The Global Climate Game

Roweno J.R.K. Heijmans

Tilburg University

January 31, 2020

Abstract

I study the private provision of public goods with strategic complementarities and preference-uncertainty. The induced game has a unique equilibrium. While the equilibrium may be first-best, there are well-identified cases for which it coincides with the worst possible outcome. I also introduce sequential global games and show that sequentiality expands the range of preferences for which the public good is provided and coordination occurs on the first-best equilibrium. A possible application of the theory is climate change. Predictions of the model square well with observed real-world climate policies. Compared to existing explanations, my theory requires weaker assumptions on incentives and has more predictive power.

1 Introduction

When public goods get under-provided, economists take free-riders to task. The basic idea is that incentives to shirk are strong and cannot be easily overcome. An important – and often implicit – assumption behind this argument is that individual contributions are substitutable. Yet it is no forgone conclusion that all public goods can be so characterized. Numerous economic interactions are subject to complementarities, invoking opposite incentives (Milgrom and Roberts, 1990). In this paper, I study the provision of a public good with strategic complementarities and preference-uncertainty. I prove existence of a unique equilibrium, in which the public good may not be provided. I identify precise conditions for which the ﬁrst-best outcome is supported. In a sequential

1 extension of the game, opportunities for efficient provision are greatly expanded. Many an application fits my general theory, but perhaps the most pressing is global warming. This, then, provides the terminology I henceforth adopt. Amid growing concerns that climate change is headed for the wrong direction, little measures to prevent it are actually undertaken. The dominant explanation dates back to Barrett (1994), who in his rightly famous analysis points at free-riding. All countries, so the logic goes, stand to benefit were climate change avoided. Yet each benefits even more letting others do the job. Every nation facing the same incentive, climate change becomes a sure thing. Barrett’s (1994) theory is intuitive and fits neatly into the classic understanding of man and state alike as homo-economic free-riders. Yet its fundamental assumptions are hard to square with the growing body of evidence that free-rider incentives aren’t that significant a driver of behavior (Andreoni, 1995; Fischbacher et al., 2001). Besides, to no unimportant extent are there complementarities – incentives to coordinate – in climate policies, begging the questions whether free-rider incentives even exist to begin with (Potters and Suetens, 2009). The time is ripe for a complementary explanation. My innovation is to take proper stock of two uncertainties that complicate climate politics. First, no country on its own can avert global warming. Unilateral policies are but a drop in the ocean. Game theorists would say that climate policy is a coordination game. Because such games are supermodular, characterized by strategic complementarities, they generally have multiple equilibria, inducing strategic uncertainty.1 It is a priori unclear which equilibrium will be selected (Van Huyck et al., 1990). Above and beyond these coordination problems, there is scientific uncertainty as to the exact consequences of global warming (c.f. Hsiang et al., 2017). Since the benefit of cutting emissions derives from damages avoided, this amounts to another potential source of equilibrium indeterminacy. It may sound reasonable that strategic and scientific uncertainty lumped together should render coordination on any outcome close to impossible, the product of two complications sounding like an even larger complication. I prove this intuition wrong. While each in isolation fuzzes optimal strategies indeed, precisely their intersection is tinder for equilibrium selection. However, the equilibrium selected may be inefficient.

1Climate policy could alternatively be interpreted as a tipping (Barrett and Dannenberg, 2017) or, yet diﬀerent, weakest link (Harrison and Hirshleifer, 1989) game, both of which incorporate what basically amounts to a discretization of the supermodularity property. All interpretations are consistent with the idea of a carbon-budget, see Allen et al. (2009).

2 How does that work? Loosely speaking, the argument is somewhat as follows. Due to the coordinative nature of best-responses, a country wants to adopt stringent policies only if sufficiently many others do. Given that the number of investing countries is not known in advance, this is what I called strategic uncertainty. Now bring in the scientific uncertainty. Being only imperfectly informed about the true consequences of climate change, countries need to second guess each other’s beliefs. This introduces the possibility that a country is willing to take measures only if, say, N countries do it also, but at the same time thinks it unlikely – as a guesstimate – that N countries will actually join in the effort. Its best response is then to do nothing. Lack of common knowledge can in such a way enforce coordination on undesirable equilibria. Mindful of this result, my theory explains the empirical observation that countries know climate change is bad yet fail to act accordingly. The Paris agreement provides a point in case. While embraced nearly universally, we are nowhere near meeting the promises laid out therein. In contrast to the existing literature, my account of this apparent contradiction does not rely on free-rider incentives, and incorporates the possibility of strategic complementarities in climate policy. The combination of coordination games and payoff uncertainty goes by the name of global games among the initiated. These were introduced to the literature by Carlsson and Van Damme (1993), gaining further momentum when Morris and Shin (1998) applied them to currency crises. The global games approach allows me to solve an important puzzle in environmental economics, if not political reality altogether: why do countries not take measures that preempt runaway global warming? Why so even if avoiding climate change is cheaper than having it? What explains the inertia when everyone knows measures ought be taken? I answer these questions without relying on free-rider incentives, while allowing for complementarities in national climate policies. As an extension of standard global games, I define a sequential global game as one where a subset of players chooses its actions before the remaining players do. This splits the original game into two subgames, where the outcome of the first is observed by players in the second. A natural interpretation in terms of environmental agreements would be technologically advanced countries being able to install renewable technologies earlier on than developing countries. These games produce sharp theoretical predictions: sequentiality facilitates coordination on the first-best. A recent branch of the literature that positively reassesses the possibilities for international environmental agreements (or public good games more generally) builds

