Game Theory and the Politics of the Global Commons

Game Theory and the Politics of the Global CommonsAuthor(s): Hugh WardSource: The Journal of Conflict Resolution, Vol. 37, No. 2 (Jun., 1993), pp. 203-235Published by: Sage Publications, Inc.Stable URL: http://www.jstor.org/stable/174522 .

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Game Theory and the Politics of the Global Commons

HUGH WARD University of Essex

I assume that politicians maximize domestic political support subject to a political feasibility constraint set by the relationship between environmental quality and some politically desirable economic performance indicator. Because pollution flows across national frontiers, the political feasibility frontier depends on other nations' environmental policies. Depending on the nature of domestic political pressures and the environmental spillovers between the countries, various game structures are possible. Knowledge of the structure of the game is helpful when addressing policy issues and the question of how to design international institutions in order to overcome international environmental collective action problems.

Hardin's seminal work on the tragedy of the commons (Hardin 1968) has resulted in the common conception that both domestic (e.g., Ostrom 1990) and international (e.g., Thomas 1992) environmental problems are collective- action dilemmas. Solutions to international environmental problems such as ozone depletion, global warming, and acid rain may be blocked by individual nations rationally pursuing their "national interests." Because some nations' policies make a big difference to outcomes and because nations clearly take account of each other's policies, game theoretic models of collective action, which allow for strategic interdependence of decision making, are applicable (e.g., Taylor and Ward 1982; Livingston 1989; Livingston and von Witzke 1990; Maler 1990; Hoel 1991).

One example is to compare the politics of global warming to a game of chicken. Although many Organization for Economic Development and Co- operation (OECD) countries are attempting to stabilize or even to reduce their emissions of C02, other important players such as the United States, China,

AUTHOR'S NOTE: I would particularly like to thank Matthew Paterson, Michael Taylor, Bob Goodin, and Michael Ward for their helpful comments on this article.

JOURNAL OF CONFLICT RESOLUTION, Vol. 37 No. 2, June 1993 203-235 ? 1993 Sage Publications, Inc.

203



204 JOURNAL OF CONFLICTRESOLUTION

and the (former) Soviet Union are currently doing nothing substantive to control CO2 emissions, even though they are acting on other greenhouse gases (Grubb 1990, chap. 2). For these three countries, controlling CO2 emissions is of prime importance because of the current high value of fossil fuels in the United States and (former) Soviet economies and the vast exploitable coal reserves in China (Morrisette and Plantinga 1991, 167). The sort of asymmetric outcome pattern that characterizes chicken is emerging (Taylor and Ward 1982; Runge 1984), in which some cooperate and others defect, freeriding on the efforts of others to provide better environmental quality. Moreover, as the theory of chicken would suggest, some nations are using commitment tactics to increase the likelihood that the eventual outcome will allow them to freeride by doing less than others. The United States claims that there is insufficient evidence to justify a policy change (Morrisette and Plantinga 1991, 165-66). Given the existence of a strong international scientific consensus, this may be calculated to make the United States appear irrational and likely to adhere to its current path at any cost. China argues that it is unfair that developing countries should be made to pay the price of containing global warming, because the problem has been caused by the rich nations. However, this ethical argument is consistent with the self-interest of a nation with large coal reserves. Further, the appeal of China's argument to other nations in the south (Morrisette and Plantinga 1991, 169-70) makes it more difficult for China to back down. Although they acknowledge their part in the problem of global warming, the Soviets argue that they cannot act given their present turmoil-at least without other nations bearing the costs (Morrisette and Plantinga 1991, 168-69). Another well-known commitment tactic in chicken games is to claim that cooperation is not feasible. Stable, asymmetric outcomes and the use of commitment tactics are empirically quite common where transboundary pollution politics is concerned, suggest- ing that chicken may be a widely applicable model. For example, the

European Community's (EC) plans to control sulfur dioxide (SO2) pollution make very different demands on the various member states (Weale, O'Riordan, and Kramme 1991, 184). Some EC members, including Britain (Wilcher 1989, chap. 4), gain by using commitment tactics.

Of course, models like prisoner's dilemma and assurance may also be

applicable (Livingston and von Witzke 1990). For instance, the ozone

depletion problem has features that correspond to an assurance game. At first some EC members, particularly Britain, France, and Italy, blocked any possibility of the EC accepting the controls on chlorofluorocarbon (CFC) production and use in nonessential products advocated by the Toronto Group, which was led by the United States (Morrisette 1989, 807-16). However, changing public perceptions, growing scientific evidence of the seriousness



Ward / THE GLOBAL COMMONS 205

of the problem, potential threats of trade sanctions, internal diplomatic pressure on the hard-line EC countries, and a growing willingness on the part of the major European producers to accept controls as long as they recipro- cally constrained competitors all led to a change in the game structure. Arguably the shift in the EC's position was the key to negotiations and the subsequent tightening up of the Montreal Protocol. Nevertheless, a number of difficult issues relating to the interests of other states, notably in the developing world, also had to be dealt with (Benedick 1991, 125-42). By focusing on the interactions between the EC and the Toronto Group, each side was eventually willing to cooperate as long as the other side did so too-a defining feature of assurance. Also, trust building and unilateral action from leader nations to demonstrate the technical feasibility of limita- tion and to encourage others to follow suit were important (Benedick 1991, 143-49). Such actions would be expected on theoretical grounds in assurance games (Ward 1989).

Rather than simply assuming that some well-known model of collective action such as prisoner's dilemma is appropriate (e.g., Dorfman 1991, 103) or using game theory as a heuristic device to fit the empirical pattern, I show below how particular game structures arise as a result of the interaction between environmental and domestic political variables. The idea that rational, self-interested politicians attempt to maximize their political support has been used before with regard to international policy issues with a collective action dimension, notably those surrounding trade and protection (see Vousden 1990, 178-92 for a survey). I extend the rational choice approach by treating transboundary pollution politics in the same way. By emphasizing domestic political support among the electorate and key economic actors, my approach necessarily abstracts from the complex political factors affecting policy. These include policy advice from scientists (Haas 1990), the con- straints and possibilities generated by domestic policy-making structures, the existence of international institutions and legal arrangements, and pressures from other nations (e.g., Young 1989; Grub 1990, chap. 2; Benedick 1991; Morrisette 1991). In reality, there may be a "two-level game" going on (Putman 1988) in which there is conflict within the administration between elected politicians and administrators, or between departments of state, over what the nation's policy should be. It seems to me, however, that domestic political support is often crucial, and that it is worth attempting to isolate its interactive effects with environmental variables.

In contrast to the assumptions made here, the existing game-theoretic literature treats policymakers as if they were maximizing some economic measure of domestic social welfare (Livingston 1989, 84; Livingston and von Witzke 1990, 165; Maler 1990, 89-90; Hoel 1991, 69). Certainly differ-



206 JOURNAL OF CONFLICT RESOLUTION

ent constituencies and organized interests consider the consequences of policy on their own welfare-and perhaps on those of others in society- when evaluating the administration. However, the use of voting or pressure group activity to reveal preferences for public goods is unlikely to lead to the political currency of support, which motivates self-interested politicians, correlating well with overall social welfare (Cornes and Sandler 1986, chap. 6). Of course, in reality politicians' principles as well as their perceptions of overall social welfare count (Wittman 1983). However, such factors are likely to be relatively unimportant unless policies are adequately supported. Thus domestic political pressures should not be treated as a special factor modifying the calculus of social welfare (cf. Hoel 1991, 59).

