Stable Linking of the Emission Permit Markets

Abstract

The linking of emission permit markets allows participants in different systems to purchase allowances from each other for the purpose of domestic compliance. A recent paper studied the efficiency gains generated in multilateral linkings between permit markets, and concluded that, despite the linking of all jurisdictions maximizing gains, it is not likely to emerge as it is not the most preferred option by all participants. We formulate the linking problem as a cooperative game and show that the linking of all jurisdictions satisfies core stability criteria. Thus, no subset of jurisdictions would benefit from creating their separate market, and the gains will be maximized. We then extend our analysis to arbitrary partitions and farsightedness level and analyze the stable linking of markets between Australia, Canada, the EU, South Korea, and the U.S. Our results indicate that the most likely stable configuration includes a market that links Australia, the EU, and the U.S., and another in which Canada is linked with South Korea. This scenario leaves about 15% of the potential gains unrealized. To mitigate this issue, we suggest that efficiency gains from market linkage be allocated according to the Shapley value, in which case our results suggest that we would see stable linking of all five jurisdictions and thus increase the efficiency.

1. Introduction

In reaction to the increasing impact of climate change on both human lives and sustainability of the Earth’s biota, many jurisdictions across different continents have employed market-based instruments such as a carbon tax or cap-and-trade (CAT) systems in an attempt to reduce the level of CO${}_2$ in the environment. For example, British Columbia, Argentina, Switzerland, Singapore, and South Africa currently have a carbon tax, while the EU, California, New Zealand, South Korea, and several local pilot programs in China use a CAT system (World Bank [1]).

Carbon taxes are fees imposed by local jurisdictions (cities, states, provinces, countries, etc.) that require companies to pay for each ton of CO${}_2$ emitted into the environment. Under this instrument, companies can release unlimited amounts of CO${}_2$, but they must pay a tax for each released ton. One of the difficulties with implementing a carbon tax is selecting the right tax level. The tax must be low enough to not slow down the economy, but high enough so that the majority of companies will find it in their best interest to reduce emissions rather than pay the tax. Another difficulty in implementing a carbon tax stems from the general unpopularity of taxes among both populations and businesses.

CAT systems, on the other hand, impose a cap on the amount of emissions that a company can release into the environment. A regulator allocates and/or sells through an auction a limited number of permits to companies generating emissions. The companies that reduce their emissions are left with some unused allowances they can sell on carbon markets, while companies that exceed their cap can turn to carbon markets to purchase additional allowances at the current market price. Under this system, the total amount of released emissions is fixed, but the allowance price depends on demand and supply. CAT systems are usually viewed as more efficient, because each company decides the means and the amount of emission reduction to meet its goals. Thus, each company faces a tradeoff—invest in emission reduction or buy additional allowances.

The focus of this paper is the stability of linked CAT markets. Cooperation among different jurisdictions in carbon regulation can exist through the linking of markets for emission permits, which happens when one jurisdiction accepts the allowances of another jurisdiction, or when two jurisdictions have common compliance rules. There are currently several examples of linked markets. At the national level, the EU Emissions Trading System is a set of linked countries. In 2014, California’s CAT Program was linked with Québec’s CAT System. Ontario’s CAT Program joined California and Québec in 2018, but it soon left the alliance. As such, no global market for carbon emissions has yet emerged. Most of the current and emerging CAT systems are unilateral efforts, in which permits issued by one jurisdiction can only be used by entities within the same jurisdiction. However, there are several linkings of permit markets in the planning stage (World Bank [1]).

Why would different jurisdictions want to link their markets? Direct linking allows participants in different systems to purchase allowances from each other for the purpose of domestic compliance. Such a trade results in a price equalization and a cost-effective allocation of abatement efforts across the linked systems. Thus, one benefit of linking is the opportunity to lower the costs of achieving emission reduction goals by shifting reductions among linked systems in a way that minimizes such costs. For example, consider a link between jurisdiction A with a low allowance price and jurisdiction B with a high allowance price. After linking, jurisdiction B benefits from having access to a lower allowance price. There are also disadvantages of linkage. For example, some jurisdictions may benefit more (or less) than others, depending on the mix of buyers and sellers that participate in the linkage, and they may want to switch to a different linked system or change the current linkage composition. Thus, we can observe dynamics in the linked markets, as illustrated in the abovementioned example of California, Québec, and Ontario, and it remains unclear which jurisdictions are eventually going to end up linked in a stable system.

Doda, Quemin, and Taschini [2] (hereafter referred to as DQT) studied the efficiency gains generated in multilateral linkings between permit markets. They decomposed the efficiency gains into two parts, effort- and risk-sharing gains, and analyzed the determinants of linkage gains and preferences. The authors provided a quantitative illustration of their model by analyzing possible linkages across permit markets covering CO${}_2$ emissions from the power sectors of five jurisdictions, assuming that each jurisdiction implements its Paris Agreement pledge. These jurisdictions—Australia, Canada, the EU, South Korea, and the U.S.—use or have considered using both emissions trading and linking. Not surprisingly, they found that the linking of all five jurisdictions can generate the highest total efficiency gains. At the same time, this group is not the most preferred option by all jurisdictions; more precisely, it is not the most preferred linking for any of the five jurisdictions. Thus, DQT state, “…we conclude that 5J [the linking of all five jurisdictions] will therefore not emerge naturally for these five jurisdictions, even though it would generate the largest gains in aggregate.” They further observe that the linkage preferences do not tally—while, for example, the U.S. may want to add the EU to its linked market with Australia, Australia might prefer to add Korea. Thus, it is not clear if any linkage would emerge as stable, and, if some stable linkages exist, how can we identify them. As increasing efficiency in carbon emission reduction is critical for achieving the Paris Agreement’s goals, it is necessary to understand the factors that impact the stability of linked markets and what can be perfomed to induce the formation of larger and more efficient linked markets. While some of the existing research found positive results for symmetric jurisdictions, the same is not true for the case with asymmetric jurisdictions, which is more likely to occur in practice. Therefore, our main goal in this paper is twofold:

• to study stability of different linking configurations with arbitrary (symmetric or asymmetric) features, and
• to identify conditions for stability of the all-inclusive linked market.