3 on work by Harstad (2012), Harstad (2016), and Battaglini and Harstad (2016). These papers are united by a focus on incomplete contracts in dynamic games of technological investment and emission reductions. A political economy justification for the focus on incomplete contracts is given by Battaglini and Harstad (2020). Whether driven by the possibility of renegotiation (Harstad, 2012), catalysed through complementary trade policies (Harstad, 2016) – see also Nordhaus (2015) –, or due to the fact that tough, long-term agreements can mitigate both hold-up and free-rider problems (Battaglini and Harstad, 2016), the common prediction in these papers is cautiously positive: countries can sustain cooperation on desirable outcomes, sometimes even the first-best. My paper shares this optimistic prediction in well-identified cases and is in that sense related. I deviate from their approach in several ways. First, I focus on one-dimensional policies, and consider static or sequential games. Second, I explicitly consider the role of uncertainty in environmental agreements. Finally, while the games studied in this branch of the literature tend to have many equilibria – the symmetric Markov perfect equilibrium is unique, but there can be many others –, the equilibrium in my analysis is globally unique. My focus on the dual face of uncertainty also nests this paper in the literature on imperfect knowledge and environmental regulation. It has long been understood that while regulating externalities under complete information is almost a triviality, informational distortions may ravel matters severely (Weitzman, 1974). Pindyck (2007) identifies three core sources of uncertainty: damage uncertainty, policy cost uncertainty, and discount rate uncertainty. The first of these is the main source of risk considered in Weitzman (2009), Weitzman (2014), Golosov et al. (2014), Hsiang et al. (2017), and Cai and Lontzek (2019). The latter was studied extensively by Weitzman (2001) and Gollier (2002). Given the general formulation of my model, each of these three core uncertainties can be interpreted as part of scientific uncertainty re climate damages. To my knowledge, I am the first to study the intersection between these well-understood sources of uncertainty and coordination problems due to strategic complementarities.

2 A Public Good Game With Strategic Comple- mentarities

Generically, a public good is characterized by only a handful of parameters. There are N ≥ 2 agents providing and enjoying the good. There are to each agent beneﬁts from

4 the good provided. And there is the individual cost of contribution. The difference between benefits and costs is the net benefit of contributing. I define a public good to exhibit strategic complementarities if the net benefit of contributing to the good is increasing in the number of agents supplying it, so best-response correspondences are upward-sloping. (If the net benefit is decreasing in the number of contributing agents, I say there are strategic substitutes, at the heart of free-rider incentives). It is important to distinguish between games of strategic complementarities versus substitutes, as they tend to ignite differential behaviors (Potters and Suetens, 2008, 2009). Denote the set of players P = {1, 2, ..., N}. Each agent i ∈ P chooses action xi ∈ {0, 1}, and all choose simultaneously. Action xi = 1 amounts to (privately) P supplying the good, while xi = 0 means no contribution. Define X = i(xi/N) as average contributions. With a slight abuse of notation, when a continuum of agents is R 1 considered, I normalize P = [0, 1] and let X = 0 xidi. In the interpretation, agents are countries, and the public good is climate change avoided. Contributing to this good can be thought of as investment in renewables, not contributing as relying on fossil fuels. Investment costs. Supplying a public good is costly. Let the gross cost of contribution – investment in the interpretation – be c, where 0 ≤ c ≤ 1. It is costly to install green technologies because this involves changes to the existing electricity network, job losses in the oil and coal industries, forsaken rents from natural resource sales, R&D in green energy generation, and so forth. At the same time, renewable technologies are subject to scale economies. On a unit-basis, it is cheaper to build one million windmills than it is to build only a thousand. Similarly, part of the knowledge created by research spills across boarders, so that the more countries undertake R&D, the less each of them needs to spend individually. These scale economies are incorporated into my model by having the net costs of investment equal to c − X. The costs of generating energy from fossil fuels are normalized to zero. Benefits or damages. Agents derive benefits from public goods, e.g. climatic damages avoided. Model-wise, let the damage due to global warming be given by δ(1 − X), with δ a damage-sensitivity parameter. The term (1 − X) incarnates the idea that higher global investment in renewables means lower emissions and milder climate change; or generally that agents like to see more of the public good provided. Some agents may enjoy the public good more than others; there is no reason to assume absolute homogeneity in preferences. For example, it stands to reason that certain countries are harmed more than others. It is possible to make the parameter

5 describing damages country-specific. I omit this more realistic specification in the general exposition of the model so as to deliver my main insight with minimal complication. Individual-specific preferences will be the main focus of Section 3. Payoffs. Finally, the payoff to agent i – country i’s welfare – is simply the benefits due to the public good minus the cost of provision if the agent is providing it. In climate change parlance, it is the sum of economic damages due to environmental degradation and the costs it incurs from investing. Denote this payoff πi(xi; X, δ), where note that R one may define X−i = j6=i xjdj as the total investment by all countries other than i. When no unclarities arise, I will write πi for simplicity of notation. The model can now be summarized as follows:

i ∈ P (1)

xi ∈ {0, 1} (2) Z X = xidi (3)

i∈P

πi = −(c − X)xi − δ(1 − X). (4)

Uncertainty and signals. Agents may be unsure how much precisely they benefit from a public good, or how much it will cost them to provide it. This could be the case, for example, when benefits realize only in the future, so many things might happen meanwhile. This brings me back to the case of climate change, where the most significant damages are expected to occur decades from now. Sea level rise is hard to predict, and meteorological systems are not well understood. Moreover, possibilities for adaptation to extreme weather are still being explored. Thus far, welfare in the model was assumed to decrease with environmental degradation via the sensitivity parameter δ. The idea that there is no general consensus on precisely how high these costs are effectively means that the true damage parameter δ is not observed. Instead, each country i receives a private noisy signal di of δ, where:

di = δ + i, (5) for i is a random variable drawn uniformly from [−ε, ε], with ε > 0 a measure of damage-uncertainty. It is clear that for any two countries i and j, signals di and dj are correlated since both have the same expected value. However, conditional on their

6 mean, signals are drawn i.i.d. It is understood that the process by which signals (and noise) are generated is common knowledge. Games of this type are generally known as global games, introduced by Carlsson and Van Damme (1993). Throughtout, I assume δ ∈ δ, δ, where δ < c − 1 and δ > 0.