I start with a two-nation model of a single-shot game in which policy decisions have to be made about environmental quality.' Countries A and B each face a trade-off between environmental quality and some politically significant economic variable. The economic variable might be unemploy- ment, inflation, loss of current consumption, loss of government expenditure programs in nonenvironmental areas, or slower economic growth-all of which could be associated with the costs of environmental cleanup under some circumstances. It is widely believed that, because of existing market distortions and lack of consumer information, certain measures can be taken to control, for example, global warming at no cost to economic growth (MacNeill, Winsemius, and Yakushiji 1991, 97). In effect, I assume that the

boundary of the feasible set for growth and environmental quality has been reached. The justification is that the hard politics begins once the feasibility frontier has been attained.2 Because the structure of the argument would be the same in each case, I will assume the trade-off is between growth and environmental quality. A and B are linked by transboundary pollution flows, so that if one nation reduces its level of pollution, this has an environmental

spillover effect, improving the other's environmental quality. Thus the trade-

1. As I show below, the model developed here can be extended to the n-nation case. Moreover, many transboundary environmental problems are bilateral, involve two negotiating blocks, or are so complex that there are pressures for bilateral deals to be struck (MacNeill, Winsemius, and Yakushiji 1991, 80). In the conclusion, I discuss repeated game models and interaction through time.

2. This assumption could be dropped and the likely effect would be that limited cooperation on improving environmental quality would be possible. In issues such as global warming, the amount that can be gained at zero or low cost may not be large relative to the magnitude of the

problem. See the survey article by Hoeller, Dean, and Nicholaisen (1990) that compares estimates of the costs of reductions in CO2 emissions that are small compared with those that most scientists believe are necessary to stabilize global temperature. More generally, there are good reasons for believing that when aggregating across various environmental problems that are likely to be linked together, growth trade-offs do exist both nationally (e.g., Christainsen and Haveman 1981) and for the world economy (e.g., Leontief 1977, chap. 11).



Ward /THE GLOBAL COMMONS 207

off each faces between economic growth and environmental quality is a function of what the other does.

Politicians in each nation attempt to maximize a political support function defined over the rate of economic growth and environmental quality. The support function reflects the fact that as both environmental quality and growth increase, the level of support will rise, because domestic constituencies ideally want both. If the rate of growth goes down, environmental quality would have to increase to maintain the same level of support. In this context, the support or opposition of key industries and unions might count alongside the voters' support.

Transboundary pollution politics often involves nondemocratic regimes too. I assume that nondemocratic regimes also need to maximize the support of certain key groups and individuals in order to remain in power, even if the mass of the population is of little political importance. Support is maximized subject to the feasibility constraint presented by the trade-off between environmental quality and growth. The assumption is that policy instruments are available to determine environmental quality, and that there is an equilibrium relationship between such policy choices and growth, so that growth is indirectly determined.

The models here are noncooperative games in continuous strategy sets defined by each nation's choice of an environmental quality level. Policies are in (Nash) equilibrium when no nation is willing to change its own policy when the other nations' policies remain the same. The aim is to determine what equilibria, if any, exist and how the equilibria alter when the variables in the model change. Of course it is unrealistic to assume that decision makers are fully informed about the environmental, economic, and political trade- offs they face, so that this equilibrium notion is problematic (Maler 1990, 88). However, equilibria may be resting points of adjustment processes in which politicians make groping attempts to maximize, although ideas from outside game theory may be needed to specify the path to equilibrium (e.g., Elster 1989, 107-12).

Treating transboundary pollution politics as a noncooperative game would be inappropriate if the nations involved could make binding agreements with each other. If such agreements were possible, ideas from cooperative game theory or bargaining theory might be used (e.g., Hoel 1991, 61-64). Of course, noncooperative equilibria could still be of theoretical importance in defining the minimum payoffs of coalitions and threat points in the bargaining model (Hoel 1991, 63). There are numerous, often long- standing, international agreements on transboundary pollution problems, some of which have the status of international law (Young 1989, part 2; Sand




1991). Undoubtedly some of these regimes are best analyzed using cooperative game theory, because all sides have invested so much in them that agreements do effectively bind. However, noncooperative game theory is still appropriate in many instances. Despite the fact that nations may avoid breaking international environmental agreements because it puts them in a bad light internationally and may lead to the breakdown of institutionalized cooperation (Young 1989, 72-76), the sanctions involved are often not strong enough to ensure compliance. It is common for noncompliance to take the form of poor implementation of environmental agreements. This is because implementation is difficult for other nations to monitor given poor data and reporting and because dispute procedures may be unclear (Bryner 1991; Sand 1991, 257-78). Moreover, there are numerous important problems, including global warming, over which international agreements are still being sought. It may take many years for an international regime that more or less binds nations to emerge where unanimity among major nations is required to get a

treaty (Sand 1991). International environmental negotiations are often strongly influenced by

who is involved, how the issues are disaggregated for bargaining purposes, the methodologies used for measuring the extent of the problem, and the limited set of solutions that are actually considered from among those that are potentially available. It is unlikely that the sort of highly abstract models

developed here will give much insight into such crucial processes. However, they can help us understand the underlying structure of interactions. Also, rational choice models often generate surprising conclusions calling into

question "obvious" arguments about environmental policy (Salazar and Lee 1990). For example, I show below that, because of the way other countries

may react, economic sacrifices made in order to improve environmental

quality are quite likely to have the counterproductive effect of worsening domestic environmental quality. Also, I show that increased domestic political support for environmental cleanup need not lead to improved environmental quality. In contrast, it has often been assumed that the greater the domestic political will to act, the more likely that cleanup will occur (e.g., Morrisette 1991, 155-58; but cf. Hoel 1991, 58-59). Much of the existing policy-related literature on transboundary pollution is founded, either explic- itly or implicitly, on the assumption that once scientists and policy analysts demonstrate that there is a common interest in achieving a solution, change will occur, despite impediments caused by national interest (e.g., Feldman 1991). This seems to me to be implausible when politicians' self-interest

generates powerful opposing motivations. Still, we should not give up hope for solutions even in such cases; both self-interest and normative pressures can be engaged in order to achieve cooperation in environmental collective




action problems (Ostrom 1990, chaps. 1,2; Young 1989). I return to this point in the conclusion.

THE ECONOMIC GROWTH VERSUS ENVIRONMENTAL QUALITY TRADE-OFF

Let QA be the measure of the dimension of environmental quality we are concerned with for country A. Let gA be the rate of economic growth in country A and gB be the rate of growth in country B.3 I assume that QA is a continuous twice differentiable function4 of the form QA = QA(gA + sA.gB), where SA is a parameter reflecting the extent to which environmental harm due to economic growth in B spills over into country A (cf. Maler 1990, 90). The parameter sA will be larger to the extent that economic growth in B

generates the sort of pollution we are concerned with and to the extent that pollution carries across the border between A and B. I assume that oo > SA > 0. As SA tends to 0, growth in B has less and less effect on environmental quality in A. If SA> 1 a given growth increase in B does more harm to A's environment than the same increase in A's growth rate.

Let GA = gA + SA.gB, where GA can be thought of as environmentally relevant "total growth." Then I assume that dQA/dGA < 0 or environmental

quality declines as total growth increases. I also assume that d2QA/dGA2 < 0. These two assumptions imply that a bigger sacrifice of growth will be needed to get a unit increase in environmental quality when quality is higher than when it is lower. One justification for this assumption is that, if methods of

improving environmental quality are used in order of their impact on growth, with low cost methods rationally being used first until their potential is fully exploited, the marginal cost of improving environmental quality must be increasing in relation to such problems as global climate change (Nordhaus 1990, 40-42).