In this paper, we used a stability concept from cooperative game theory—the core—to explore if the linkage of all jurisdictions satisfies stability criteria; while the underlying game is superadditive, it is not convex, hence its core may be empty. Moreover, the efficiency gains observed by each jurisdiction do not satisfy properties of a population monotonic allocation scheme, which would be a sufficient condition for core membership. However, we were able to show that the allocation that exactly assigns to each jurisdiction exactly its efficiency gains belongs to the core. This is good news, because it implies that the all-inclusive linked market, which maximizes efficiency gains, is stable. However, it may not be the only potentially stable outcome.

To extend our analysis to arbitrary coalition structures and different levels of farsightedness, we used data from DQT to calculate the emissions for all five jurisdictions. We used a myopic stability concept and a farsighted stability concept—the core and the equilibrium process of coalition formation (EPCF), respectively—to identify stable linked markets for the five jurisdictions. Both approaches identify the same sets of linkages as being potentially stable, but the EPCF provides some additional information on the level of farsightedness required to achieve the stability of a particular linkage, which may help in identifying linkings that are more likely to actually emerge in practice. However, by analyzing the efficiency gains data, we can observe that the largest market, the U.S., tends to receive the smallest gains in all the linkings in which it participates. Thus, we also considered another approach, in which each jurisdiction is allocated a share of gains based on its contribution to achieved gains, through the Shapley value allocation. The Shapley value is usually perceived as fair, and it has been considered as an allocation method in a variety of settings. In our case, the use of this more “fair” allocation increased the number of potentially stable linkings from three to six. We note that the linking of all five jurisdictions is identified as stable under both approaches, but it seems to be easier to implement under the Shapley value allocation. Without a redistribution of gains, the most likely stable outcome includes two groups—one in which Australia is linked with the EU and the U.S., and another in which Canada is linked with South Korea. As the latter case leaves about 15% of efficiency gains unrealized, it would seem opportune to implement the Shapley value allocation of efficiency gains among members of a linked group.

The plan of the paper is as follows. In Section 2 we present a review of the related literature, followed by Section 3, in which we describe the DQT model and the quantitative example introduced in their paper. Section 4 summarizes some concepts from game theory that are used in Section 5 to show the stability of the all-inclusive linking, and, in Section 6 to identify other stable linkings between the five jurisdictions from the DQT example. We analyze two cases: the original one, in which gains are not redistributed and its variation in which gains are distributed according to the Shapley value allocation. We conclude in Section 7. The proof of Proposition 2 is provided in Appendix A.

2. Literature Review

This paper is related to several different streams of literature. We list some papers in each stream below.

There is a vast body of literature analyzing the economics of the linking of emission permit markets. Stevens and Rose [3] presented a generalized dynamic model of GHG emissions trading under constraints on the volume of transactions. Their results indicate that the greatest gains would stem from extending permit trading among industrialized nations, but some sizable gains would also emerge from the inclusion of developing countries. Flachsland et al. [4] concluded that international emissions trading may not be welfare-enhancing for all countries due to the presence of market distortions or terms-of-trade effects. Doda and Taschini [5] showed how, depending on jurisdiction characteristics, one partner in a two-jurisdiction linked market can prefer autarky to linkage. Rose et al. [6] examined a stepwise approach to implementing a global system of GHG emissions trading. They conducted numerical simulations that indicate that a CAT system covering the power and industry sectors in all countries that made pledges to the 2015 Paris Climate Agreement could significantly reduce the associated mitigation costs. These papers have mainly focused on three sources of gains from linking agreements: price convergence, a cost-effective reallocation of abatement efforts, and a reduction of price volatility. Unlike this paper, they do not study stability of linked markets.

Another set of papers have analyzed strategic aspects of permit trading. In this set, papers can be divided into two groups with different conclusions. Earlier papers pointed primarily to potential negative consequences of linkage, such as domestic emission cap relaxation. Carraro and Siniscalco [7] showed that linked markets tend to involve only a fraction of negotiating jurisdictions, and that financial transfers and additional commitments are required to induce other jurisdictions to join. Helm [8] compared endogenous choices of tradable and non-tradable emission allowances by jurisdictions and found that the cost savings of trading do not necessarily lead to less pollution. Pan et al. [9] analyzed the experiences, lessons, and insights from three key global carbon markets—North America, the EU and China—in terms of the barriers to linking the global carbon market. Our work focuses on identifying stable linked markets with positive consequences.

The second group found that permit market linkage can induce technology transfers and greater low-carbon investments, leading to reduced emissions and lower caps. Helm and Pichler [10] analyzed technology transfer between more developed and less developed countries, and concluded that a joint system of technology transfer and permit trading can lead to lower overall emissions and the efficiency gains on the permit market. Holtsmark and Midttømme [11] showed that linking of permit markets leads to an increased investment in green energy capacity in all permit market jurisdictions, and jurisdictions with a higher green energy capacity respond by issuing fewer permits in the future. Antoniou et al. [12] showed benefits of permit markets linkage in a two-jurisdiction model under a symmetric scenario; however, in the asymmetric case the linkage may fail to form. Li et al. [13] compared three linked markets (China; China and the U.S.; China, the U.S., and the nations in Southeast Asia) and concluded that emissions and energy consumption outcomes would be similar in the bilateral and multilateral scenarios, but economic outcomes are more favorable in the multilateral linkage. These papers in general do not consider stability; instead, they analyze the effects of linkage in a given set of jurisdictions, allowing that in some instances some jurisdictions may benefit from defection.