Now suppose country i receives signal di. What does i infer? First, that E[δ|di] = di. In words, a country’s best guess for the environmental damage parameter is the signal it receives. Second, the country learns that E[dj|di] = di. Thus, any country i’s best guess about the signal dj received by country j is simply its own signal, di. There is no need dwelling further in such trivialities, with one exception. Receiving signal di, country i learns that with probability 1/2, the true damage parameter δ is smaller than di (and, symmetrically, that with same probability it is larger). Moreover, given signal di, country i knows that with probability 1/2 neighbor j has received a higher (or lower) signal dj. Formally, Pr[di > dj|di] = Pr[di < dj|di] = 1/2. These are all the building blocks necessary for an analysis of the Global Climate Game, to which I now proceed.

2.1 Two Countries

Intuition may be fostered – and necessary game theoretic techniques may be most simply equipped – by discussing a two-country world ﬁrst, i.e. P = {1, 2}. In this case, the game outlined above can be represented by Figure 1.

Country 2

0 1

1 1 1 0 (−δ, −δ) − 2 δ, 2 − c − 2 δ Country 1

1 1 1 1 ( 2 − c − 2 δ, − 2 δ) (1 − c, 1 − c)

Figure 1: Payoﬀ (π1, π2) matrix for the game γ(δ) with two countries.

For 2 − 2c < δ < 1 − 2c, this game is a coordination game with strategic complementarities. Given the action played by country 1, country 2 is best oﬀ mimicking this action. The game therefore has two pure strategy Nash equilibria, one where both do not invest,

7 ∗ ∗ ∗ xi (δ) = 0 xi (δ) = xj xi (δ) = 1

δ 2c − 2 2c − 1 δ

∗ Figure 2: Partitioning of the damage parameter space δ ∈ [δ, δ]. xi (δ) denotes the best response of country i, given δ. De facto, Barret’s (1994) world consists of games for which δ < 2c − 2 only.

(x1, x2) = (0, 0), another where both do, (x1, x2) = (1, 1). There also exists a mixed strategy equilibrium. Without further ad hoc assumptions on how countries choose their actions, there is no a priori reason to favor any of these three Nash equilibria over the others. Some people find it intuitive that the Pareto-dominant equilibrium is chosen, since this makes both countries strictly better off. Others might argue that countries will coordinate on the, in a sense, less risky no-investment equilibrium, incurring a sure loss as compared to a lottery between a bigger loss and a possible gain. In reality, neither of these equilibria is theoretically defensible as more, or less, likely than any other. Anything could happen. In the complete information version of this game (i.e. where δ is observed), the support of all possible damage parameters δ is partitioned in an intuitive way. For δ < 2c − 2, damages are relatively low. So low, in fact, that it never pays to incur the comparatively high cost of investment to avoid them. In this case, it is clear what will happen: not investing is a dominant strategy for either country, and so neither will invest. Similarly, when δ > 2c − 1, damages are relatively high when compared to investment costs, and countries will always want to invest, given how big of a deal global warming really is. Hence, when δ > 2c − 1, investing is a dominant strategy, and both countries will invest. Finally, when δ ∈ [2c − 2, 2c − 1], the game has no dominant strategies. Instead, countries are playing a coordination game, where each is best off mimicking the action played by the other. In such a game, the outcome is unpredictable. Figure 2 summarizes. However, recall that countries are not playing a complete information game. Instead, actions are chosen based on a perturbed version of the game presented in Figure 1, where climate damages δ are unobserved and countries observe private signals di, see equation (5) and the surrounding discussion. It is according to this perturbed game that countries maximize expected welfare. I obtain the following surprising result:

Proposition 1. Consider the two-country model. Let ε be suﬃciently small. Then

8 ∗ each country i invests if and only if its signal di is high enough, di > δ , where:

4c − 3 δ∗ = . (6) 2

The proof is long and given in appendix A. While the remainder of this paper can be understood without mastering the rather technical proof of Proposition 1, daring readers may foster further understanding of my result by plotting through its argument. Proposition 1 establishes two counter-intuitive results. First, as was discussed, under perfect information of damages δ, equilibrium outcomes are unpredictable. This is so because, at least for a range of possible δs, countries play a coordination game. Global investment in renewables is a Nash equilibrium of this game, but so is global reliance on fossil fuels, as well as a randomization between the two. Paradoxically, Proposition 1 says that turning from a game of complete information to a game of incomplete information (even with vanishing uncertainty), equilibrium ambiguity is removed altogether. Instead, equilibrium strategies are unique for any signal received. This finding echoes the conclusion in Rubinstein (1989) that games of “almost common knowledge” can be very different from games of common knowledge proper. 2 The second counter-intuitive implication of Proposition 1 is that coordination may be on the Pareto-dominated equilibrium. This means countries are forced to play an equilibrium of which they know it yields the lowest payoff, and of which they know they both know it. The precise conditions under global which coordination on the Pareto-dominated equilibrium occurs are specified in Corollary 1.

Corollary 1. Let ε be suﬃciently small and c ≥ 1/2. Then countries coordinate on the Pareto-dominated no-investment equilibrium for any δ ∈ [c − 1, δ∗).

Figure 3 illustrates graphically how the support of signals di, for each country i, is partitioned in the incomplete information case. Especially when ε → 0, so that di ≈ δ and we have a game of “almost common knowledge”, the contrast with Figure 2 is striking.

2.2 Many Countries

In the previous subsection, the world was assumed to consist of two countries, a blatant falsity. Luckily, the techniques I presented in this simplest possible case are can be

2To be precise, an event E is said to be common knowledge if all players know E, all players know that all know E, all know that all know that all know E, and so forth (Aumann, 1976).

9 ∗ ∗ xi (di) = 0 xi (di) = 1

δ 4c−3 2 δ

∗ Figure 3: Partitioning of the signal space di ∈ [δ, δ] (for ε > 0). xi (di) denotes the best response of country i, given di. generalized to more than two agents – countries in the interpretation. Consider then a world composed of N > 2 countries. Each simultaneously and independently chooses whether or not contribute to the global climate good by investing in renewables but is assumed to be imperfectly informed about the true damages to result from climate change. In this setup, the following result – a generalization of Proposition 1 – obtains:

Proposition 2 (Finite N countries). Consider the model with a ﬁnite number N > 2 of countries. Then there exists a unique δ∗ such that each country i invest if and only ∗ if di > δ .