It will be convenient here to measure growth and environmental quality in relation to the status quo, because the interesting question is whether one or both nations will sacrifice growth in order to improve environmental

quality over its current level. Given this focus, it is a convenience with no

analytical cost to transform the growth and environmental quality measures in such a way that current growth is 0 and current environmental quality is 0

3. Although rates of growth are likely to be bounded, I ignore this point in order to avoid the extra complication of corer solutions. I also assume that the economic growth rates of the two countries do not covary.

4. There are some areas in which environmental quality is a discontinuous function of growth (Taylor and Ward 1982).




for each country. Thus I require that QA(O) = 0, so that the status quo is feasible. I will denote the status quo by 0 in the rest of this article.

In diagram 1 the solid curve through 0 is the contour of QA corresponding to gB = 0, which I will denote by QAO. If B sticks to its status quo level of growth, which is 0, changes in gA move A along QAO which, thus, represents A's political feasibility frontier for support maximization, given B's status quo level of growth. In line with the assumptions made about the derivatives of QA with respect to GA: as gA decreases QAO increases from 0, but at a slower and slower rate as it becomes harder and harder to improve environmental

quality; as gA increases, QA? falls at a faster and faster rate for each extra increment in growth. The steeper the slope of QAO, the smaller the sacrifice of economic growth that has to be made to get a given improvement in environmental quality.

Knowing the shape of QAO, it is easy to determine what the shape of contours of QA will be for any other value of gB. Suppose gB falls from 0 to -1. There is an environmental spillover effect equal to that of A cutting its growth rate by SA. In Figure 1, if A was at x on QAO, the effect of a unit fall in gB would be to move A to z. At z, the level of QA is the same as at y, the

point on QAO corresponding to a value of gA which is SA units less than at x. A ends up with the same rate of growth, but better environmental quality. Applying this argument to each point on QAO, the effect of a unit fall in gB is to move A's possibility frontier SA units to the right, generating the possibility frontier QA-1 shown as a dashed curve in Figure 1.

The argument easily generalizes. As shown in Figure 1, the net effect of a change in gB of-u (u > 0) is to move A's feasibility frontier u.sA units to the

right; and the net effect of a change in gB of +v (v > 0) is to shift A's feasibility frontier v.sA units to the left. The larger the value of sA, the bigger the effect on A's feasibility frontier of given change in gB. Notice that, because the effect of changes in gB is to shift the feasibility frontier to the right or to the left, the slope of contours of QA will be the same along any line parallel to the gA axis. Analogous assumptions will be made about the functional form and relevant derivatives of QB which gives environmental quality in B as a function of growth in B plus growth in A "weighted" by a spillover parameter SB. There are no good general reasons to assume that SA = SB.

THE SUPPORT FUNCTION

In Figure 2 the dashed curves are iso-support contours for a continuous, twice differentiable support function VA(QA, gA). If both QA and gA increase, a higher iso-support line is attained. Thus support along the iso-support curve



QU

1 Figure 1: A's Political Feasibility Frontier




vA +

N

- V0 A

0 "Ai

Figure 2: A's Adjustment Path

VA? is lower than along the iso-support curve VA+. The slope of a particular iso-support curve at any point is the marginal rate of political substitution (mrps)-the limit of the ratio of the marginal decrease in environmental quality to the marginal increase in growth that would maintain support at the same level.

The mrps is a measure of the relative political importance of environmental quality compared to growth, for it measures how much environmental quality a politician can sacrifice at the margin to get more growth while maintaining a given level of support. If a lot of environmental quality can be sacrificed as long as a given increase in growth occurs, environmental quality is relatively unimportant. If environmental quality were generally relatively unimportant, this would be reflected in the iso-support curves through any point being steeper. I assume that iso-support curves are concave to the origin.




What the assumed concavity of iso-support curves amounts to is the idea that holding the level of support constant the greater the level of growth, the less important it becomes politically when compared with environmental quality. B's support function VB(QB, gB) will be assumed to have the same general form as VA.

In Figure 2, VA has the properties that for any given level of gA, the mrps decreases as QA increases, and for any given level of QA, the mrps increases as gA increases. For instance, VA? is flatter at 0 than the iso-support curve passing through the point x is at x, where QA is higher but gA = 0. Thus VA has the declining marginal political significance (dmps) property for QA: as environmental quality increases, holding growth constant, it becomes less relatively politically significant. Also VAO is steeper at 0 than the iso-support curve through y is at y, where gA is higher but QA = 0. Thus VA also has the dmps property for gA: as growth increases, holding environmental quality constant, it becomes less relatively politically significant compared to environmental quality.5 Where VA has the dmps property for both QA and gA, I will simply say that it has the dmps property.

Figure 2 also shows QAO. A will maximize support at the point at which the feasibility frontier is tangential to the highest attainable iso-support curve. If gB = 0, VAO is the highest attainable iso-support curve, so that A would choose gA = 0 and QA = 0.

REACTION FUNCTIONS

It is a plausible assumption in relation to most transboundary environmental problems that 0 is an equilibrium, because countries often persist with existing policies for a considerable time after awareness of the environmental

5. Some authors have noticed that there is an issue-attention cycle whereby actual or potential improvements in environmental quality take the political heat from the issue relative to other issues such as the economy (e.g., Downs 1972). This lends some empirical credibility to the dmps property. Let mA be the mrps. If VA has the dmps property for QA, 6mA/6QA < 0. By a standard result,

5VA/5QA mA VA/6gA

Then 6mA/8QA < O if

(A/A VA/QA2 VA/gA VA/QA VA/QA * VA/6gAQA) < 0

Declining absolute political returns to environmental quality (e.g., 62VA/QA2 < 0) are neither necessary nor sufficient for this, emphasizing that the dmps property should not be confused with declining absolute political significance.




problem has become widespread. The interesting question is what other equilibria, if any, exist? A's adjustment path traces out how A's support- maximizing policy choice varies as B's growth rate alters and can be found by connecting points of tangency between the feasibility frontier and highest attainable iso-support curves. A's reaction function rA gives the support- maximizing level of gA for any value of gB, and can be derived from A's adjustment path. Because I assume 0 is an equilibrium, r^ = 0 for gB = 0. Because of the assumed form of VA and QA, it is guaranteed that rA is a

continuous, differentiable function of g. The qualitative features of VA can easily be linked to the slope of the adjustment path and reaction function.

Proposition 1. If VA has the dmps property and if gB decreases, then A will adjust its policy to maximize support by strictly increasing both gA and QA.

To see this, consider what happens when gB drops from, say, 0 to -dgB in

Figure 2. This change moves A's feasibility frontier dgB.sA units to the right, as explained above. Thus the slope of QA-gB where it cuts the gA axis at the point y is the same as the slope of QA? where it goes through 0. Because VA has the dmps property for gA, the iso-support curve through y is flatter than VA? at 0. Thus it must be flatter than QAdgB at y, as shown in Figure 2. Hence the support-maximizing choice for A given gB = -dgB cannot be to leave QA at 0. Moreover, if QA were reduced below 0 by moving along QA-dgB this would put A on an even lower iso-support curve than the one through y, because of the concavity of the iso-support curves to the origin and the convexity of the feasibility frontier to the origin. The conclusion is, then, that the new support-maximizing policy is such that QA > 0.

