tit-for-tat strategy in which tariffs will be imposed upon China in the second round if China does not curb its emissions.

With the new tariff, by just round two China would agree to also cut emissions as it would receive the highest utility from this decision.

Treaty Participation Game

Let's take into account the Kyoto Protocol - an international treaty with the United Nations Framework Convention on Climate Change in which members pledge to reduce emissions. Players (in this case countries) decide where to sign or not-sign the treaty. In this game there are three stages:

Stage 1: All players simoltaneously sign/don't sign.

Stage 2: Signatories decide whether to collectively curb emissions or continue polluting.

Stage 3: Non-signatories decide whether to curb emissions or continue polluting.

In Stage 1, if country A decides not to sign. then country B is indifferent about signing. If country A decides to sign, then country B is better of by signing.

In Stage 2, if both countries sign the will maximize their utility. If only one country signs, then the non-signer will pollute, so they will also pollute.

In Stage 3, non-signatories will reach their Nash-Equilibrium by choosing to continue polluting.

Let there be n players. 

Let  \alpha = porportion of countries that sign.

So there are \alpha n signers.

Then \pi_n (\alpha) = payoff for non signers.

Then \pi_s (\alpha) = payoff for signers. 

Countries will agree to self enforce iff

\pi_n(\alpha-\fraq{1}{n})\leq \pi_s\alpha

and

\pi_n(\alpha)\geq \pi_s(\alph+\fraq{1}{n})

Work Cited

http://ageconsearch.umn.edu/bitstream/95061/2/Climate%20Change%20and%20Game%20Theory.pdf

http://opim.wharton.upenn.edu/~clemons/blogs/prisonersblog.pdf

http://www.pnas.org/content/108/38/15739.full.pdf

http://www.nasa.gov/offices/emd/home/ggm.html#.Vk0ae0ZEyap

http://unfccc.int/kyoto_protocol/items/2830.php