Proposition 2 is encouraging in light of my intended goal to reconcile the apparent global failure to take actions that preempt global warming with the theoretical predictions of the model I propose, given the world is known to be composed of more than two nations.

2.3 Continuum of Countries

Having examined the case of two respectively many countries, I conclude my analysis of the basic model by discussing an infinity, or continuum, of agents. There are only finitely many countries in the world, so arguably a result on infinitely many of them does not make much sense from that viewpoint. This observation leads back to my intended goal of analyzing public goods with strategic complementarities generally. Though there are only some 200 countries in the world, a general model of private public good provision should treat that – or any other – number as a special case. It is therefore encouraging that Propositions 1 and 2 extend to continuous player-sets. To stay with the example of global warming, while one could indeed consider the main providers of the public good to be nation states, it is equally well a good provided by all citizens, each of whom on its own decides to drive a car or not, book a flight or not, eat meat or not, et cetera. Yet the global population is gargantuan. In many senses,

10 a player-set composed of billions of agents is most accurately abstracted as a continuum, of which one may then normalize the mass in a fashion one deems appropriate.

Proposition 3 (Continuum of countries.). Consider the model with a continuum of countries, indexed i ∈ [0, 1]. Then there exists a unique δ∗ such that each country i ∗ invest if and only if di > δ .

Two-action global games with a continuum of players are at the heart of Morris and Shin (1998) on currency crises. My result echoes theirs in that countries (speculators) play a threshold-strategy: for any private signal received below the threshold δ∗, do not invest; above, do.

3 Individual-Speciﬁc Payoﬀ Parameters

In the preceding analysis, it was assumed that the climate-damage parameter δ was equal for all countries. This assumption is evidently not supported by physical reality. In the present section, I discuss a situation where damages are allowed to vary across countries. One could equivalently interpret this model as one where preferences diﬀer between agents (Layton and Brown, 2000). The welfare of country i ∈ P becomes:

πi = −(c − X)xi − δi(1 − X), (7)

where δi is now a country-speciﬁc damage parameter. The following result obtains:

∗ Lemma 1. Let xi (di) denote country i’s equilibrium strategy conditional on receiving ∗ signal di. Then there exists, for each country i, a unique minimum signal δi such that ∗ the country invests if and only if di > δi . Formally:

∗ ∗ ∗ ∗ ∗ ∗ ∃! δ = (δ1, δ2, ..., δN ): xi (di) = 1 ⇐⇒ di > δi ∀i ∈ P (8)

Lemma 1 takes away the concern that my equilibrium-uniqueness results are driven by the untenable assumption of perfectly homogeneous agents, i.e. countries. It is not implies that the equilibrium will always coincide with that in a homogeneous-country world. To see this, consider the following model-speciﬁcation. Let the damage-parameter for each country i be given by:

δi = δw + µi, (9)

11 where δw is the worldwide ‘average’ damage-sensitivity and µi a term capturing the national deviation from the global average. For simplicity, I let µi be i.i.d draws from the uniform distribution on [−µ, µ], so µ is a measure of damage- or preference-heterogeneity among nations. Note that one can interpret the model with homogeneous damages as a trivial case of the heterogeneous country model, with µ = 0. Another way of saying the same thing is that the model of country-speciﬁc damages induces a mean-preserving spread of δis around the homogeneous-country model with δ = δw. It can be shown that heterogeneity generally narrows the range of parameters for which there will be global coordination on public good provision, such as investment in renewables.

Proposition 4. Let EX(δi|δw) denote expected global contributions to the public good – investment – in the heterogeneous-agent model given δw, and let X(δw) denote global contributions in the homogeneous-agent model for δ = δw. Then:

EX(δi) ≤ X(δw). (10)

4 Committed Climate Coalitions: A Sequential Global Game

Nordhaus (2015) famously envisaged that ‘climate clubs’ could be convened to overcome free-rider plagued inefficient abatement. Similarly, Barrett (2006) and also Acemoglu et al. (2012) suggest that targeting R&D, rather than emission abatement directly, might improve global cooperation. In a related spirit, I now discuss the potential of a Committed Climate Coalition to foster global coordination on mitigation. One interpretation of such a coalition is the group of technologically advanced countries, to whom new technologies and investment opportunities arise earlier in time than to those countries lagging behind the frontier. This structure induces a sequential global game, to be formally defined below. Define the subset of technologically advanced early-movers – the countries that can form a Committed Climate Coalition – as PC ⊆ P. The remaining countries are followers and belong to PF ⊆ P. Each country i is assumed to be either technologically advanced or lagging, so PC ∪PF = P and PC ∩PF = ∅. Define the size of the Committed Climate Coalition as C := |PC |/|P|. The game proceeds as follows.

1. Providence draws a true δ ∈ [δ, δ].

12 2. Each country i ∈ P receives private signal di of δ.

C 3. All countries i ∈ P simultaneously choose (commit to) action xi ∈ {0, 1}.

4. All countries i ∈ P observe (xi)i∈PC .

F 5. All countries i ∈ P simultaneously choose xi ∈ {0, 1}.

6. Payoﬀs are realized according to δ and the actions chosen by all players.

Both for PC = P and PC = ∅, this sequential game reduces to the standard, non-sequential game studied in Section 2. In all but these extreme cases, the possibility of technologically advanced countries to form a Committed Climate Coalition increases global investment. Global investment is highest when half of the countries is technologically advanced, i.e. C = 1/2. Proposition 5 formalizes these observations.

Proposition 5. Let EX(C) denote expected global investment in renewables given the subset of technologically advanced countries is of size C ∈ [0, 1]. Then:

∂ X(C) 1 E 0 ⇐⇒ C . (11) ∂C Q R 2

The intuitive reason that sequentiality fosters coordination on investment is most easily conveyed for a two-country world. While theoretically not the most interesting of cases, the underlying logic is robust to increased numbers of players. Let country 1 be the technologically advanced first mover and let country 2 be the follower. If country 1 invests in the first stage, it takes away the strategic uncertainty for country 2. That is, instead of second-guessing what country 1 believes and therefore how likely it is to invest, country 2 can simply observe that country 1 has invested. Seeing this, it becomes much more attractive for country 2 to invest for any private signal it receives, given the coordinative nature of best-responses. Backward inducing this effect on country 2, investment becomes more attractive for the technologically advanced country 1 for any private signal it receives.