At the point x where QAdgB cuts the QA axis it is flatter than QA? where

QAO goes through 0. The reason is that the slope of QAdgB at x is the same as the slope of QAO for gA = -SA.dgB, as explained above. Because VA has the

dmps property for QA, the slope of the iso-support curve through x is steeper than that of VAO at 0. Hence, x cannot be a support-maximizing choice for A, because the iso-support curve must be steeper than QAgB. Moreover, if A chose a value of gA less than 0 by moving along QAgB, this would move it onto a lower iso-support curve than the one through x, by the convexity of the feasibility frontier with respect to the origin and the concavity of iso-support curves. Hence the new support-maximizing choice for A along QAdgB must be such that QA > 0 and gA > 0, at point like z on a higher iso-support curve VA+.

This argument obviously generalizes, because it relies on assumed properties of QA and VA that hold whether we start at 0 or at some other point at which A is making a support-maximizing choice for the given value of gB: if



Ward/THE GLOBAL COMMONS 215

gB is reduced, A will move to a new support-maximizing point at which both QA and gA are strictly increased.

Suppose that A was maximizing support at the point z in Figure 2 but gB increased by a finite amount 6gB. The feasibility frontier shifts inwards to QA?, leading A to maximize support at 0. Let the change in A's support- maximizing value of gA be denoted by 6rA. As shown in the proof of proposition 1, if VA has the dmps property 0 > 8rA > -gB.sA, because z is to the left of y (where gA = 6gB.sA) and to the right of 0. Dividing through each term in this inequality by 8gB, 0 > rA/6gB > -SA. In the limit as 8gB goes to

zero, the ratio of the finite change in rA to the finite change in gB is equal to the differential of rA with respect to gB evaluated at gB = 0; hence the differential for gB = 0 is strictly less than 0 and strictly greater than -SA. Again this argument relies on properties of VA and QA that are independent of the particular support-maximizing value of gA in question. Notice that if the weaker assumption is made that VA has the dmps property for QA, the same argument used above shows that 0 > drA/dgB for all values of gB. Hence:

Proposition 2. If VA has the dmps property, 0 > drA/dgB > -SA for all values of gB; and if VA has the dmps property for QA, 0 > drA/dgB.

In Figure 3, rA is drawn as a solid curve. It is assumed that both VA and VB have the dmps property. Because of proposition 2, any point on rA lies in the interior of the horizontally shaded, cone-shaped sets bounded by the gB axis and the line rA = -SA.gB- Moreover, for any value of gB, rA must be flatter than the line rA = -SA.gB. B's reaction function, rB, is shown as a dashed curve.

By proposition 2 any point on rB must lie in the interior of the vertically shaded, cone-shaped sets bounded by the gA axis and the line gA = -gB/sB. Moreover, for any value of gB, rB must be steeper than the line gA = -gB/sB. Nash equilibria occur where the two reaction functions intersect, because at any other point in the space, at least one country is not choosing a support- maximizing growth rate given what the other is doing. In this case 0 is the only equilibrium.

CHARACTERISTICS OF GAME STRUCTURES

In order to show that a collective action game is being played over improvements in environmental quality, I need to show that there exist outcomes in which both nations improve their environmental quality over the status quo that leave politicians in both nations with greater political support. The proof of proposition 3 demonstrates that this is, indeed, the case.




10

Figure 3: A and B's Reaction Functions

Proposition 3. So long as 0 is a Nash equilibrium, there exists an outcome that is Pareto-superior to 0.

If both A and B cut their growth by -dgA, A's environmental quality is that associated with the "total growth" level of (-dgA - dgA.sA), where the first term is the cut in growth A makes and the second term is cut in growth B makes in terms equivalent to A's growth rate. Hence the effect of each nation making a cut in its growth rate of -dgA is to move A to the point (-dgA, QAO[-dgA.{ 1 + SA}]). The point A attains under equal cuts of dgA is on a higher iso-support curve than VAO if

VA (-dgA)

QA [-dgA.(+ <A)] QA 0[ dgA.(1 + SA)]




because VA?(-dgA) is the distance of the relevant iso-support curve from the gA- axis when gA = -dgA. By assumption the first derivatives of VAO and QAO are defined everywhere; hence, by the function of a function rule, the first derivatives of VAO(-dgA) and QA?(-[ + SA].dgA) are also defined. By assumption, dQA?/dgA is not equal to zero at the origin; hence, by the function of a function rule, dQA?(-dgA. [1 + SA])/dgA is not equal to zero at the origin. Also VAO(O) = QAO(O) = 0. Thus we can apply l'Hopital's rule:

ligAO VA 0(- dgA) dVA (- dgA)/dgA -dVA /dgA lim dgA--> 0 0[

QA [-dgA.( + SA)] dQA [-dgA.( + SA)]/dgA -(1+ SA).dQA /dgA

evaluated at gA = 0. Because 0 is an equilibrium, dVA?/dgA = dQA?/dgA for gA = 0, or else A would not be maximizing its support given B's policy choice. Thus in the limit the ratio is 11(1 + SA), which is less than 1, since SA > 0. The same

argument applies to B. Thus there is some value of-dgA that is small enough to ensure that both A and B are on higher iso-support curves than those through 0.6

Continuous strategy prisoner's dilemma games have one equilibrium outcome that is Pareto-inferior to at least one nonequilibrium outcome (Taylor and Ward 1982). The game shown in Figure 3 has a single equilibrium at 0. Proposition 3 demonstrates the existence of an outcome Pareto-superior to 0, justifying the characterization all games such that 0 is the only equilibrium as continuous strategy two-nation prisoner's dilemma.

The following proposition starts to narrow down the possible sorts of

game structure in the model:

Proposition 4. If VA and VB have the dmps property for QA and QB respectively, the game is either prisoner's dilemma or chicken.

By proposition 2, both reaction functions are strictly negatively sloped. If the reaction functions only intersect at 0, the game is prisoner's dilemma, as shown by proposition 3; otherwise it is chicken. Ageneral two-player chicken game in finite or continuous strategies can be defined as one where there are at least two equilibria; the highest ranked equilibrium of one player differs from the highest ranked equilibrium of the other, so that each player may have incentives to commit to a strategy corresponding to his favored equilibrium; and the outcome, if both players adhere to their commitments, is Pareto-inferior (cf. Taylor and Ward 1982).

6. Of course, proposition 3 does not imply that the only all-round improvements result from small and equal cuts in growth.




If VA and VB have the dmps property for QA and QB respectively, by proposition 2 the reaction functions must be strictly negatively sloped. Hence, if there are other equilibria besides 0, then they must lie either to the northwest or to the southeast of 0. There are two relevant variants of chicken. Figure 4a represents the variant in which there is at least one equilibrium both to the northwest and to the southeast of 0. A's preference ranking of equilibria is x > 0 > y > z, where > denotes the strict preference relation. The reason is that any reduction in ga puts A on a higher feasibility frontier, allowing a greater level of support to be enjoyed given an optimal choice of gA is made, as will be the case for points on rA. By the same reasoning, B's ranking is z > y > 0 > x. Given this is so, A might commit to the level of gA corresponding to x to make B accept its preferred equilibrium. Whether such tactics would be chosen and whether commitments would be adhered to would depend on perceptions of the payoff structure and B's reputation for sticking to its commitments (e.g., Ward 1987). Notice that equilibrium z is not locally stable: a Cournot adjustment process would lead away from it, as indicated by the arrows. However, by firmly committing itself to the corre-

sponding level of gB, B would both render the equilibrium locally stable and

put pressure on A to accept its preferred equilibrium.7 If each nation adhered to a commitment to increase its growth, the result would be an outcome Pareto-inferior to 0. Both nations would increase their growth rates resulting in each nation being forced onto a lower feasibility frontier than the one that

passes through 0, the level of support enjoyed at 0 becoming unattainable and environmental quality deteriorating in both nations. Notice that by proposition 3 the outcome arrived at if both nations stuck to their commitments would also be Pareto-inferior to nonequilibrium outcomes in which both nations reduced their growth rates.