The literature focusing on the stability of linked markets is of special interest to us. The first paper in this area is Biancardi and Villani [14], which used a two-stage, non-cooperative game theoretic model to study international environmental agreements in a pollution abatement model with a quadratic cost function and symmetric jurisdictions. Each jurisdiction first decides whether to join the coalition and then chooses its quantity of pollution abatements. The authors used a farsighted stability concept, the largest consistent set (LCS), and derived sufficient conditions for the stability of the grand coalition (the linking of all permit markets). Their model does not assume that emissions are traded between jurisdictions, but rather a jurisdiction that individually joins a coalition, chooses its abatement level and sees the benefits from the aggregate abatement levels of all coalition members. This differs from our model assumptions. In addition, we analyze a more realistic setting with asymmetric jurisdictions. The paper most closely related to our work is Heitzig and Kornek [15], which analyzed linking CAT markets by using a probabilistic EPCF. The authors considered a scenario with six jurisdictions—China, Europe, former Soviet Union, India, Japan, and the U.S.—and assumed that each jurisdiction has the same level of farsightedness. They assign weights to each jurisdiction, which in turn determines the jurisdictions’ bargaining powers, and are used to determine the transition probabilities in the EPCF. The authors assumed that a jurisdiction can propose its favorite move, which does not have to include itself: for instance, China can propose a move that requires India and the U.S. to link their markets, which does not seem realistic. Another questionable assumption of this paper is that payoffs in a coalition depend on a sequence of moves. Thus, a payoff that, say, Japan receives when all six jurisdictions are linked is different if Japan joined the linkage second or if Japan joined the linkage last. This type of cooperative-gain allocation can provide an incentive for jurisdictions to delay their participation in linkage and thus increase their gains, which is not a desirable scenario. Finally, the authors assumed that when a single jurisdiction leaves a coalition, the coalition dissolves. As previously illustrated in the example of California, Québec, and Ontario, when Ontario left the linked markets group, California and Québec remained linked in a common group, hence this assumption, too, does not seem realistic. In our analysis, we use the deterministic EPCF, and assume that each coalition proposing a specific move participates in that move and a defection from a linkage does not lead to the dissolution of the linkage between the remaining jurisdictions.

In the operations management (OM) literature, several studies of CAT markets have investigated how companies should modify their production decisions in reaction to carbon control mechanisms. For example, Subramanian et al. [16] studied a three-stage game in a symmetric oligopoly, Benjaafar et al. [17] looked at simple inventory control models that account for both cost and carbon footprint, and Gong and Zhou [18] integrated emissions trading and dynamic inventory management. More recently, Park et al. [19] studied the impact of carbon costs on supply chain structure, Drake et al. [20] studied the impact of carbon tax and CAT regulation on a company’s technology choice and capacity decisions, Yuan et al. [21] studied how companies participating in CAT markets should balance permit trading and production faced with fluctuating prices and random product demand, and Anand and Giraud-Carrier [22] analyzed the impact of carbon taxes and CAT systems on oligopolistic firms. We were not able to identify any OM papers that analyzed the linking of CAT markets, which can have a significant impact when firms establish their facility location and perform sourcing decisions.

3. The Model

DQT study a model with n jurisdictions, $N=\{1,\dots ,n\}$, in which emissions by jurisdiction i are defined by level $q_i\ge 0$ and a random shock ${\theta }_i$. It is assumed that, for this random shock,

$$\begin{array}{c}\hfill \mathbf{E}\left[{\theta }_i\right]=0,\mathrm{Var}\left[{\theta }_i\right]={\sigma }_i^2,\mathrm{and}\mathrm{Cov}[{\theta }_i,{\theta }_j]={\rho }_{ij}{\sigma }_i{\sigma }_j,{\sigma }_i\ge 0,{\rho }_{ij}\in [-1,1].\end{array}$$

The benefits stemming from emissions for jurisdiction i are calculated as

$$\begin{array}{c}\hfill B_i(q_i,{\theta }_i)=({\beta }_i+{\theta }_i)q_i-q_i^2/\left(2{\gamma }_i\right),\end{array}$$

(1)

where ${\beta }_i>0$ and ${\gamma }_i>0$. ${\gamma }_i$ represents marginal abatement technology; thus, ${\gamma }_i>{\gamma }_j$ implies that i can use a lower cost abatement technology than j. It is easy to see that (1) is maximized at $q_i^{*}={\gamma }_i({\beta }_i+{\theta }_i)$.

DQT assume that, under autarky, each jurisdiction has an exogenous and fixed emissions cap, ${\omega }_i$, and they express this cap as proportional to the jurisdiction’s technology, ${\omega }_i={\alpha }_i{\gamma }_i,{\alpha }_i\in (0,{\beta }_i)$. Autarky permit prices are positive, provided by $p_i^{*}={\overline{p}}_i+{\theta }_i>0$, where ${\overline{p}}_i={\beta }_i-{\alpha }_i$. DQT assume that ${\theta }_i>{\alpha }_i-{\beta }_i$ to focus on the interior equilibria.

Suppose that jurisdictions in $S\subseteq N$ link their permit markets. DQT assume that the equilibrium permit price, $p^S$, and equilibrium emission levels for each jurisdiction, $q_i^S$, are characterized by the equalization of marginal benefits across jurisdictions in S and market clearing:

$${\beta }_i+{\theta }_i-q_i^S/{\gamma }_i=p^S\mathrm{for}\mathrm{all}i\in S,\mathrm{and}\sum _{i\in S}{q}_i^S=\sum _{i\in S}{\omega }_i.$$

Let

$${\Gamma }_S=\sum _{i\in S}{\gamma }_i,{\widehat{\mathsf{\Theta }}}_S=\frac{1}{{\Gamma }_S}\sum _{i\in S}{\gamma }_i{\theta }_i,\mathrm{and}{\overline{p}}^S=\frac{1}{{\Gamma }_S}\sum _{i\in S}{\gamma }_i{\overline{p}}_i;$$

then

$$\begin{array}{c}\hfill p^S={\overline{p}}^S+{\widehat{\mathsf{\Theta }}}_S,q_i^S-{\omega }_i={\gamma }_i(p_i-p^S).\end{array}$$

Proposition 1 in DQT shows that the expected efficiency gains for jurisdiction i with respect to its benefits in autarky can be expressed as

$$\begin{array}{c}\hfill \mathbf{E}\left[{\delta }_{S,i}\right]=\mathbf{E}[B_i(q_i^S,{\theta }_i)-p^S(q_i^S-{\omega }_i)-B_i({\omega }_i,{\theta }_i)]={\gamma }_i\frac{{({\overline{p}}_i-{\overline{p}}^S)}^2+\mathrm{Var}({\theta }_i-{\widehat{\mathsf{\Theta }}}_S)}{2}.\end{array}$$

(2)

The authors refer to ${({\overline{p}}_i-{\overline{p}}^S)}^2$ as the effort-sharing gains, while they refer to $\mathrm{Var}({\theta }_i-{\widehat{\mathsf{\Theta }}}_S)$ as the risk-sharing gains. Effort-sharing gains are related to the intra-group variation in the expected autarky prices and are independent of the shock structure. The risk-sharing gains are related to jurisdictional shock characteristics, and are independent of jurisdictions’ autarky prices.