5 Discussion and Conclusions

This paper studies a game in which any number of agents privately contribute to a public good and individual actions exhibit strategic complementarities. If there is uncertainty

13 about payoffs, or preferences, the game has a globally unique and well-characterized equilibrium. Under reasonable conditions, the equilibrium selected is Pareto-dominated by other potential equilibria. I also define sequential global games, where a subset of players decides whether to supply the public good before all remaining players take their decisions. This type of sequentialty greatly expands the range of payoff-parameters for which the unique equilibrium is first-best. While the model is general, its underlying assumptions and structure fit an interpretation in terms of climate policy or international environmental agreements. This setup allows for a natural meaning of payoff uncertainty: scientific uncertainty about the damages from climate change. In a theme and variations of this framework, I explore the possibilities for international climate change mitigation. While in some instances runaway global warming can be avoided, for non-negligible cases will countries coordinate on the Pareto-dominated equilibrium where climate change is allowed to happen and all are worse off. This result rationalizes lackluster climate policies as observed in everyday reality. Compared to existing explanations, my analysis has the major advantage of not relying on free-rider incentives. This is desirable because a growing body of evidence casts doubts on the importance of free-rider incentives as drivers of behavior. Moreover, in games of strategic complementarities it is not even obvious that free-rider incentives exist at all. In offering a theory that squares both of these observations with the empirics of climate policies, the paper contributes to understanding international environmental agreements as they play out in the real world. To summarize my results, Propositions 1–3 establish the existence of a unique equilibrium strategy for all countries. According to this strategy, each country invests in renewables for any estimate of climate damages above a threshold, and does not invest for all estimates below it. Corollary 1 highlights a paradoxical and perhaps undesirable property of the threshold-strategy: for a range of damages, countries will coordinate on the Pareto-dominated no-investment equilibrium even though they know coordination on investment would yield higher welfare. This result rationalizes the empirical observations that countries agree runaway climate change should be avoided but at the same time fail to curb emission on any scale sufficient to meet that goal. Lemma 1 generalizes Propositions 1-3 to the case of heterogeneous agents, interpreted as country-specific damages from climate change. Proposition 5 underlines the ability of technologically advanced countries to stimulate

14 global investment in renewables by forming a Committed Climate Coalition that invests ahead of other countries lagging behind the technological frontier. Formalized as a Stackelberg sequential game, the possibility of forming a Committed Climate Coalition boosts the incentive to invest for all countries compared to the static, non-sequential game of simultaneous investment. Proposition 5 is especially important in light of real-world policy. While conventional wisdom has it that countries best wait out others to invest, my result puts the logic upside down: by investing in renewables early on, a country creates strong incentives for other countries to invest, making it far more likely they will follow suit. It would be interesting to test me theoretical predictions on sequential global games in the laboratory. Applications of my model extend beyond the Global Climate Game. Speculative- attacks (Morris and Shin, 1998), vaccination programs to exterminate infectious diseases, private dyke maintenance (as in the Dutch polder model), ﬁrms in oligopolistic markets, and even potluck lunches can all be thought of situations where several agents jointly provide a public good, individual actions are complements, and some form of beneﬁt or preference uncertainty is likely to exist.

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A Two Country Global Games in Technical Detail

A.1 Basic Deﬁnitions

When denoting a single country, we use label i. When denoting the other country, labels j and −i will be used interchangeably. We write πi(xi, xj|δ) for the payoff to country i when i plays xi and j plays xj, given δ. The notation πi(xi, x−i|δ) has the exact same meaning. Label i does not identify the first country. Definitions below apply to both countries; there is no need to duplicate equations in reversed notation. The main solution concept in my analysis will be iterated dominance. It is based on the more basic concept of a dominant strategy.

Deﬁnition 1 (Dominant strategy). Given δ, xi ∈ {0, 1} is a dominant strategy (for each country i) if and only if

πi(xi, xj|δ) > πi(1 − xi, xj|δ), (12)

for each xj ∈ {0, 1}.

An action is a dominant strategy if it is a best-response to any strategy played by the opponent. Clearly actions x and 1 − x cannot be simultaneously dominant; if action

18 x is dominant, then action 1 − x is said to be dominated. In traditional analyses such as Barrett’s (1994), countries do not invest in renewables because investment is assumed to be a dominated strategy. Not all games have a dominant strategy. Such games are not solvable with reference to strict dominance. A point in case are coordination games.

Deﬁnition 2 (Coordination game). Given δ, the game is a coodrination game if and only if, for each country i and for both x ∈ {0, 1}:

πi(x, x|δ) > πi(1 − x, x|δ). (13)

Countries play a coordination game if any one country’s payoff is highest were it to play the same action as the other. Note the contrast with traditional models, which posit that no investment is an unconditionally unique best response, a dominant strategy (Barrett, 1994). In a coordination game, no investment is a best response only if other countries do not invest. If they do, the best response is to, in fact, invest as well. Coordination games are defined in terms of mutual best-responses. However, note that in symmetric games (with symmetric payoffs), it suffices to identify the necessary conditions for one country only, since if they’re met by one, then they’re met by the other (i.e. πi(·) = πj(·)). Henceforth, I shall assume to be dealing with a symmetric game, or formally that πi(·) = πj(·). This helps simplify notation without loss of insight. Dominance is a way of saying that playing one action yields a (conditionally) higher payoff than playing the alternative. An isomorphic statement is that the gain from playing one action (investment) instead of the other (fossil fuel) is positive, or negative. It is worthwhile to formally define the gain from investment; this will simplify notation later on in the analysis.