Figure 4b represents the variant of chicken in which, relative to 0, the

asymmetric equilibria all favor one nation and disfavor the other nation. That is, there is at least one equilibrium to the northwest (southeast) of 0 but no

equilibria to the southeast (northwest) of 0. A's ranking of equilibria is x > y > 0, whereas B's ranking of equilibria is 0 > y > x. A might commit to bring about and to stabilize x, and B might commit to 0. If both A and B stuck to their commitments, the outcome would, again, be Pareto-inferior to 0 and to outcomes where both nations reduced their growth rates: B would be forced onto a lower feasibility frontier, and A would move to a non-support- maximizing point on the feasibility frontier through 0.

7. Locally unstable equilibria are implausible predictions of game outcomes unless, as in the case under discussion, players might have incentives to choose meta-strategies, such as commitments, that stabilize them.




(a) r rB / A

xX

N

\ "; (\)

\ \

(b)'f

(b)

Figure 4: Variants of Chicken




A sufficient condition for the game to be prisoner's dilemma follows from proposition 2:

Proposition 5. If VA and VB have the dmps property and if sA*sB < 1 the game is prisoner's dilemma and 0 is a stable equilibrium.

As Figure 3 suggests, 0 is the only equilibrium if the line gA = -sA.gB is flatter than, or equal in slope to, the line gA = -gA/sB, because drA/dgB > -A and drB/dgB < -I/Ss. The sufficient condition is, then, -SA > -1/SB or SA * SB < 1. The status quo 0 is stable because any disturbance from it will lead back to it via a Coumot action/reaction process in which at each stage both players entertained the zero conjecture that the other nation's strategy choice would not change in response to their own. The convergence of such a process on 0 is guaranteed by the fact that, for any value of gB, drA/dgB > drB/dgB if the condition holds.

By proposition 5, if environmental quality and growth are of declining marginal political significance, and if there is no amplification of environmental harm as pollution crosses between the two countries, so that sA < 1 and SB < 1, the structure of the game is prisoner's dilemma.

Chicken is only possible if the reaction functions intersect at other places besides 0. This requires that the cone-shaped sets in which rA and rB lie should intersect, although the sets may intersect without the reaction functions

intersecting at some point other than 0. Thus:

Proposition 6. If VA and VB have the dmps property a necessary condition for the game to be chicken is that SA'Sg > 1.

If environmental factors result in a majority of the pollution in A coming from B, and if sacrificing a unit of growth brings at least as large a cut in emissions in B as in A, SA will plausibly be greater than 1. Many experts accept that, because of the direction of prevailing winds, 50% of the acid rain

falling in Canada originates in the United States, largely from electricity utilities burning high-sulfur coal in the Midwest (Wilcher 1989, 13-14). Moreover, many Canadians believe that acid rain originating in the United States does more harm to especially sensitive Canadian lakes and forests than that originating in Canada itself (Wilcher 1989, 21). Evidence from the U.S. Environmental Protection Agency shows that the direct economic costs of

reducing emissions in U.S. electricity utilities are similar to those from industrial sources (Regens and Rycroft 1988, 92). Because the most significant Canadian sources are industrial, this suggests little cross-border varia-




tion in direct control costs (cf. Moller 1989, 1212).8 Acase can be made, then, that the spillover parameter from the United States to Canada is greater than 1, although there is a great deal of uncertainty. Given that only around 15% of the acid rain falling in the United States originates in Canada, it is highly likely that the spillover parameter from Canada to the United States is less than 1. Clearly we do not have enough information to tell whether the spillover from Canada to the United States is great enough for the necessary conditions for chicken to be fulfilled. However, the interaction has chicken- like features. This is partly because of the congressional logjam over amend- ing the Clean Air Act and partly because the Reagan administration was unwilling to act in a leadership role. The United States was committed to a policy that Canadian government agencies believed would lead to increased acid deposition because of increased U.S. coal burning (Wilcher 1989, 28). The 1985 federal/provincial pact on acid rain (Wilcher 1989, 28) can be seen as a forced unilateral move to bring Canada back into equilibrium in the face of the perception that U.S. emissions will increase. We might infer from this that the necessary conditions for chicken were met, although this requires that the dmps assumption, among others, be satisfied. Although this example brings out the difficulty of applying the model empirically, it also illustrates that by bringing together formal results and empirical observation, light can be shed on the way in which the structure of the game is generated.9

By referring back to Figure 3, the question of what happens to environmental quality if VA and BB have the dmps property, the game is chicken, and there is a departure from 0 can easily be answered. The line gA = -sA.gB or gA +

SA.gB = 0 is the set of points for which total growth as it relates to A's environmental quality is equal to 0. Points below (to the southwest of) this line are points at which total growth is lower than 0 and A's environmental

quality is higher than 0; and points above (to the northeast of) the line are

points at which total growth is higher than 0 and A's environmental quality is lower than 0. Any equilibrium to the northwest of 0 must be on rA and must, by proposition 2, be below the line gA = -sA.gB; hence it must be a point such that QA > 0. An equilibrium to the southeast of 0 must be on rA and must, by proposition 3, be above the line gA = -SA.gB; hence it must be a point such that QA < 0. The line gA = -gB/sB or gB + sB.gA = 0 is the set of points for which

8. Even if the direct financial costs per unit of reduced emission were the same in the U.S. and Canadian economies, the impacts on growth rate could differ. My argument presupposes similar impacts.

9. Of course, it would also be necessary to account for the measures included in the Clean Air Act Amendment of 1990 to reduce U.S. emissions-an unanticipated development unless the incentive structure in the United States shifted.




total growth as it relates to B's environmental quality is equal to 0. Repeating the arguments applied to A's environmental quality: any equilibrium to the northwest of 0 must be a point at which QB < 0; and any equilibrium to the southeast of 0 must be a point at which QB > 0. These results are summarized in proposition 7, which also gathers together several other points already made about asymmetric equilibria.

Proposition 7. Assume VA and VB have the dmps property. Then any equilibria to the northwest of 0: g > 0 and gB < 0; QA > 0 and QB < 0; VA is greater than at 0 and VB is less than at 0. Similarly any equilibrium to the southeast of 0 is such that: gB > 0 and g, < 0; QB > 0 and QA < 0; VB is greater than at 0 and VA is less than at 0.

Thus if VA and VB have the dmps property it is impossible to get an equilibrium in which the environmental quality of both nations is improved over 0 as long as the game is strategically equivalent to a one-shot game. When the game is the first variant of chicken discussed above, the nation managing to get its preferred equilibrium wins in terms of growth environmental quality, and support for the government. The other nation pays the whole price through losses in each of these three dimensions. If the game is the second variant of chicken, the best that the disadvantaged nation can get is 0; and if the other nation gets its preferred equilibria, it will improve its environmental quality, growth rate, and support at the cost of the disadvan-

taged nation. Some authors have argued that there are normative pressures in transboundary pollution politics for symmetric outcomes under which

equal cuts or equal sacrifices are made all round (e.g., Grubb 1990, 277-91). One reason for this may be domestic political pressure. Popular resentment in a nation stuck in an unfavorable equilibrium could destabilize the situation

by altering the domestic political trade-offs. Politicians might even attempt to foster such resentment for tactical reasons.

