تئوری بازی ها و دیپلماسی آب و هوا

Starting with the “New Periodic Table” (NPT) of 2 × 2 order games introduced by Robinson and Goforth (2005), we provide an exhaustive treatment of the possible game-theoretic characterizations of climate negotiations between two players (e.g., Great Powers or coalitions of states). Of the 144 distinct 2 × 2 games in which the players have strict ordinally ranked utilities, 25 are potentially relevant to climate problem. The negotiations may be characterized as a No-Conflict Game, Prisoner's Dilemma, Coordination Game, Chicken, Type Game, or Cycle, depending on the payoff matrix. Which game corresponds to the actual state of the world depends both on the severity of risks associated with climate change and the perceptions of the governments engaged in the negotiations. Nash equilibrium or Maxi-min equilibrium (or neither) may be the outcome. Achieving universal abatement of greenhouse gas emissions may require side payments or enforcement mechanisms outside the game framework, but we show how the negotiations themselves may offer opportunities to select between Nash equilibria or alter the payoff rankings and strategic choices of the players. In particular, scientific information pointing to the severity of the risks of climate change suggests characterization of the negotiations as a Coordination Game rather than a Prisoner's Dilemma.

Game-theoretic models provide an elegant formalization of the strategic interactions that underlie the international climate negotiations. Needless to say, there is a long and lively tradition of applying game theory to problems of international relations, including global environmental protection. We will not attempt to give a comprehensive survey of this literature.1 Instead, we will comprehensively examine all of the 2 × 2 order games2 that might be relevant to the climate negotiations, and that show how the payoff structure depends on interpretation of the scientific evidence. We will argue that assessment of the magnitude of the global climate risk is the key determinant of the kind of “game”3 being played. This in turn affects the feasibility of reaching an agreement, and the possible role of equity considerations in facilitating an agreement. Game theory incorporates key elements of both the realist and liberal views of international politics (Stein, 1990). It is consistent with realism because the players are assumed to have a unitary will, that is, each government acts as a single agent rather than as some kind of complex organization whose decisions result from domestic political interactions.4 At the same time, it shows how self-interested behavior can lead to order and welfare-improving outcomes (though it need not necessarily do so), just as the market economy can. The game-theoretic approach does require that governments are able to rank-order outcomes in a manner that is consistent with agent rationality (i.e., a ranking can be assigned to each outcome and the rankings are transitive). Note that the perceived interests of the governments can allow for some weight being given to the well-being of other nations; all that is required is that the outcomes be ranked. In general, the payoffs of a game can be either ordinal (only a rank ordering is possible) or cardinal (different outcomes can be compared on an absolute scale, such as in monetary units). We will focus most of our attention on ordinal rankings. Conclusions based only on ordinal rankings are more general; cardinal evaluations of outcomes require much stronger assumptions about the utilities of the agents. Ordinal ranking of outcomes allows us to bypass the comparisons of utility across countries with very different levels of income that plague conventional cost–benefit analysis.5 Because ordinal rankings are preserved under any positive monotonic transformation, any conclusions based only on the ordinal ranking of outcomes carry over if the game payoffs are expressed in monetary or utility units. Consider first the simplest possible games. There are only two players, who will be identified as “Row” and “Column.” We will consider games with more players below, but we concur with Barrett that the essence of many international relations situations can be captured by the simple 2 × 2 framework.6 We use this nondescript “Row” and “Column” terminology for the players because the games will be interpreted as representing different kinds of international relations — sometimes Great Power rivalry (as between the United States and China, for example) and sometimes other strategic interactions (as between the relatively rich OECD countries and the relatively poor developing nations). Each player chooses one of two strategies, “Abate” or “Pollute.” The payoffs of the games are given in a simple matrix, each cell of which has two elements, the payoff for Row followed by the payoff for Column. The generic payoff matrix is shown in Fig. 1. Thus, a in the upper left cell of the matrix is the payoff for Row if Row chooses the strategy Abate, and Column chooses the strategy Abate. Similarly, u is Column's payoff from this pair of strategy choices. The payoffs to each player are measured in ordinal terms, so {a, b, c, d} and {u, v, w, x} can take on values {4, 3, 2, 1}, with 4 corresponding to the best outcome, 3 and second-best outcome, 2 the next-to-worst outcome, and 1 the worst outcome for each player. There are 144 distinct games of this simplest type. The number of distinct 2 × 2 order games has been known since the 1960s, but recent work by Robinson and Goforth (2005) shows that these games can be organized in a unified topological framework based on a natural measure of the “distance” between payoff structures.7 Robinson and Goforth's book synthesizes what has been known about the classification of 2 × 2 games to date, and their “New Periodic Table” (NPT) of the 2 × 2 games efficiently organizes the information and leads to new insights about the nature of the games. We will use the Robinson and Goforth NPT to provide an exhaustive and theoretically unified treatment of all 2 × 2 games that might be relevant to the climate negotiations. We will be able to cover every