Deﬁnition 3 (Gain from Investment Conditional on Observation). Let xj = α for α ∈ {0, 1}. Then deﬁne:

α gi (di) := E [πi(xi = 1, xj = α|δ) − πi(xi = 0, xj = α|δ)|di] . (14)

α In words, gi (di) is the conditionally (on di) expected gain in payoﬀ to country i if it switches from xi = 0 (no investment) to xi = 1 (investment), given country j plays xj = α and given di. By the linearity of payoﬀs in δ, see in particular

19 equation (4) or (7), we may write the expected gain from investment as a function of

δ (given di) as the gain from investment as a function of the expected δ (given di), α or formally gi (di) = E [πi(xi = 1, xj = α|δ) − πi(xi = 0, xj = α|δ)|di] = πi(xi = 1, xj =

α|di) − πi(xi = 0, xj = α|di).

Investment (xi = 1) is (in expectations) a dominant strategy if and only if

0 1 gi (di) > 0 ∧ gi (di) > 0, (15) i.e. the gain from investment relative to no investment is unconditionally positive. If both inequalities are reversed, investing is a dominated strategy. Coordination games are naturally identiﬁed by the condition, for each country i:

0 1 gi (di) < 0 ∧ gi (di) > 0, (16) saying a country’s gain from investment is negative when the other country is not investing, but positive when the other is, for both countries simultaneously. Each country i wants to copy the action played by the other country −i. Stretching notation, deﬁne:

Deﬁnition 4 (Expected Gain from Investment).

p 1 0 gi (di) := p · gi (di) + (1 − p) · gi (di), (17) for p ∈ [0, 1].

p This says gi (di) is the conditionally (on di) expected gain to country i from investing, given country j invests (plays xj = 1) with probability p (and does not invest with probability 1 − p). Simple calculus shows that in this game:

1 + d − 2c p gp(d ) = i + . (18) i i 2 2

p p p Note here that by the symmetry of payoﬀs, one may write gi = gj = g , dropping the subscript i for notational simplicity. It is another matter of simple calculus to establish:

∂gp > 0 (19) ∂di ∂gp > 0. (20) ∂p

20 There is economic signiﬁcance to these signs. When the relative gain from not emitting rises, in terms of damages avoided, the incentive to invest increases as well, which is what (19) tells us. Similarly, the net cost of investment is lower when the other country invests, due to spillovers. As the other country becomes more likely to invest (p increases), the anticipated spillover, and thus investment gain, therefore rises. This is what (20) says. By (20), the expected gain from investment to country i is increasing in the probability with which country j invests. The following implication thereof is immediate:

Lemma 2.

p q ∀q ≤ p : g (di) < 0 =⇒ g (di) < 0 (21) p q ∀q ≥ p : g (di) > 0 =⇒ g (di) > 0

Thus far, the probability that the other country invests was taken as exogenously given. But that’s no tenable assumption. Any country’s strategy will depend on its expected payoﬀs, which are a function of its private signal. The question arises what probability country i can rationally attach to country j playing some action, and vice versa. This motivates the related question what country i can infer, given its signal di, about the signal dj observed by country j.

A.2 Conditional Signal Distributions

Fix δ. The highest possible signal country i can receive is di = δ + ε. Similarly, the lowest possible signal observable by country j would be dj = δ − ε. It follows that di is at most 2ε greater than dj. By the same argument, di can never be more than 2ε below dj either. Given di, country i believes dj to be distributed on [di − 2ε, di + 2ε] according to density f(dj|di), which satisﬁes

Observation 1. Let di ≤ z ≤ di + 2ε. Then:

1 |d − z| f(z|d ) = − i . (22) i 2ε 4ε2

It follows:

di+2ε Z 1 (d + 2ε) − z 2 Pr[d ≥ z|d ] = f(d = t|d )dt = i , (23) j i j i 2 2ε z

21 and symmetrically for dj < di.

All technicalities are now in place to proceed with the formal analysis.

A.3 Iterated Dominance: Extending the Domain of Domi- nance

Inspection of the model reveals that investment, xi = 1, is a dominant strategy for each country i when δ > 2c − 1. Since signals are on average accurate, i.e. E[δ|di] = di, country i rationally invests whenever di > 2c − 1. ¯ Deﬁne d0 = 2c − 1 as the initial, or zero-th, threshold level such that country i ¯ ¯ invests for all observation di > d0, by strict (expected) dominance. Observing di = d0, ¯ country i infers that dj > d0 with probability 1/2, in which case country j invests, by dominance. It follows:

¯ ¯ di = d0 =⇒ Pr[dj > d0|di] = 1/2 =⇒ Pr[xj = 1] ≥ 1/2. (24)

Since the gain to investment is increasing in the probability with which the other country invests, see (20), the least country i can be sure to gain in expectations when ¯ di = d0 is: 1 g1/2(d¯ ) = , (25) i 0 4 ¯ where it has been plugged in that d0 = 2c − 1. Since this gain is strictly positive, ¯ ¯ ¯ country i stands to gain from investment even for some signals di below d0. Let d1 < d0 ¯ be the lowest value such that, given country j invests when dj > d0, country i can ¯ ¯ expect to gain from investing only if di > d1. The point d1 is seen to solve:

d¯ − d¯ p p1 ¯ 1 0 1 g (d1) = + = 0, i 2 2 (26) ¯ ¯ p1 = Pr[dj > d0|di = d1],

¯ ¯ Note that p1 and d1 are jointly determined. That is, given d0, I deﬁne p1 as the ¯ ¯ ¯ probability that dj > d0, given di = d1. Plugging in (23) for p1, d1 solves:

d¯ − d¯ 1 d¯ − d¯ 2 0 1 = 1 − 0 1 . (27) 2 4 2ε

¯ ¯ Note that for d1 = d0, the left-hand side of (27) is larger than the right-hand side, so

22 ¯ ¯ this won’t do. Moreover, for d1 = d0 − 2ε, the right-hand side of (27) is larger than the ¯ left-hand side, which is similarly problematic. It follows that there exists a d1 between ¯ ¯ d0 and d0 − 2ε such that (27) is satisﬁed.