If the dmps property holds for both countries, domestic environmental

quality will not go up if a country makes unilateral economic sacrifices to

get a cleaner environment; policies eventually converge on equilibrium outcomes underpinned by the need to maximize domestic political support. If the game is prisoner's dilemma, such a move would not result in a new

equilibrium, and things would eventually gravitate back to the status quo. If the game is chicken, and if there is any eventual change from the status quo, the nation foolhardy enough to attempt to make a unilateral sacrifice would be driven by domestic political forces to a new equilibrium with worse environmental quality, as proposition 7 shows. If growth is of increasing marginal political significance (imps) around its status quo level for the




nation making the unilateral sacrifice, equilibria in which both nations' environmental quality is improved may be attainable.10 Bearing this qualifi- cation in mind, policy makers should be wary of unilateral sacrifices.

This is not to suggest that economic sacrifices will never be reciprocated or that unilateral moves have no significant advantages in other respects (Benedick 1991, 145-46): but they require special political circumstances if they are to be reciprocated and if domestic environmental and political benefits are to flow. The requirement is that an equilibrium should exist to the southwest of 0 in the reaction function diagram, implying that both nations have made economic sacrifices. Because the reaction functions pass through 0, this further requires that each reaction function be positively sloped for some range of negative values of the other nation's growth, indicating a willingness to follow suit when the other cuts its growth rate. As proposition 2 shows, this is impossible if VA and VB have the dmps property for QA and QB respectively. It is possible to imagine circumstances in which environmental quality is of imps. For instance; an increase in individuals' perceptions of the importance of cleanup may accompany cleanup itself as the result of the work of concerned scientists-something that probably occurred in the late 1980s in relation to ozone depletion. The effect of this

dynamic preference shift could be modeled by the assumption that environmental quality is of imps within the context of the static support function adopted here.

Figure 5a shows an adjustment path that leads to r^ being positively sloped for some negative values of gB. As gB falls from +x > 0 to -y < 0, A reacts by reducing gA and increasing QA. Take any point p on the reaction function

corresponding to a value of gA in the range (-z, w) within which the

adjustment path is negatively sloped. Then for the values of g^ and QA it is

10. Suppose that VA and VBhave the dmps property for QA and QB respectively, but that gB was of imps in the vicinity of 0. Any point below (to the southwest of) both gA = -SA-gB and gA = -gB/sB is a point at which both A and B's environmental quality is greater than in the status quo. If A's growth rate increased by dgA, B might decrease its own growth rate by an amount greater than dgA.sB. Thus to the northwest of 0 and in its vicinity rB could lie below the line gA = -gB/sB. Assuming the dmps property holds for VA, rA will be below the line g^ = -SA-gB There are two interesting cases. If SA.sB < 1, the reaction functions could intersect in the region to the northwest of 0 where environmental quality is higher then at 0 for both nations if rB is sometimes below gA = -gB/sB and also below gA = -SA.gB (in the region where rA lies). Second, if SA.SB > 1, the reaction functions could intersect in the region to the northwest of 0 where environmental quality is higher than at 0 for both nations if rB is sometimes below gA = -gB/SB (and thus below gA = -SA.gB) and rA is sometimes below gA = -g/sB (as well as being below gA = -sA.gB). At such equilibria gA will always be higher than 0, though. Since the conclusions in proposition 7 about support levels in asymmetric equilibria still hold, being dependent only the dmps property holding for environmental quality in each case, B will end up with lower political support and A with higher political support. If VA showed imps for gA the same effect could occur to the southeast of 0.




(a) QA

A's adjustaent path

(b)

\ s rB .^A

r-_s

{ \ rA

Figure 5: An Assurance Game

necessary that VA shows imps for Q^, or else the adjustment path could not be negatively sloped.

The continuous-strategy generalization of assurance is that there should be at least two equilibria, one of which is Pareto-superior to all the others. Figure 5b shows a pair of reaction functions rA and rB corresponding to the



Ward/ THE GLOBAL COMMONS 225

adjustment path shown in Figure 5b. There are two equilibria besides 0: at p both nations' growth rates are below 0; and at n both nations' growth rates are above 0. Each nation's preference ordering over equilibria is p > 0 > n, because a cut in the other nation's growth rate always leads to a point on the reaction function corresponding to a higher level of support. Equilibrium p is Pareto-superior. Thus the game is assurance. Each nation would reduce its growth rate so long as it was assured that the other would do the same. The problem is that neither side may know for certain that the other will recipro- cate, because there may be incentives to lie about preferences. In this context, the unilateral cuts in pollution levels that some now advocate may be a useful way of testing the waters to see if the other side will follow suit (Ward 1989). Summarizing the argument:

Proposition 8. Equilibria in which QA > 0, QB > 0, gA < 0, and gB < 0 are only possible if QA and QB are of imps for some values of gA and gB less than 0 and values of QA and QB greater than O.1

SOME COMPARATIVE STATISTICS RESULTS

As SA falls toward zero, rA gets flatter, approaching the gB axis; and as SA

goes to infinity, rA gets steeper, approaching the QA axis. Indexing reaction functions by the spillover parameter, rA(sA") will take on the same value when

gB = gB*-SA"/SA' as rA(sA') takes on for gB = gB*. The reason is that when SA = SA",

gB has to change SA'/SA" times as much to bring about a given shift in A's feasibility frontier as when SA = SA'; but the policy position chose by A is determined by where the feasibility frontier lies and would, thus, be the same in both cases.

One implication of this is that A's environmental quality need not increase as SA falls, indicating that A was less vulnerable to flows of pollution from B. Suppose that VA and VB have the dmps property. By proposition 7, if SA

falls far enough, then the game becomes prisoner's dilemma. Assuming that the game was the first variant of chicken discussed above, a large enough fall in SA eliminates equilibria-and thus potential outcomes-in which A's environmental quality is higher than 0. Clearly if the fall in SA is small enough, the number of equilibria remains the same, and the strategic structure is still

11. Clearly, the condition in proposition 8 is necessary but not sufficient. For one thing, there may be equilibria in the northeast or southeast quadrants. Suppose, for example, that the reaction functions in Figure 5b were rA' and rB', generating equilibria at p, 0, and n but an additional equilibria at q. Because the first preferences over equilibria differ, this is another version of chicken.




chicken, because the transformation of rA with shifts in SA is continuous, even though the strategic structure can change discontinuously with large enough shifts in SA. Where the number of equilibria and the strategic structure of the game is constant, we can talk of the game being preserved under the fall in SA. Consider what happens to QA as the equilibrium q shifts to the equilibrium q' in Figure 6 as a consequence of SA falling from SA' to SA". Denote the coordinates of q by (gB*, gA*). Then following the discussion above about the way in which shifts in sA move rA, the coordinates of q", the point on rA(sA") due east of q, are (gB*.SA'/SA", gA*). Total growth at q is gA* + SA'.gB; and at q" it is gA* + sA't.gBSA/SA"; so A's environmental quality is the same at q" as at q'. By proposition 2, the slope of rA(sA") at q" and to the right of q" along the reaction function is greater than -SA". Therefore the equilibrium point q' must be above (to the northeast of) a line drawn through q" with

slope -SA". But points above this line are points associated with higher total

growth than that at q". Hence total growth is higher at q' than at q and A's environmental quality is lower at q' than at q. If q was the outcome of the

original chicken game and q' the outcome of the transformed game under a fall in SA, A's environmental quality would be worse when its vulnerability was lower. Because decreases in SA may not increase A's environmental

quality in equilibrium, the effect of measures to insulate A from B's pollution may be counterproductive.12