possible case of preferences and strategic interactions, and drawing on the NPT topology we will show how different subsets of the climate-relevant games fall into categories with specific characteristics. Determining just which situation is most descriptive of the actual state of play in the negotiations then depends on how the preference rank-orderings of the players are assessed. First we must establish which of the 2 × 2 games are potentially applicable to the climate problem. We narrow down the number of games by requiring that the payoff structures satisfy two “climate relevant” restrictions: (1) The outcome (Abate, Abate) is preferred by both players to the outcome (Pollute, Pollute), and (2) Each player's pollution imposes a negative externality on the other. The first of these two restrictions amounts to assuming that there is no economic or geopolitical advantage to be gained by either party if both pollute instead of both abating, and that the climate problem is real. It does not require that climate is either party's top priority. The second restriction amounts to the presumption that neither party's pollution benefits the other party. In the generic payoff matrix of Fig. 1, the first restriction says that a > d and u > x. The second restriction requires a > b, c > d, u > w, and v > x. These two restrictions reduce the number of climate-relevant 2 × 2 games to 25.8 It should be noted that our two climate-relevance conditions apply to countries' greenhouse gas emissions. A small country rich in oil or gas reserves may derive much of its national income from the export of its fossil fuel resources. From a short-term perspective, the government of such a country might prefer that the rest of the world adopt the Pollute strategy, violating our “negative externality” condition (2). This situation might apply to a few countries, but not to the major powers.9 In addition, we do not consider the thinly-supported claims that some countries or regions would benefit from global warming (see the critique of these claims in Ackerman et al. (2009a)). The world is already committed to some amount of warming because of cumulative emissions to date, and the pending policy question is how much more warming can be allowed if we are to avoid “dangerous anthropogenic interference with the climate,” even if there may be some relatively minor increases in agricultural productivity in a few regions stemming from the warming, changes in precipitation patterns, and CO2 fertilization. A recent paper by Perlo-Freeman (2006) defines a class of “Co-operate–Defect” (C–D) games “which are characterized by each player having a dominant preference for a particular strategy by the other player” (p. 1). This class of games corresponds to the games satisfying our second climate-relevance criterion. There are exactly 36 C–D games. Eleven of these fail to satisfy our first climate-relevance criterion. Perlo-Freeman's paper shows how the C–D games are applicable to arms race games and collective action problems, and that they form an interesting subset of the NPT with topological properties related to those of the NPT. The players of the C–D games can be described by one of six distinct preference-ordering types, and Perlo-Freeman demonstrates how slight changes in the preference ordering of one of the players can result in sharp changes in the Nash equilibrium of the game.10

We have examined the entire set of 25 climate-relevant 2 × 2 order games — eight of the “no conflict” variety, the Prisoner's Dilemma and its two related Alibi games, the Coordination Game, Chicken and its four related asymmetric relatives, the six “unhappy” games in which it is difficult to reach an agreement to Abate, and the two Cycle games that have no pure-strategy Nash equilibrium. Instead of picking one or more examples on an ad hoc basis, we have covered every possibility for describing the strategic interactions that might characterize the international climate negotiations. Our exhaustive treatment of the climate-relevant 2 × 2 order games suggests several general conclusions. The first, and perhaps the most important, is that there is no reason to assume that the Prisoner's Dilemma is the best description of the climate negotiations. The most favorable situation would be if the climate game were of the “no conflict” variety. If it were, there should be no barrier to an agreement, although negotiations might still be necessary to overcome Maxi-min tendencies and achieve the Pareto-optimal outcome. The absence of progress in the negotiations suggests, however, that the “no conflict” payoff structure is not the real-world situation. However, a payoff structure that is entirely consistent with the current state of scientific knowledge is that of the Coordination Game. If this is the true state of the world, reaching a global agreement boils down to a problem of equilibrium choice. While economics can offer some help in this (by ruling out wasteful, inconsistent, or self-defeating approaches), solution to the equilibrium choice problem falls at least partly outside the realm of economics. In particular, fairness could have a bearing on selecting the equilibrium. Second, the Nash equilibrium is not the only possible solution concept. Risk aversion and a lack of trust can lead countries to adopt a Maxi-min strategy, although under some payoff configurations this runs the risk of being exploited by the other player. Again, there is no hard-and-fast guideline here. Economists have tended to favor the Nash equilibrium because of its correspondence to Walrasian general equilibrium, but the Nash solution is prone to exhibit multiplicities. Maxi-min strategies often reach a Nash equilibrium, but sometimes they do not. In any case, Nash equilibrium should not simply be equated with “rationality” on the part of the players. Third, for the climate problem it may be that different countries are operating in different moral universes, or that their material circumstances are so divergent that cooperation on climate is very difficult. The existence of climate-relevant Type games raises this possibility. If this is indeed the situation, or if the climate game really