The authors refer to the linked market between jurisdictions in set S as S-linkage, and, in their Proposition 2 they show that any S-linkage can be decomposed into its internal bilateral linkages. In other words, if we denote

$${\Delta }_{\{i,j\}}={\delta }_{\{i,j\},i}+{\delta }_{\{i,j\},j},$$

then the gains that jurisdiction i generates in the S-linkage can be expressed as

$$\begin{array}{c}\hfill {\delta }_{S,i}=\frac{1}{{\Gamma }_S^2}\sum _{j\in S\backslash \left\{i\right\}}\left\{{\Gamma }_{S\backslash \left\{i\right\}}{\Gamma }_{\{i,j\}}{\Delta }_{\{i,j\}}-\frac{{\gamma }_i}{2}\sum _{k\in S\backslash \{i,j\}}{\Gamma }_{\{j,k\}}{\Delta }_{\{j,k\}}\right\}.\end{array}$$

(3)

Further, adding equation (3) over all $i\in S$ provides

$$\begin{array}{c}\hfill {\Delta }_S=\sum _{i\in S}{\delta }_{S,i}=\frac{1}{{\Gamma }_S^2}\sum _{i,j\in S^2,i<j}{\Gamma }_{\{i,j\}}{\Delta }_{\{i,j\}},\end{array}$$

(4)

which means that the aggregate S-linkage gains can be expressed as a weighted sum of gains of all the bilateral linkages in S.

In order to quantitatively illustrate their model, DQT obtained estimates of the annual baseline power emissions and marginal abatement cost curves in 2030 from Enerdata, a private research and consulting firm whose clients include several national governments, UNDP, and the European Commission, for the power sector of five jurisdictions with similar levels of development. The five jurisdictions either use or have considered emissions trading and linking and include Australia (A), Canada (C), the European Union (E), South Korea (K), and the United States (U) (In the text, we will also use the EU and the U.S. to refer to the European Union and the United States, respectively). Enerdata also provided the authors with estimates of the annual emission caps consistent with the 2030 target achievement as defined in the Nationally Determined Contributions and announced at the Conference of Parties in Paris. According to the authors, the POLES model of Enerdata, used to obtain estimates of the annual emission caps, is a black box for them, and they used its results as provided. DQT used these data to compute the parameters of their model and calibrated the shock properties using the residuals from the regression of historical emissions on time and time squared with data from the International Energy Agency. Their results are provided in

Table 1. Annual baseline power emissions, emission caps, expected autarky permit prices, calibrated flexibility coefficients, linear intercepts, and standard deviations of autarky prices.

	Unit	E	A	U	C	K
${\overline{q}}_i$	${10}^6{\mathrm{tCO}}_2$	841.8	171.3	1946.8	90.2	287.5
${\omega }_i$	${10}^6{\mathrm{tCO}}_2$	724.1	150.1	1469.3	66.3	225.8
${\overline{p}}_i$	2005 USD/${\mathrm{tCO}}_2$	89.8	27.1	92.8	113.7	92.6
${\gamma }_i$	${10}^3{\left({\mathrm{tCO}}_2\right)}^2/$$	1309.9	784.1	5146.4	210.2	665.3
${\beta }_i$	2005 USD/${\mathrm{tCO}}_2$	642.5	218.5	378.2	428.9	432.0
${\sigma }_i$	2005 USD/${\mathrm{tCO}}_2$	31.4	11.8	21.9	56.3	48.4

and

Table 2. Pairwise correlation coefficients ${\rho }_{ij}$.

	E	A	U	C	K
E	1.00
A	0.36	1.00
U	0.07	0.42	1.00
C	0.18	0.18	0.43	1.00
K	−0.15	0.24	0.51	0.00	1.00

.

DQT used (2) to calculate the gains obtained in all possible partitions of the set of five jurisdictions, and they provided the information for three of them: $\left\{\mathit{EAUCK}\right\}$ (all five jurisdictions linked together); $\{\mathit{ECK},\mathit{AU}\}$ (the EU, Canada, and South Korea linked in one market, Australia and the U.S. in another), which generates the highest gains if no jurisdiction is an autarky; and $\{\mathit{UCK},\mathit{EA}\}$ (the U.S., Canada, and South Korea in one market, the EU and Australia in the other), which generates the lowest gains if no jurisdiction is an autarky. The authors observed that the total gains are maximized if all five jurisdictions are linked, in $\left\{\mathit{EAUCK}\right\}$. At the same time, $\left\{\mathit{EAUCK}\right\}$ may not generate the largest gains for any specific jurisdiction. For instance, South Korea would prefer to be linked with the EU and Canada only, as it would achieve higher gains. DQT also noted that a monotonic relationship between the magnitude of efficiency gains and cardinality of a group does not exist: while adding the EU to $\left\{\mathit{AU}\right\}$ increases the U.S. gains, adding South Korea or Canada decreases them. At the same time, Australia would rather join South Korea or Canada than the EU. Thus, DQT concluded that $\left\{\mathit{EAUCK}\right\}$ will not emerge naturally for these five jurisdictions. We show, in this paper, that the all-inclusive linking is, in fact, stable. In addition, DQT do not explore what linkings may eventually emerge as stable. In this paper, we first calculate the jurisdictions’ gains for all possible linkings, and then identify linkings that could be stable and the conditions under which such stability may happen.

In

Table 3. EPM-DQT game: gains indifferent partitions (in billions of 2005 USD per year).