Suppose that the iso-support contour of VA through each point in the gA/QA

space got steeper, implying that environmental quality became relatively less

important politically. Referring back to Figure 2, it is clear that the effect of this will be to move the points of tangency between the feasibility frontier and the highest attainable iso-support curve southeast along the feasibility frontier: the iso-support curve would now intersect the feasibility frontier at the point at which it had been at a tangent to it, and to move to the northwest

along the frontier would be to move to a lower iso-support curve. The

implication of this is that for a given level of gB A will choose a higher value of gA and lower value of QA, so that A's adjustment path is shifted to the southeast.13

One might expect that if environmental quality were of lower political significance, this would be reflected in equilibrium outcomes in which QA

12. Using similar methods, it can be shown that so long as dmps holds for VA and VB and the chicken game is preserved as SA falls: A's environmental quality will increase at locally unstable equilibria to the northwest of 0 and at locally stable equilibria to the southeast of 0; A's environmental quality will decrease at locally unstable equilibria to the southeast of 0 and at

locally stable equilibria to the northwest of 0. 13. If the cost of environmental quality went up so that the feasibility frontier through each

point in the gA/QA space becomes flatter, a qualitatively similar shift in rA would occur.




Figure 6: The Effect on Equilibria of a Fall in SA

was lower. However, this is not always true. I assume that VA and VB always have the dmps property and that the change in the slope of the iso-support curves is small enough to ensure that the game is preserved. A's reaction function before the shift is rA and its reaction function after the shift is rA'. Consider a locally stable equilibria like p (or q) in Figure 7. So long as drB/dgB> -SA between p and p' along rB, p' will be a point where total growth is lower than at p. None of the assumptions made here rule out this possibility: because VA and VB have the dmps property and the game is chicken, -SA<

-1/SB, but drB/dgB < -1/SB which does not preclude drB/dgB > -SA between p and p'. The mechanism here is that B may react to A's increase in growth rate and the consequent reduction in its own environmental quality by reducing its own growth rate to compensate; and B's reduction in growth rate can more than compensate for the higher growth in A if it is large enough, as it will be if rB is flat enough. If the outcome of the untransformed game was p and that




^~~B~ rA'

rA

Figure 7: The Effect on Equilibria of a Decrease in the Significance of Environ- mental Quality

of the transformed game was p', the fall in the political significance of environmental quality for A could, paradoxically, result in higher environmental quality in equilibrium.14 Thus we should be cautious about concluding that environmental quality will automatically go up as the importance of the environment as an issue increases.

14. It can be shown that shifts in locally unstable equilibria like n and 0 to the transformed equilibria n' and 0', respectively, will be accompanied by falls in QA.




CONCLUSION

The model generalizes to the n-nation case, as I show in the appendix. The most important results about the links between the support function and the structure of the game continue to hold. In particular: if 0 is an equilibrium, there is a collective action problem; if each nation's support function has the dmps property, the game is either n-person prisoner's dilemma or n-person chicken; and the existence of equilibria in which all nations cut their growth rates is only possible if the dmps property is violated for environmental quality in each nation.

Transboundary pollution politics is actually a repeated game in which national politicians have to make a sequence of decisions through time about policy in relation to the same problem.15 This suggests that solutions enhanc-

ing the environmental quality of all the nations concerned may be attained through conditional cooperation in which each nation cooperates by reducing its emissions as long as the other nation has also done so in the past (Livingston 1989, 89-90; Maler 1990, 92-93). If players discount the future lightly enough, any infinitely repeated noncooperative game with finite strategy sets has such conditionally cooperative, subgame perfect equilibria. The reason is that credible threats of retaliation can be mounted. These allow Pareto-inferior, noncooperative outcomes to be avoided (e.g., Fudenberg and Maskin 1986). Conditional cooperation may also be achieved through inter- linking, cooperation in one game being conditional on others' current or past cooperation in some other game.

Infinitely repeated 2 x 2 prisoner's dilemma games in which players use

pure strategies may have a conditionally cooperative equilibria as well as always having an equilibrium where both players defect in each round (Taylor 1987, chap. 3). There are never asymmetric equilibria in which one

player defects throughout and the other player cooperates throughout. In contrast, in infinitely repeated 2 x 2 chicken, there may be a conditionally cooperative equilibrium, but two such asymmetric equilibria always exist (Ward 1992). This means that the existence of a conditionally cooperative equilibrium does not solve the problem in the case of chicken: because of the existence of asymmetric equilibria, there are still incentives to commit, and the "collision" outcome when both players commit is still possible in the aggravated form in which it is repeated through time (Ward 1992).16 In

15. To say that the game has a time dimension is not to say that it is repeated in exactly the same form (Taylor 1987, 107): in this context environmental damage may be irreversible, so that the current payoffs depend on the past pattern of play.

16. Infinitely repeated n-person prisoner's dilemma games may also have asymmetric equilibria (Taylor 1987, 92-93), so that the same problem arises as in repeated 2 x 2 chicken.




relation to the case depicted in Figure 4a, for example, conditional cooperation might stabilize an outcome mutually preferred to the status quo in which both nations cut their growth rates and both nations improve their environmental quality; but the most asymmetric equilibria, x and z in which one nation increases its growth rate and the other nation reduces its growth rate, would still be stable outcomes through time. The danger is that A would commit to repeatedly choosing a value of gA corresponding to x and B would commit to repeatedly choosing a value of gB corresponding to z, leading to repetition of the Pareto-inferior "collision" outcome of the single-play game.17

Leaving aside the problem of commitment in repeated chicken games for the moment, conditional cooperation will not always be possible: politicians often have short time horizons and know that they have finite political careers; the short-term benefits from breaking away from conditional cooperation may be high; players may not be assured that others will actually conditionally cooperate where there are incentives to bluff about preferences (Ward 1989). Thus it needs to be recognized that the necessary conditions for solutions based on conditional cooperation do not always exist. However, international regimes of cooperation may help promote conditional cooperation where it would otherwise be impossible. Of course, building a new regime of cooperation to help solve a collective action dilemma may generate a second-order collective action problem (Bates 1988). However, once a regime has arisen through negotiation, spontaneous coordination, or imposi- tion (Young 1989, chap. 4) it may foster conditional cooperation in a number of ways: by encouraging trust and longer time horizons; by lowering the incentives to freeride in any particular round through normative, legal, and economic means; by generating incentives to maintain institutional structures that would be destroyed by freeriding; by generating possibilities for issue interlinkage which were not previously available (e.g., Young 1989, chap. 3).

It has not been widely appreciated in the literature that the sort of regime required to solve chicken games may be somewhat different from that needed to solve prisoner's dilemma or assurance games.18 In order to solve interna-

17. Strictly speaking, the arguments below apply to a simplified form of the continuous game in which the one-shot game is such that players can choose a reduced growth strategy, giving a Pareto-improvement on the status quo if both players choose it, and a strategy corresponding to their preferred asymmetric equilibrium.