Lemma 3. For each country i,

¯ ¯ ∗ ¯ ∃d1 < d0 : xi (di) = 1 ∀di > d1, (28)

∗ where xi (di) denotes country i’s expected payoﬀ-maximizing action, given di.

How far can this argument be stretched? Is it possible to deﬁne a decreasing sequence ¯ of thresholds {dk} such that, after k repritions of the above argument, each country i ¯ ¯ will invest for any signal di > dk? If so, where is this sequence {dk} heading? I will now answer these questions.

A.4 Iterative Investment Threshold

First, some basics. For di < 2c − 2, it is strictly dominant not to invest, see (4). Label ˆ this point d0 = 2c − 2 (for later use). Moreover, for di > 2c − 1 it is strictly dominant to ¯ ¯ invest. Label this point d0 = 2c − 1. Hence, supposing {dk} exists, it must be the case ¯ ˆ ¯ ¯ 3 that dk ∈ [d0, d0] for any k, so dk would be deﬁned on a sequentially compact set. ¯ I am now going to show that the sequence {dk} indeed exists and is decreasing. I will do so by induction on k, startiong from k = 1. ¯ ¯ If dk exists, then dk+1 is deﬁned to be the point such that country i gains from ¯ investing for all di > dk+1, given it has already been established by iterated dominance ¯ ¯ that for all dj > dk country j will invest. Hence, convniently rewriting (18), dk+1 solves:

k ¯ ¯ X ds+1 − ds pk+1 gpk+1 (d¯ ) = + = 0, k+1 2 2 s=0 (29) ¯ ¯ pk+1 = Pr[dj > dk|di = dk+1], where pk+1 is the lower bound for the probability that country j invests. Combined,

3A set is sequentially compact if ever sequence of points in the set has a convergent subsequence converging to a point in the set. By the Bolzano-Weierstrass Theorem, a set is sequentially compact if and only if it is closed and bounded. In a metric space, which a closed interval of the real numbers naturally is, a set is sequentially compact if and only if it is compact.

23 ¯ dk+1 must solve:

k ¯ ¯ ¯ ¯ X ds+1 − ds Pr[dj > dk|di = dk+1] gpk+1 (d¯ ) = + = 0. (30) k+1 2 2 s=0

¯ ¯ Since d1 was already found to exist, solving for k = 1 – or d2 – yields:

d¯ − d¯ 1 d¯ − d¯ 2 d¯ − d¯ 1 0 + 1 − 1 2 = 1 2 . (31) 2 4 2ε 2

¯ ¯ Now recall that d1 < d0 solves (27), reproduced here for ease of reference:

d¯ − d¯ 1 d¯ − d¯ 2 1 0 + 1 − 0 1 = 0. 2 4 2ε

¯ ¯ Assume d2 = d1. Then, with (27) in mind, it is immediate that the left-hand side of ¯ ¯ (31) is greater than the right-hand side. Similarly, for d2 = d1 − 2ε, the left-hand side of ¯ ¯ (31) is smaller than the left-hand side. Hence, there exists a d2 < d1 that solves (31). ¯ ¯ ¯ From d2 < d1, one can similarly obtain d3. Repeating the argument, by induction ¯ ¯ ¯ on k, it follows that dk+1 < dk for all k > 0. This implies {dk} must exist, and is decreasing.

¯ Lemma 4. The sequence {dk} exists and is decreasing. ¯ Being deﬁned on a sequentially compact set, {dk} converges to a point in the set. ¯ ¯ Lemma 5. The sequence {dk} converges to a point d ∈ [2c − 2, 2c − 1]. ¯ ¯ ¯ Since the sequence {dk} is converging, it is true by deﬁnition that limk→∞ dk+1 −dk =

0, from which one can be conclude that limk→∞ pk+1 = 1/2. It follows that the limit is characterized by solving (29) for pk+1 = 1/2: ¯ ¯ Proposition 6. The limit d of sequence {dk} is given by:

2 · d¯ − 1 4c − 3 d¯= 0 = . (32) 2 2 ¯ Proposition 6 shows that for any signal di > d, each country i will invest. Remarkably, ¯ ¯ this includes all signals di ∈ (d, d0), for which in the perfect information game it is unclear which action a country will play. Introducing scientiﬁc uncertainty about the

24 true damage sensitive of the economy paradoxically helps reduce the region where there is equilibrium-indeterminacy as a result of strategic uncertainty. The logic of iterated dominance leads further. Thus far, I have shown how the fact that investment is a dominant strategy for a region of high damages “infects” the neighboring area of lower damages, so that in this area investment becomes an (expected) dominant strategy also, even though it was not before the iterated dominance was applied. In a sense, starting for damages where investment is a strictly dominant strategy proper, iterated dominance invades the lower damages neighborhood and takes it hostage. But the same argument can be made when starting from the region where not investing is strictly dominant. To this analysis I shall now proceed.

A.5 Iterative No-Investment Threshold ˆ ˆ Recall that I previously deﬁned d0 = 2c − 2. For all signals di < d0, not investing is a ¯ dominant strategy for country i. Similarly, by Proposition 6, for all di > d, country i invests. Thus, the set of signals for which it could be iteratively dominant to not invest ˆ ¯ is [d0, d], which is compact. ˆ Observing di = d0, country i infers that country j will not invest with at least probability 1/2:

ˆ ˆ di = d0 =⇒ Pr[dj < d0] = 1/2 =⇒ Pr[xj = 0] ≥ 1/2. (33)

Since: 1 g1/2(dˆ ) = − , (34) i 0 4 ˆ country i will deﬁnitely gain, in expectations, by not investing when di = d0. One could ˆ ˆ ˆ therefore hypothesize the existence of a point d1 > d0 such that for all signals di < d1, country i is better oﬀ – in expectations – not investing. Given that country j does not ˆ ˆ invest when dj < d0, this d1 solves:

dˆ − dˆ 1 − p gp1 (d ) = 1 0 − 1 = 0 (35) i i 2 2 ˆ ˆ 1 − p1 = Pr[dj < d0|di = d1] (36)

25 ˆ Plugging in the conditional cdf of signals, (23), one ﬁnd that d1 is the solution to:

" #2 dˆ − dˆ 1 dˆ − dˆ 1 0 = 1 − 1 0 . (37) 2 4 2ε

ˆ ˆ If d1 = d0, the right-hand side of (37) is greater than the left-hand side. If, alternatively, ˆ ˆ d1 = d0 + 2ε, the right-hand side of (37) is smaller than the left-hand side. Hence, there ˆ ˆ is a (unique) d1 > d0 that solves (37).