Partition	E	A	U	C	K	Total
$\{UCK,EA$}	439.85	734.81	51.48	311.98	466.42	2004.54
$\{ACK,EU$}	573.58	697.02	145.99	548.22	551.84	2516.64
$\{AUK,EC$}	51.23	1445.08	214.76	319.23	486.82	2517.12
$\{AUC,EK$}	281.79	1442.02	175.30	341.31	554.81	2795.23
$\{ECK,AU$}	291.89	1392.99	212.23	332.45	580.08	2809.65
$\{EUK,AC$}	645.67	185.19	127.92	690.81	524.22	2173.81
$\{EUC,AK$}	579.59	536.75	149.67	297.59	632.60	2196.20
$\{EAK,UC$}	491.23	907.63	11.99	293.54	705.67	2410.06
$\{EAC,UK$}	351.21	870.45	58.00	468.24	448.62	2196.52
$\{EAU,CK$}	565.95	1405.88	324.36	361.57	114.24	2771.98
$\{EAUC,K$}	564.50	1446.56	294.93	337.56	-	2643.55
$\{EAUK,C$}	627.50	1442.18	277.21	-	551.89	2898.78
$\{EACK,U$}	436.89	1011.47	-	440.69	689.74	2578.80
$\{EUCK,A$}	646.86	-	123.10	310.47	537.01	1617.44
$\{AUCK,E$}	-	1486.56	173.84	352.12	497.47	2510.01
$\{EAUCK$}	623.40	1477.89	246.82	345.42	559.85	3253.39

, we present gains for the partitions that include all possible linkings (Note that, if a specific partition is not included in the table, its payoffs can be easily obtained from Table 3, as the gains generated by members linked in one market do not depend on the linkages of other jurisdictions. Thus, if we know the gains that each jurisdiction generates in $\{\mathit{UCK},\mathit{EA}\}$ and in $\{\mathit{EAK},\mathit{UC}\}$, we can easily find gains in $\{\mathit{EA},\mathit{UC},\mathit{K}\}$). We can see from the table that the partitions with highest gains all include the linkages of Australia, the jurisdiction with the highest price, and the U.S., the jurisdiction with the largest market.

We also calculated the linkage equilibrium prices for each group of linked markets, which are presented in

Table 4. Linkage equilibrium prices for different linked markets (in 2005 USD/tCO${}_2$).

Market	Price	Market	Price	Market	Price	Market	Price
$\left\{\mathit{EA}\right\}$	66.3	$\left\{\mathit{UC}\right\}$	93.6	$\left\{\mathit{EUC}\right\}$	92.9	$\left\{\mathit{EAUC}\right\}$	85.9
$\left\{\mathit{EU}\right\}$	92.2	$\left\{\mathit{UK}\right\}$	92.8	$\left\{\mathit{EUK}\right\}$	92.2	$\left\{\mathit{EAUK}\right\}$	85.8
$\left\{\mathit{EC}\right\}$	93.1	$\left\{\mathit{CK}\right\}$	97.7	$\left\{\mathit{ECK}\right\}$	93.0	$\left\{\mathit{EACK}\right\}$	75.6
$\left\{\mathit{EK}\right\}$	90.7	$\left\{\mathit{EAU}\right\}$	85.1	$\left\{\mathit{AUC}\right\}$	85.1	$\left\{\mathit{EUCK}\right\}$	92.8
$\left\{\mathit{AU}\right\}$	84.1	$\left\{\mathit{EAC}\right\}$	70.6	$\left\{\mathit{AUK}\right\}$	85.0	$\left\{\mathit{AUCK}\right\}$	85.9
$\left\{\mathit{AC}\right\}$	45.4	$\left\{\mathit{EAK}\right\}$	72.7	$\left\{\mathit{ACK}\right\}$	64.3	$\left\{\mathit{EAUCK}\right\}$	86.5
$\left\{\mathit{AK}\right\}$	57.2			$\left\{\mathit{UCK}\right\}$	93.5

.

As expected, we can see from the table that Australia’s participation in a linked market is critical for a significant price reduction in that market, as Australia’s price is significantly lower than any other autarky price (see Table 1), and the price wedge determines the effort-sharing gain (see Table 3). Thus, linkages including Australia realize higher effort-sharing gains.

Table 3 also shows significant differences between the gains observed by different jurisdictions. For example, Australia’s gains significantly dominate those realized by any other jurisdiction; most of these gains are captured in effort sharing, as previously mentioned. On the other hand, the U.S. has the lowest shares among all jurisdictions. When analyzing the details of the U.S. gains, we can observe that in all linkages that include the EU, the biggest portion of the gains observed by the U.S. come from risk-sharing efforts, as the EU and the U.S. have a very low correlation coefficient (0.07). As the U.S. represents the largest emissions markets (exceeding the remaining four jurisdictions combined), it significantly contributes to the gains observed by other linkage partners. As such, it may be in the interest of other jurisdictions to transfer some of their gains to the U.S., in order to incentivize its participation in linkage. We will discuss alternative possible gain allocations in the next section, and then apply it to our data.

We next show in

Table 5. Jurisdictions’ preferences for different linkages and corresponding gains (in billions of 2005 USD per year).

