In game theory, there are a number of solution concepts, such as the Nash equilibrium, and the subgame perfect equilibrium, that help us to understand strategic behaviour. What role do these concepts have when looking at how to facilitate international cooperation on climate change?

When using a model to help understand a problem, it is important to be aware of the limitations of the model. Many applications of game theory require that decision makers are rational. That is, they have clear preferences, form expectations about unknowns, and make decisions that are consistent with these preferences and expectations. These assumptions may not be consistent with experimental psychology. Elinor Ostrom has considered the the role that human behaviour considerations relate to cooperation problems, and applied this to climate change. She found that a `surprisingly large number of individuals facing collective action problems do cooperate’. She also found that cooperation is more likely if people gain reputations for being trustworthy reciprocators; reliable information is available about costs and benefits of action; individuals have a long-term time horizon; and are not in a highly competitive environment.

So the application of game-theoretic solutions concepts should be taken with a pinch of salt. For example, there is Nash equilibrium that arises from a basic model where countries make a continuous choice about how much to reduce their emissions. As one would expect, this involves small amounts of emission reductions (that reflect the damage that a country will do to itself from its greenhouse gas emissions), but much less than would occur in a fully cooperative situation. But what if one country were to go first, and reduce its emissions by more than the Nash equilibrium choice? If the marginal damage from a tonne of emissions increase with respect to total emissions, then the Nash equilibrium response of other countries would be for them to reduce their emissions by less than they otherwise would (see e.g. Finus, 2001, Chapter 9). But behavioural considerations suggest that other countries would be likely to reciprocate, and reduce emissions by more than they otherwise would.

Eric Maskin, in a paper published in 2009, argues that “the principal theoretical and practical drawbacks of Nash equilibrium as a solution concept are far less troublesome in problems of mechanism design than in most other applications of game theory”. Mechanism design is focused on how to design games whose solution concepts lead to cooperative outcomes. One reason why game theoretic solution concepts are less troublesome in mechanism design, is that the rules of the game are clear to players, and to analysts. Another reason given by Maskin is that one can design games that do not have multiple equilibria or have equilibria that are stronger than the Nash equilibrium.

If humans are more cooperative than assumed in our models, the models could work as a ‘lower benchmark’, and at least as much cooperation as predicted by the models could be observed. When mechanisms have game theoretic solution concepts that could lead to more cooperation on climate change, such mechanisms ought to be given serious consideration.

In the previous blog post, the question of how much a country can make a commitment was discussed. An issue of interest is how much a Party can commit to increase their emission reduction if certain conditions are met. This is of interest because many mechanisms for providing public goods are based on players matching each others commitments in some sense.

One of the most interesting papers on matching commitments is

Boadway, R., Song, Z., Tremblay, J.-F., 2009, The Efficiency of Voluntary Pollution Abatement when Countries can Commit, Queen’s Economics Department Working Paper No. 1205.

They investigate a process where countries can choose a ‘matching rate’ at which they will increase their abatement based on other countries’ abatement. This process is a game whose solution (a subgame perfect equilibrium) is also a socially optimal outcome. This requires whether countries can commit to their matching rates.

This is why it is interesting whether a pledge announced at a climate meeting (and possibly included in a UNFCCC text) is a strong commitment or not. In the Kyoto process, the pledge and review process is repeated, which means that if a player goes back on a commitment, they could face consequences later.

After an international treaty is negotiated, it then has to be ratifed by its participants. This can be modelled as a two stage extensive form game. In Stage 1, the players negotiate the treaty; in Stage 2, each country decides whether to ratify the treaty. For some countries, for example the United States, ratification can be difficult. The United States requires 67 out of 100 Senate votes in order to ratify a treaty.

The most important solution concept for an extensive form game is known as a subgame perfect equilibrium. Each stage of the game is treated as a subgame. The subgame perfect equilibirum is an equilibirum which is also a Nash equilibirum for each subgame.

The main technique for calculating subgame perfect equilibria is known as backwards induction. In this technique the subgame perfect equilibria for the “last” subgames are calculated first. Then taking these actions as given, we calculate the equilibria for preceeding subgames and so on.

By backwards induction, for negotiators in Stage 1 to play the subgame perfect equilibrium, they will take into account that a treaty will have to be sufficiently aligned with the domestic interests of the United States, in order for it to be ratified by the United States.  The US Senate has two representatives from each state, so states with low populations (such as those in the midwest) are disproportionately represented. Coal is widely used in the midwest, and agriculture is an important industry. The US Senate is likely to want to see commitments from major developing countries. All of these issues are therefore likely to be important in international negotiations.

The US Senate will most probably consider the Waxman-Markey bill before the negotiations in Copenhagen commence. The Waxman-Markey bill will most likely require 60 out of 100 votes to avoid a filibuster. Issues that will affect the passage of the Waxman-Markey bill through the Senate will also be important for treaty ratification.

In the ultimatum game, there are two players and a sum of money. The first player proposes how to divide up the sum of money, and the second player chooses whether to accept or reject the proposal. If the second player rejects the proposal, neither player receives anything. This game has a unique subgame perfect equilibrium where the first player receives all of the money, or almost all of the money when payoffs are discrete.

Experiments where people have played the ultimatum game have consistently found that the first player will usually offer significantly more money to the other player than the subgame perfect equilibrium, and the second player will be unlikely to accept the offer if they are offered less than 30% of the total amount.[1]

It has been argued by Fehr and Gächter that the ultimatum game provides evidence that economic agents don’t just base their decisions on pure self interest, and reciprocal considerations play an important role in people’s actions. It has been argued by Barrett that the ultimatum game also provides evidence that an international environmental agreement is more likely to be self-reinforcing if it is perceived by its parties to be fair. [2]

[1] Güth et al. (1982), An Experimental Analysis of Ultimatum Bargaining, Journal of Economic Behavior and Organization, 3, pp. 367-388

[2] Fehr and Gächter (2000), Fairness and Retaliation: The Economics of Reciprocity, The Journal of Economic Perspectives, 14 (3), pp. 159-181;  Barrett (2003), Environment and Statecraft – The Strategy of Environmental Treaty-Making, pp. 299-301.