International Environmental Agreement as an
Equilibrium Choice in a Di¤erential GameK. Akao, H. Sakamoto, K. Uchida. Y. Wasa
May 24, 2019
In the absence of global government, international negotiation and agreement
are the core for the solution of global environmental issues. Since the …rst
international United Nations Conference on the Human Environment in 1972,
several environmental conventions and protocols have been adopted. Some agree
long term targets and some agree to the schedules of pollution control. The
latest former example is the Paris Agreement of the United Nations Framework
Convention on Climate Change (UNFCCC) which sets a long term target thatholding the increase in the global average temperature to well below 2 C above
pre-industrial levels. A prominent and successful latter example is the Montreal
Protocol on Substances that Deplete the Ozone Layer, which has set the years toend the production and the consumption of the ozone depleting substances. The
other latter examples include the Protocol on the Reduction of Sulphur Emissionof the Convention on Long-range Trans-boundary Air Pollution and the Kyoto
Protocol of UNFCCC, although in the Kyoto Protocol, the commitment wasfragile and some countries seceded.
How can we understand these agreements? Dockner and Long (1993) model
an international environmental negotiation as a di¤erential game. Showing theexistence of multiple Markov Nash equilibria, they propose a compelling interpretation about an international environmental negotiation: It may be a preplay
communication to choose a better Nash equilibrium. This interpretation is supported by the notion of payo¤ dominance (Harsanyi and Selten, 1988). The
negotiation can be seen as a process which reduces the uncertainty about players’behaviors and prevents players from going to a less e¢ cient equilibrium.
By following their idea, this paper investigates what equilibrium is chosen as
an international environmental agreement. Formally, this paper studies Docknerand Long’s (1993) symmetric two player di¤erential game model about pollution control. We consider symmetric stationary continuous and discontinuousMPNEs. We also consider symmetric open-loop Nash equilibria (OLNEs).1 We
rank their payo¤s for each pollution level, with special interest in the payo¤
dominant equilibrium. Furthermore, we examine the e¢ ciency of the steadystate of the payo¤ dominant equilibrium.
1 Since we only consider a symmetric equilibrium and a stationary Markov strategy, weomit the terms “symmetric” and “stationary” hereafter.

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In the model, there is a continuum of MPNEs and a unique OLNE. As wewill show, an MPNE, whether it is continuous or discontinuous, has a uniquesteady state which is globally asymptotically stable. Therefore there is a oneto-one correspondence between an MPNE and its long-run consequence. Thisimplies that the agreement of a long-term target such as the 2 C target in Paris
Agreement can be interpreted as a choice of an MPNE. An OLNE, the strategyof which is a function of time, corresponds to an agreement which commits aschedule of actions like the Montreal Protocol.
Dockner and Long (1993) show that, for a speci…c Markov equilibrium thatthey term as the most e¢ cient Markov equilibrium, its steady state, hereafterreferred to as the DL steady state, converges to a Pareto e¢ cient steady stateas the discount rate approaches zero. Therefore, when the discount rate is verylow, we may have an approximately e¢ cient outcome in the long-run even in
the noncooperative circumstances. This result has an important implication,
because, in the cooperative game theory, many studies …nd that a coalition by
many countries is di¢ cult in an international environmental agreement (Barrett,2005).
The DL steady state, however, has been argued in two points. First, theequilibrium strategy is not de…ned over the state space, i.e. it is not subgame
perfect.2 Second, as Rubio and Casino (2002) argue, the steady state is notstable. With a small perturbation, the state variable moves away from thesteady state and reaches to the state where the strategy is not de…ned.
In this paper, we reproduce the DL steady state as the globally asymptotically stable steady state of MPNE, by allowing a discontinuous strategy. Therefore we resolve the criticisms and strengthen the Dockner and Long’s e¢ ciency
result.
In the application of noncooperative dynamic games to international environmental regulation, it is typical to compare the cooperative solution, an OLNE
and a Markov Nash equilibrium (See, for example, Levhari and Mirman, 1980and Ploeg and de Zeeuw, 1992). The most studies examine a speci…c Markov
Nash equilibrium, the one constituted by linear strategies (linear Markov Nashequilibrium). Tsutsui and Mino (1990) showed a continuum of nonlinear Markov
Nash equilibria in a linear quadratic di¤erential game model. Dockner andLong (1993) showed this multiplicity in a pollution control game. Besides theaforementioned result, they showed that the Markov Nash equilibria are moreenvironmentally conservative than the linear Markov Nash equilibrium.3 Rowat(2000) analyzes a more general linear quadratic models encompassing Tsutsuiand Mino (1991) and Dockner and Long (1993) and studies MPNEs of n player
asymmetric games in the context of global warming. In the literature, the pay2 The problem was recognized by Tsutsui and Mino (1991) who showed the multiplicity ofMarkov Nash equilibria in a similar model to Dockner and Long (1993). Dockner and Wagener(2007) use the term “local Markov perfect Nash equilibrium” when a Markov equilibrium islocally subgame perfect.
3 A property of the linear quadratic model as used by Dockner and Long (1993) is thatthe elasticity of marginal utility in control variable is increasing. Wirl (2007) shows that,
if the elasticity of marginal utility is decreasing, the nonlinear Markov strategies are lessenvironmentally conservative than the linear Markov strategy.

2o¤ ranking of Nash equilibria is limited to the comparison of steady states and,except for a few papers such as Dockner and Long (1993), is limited to thecomparison among speci…c equilibria, typically between a linear OLNE and alinear Markov Nash equilibrium.
We rank all continuous and discontinuous MPNEs as well as a unique OLNE
over the state space. Our ranking results are summarized as follows. The DL
steady state divides the state space into two intervals. In the larger interval,the payo¤ ranking between two MPNEs is preserved. Therefore, there is an
MPNE which is the payo¤ dominant equilibrium over the interval. The steadystate of the MPNE may coincide with the DL steady state and it is globally
asymptotically stable. In the smaller interval, the payo¤ dominant MPNE varies
depending on the pollution level. At the pollution level of the DL steady state,keeping the level is the payo¤ dominant. For the payo¤ comparison between
an MPNE and the OLNE, there may be a unique level of pollution stock suchthat if pollution is greater than it, the OLNE dominates the MPNE and viceversa. Even the payo¤ dominant MPNE may be dominated by the OLNE if thepollution stock level is high.
These results indicate that, assuming that an international environmental
negotiation is a process to choose the payo¤ dominant equilibrium, eventually
the DL steady state is chosen as the long-run equilibrium. If the pollution levelis heavier enough than the DL steady state, a renegotiation may occur once toswitch from the OLNE to the MPNE of the DL steady state. If the pollution
level is smaller than the DL steady state, every time a renegotiation occurs, anddue to the cost of negotiation, although we do not incorporate it in the model,
this may imply that no negotiation starts.

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