Friday, 18 November 2016

Reason to be surprised, or not, about the Paris Agreement on climate change coming into force

The latest United Nations Climate Change Conference (COP22 in Marrakech) finishes today. The Paris Agreement, negotiated at the previous conference in Paris last year, officially came into force on 4 November. Under the agreement, countries commit to reduce their greenhouse gas emissions, in order to limit the increase in the global temperature to well below two degrees above pre-industrial levels.

Some basic game theory gives us reason to be surprised that this agreement has been successfully negotiated. Consider this: If a country reduces its emissions, that imposes a large cost on that country and provides a small benefit to that country, which is also shared by other countries. If a country doesn’t reduce its emissions, that imposes a small cost on that country and on other countries, while providing a small benefit only to the country that didn’t reduce emissions.

Now consider the negotiations as involving only two countries (Country A and Country B), as laid out in the payoff table below [*]. The countries have two choices: (1) to reduce emissions; or (2) to not reduce emissions. A small benefit is recorded as "+", a small cost is recorded as "-", and a large cost is recorded as "--" [**]. The costs and benefits (noted in the previous paragraph) imposed by the choice of Country A are recorded in red, and the costs and benefits imposed by the choice of Country B are recorded in blue. So, if both countries choose to reduce emissions the outcome will be in the top left of the payoff table. Country A receives a small benefit from their own action to reduce emissions (red +), a small benefit from Country B's action to reduce emissions (blue +), and a large cost of reducing emissions (red --). Other payoffs in the table can be interpreted similarly.


The problem lies in where the Nash equilibrium is in this game. Consider Country A first. They have a dominant strategy to not reduce emissions. A dominant strategy is a strategy that is always better for a player, no matter what the other players do. Not reducing emissions is a dominant strategy because the payoff is always better than reducing emissions. If Country B reduces emissions, Country A is better off not reducing emissions (because ++- is better than ++--). If Country B does not reduce emissions, Country A is better off not reducing emissions too (because +-- is better than +---). So Country A would always choose not to reduce emissions, because not reducing emissions is a dominant strategy.

Country B faces the same decisions (and same payoffs) as Country A. They also have a dominant strategy to not reduce emissions. If Country A reduces emissions, Country B is better off not reducing emissions (because ++- is better than ++--). If Country A does not reduce emissions, Country B is better off not reducing emissions too (because +-- is better than +---). So Country B would always choose not to reduce emissions, because not reducing emissions is a dominant strategy.

Both countries will choose their dominant strategy (to not reduce emissions), and both will receive a worse payoff (+--) than if they had both chosen to reduce emissions (++--). This game is an example of the prisoners' dilemma. There is a single Nash equilibrium, that occurs where both players are playing their dominant strategy (to not reduce emissions). So, based on this we might be surprised that the Paris Agreement has come into force, since all countries are better off if they choose not to reduce emissions, and instead just free ride on the emission reductions of other countries.

However, that's not the end of this story, since there are two alternative ways to look at the outcome. First, the payoff table above makes the assumption that this is a simultaneous, non-repeated game. In other words, the countries make their decisions at the same time (simultaneous), and they make their decisions only once (non-repeated). Of course, in reality this is a repeated game, since countries are able to continually re-assess whether to reduce emissions, or not.

In a repeated prisoners' dilemma game, we may be able to move away from the unsatisfactory Nash equilibrium, towards the preferable outcome, through cooperation. Both countries might come to an agreement that they will both reduce emissions. However, both countries have an incentive to cheat on this agreement (since, if you knew that the other country was going to reduce emissions, you are better off to not reduce emissions). So the countries need some way of enforcing this agreement. Unfortunately, the Paris Agreement has no binding enforcement mechanism. So there is no cost to a country reneging on their promise to reduce emissions.

In the absence of some penalty to non-cooperation in a repeated prisoners' dilemma, a tit-for-tat strategy is the most effective means of ensuring cooperation in the two-person game. In a tit-for-tat strategy, the player starts out by cooperating. Then they choose the same strategy that the other player chose in the previous play of the game. However, it isn't clear how a tit-for-tat strategy works when you have more than two players (which we do). So, we probably can't rely on this being a repeated game to ensure cooperation between countries.

Second, the prisoners' dilemma looks quite different if the players have social preferences. For example, if players care not only about their own payoff, but also about the payoff of the other player. Consider the revised game below, where each of the countries receives a payoff that is made up of their own payoff in the base case (coloured red or blue as before) plus the other player's payoff in the base case (coloured black). This is the case of each country having highly altruistic preferences.


The game now changes substantially, and reducing emissions becomes a dominant strategy for both players! Consider Country A first. If Country B reduces emissions, Country A is better off reducing emissions (because ++++---- is better than +++----). If Country B does not reduce emissions, Country A is better off reducing emissions too (because +++---- is better than ++----). So Country A would always choose to reduce emissions, because reducing emissions is now their dominant strategy.

Country B faces the same decisions (and same payoffs) as Country A. If Country A reduces emissions, Country B is better off reducing emissions (because ++++---- is better than +++----). If Country A does not reduce emissions, Country B is better off reducing emissions too (because +++---- is better than ++----). So Country B would always choose to reduce emissions, because reducing emissions is now their dominant strategy.

Of course, this is the extreme case, but you get similar results from a variety of intermediate assumptions about how much each country cares about the payoff of the other country (for brevity I'm not going to go through them). We get similar (and stronger) results if we consider our social preferences towards the wellbeing of future generations. The takeaway message is that if we care about people living in other countries (or future generations), we might not be surprised that the Paris Agreement has come into force.

So, game theory can show us both why we might be surprised about the success of the Paris Agreement (because countries have incentives not to reduce emissions), or not surprised about its success (because we care about other people - in other countries, or in future generations).

*****

[*] If we assume more than two countries, we would get pretty much the same key results, but it is much harder to draw a table when there are more than two players, so I'm keeping things simple throughout this post by considering only the two-country case.

[**] Of course, the size of the +'s and -'s will matter, and they probably differ substantially between countries, but for simplicity let's assume they are equal and opposite.

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