E		A		U		C		K
Partition	Gain	Partition	Gain	Partition	Gain	Partition	Gain	Partition	Gain
$\{\mathit{EUCK},\mathit{A}\}$	646.86	$\{\mathit{AUCK},\mathit{E}\}$	1486.56	$\{\mathit{EAU},\mathit{CK}\}$	324.36	$\{\mathit{EUK},\mathit{AC}\}$	690.81	$\{\mathit{EAK},\mathit{UC}\}$	705.67
$\{\mathit{EUK},\mathit{AC}\}$	646.67	$\left\{\mathbf{EAUCK}\right\}$	1477.89	$\{\mathit{EAUC},\mathit{K}\}$	294.93	$\{\mathit{ACK},\mathit{EU}\}$	548.22	$\{\mathit{EACK},\mathit{U}\}$	689.74
$\{\mathit{EAUK},\mathit{C}\}$	627.50	$\{\mathit{EAUC},\mathit{K}\}$	1446.56	$\{\mathit{EAUK},\mathit{C}\}$	277.21	$\{\mathit{EAC},\mathit{UK}\}$	468.24	$\{\mathit{EUC},\mathit{AK}\}$	632.60
$\left\{\mathbf{EAUCK}\right\}$	623.40	$\{\mathit{AUK},\mathit{EC}\}$	1445.08	$\left\{\mathbf{EAUCK}\right\}$	246.82	$\{\mathit{EACK},\mathit{U}\}$	440.69	$\{\mathit{ECK},\mathit{AU}\}$	580.08
$\{\mathit{EUC},\mathit{AK}\}$	579.59	$\{\mathit{EAUK},\mathit{C}\}$	1442.18	$\{\mathit{AUK},\mathit{EC}\}$	214.76	$\{\mathit{EAU},\mathit{CK}\}$	361.57	$\left\{\mathbf{EAUCK}\right\}$	559.85
$\{\mathit{ACK},\mathit{EU}\}$	573.58	$\{\mathit{AUC},\mathit{EK}\}$	1442.02	$\{\mathit{ECK},\mathit{AU}\}$	212.23	$\{\mathit{AUCK},\mathit{E}\}$	352.12	$\{\mathit{AUC},\mathit{EK}\}$	554.81
$\{\mathit{EAU},\mathit{CK}\}$	565.95	$\{\mathit{EAU},\mathit{CK}\}$	1405.88	$\{\mathit{AUC},\mathit{EK}\}$	175.30	$\left\{\mathbf{EAUCK}\right\}$	345.42	$\{\mathit{EAUK},\mathit{C}\}$	551.89
$\{\mathit{EAUC},\mathit{K}\}$	564.50	$\{\mathit{ECK},\mathit{AU}\}$	1392.99	$\{\mathit{AUCK},\mathit{E}\}$	173.84	$\{\mathit{AUC},\mathit{EK}\}$	341.31	$\{\mathit{ACK},\mathit{EU}\}$	551.84
$\{\mathit{EAK},\mathit{UC}\}$	491.23	$\{\mathit{EACK},\mathit{U}\}$	1011.47	$\{\mathit{EUC},\mathit{AK}\}$	149.67	$\{\mathit{EAUC},\mathit{K}\}$	337.56	$\{\mathit{EUCK},\mathit{A}\}$	537.01
$\{\mathit{UCK},\mathit{EA}\}$	439.85	$\{\mathit{EAK},\mathit{UC}\}$	907.63	$\{\mathit{ACK},\mathit{EU}\}$	145.99	$\{\mathit{ECK},\mathit{AU}\}$	332.45	$\{\mathit{EUK},\mathit{AC}\}$	524.22
$\{\mathit{EACK},\mathit{U}\}$	436.89	$\{\mathit{EAC},\mathit{UK}\}$	870.45	$\{\mathit{EUK},\mathit{AC}\}$	127.92	$\{\mathit{AUK},\mathit{EC}\}$	319.23	$\{\mathit{AUCK},\mathit{E}\}$	497.47
$\{\mathit{EAC},\mathit{UK}\}$	351.21	$\{\mathit{UCK},\mathit{EA}\}$	734.81	$\{\mathit{EUCK},\mathit{A}\}$	123.11	$\{\mathit{UCK},\mathit{EA}\}$	311.98	$\{\mathit{AUK},\mathit{EC}\}$	486.82
$\{\mathit{ECK},\mathit{AU}\}$	291.89	$\{\mathit{ACK},\mathit{EU}\}$	697.02	$\{\mathit{EAC},\mathit{UK}\}$	58.00	$\{\mathit{EUCK},\mathit{A}\}$	310.47	$\{\mathit{UCK},\mathit{EA}\}$	466.42
$\{\mathit{AUC},\mathit{EK}\}$	281.79	$\{\mathit{EUC},\mathit{AK}\}$	536.75	$\{\mathit{UCK},\mathit{EA}\}$	51.48	$\{\mathit{EUC},\mathit{AK}\}$	297.59	$\{\mathit{EAC},\mathit{UK}\}$	448.62
$\{\mathit{AUK},\mathit{EC}\}$	51.23	$\{\mathit{EUK},\mathit{AC}\}$	185.19	$\{\mathit{EAK},\mathit{UC}\}$	11.99	$\{\mathit{EAK},\mathit{UC}\}$	293.54	$\{\mathit{EAU},\mathit{CK}\}$	114.24
$\{\mathit{AUCK},\mathit{E}\}$	-	$\{\mathit{EUCK},\mathit{A}\}$	-	$\{\mathit{EACK},\mathit{U}\}$	-	$\{\mathit{EAUK},\mathit{C}\}$	-	$\{\mathit{EAUC},\mathit{K}\}$	-

the rankings of jurisdictions’ preferences for different linkages. The linkage of all five jurisdictions is shown in bold. Note that this is not the most preferred outcome for any of the jurisdictions. We also observe that some of DQT’s intuition when considering risk sharing only does not hold in this example, because of the effort-sharing component. For instance, one intuition was that a jurisdiction with low $\sigma$ prefers to be linked with a single jurisdiction with a low $\sigma$. However, the jurisdiction with the lowest $\sigma$ value, Australia, has the first linkage between the two jurisdictions ranked only at the eighth place, hence it prefers to be linked with more than one additional jurisdiction. However, Australia has the lowest autarky price, which is important for the effort-sharing component and drives it to prefer linking with multiple jurisdictions. Similarly, the U.S. has a low $\sigma$ value, which should drive it to prefer two-member linkings, but, at the same time, it has the highest $\gamma$, for which linkage with multiple jurisdictions works better. Thus, it can be difficult to infer the preferences of different jurisdictions a priori, as multiple parameters must be considered simultaneously. In addition, it can be seen that preferences vary significantly across different jurisdictions, so it is not obvious what linkage may emerge as stable.

As we want to gain a better understanding of these dynamics, we will introduce some concepts from game theory in the next section, and apply them to our model.