18. Stein has also argued that the sort of regime needed to achieve solutions in the case of chicken is different. The key point in chicken is that both players wish to avoid a certain collision outcome, so that what is required is a coordination mechanism to ensure that one of the asymmetric equilibria arises (Stein 1982, 309-11). One problem with this suggestion in the context being dealt with here is that such a regime would be likely to deliver lower environmental quality for one nation, as I showed above. Also, Stein ignores repeated play.




tional environmental problems structured like chicken games, besides con- solidating conditional cooperation, the regime would also need to prevent nations committing to increased growth, at least if all-round environmental improvement were the goal. The reason is, as I suggested above, that asymmetric equilibria with stable freeriding will always exist. It has been argued that the way forward to solve global warming is to start with agreements among those countries in the north and the south that are currently willing to proceed. Standouts like the United States, the former Soviet Union, and China will gradually be drawn into the process for fear that they will be left on the sidelines (Paterson and Grubb 1992). This seems to me to be a procedure with a high probability of institutional freeriding, because the asymmetries of the underlying chicken equilibria are reinforced by the exclusion of the freeriders from the international regulatory regime. In contrast, I would argue that nations ought to be subject to conditional future penalties for the use of commitment tactics, not just for noncooperation as such. In order for this to work, the regime ought to provide clear definitions of what commitment means and which commitments will provoke retaliation. Of course, domestic political pressures may shift, changing the nature of the game and eliminating incentives for freeriding and commitment. Such changes were crucial with regard to ozone depletion, as I argued above. However, given that the economic and political stakes are much higher, it is not easy to imagine that this will happen in the short to medium term as far as global warming is concerned. If I am right to suggest that global warming is like a chicken game, any regime that emerges will have to deal with the problem of commitment, and this will pose enormous legal and institutional problems that did not, in the end, arise in the issue of ozone depletion.




APPENDIX The n-Nation Model, n > 2

Let the n nations be denoted by A, B..., I,.., N. Then assume Qi = Qi(GI) where

GI = gi + . . si+ SI - 1.gI - 1 + . . . + SI + 1.gI + 1 + . . + SIN.gN;

QI is a continuous, twice differentiable function of G such that dQi/dGI < 0, d2Qi/dGI2 < 0 and Qi(0) = 0; and si, o > Su > 0 is the spillover parameter from growth in J to I's environmental quality.

Let G' = Gi - gI. Given the assumed form of Qi and VI, for any given level of GI', there exists a unique value of rq, say rI = ri(G,'), that maximizes A's support, generating a reaction function defined over the vector of growth rates {gA, .. ., g- , g + 1 ... gN}.

In just the same way that changes in gB affect A's political feasibility frontier in the two-nation model, increases in GI' move I's political feasibility frontier to the left and decreases in G' move I's political feasibility frontier to the right, a change of x, x > 0, in GI' shifting the frontier -x units to the left, and so on. Thus, using exactly the same methods employed to prove proposition 1, if VI has the dmps property for

Qi and if the growth rates of all other countries strictly increase, r, will strictly decrease; and if the growth rates of all other countries strictly decrease, ri will strictly increase. Proposition 2 also generalizes: if VI has the dmps property, -su < 8rj/8gJ < 0

everywhere for all I ? J. Nash equilibria occur at points common to all reaction functions. Supposing that

0 is an equilibrium. Because the proof of proposition 4 also generalizes, there exists an equal reduction in the growth rate of each nation that leads to an outcome

Pareto-superior to the status quo.19 Hence, if 0 is an equilibrium of the n-nation game, there is a collective action problem; and if 0 is the only equilibrium of the n-nation

game, the structure is n-nation prisoner's dilemma.

Suppose that VI has the dmps property for Qi for each nation I. Then if there is

any other equilibrium, e, besides the status quo, 0, e must be such that there exists at least one nation I for which ri < 0 and at east one nation J for which r > 0. Suppose, first, that rj > 0 for all J. Then the growth rates of all other countries increase as well as I's. But if VI has the dmps property for QI, an increase in the growth rates of all other countries must lead to r < 0: GI' > 0, so I's feasibility frontier moves to the left. This contradicts the original assumption. Suppose, second, that rj < 0 for all J. Then the growth rates of all other nations except I decrease, so that r > 0, which contradicts the original assumption. Hence, if the status quo is not the only equilibrium and environmental quality is of declining relative political importance for each nation, other equilibria will be asymmetric in the sense that at least one nation's growth rate

19. The limit of the ratio of the QA coordinate of the point on V?A to the QA coordinate of the point attained under equal growth cuts all round as the growth cut goes to zero is

1/(1 + SAB + ... + SAI +... + SAN) < 1-




will go up and at least one nation's growth rate will go down if the system moves to the new equilibrium. There is no possibility of each nation cutting its growth rate to achieve better environmental quality unless the dmps property for environmental quality is violated for at least one nation.

Moreover, if VI has the dmps property for both Qi and gi for each nation I, at any alternative equilibrium to the status quo, e, the environmental quality of at least one nation will have fallen. There is at least one nation I such that ri < 0 at any alternative equilibria e. But, as VI has the dmps property for QI, the fall in the chosen value of gi must be a consequence of an increase in G' and an associated shift of I's feasibility frontier to the left. Because VI also has the dmps property for gi, such a leftward shift in I's feasibility frontier provokes I to choose a lower value of Qi, the proof being directly analogous to that for the first part of proposition 1.

If VI has the dmps property for Qi for each nation I and 0 is not the only equilibrium, the game is n-nation chicken. In order to demonstrate this, I first need to show that there are no equilibria that are highest ranked in the preference ordering of all nations, so that some nations may have incentives to commit in order to bring about their preferred equilibria. Let E be the set of equilibria, where E contains at least 0 and one other equilibrium. I prefers equilibrium e' to equilibrium e" if and only if Gi'(e') <

Gi'(e"), where Gi'(e) is the weighted sum of the growth rates of the other nations at equilibrium e. The reason is that increases in GI' move I's feasibility frontier to the left, so that lower support is associated with the new best policy mix chosen. Suppose that e is J's highest ranked equilibrium (or one of J's highest ranked equilibria) and that e is not the status quo, 0. Then there exists some other nation I for which e is not its highest ranked equilibrium. As I showed above, there exists an I such that rI < 0 at e. But this means that I at least prefers 0 to e (and may prefer some third equilibrium to both 0 and e). The reason is that if VI has the dmps property, I would not choose r < 0 at e if Gi'(e) < GI'(0) = 0, because a decrease in GI' leads I strictly to increase r. Hence Gi'(e) > GI'(0), so that I prefers 0 to e. Now suppose 0 is the highest ranked equilibrium (or one of the highest ranked equilibria) of some player I. Then there must be at least one other player J such that there exists an equilibrium e that J strictly prefers to 0. For any equilibrium, e, other than the status quo there must exist some player J such that rj > 0. Vj has the dmps property for Qj so that rj > 0 only if Gj'(e) < Gj'(0), and this implies J strictly prefers e to 0.

To show the game is chicken, I still have to show that if all nations make a commitment to their preferred equilibrium outcome, then the result will be Pareto- inferior to the status quo. Let us call the outcome generated if each nation sticks to its commitment c. There exists at least one nation I that commits to gi > 0, because the status quo is not the preferred outcome of all players. For every other player, J, Gj(c) > 0, because at least I commits to an increase in growth over the status quo and no player commits to a decrease in growth, because each player strictly prefers 0 to any equilibria in which its growth rate falls below 0. For I, at best GI'(c) = 0 if all other nations commit to the status quo and GI'(c) < 0 if at least one other nation commits to increase its growth. Hence for each nation, K, GK'(C) > 0 and for at least one nation, L, GL'(C) > 0, so that C is Pareto-inferior to 0.




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