is a Prisoner's Dilemma, then a range of incentives and inducements to cooperate may be needed to reach an acceptable international agreement. Even in the games in which the players are as oppositely motivated as possible (the Type games) it is possible that a well-thought-out negotiating strategy coupled with suitable side payments and negative incentives could bring about an agreement to Abate. Fourth, slight changes in the preferences of the players can dramatically change the prospects for a climate agreement. This is equivalent to Perlo-Freeman's conclusion that “a small change in the preference ordering by one player can lead to a radical change in the resulting Nash equilibrium” (2006, p. 2). In the context of climate, we have shown how slight modification of the preferences giving rise to the Prisoner's Dilemma can transform the situation into a Coordination Game. As long as the preference orderings are subject to modification through the diffusion of scientific information and diplomatic negotiations, there is room for optimism that a solution to the climate problem can be reached. Given the imperatives of economic development and poverty reduction, it seems likely that transfers from rich countries to poor countries, in the form of low-carbon or zero-carbon energy supply technologies, will be essential to induce the poorest developing countries to participate. However, there is no reason that the investments in low-carbon energy technologies made in these countries have to disrupt or destabilize Great Power relationships. Such investments could be part of a relatively healthy form of Great Power competition, because diffusion of the technologies could contribute to the spread of a powerful nation's influence without resort to coercion or war. The game-theoretic examples do demonstrate, however, that a unilateral approach to emissions reduction may not be the best strategy for a country like the United States. In both the PD and the Coordination Game, the least-preferred outcome for either party is if it abates while the other pollutes. Of course, it is possible that early action to abate emissions will confer an advantage in the development of clean energy technologies. In the Coordination Game, unilateral adoption of the Abate strategy by one of the major countries or blocs may increase the incentive for others to abate or be a signal to others that moving to the (Abate, Abate) Nash equilibrium would beneficial for all. But if one of the Great Powers (or set of GPs) holds as its highest goal the short-run maximization of its relative power, domestic abatement measures by the others will be in vain. Business as usual by either the US or China (or, for that matter, the EU, India, or the rest of the world collectively) will threaten climate stability no matter what actions are taken by any one country or subset of countries. In the Chicken family of games, prior commitment to Abate by the nation whose dominant strategy is Abate guarantees that the other will Pollute. Only if the other nation follows a Maxi-min strategy can the (Abate, Abate) outcome be obtained, and this will not happen if the first party announces in advance of the play of the game that it will Abate. The most serious difficulties in reaching a global climate protection agreement arise if one of the major countries (the ones whose emissions alone are enough to produce dangerous anthropogenic interference with the climate) ranks highest the outcome in which it pollutes while the rest of the world abates. All of the “unhappy” games as well as the PD and the Chicken family have this feature. In the Chicken games, all parties playing Maxi-min can lead to mutual Abatement, although one party's playing Maxi-min can be exploited by the other. In the “unhappy” games both the Nash equilibrium and the Maxi-min equilibrium risk climate catastrophe. The NPT for order games may be helpful in pointing the way to resolving the diplomatic impasse. Bruns, 2010 and Bruns, 2011 notes how “swaps,” defined as switching the ranks of outcomes for a player, can transform one climate-relevant game into another. Some games are transformed rather easily; all of our “unhappy” games can be converted into games with a (4, 4) Nash equilibrium by swapping the payoffs 3 and 4 for either one or both players. For example, the Type game 261 becomes the No-conflict game 362 if Column's rankings 3 and 4 are swapped, and Chicken becomes No-conflict game 311 if payoffs 3 and 4 are swapped for both Row and Column. As noted earlier, one possible way a government's rankings can change is through greater understanding of the science that underlies assessment of the risks of climate change. Side payments or negative incentives such as trade sanctions are also feasible instruments, but because we are discussing only ordinal games we cannot say how large the side payments (or side-punishments) would have to be to make these transformations. Of course, there is no way to force sovereign governments to accept the science. As a character in one of Schiller's (1801) plays said long ago, “Mit der Dummheit kämpfen Götter selbst vergebens.” Practical policy will come down to (1) how serious the long-term climate risk is considered to be, (2) whether the governments of the Great Powers (i.e., the nations that decisively shape the negotiations) genuinely care about future generations, (3) whether climate abatement policies can be devised that do not give an advantage to one or a subset of the Great Powers, and (4) whether suitable instruments can induce the necessary investments in clean energy and efficient end-use technologies while simultaneously promoting economic growth in the poorest developing countries. The greatest reason for optimism is that science is universal; understanding the risks of climate change can eventually bring all major governments to realize that Abatement is in their long-run interest. Diplomacy also can work to create incentives that will push the governments towards cooperation. Like economics more generally, game theory can offer guidance for the successful navigation of these diplomatic shoals, but is not by itself able to demonstrate the solution.