4. Game Theoretical Concepts

In this section, we first introduce some general terminology from cooperative game theory and then discuss some stability concepts used to analyze the stability of linked permit markets. We first discuss myopic stability, captured through the concepts of the core. We then present the equilibrium process of coalition formation (EPCF), introduced by Konishi and Ray [23], as an alternative farsighted stability concept that enables us to measure the level of farsightedness required to obtain the stability of specific outcomes.

Let $N=\{1,2,\dots ,n\}$ be the set of players. A subset $S\subseteq N$ is called a coalition, and N is called the grand coalition. A pair $(N,v)$, where $v:2^N\to \mathbf{R}$ is a function such that $v(\varnothing )=0$, is called a cooperative game, and v is called the characteristic function of the game. A coalition structure, $\mathcal{Z}$, is a partition on N. That is, $\mathcal{Z}=\{S_1,\dots ,S_m\},{\cup }_{i=1}^m{S}_i=N,S_j\cap S_k=\varnothing ,j\ne k$. For a given $\mathcal{Z}$, we denote $S_{\mathcal{Z}}\left(i\right)=\{S_k\in \mathcal{Z}:i\in S_k\}$. We denote by $\mathbf{Z}$ set of all coalition structures on N.

In our model, we will let N denote the set of all jurisdictions, and $v\left(S\right)$ will denote the total efficiency gains when the jurisdictions in S link their permit markets, $v\left(S\right)={\sum }_{i\in S}\mathbf{E}\left[{\delta }_{S,i}\right]$. We will refer to $(N,v)$ as the emission permit markets game, or the EPM game; the specific instance of the game that considers the above-mentioned five jurisdictions will be referred to as the EPM-DQT game. For every jurisdiction $i\in N$, let ${\Pi }_i^{\mathcal{Z}}$ denote i’s gain in structure $\mathcal{Z}\in \mathbf{Z}$. Thus, ${\Pi }_i^{\mathcal{Z}}=\mathbf{E}\left[{\delta }_{S_{\mathcal{Z}}\left(i\right),i}\right]$. Denote by ${\rightharpoonup }_S$ defection by $S\subseteq N$: ${\mathcal{Z}}_1{\rightharpoonup }_S{\mathcal{Z}}_2$ if structure ${\mathcal{Z}}_2$ is obtained when S defects from ${\mathcal{Z}}_1$.

4.1. Allocation Rules and the Core

A mapping $\Phi$ that assigns to every cooperative game $(N,v)$ a vector $\phi =({\phi }_1,\dots ,{\phi }_n)\in {\mathbf{R}}^N$ is called an allocation rule, and $\phi$ is called an allocation. Each player has preferences over the possible coalition structures that are determined by the coalition to which they belong and the allocation rule that is used. We will represent preferences by a complete, reflexive, and transitive binary relation ${⪰}_i$ defined on the set $\{S\subseteq N:i\in S\}$. We will use ${\succ }_i$ to denote the associated asymmetric relation of the strict preferences. If ${\mathcal{Z}}_1{\succ }_i{\mathcal{Z}}_2$ for all $i\in S$, we write ${\mathcal{Z}}_1{\succ }_S{\mathcal{Z}}_2$.

We say that an allocation $\phi$ is a member of the core of $(N,v)$ if it satisfies

$$\begin{array}{cc}\hfill \sum _{i\in S}{\phi }_i\ge v\left(S\right)& \forall S\subseteq N,{\sum }_{i=1}^n{\phi }_i\left(v\right)=v\left(N\right).\end{array}$$

(5)

When core allocations are used, no subset of players has an incentive to secede from the grand coalition and form subcoalitions. However, the core can, in general, be empty. A game is said to be superadditive if for each $S,T\subseteq N$ such that $S\cap T=\varnothing$, it holds that $v(S\cup T)\ge v\left(S\right)+v\left(T\right)$. Thus, as the value of two disjoint coalitions after a merger is never less than the sum of their values before a merger, when a game is superadditive, it is reasonable to expect a grand coalition formation. However, this is not always the case. A particular class of superadditive games is the class of convex games. A game is said to be convex if, for each $S,T\subseteq N$, it holds that $v(S\cup T)+v(S\cap T)\ge v\left(S\right)+v\left(T\right).$ The core of a convex game is non-empty.

Sprumont [24] introduced the concept of Population Monotonic Allocation Scheme (PMAS) in cooperative games. A vector ${\left(y_i^S\right)}_{i\in S\subset N,S\ne \varnothing }$ is a PMAS of the game $(N,v)$ if and only if it satisfies the following conditions:

(i)

${\sum }_{i\in S}{y}_i^S=v\left(S\right)$ for all $\varnothing \ne S\subseteq N$, and

(ii)

For all $\varnothing \ne T\subseteq S\subseteq N$ and for all players $i\in T$, $y_i^S\ge y_i^T$.

Sprumont [24] shows that a core allocation, $\phi$, is reached through a PMAS of the game if there exists a PMAS $y={\left(y_i^S\right)}_{i\in S\subset N,S\ne \varnothing }$ such that $y_i^N={\phi }_i$. However, there are games with nonempty cores without PMAS.

The Shapley value (Shapley [25]) is an allocation rule that satisfies the following axioms:

• Symmetry: for all permutations $\pi$ of N, ${\Phi }_{\pi \left(i\right)}\left(\pi v\right)={\Phi }_i\left(v\right)$, where $\pi v$ is defined as follows: for any $S=\{i_1,i_2,\dots ,i_s\}$, $\pi v\left(\{\pi \left(i_1\right),\pi \left(i_2\right),\dots ,\pi \left(i_s\right)\}\right)=v\left(S\right)$;
• Null player: if i is a null player, i.e., $v(S\cup \{i\left\}\right)=v\left(S\right)$ for all $S\subset N$, then ${\Phi }_i\left(v\right)=0$;
• Efficiency: ${\sum }_N{\Phi }_i\left(v\right)=v\left(N\right)$;
• Additivity: ${\Phi }_i(v+w)={\Phi }_i\left(v\right)+{\Phi }_i\left(w\right)$ for any pair of cooperative games $(N,v)$ and $(N,w)$.