Lemma 6. For each country i,

ˆ ˆ ˆ ∗ ∃d1 > d0 : ∀di < di : xi (di) = 0, (38)

∗ where xi (di) denotes country i’s expected payoﬀ-maximizing action, given di. ˆ By Lemma 6, each country i does not invest for any signal di < d1. Granted that, ˆ the maximum expected gain from investment to country i when observing di > d1 is:

d − dˆ dˆ − dˆ 1 − Pr[d < dˆ |d ] i 1 + 1 0 − j 1 i . (39) 2 2 2

As long as this expected gain is negative, country i will not invest. It follows that ˆ ˆ country i will not invest for any signal di < d2, where d2 is implicitly determined by:

" #2 dˆ − dˆ dˆ − dˆ 1 dˆ − dˆ 1 2 = 1 0 − 1 − 2 1 . (40) 2 2 4 2ε

ˆ ˆ When d2 = d1, the left-hand side of (40) is greater than the right-hand side – for recall ˆ ˆ ˆ that d1 solves (37) – while for d2 = d1 + 2ε, the left-hand side of (40) is smaller than the ˆ ˆ right-hand side. Hence, there exists a d2 > d1 such that each country i does not invest ˆ for any di < d2. From this, after iterating the above argument once again, it follows ˆ ˆ there exists a d3 > d2 below which neither country invest. Repeating the argument over ˆ and over, after k iterations, dk+1 solves:

k ˆ ˆ X ds+1 − ds 1 − pk+1 − = 0 2 2 s=0 (41) ˆ ˆ pk+1 = Pr[dj < dk|di = dk+1]

26 ˆ ˆ ˆ where by the logic that established d2 > d1 > d0 and induction on k it follows that ˆ ˆ ˆ ˆ ˆ dk+1 > dk > ... > d1 > d0, for all k. Hence, there exists an increasing sequence {dk} of points such that, after k repetitions of the argument above, it has been established that ˆ country i will not invest for any signal di < dk. Remember that this sequence is deﬁned on a compact set. An increasing sequence deﬁned on a compact set must converge.

ˆ ˆ ˆ ¯ Lemma 7. The sequence {dk} converges to a point d ∈ [d0, d]. ˆ ˆ Since by convergence limk→∞|dk+1 − dk|= 0 and therefore limk→∞ pˆk = 1/2, one

ﬁnds the limit of this sequence by solving (41) for pk+1 = 1/2: ˆ ˆ Proposition 7. The limit d of sequence {dk} is:

4c − 3 dˆ= = d.¯ (42) 2 ¯ By Proposition 6, each country i will invest for any signal di > d. By Proposition 7, ˆ ˆ ¯ each country i will not invest for any signal di < d. Moreover, d = d. The result in Proposition 1 follows.

A.6 Risk-dominance and equilibrium selection

The main result in Carlsson and Van Damme’s (1993) seminal paper is that, in any 2 × 2 global game, coordination will be on the risk-dominant equilibrium (if the noise in signals is suﬃciently small).

Deﬁnition 5 (Risk Dominance (Harsanyi and Selten, 1988)). Consider a 2 × 2 game with two strict Nash equilibria, (x1, x2) = (1, 1) and (x1, x2) = (0, 0). Then the (strict Nash) equilibrium (x∗, x∗), x∗ ∈ {0, 1}, is risk-dominant if and only if:

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ π1(x , x ) · π2(x , x ) ≥ π1(1 − x , x ) · π2(1 − x , x ). (43)

The formal deﬁnition above is due to Harsanyi and Selten (1988). In symmetric games, it is easily seen to be equivalent to:

1 1 1 1 · π (x, x) + · π (x∗, x∗) ≥ · π (x∗, x∗) + · π (x∗, x∗) (44) 2 1 2 2 2 1 2 2

With this in mind, deﬁne the following:

27 Deﬁnition 6 (λ-Dominance). Action xi ∈ {0, 1} is λ-dominant for player i if and only if

λ · πi(xi, 1|δ) + (1 − λ) · πi(xi, 0|δ) > (45)

λ · πi(1 − xi, 1|δ) + (1 − λ) · πi(1 − xi, 0|δ), for λ ∈ [0, 1].

Thus, action xi is λ-dominant for i if it yields the higher payoff given j plays xj = 1, i.e. invests, with probability λ. Action xi is dominant if it is λ-dominant for every λ ∈ [0, 1]. As defined here, λ-dominance is related to but more restrictive than what Morris et al. (1995) call p-dominance. [Explain here relation.] For the present purposes, the simpler definition of λ-dominance suffices.

Corollary 2. An equilibrium is risk-dominant iﬀ it is 1/2-dominant.

Proposition 8 (Carlsson and Van Damme (1993)). Let ε be suﬃciently small. Then, countries coordinate on the (expected) risk-dominant equilibrium.

¯ Proof. The sequence of iterated dominance threshold levels {dk} is a Cauchy sequence. ¯ Hence any subsequence of {dk} converges to the same point, which is the limit of the sequence itself. Hence, in particular the sequence converges to the limit point of the ¯ ¯ ¯ ¯ sequence where pk = Pr[dj > dk|dk+1]. In the limit, dk+1 → dk, so pk → 1/2. Since for ¯ any di > d, for each i, country i invests, the result follows. Q.E.D.

Proposition 8 may seem heavily theoretical and therefore of doubtful value in terms of predictive power. Surprisingly, this is not true. There is strong experimental evidence in support of its general prediction that in coordination games with two strict Nash equilibria, subjects play the risk-dominant equilibrium, see Heinemann et al. (2004).