The Shapley value is obtained by averaging the marginal contributions for all possible orderings,

$$\begin{array}{c}\hfill {\Phi }_i\left(v\right)=\sum _{\{S:i\in S\}}\frac{\left(\right|S|-1)!(n-|S\left|\right)!}{n!}(v\left(S\right)-v(S\backslash \left\{i\right\}).\end{array}$$

(6)

Shapley [25] showed that (6) is the unique allocation rule that satisfies the above four axioms. Hereafter, ${\varphi }_i$ is used to denote the allocation that jurisdiction i receives according to the Shapley value. Although Shapley value allocations in some instances may satisfy the core constraints, this is not the case in general. However, the Shapley value is always contained in the core of a convex game (Shapley [26]. The Shapley allocation provides a direct link between players’ marginal contributions and their allocations, and is therefore consider to be“fair.” As such, the allocation is used as a profit or cost-sharing mechanism in trans-shipment problems (e.g., Granot and Sošić [27]; Sošić [28]), inventory pooling (Kemahlioğlu-Ziya and Bartholdi [29]), and supply chain information sharing (Leng and Parlar [30]), among other examples.

In what follows, we also want to analyze the stability of coalition structures other than the grand coalition. Given a coalition structure $\mathcal{Z}$ and allocation $\phi$, if there is a $T\subseteq N$ such that $T{\succ }_i{S}_{\mathcal{Z}}\left(i\right)$ for all $i\in T$, we say that T blocks $\mathcal{Z}$. If $\nexists T\subseteq N$ which blocks $\mathcal{Z}$, we say that $\mathcal{Z}$ is the core coalition structure for $\phi$.

4.2. The EPCF

The equilibrium process of coalition formation (EPCF), introduced by Konishi and Ray [23], considers the infinite horizon discounted value of players’ payoffs. One of the benefits of this concept is that we can investigate the level of farsightedness required to achieve each of the stable outcomes.

Let ${\delta }_i\in [0,1]$ denote the discount factor for i’s future payoffs. Then, i’s payoff from a sequence of structures $\left\{{\mathcal{Z}}_t\right\}$ can be written as ${\sum }_{t=0}^{\infty }{\delta }_i^t{\Pi }_i^{{\mathcal{Z}}_t}$. When ${\delta }_i=0$, only immediate payoffs are considered, which corresponds to myopic stability concepts such as the core. In this paper, we are interested in evaluating the lowest levels of the discount factors under which specific outcomes are stable.

A process of coalition formation (PCF) is a transition probability $\psi :\mathbf{Z}\times \mathbf{Z}\to [0,1]$ such that ${\sum }_{\mathcal{Y}\in \mathbf{Z}}\psi (\mathcal{Z},\mathcal{Y})=1$ for $\forall \mathcal{Z}\in \mathbf{Z}$. A PCF $\psi$ induces a value function $V_i$ for every firm i, which represents i’s infinite horizon payoff starting from the structure $\mathcal{Z}$ under $\psi$ and is the unique solution to the equation $V_i(\mathcal{Z},\psi )={\Pi }_i^{\mathcal{Z}}+{\delta }_i{\sum }_{\mathcal{Y}\in \mathbf{Z}}\psi (\mathcal{Z},\mathcal{Y})V_i(\mathcal{Y},\psi ).$

We say that a defection $\mathcal{Z}{\rightharpoonup }_S\mathcal{Y}$ is profitable under $\psi$ if $V_i(\mathcal{Y},\psi )\ge V_i(\mathcal{Z},\psi )$ for $\forall i\in S$; we further say that the defection is strictly profitable if the above inequality is strict. We say that the defection is efficient if there is no other move $\mathcal{Z}{\rightharpoonup }_S\mathcal{V}$ such that $V_i(\mathcal{V},\psi )>V_i(\mathcal{Y},\psi )$ for $\forall i\in S$. Konishi and Ray [23] considered that a defection from one structure to another occurs only if all members of the deviating set agree to move and they cannot find a strictly better alternative structure. In addition, a defection from a structure must occur if there is a strictly profitable move. The Equilibrium PCF (EPCF) is then defined as follows. A PCF $\psi$ is an Equilibrium PCF if the following holds:

• Whenever $\psi (\mathcal{Z},\mathcal{Y})>0$ for some $\mathcal{Y}\ne \mathcal{Z}$, there exists $S\subseteq N$ such that $\mathcal{Z}{\rightharpoonup }_S\mathcal{Y}$ is profitable and efficient;
• If there is a strictly profitable defection from $\mathcal{Z}$, then $\psi (\mathcal{Z},\mathcal{Z})=0$ and there exists a strictly profitable and efficient defection $\mathcal{Z}{\rightharpoonup }_S\mathcal{Y}$ such that $\psi (\mathcal{Z},\mathcal{Y})>0$.

We say that a PCF $\psi$ is deterministic if $\psi (\mathcal{Z},\mathcal{Y})\in \{0,1\}$ for $\forall \mathcal{Z},\mathcal{Y}\in \mathbf{Z}$. A structure $\mathcal{Z}$ is absorbing if $\psi (\mathcal{Z},\mathcal{Z})=1$, while a PCF $\psi$ is absorbing if, for every structure $\mathcal{Y}$, there is some absorbing structure $\mathcal{Z}$ such that ${\psi }^{\left(k\right)}(\mathcal{Z},\mathcal{Y})>0$, where ${\psi }^{\left(k\right)}$ denotes the k-step transition probability. We consider the set of all absorbing states, under all deterministic absorbing EPCFs, as the set of farsighted stable outcomes. We denote the lower bound of the discount factor required for an outcome to be stable for player i by ${\tilde{\delta }}_i$.

Theorem 4.1 in Konishi and Ray [23] states that, for any core coalition structure $\mathcal{Z}$, we can find $\tilde{\delta }\in (0,1)$ such that $\mathcal{Z}$ is farsighted stable for $\tilde{\delta }<\delta <1$. However, as shown in their Theorem 4.2, there can be farsighted stable outcomes that are not core structures.

5. Stability of Linked Markets for the EPM Game

As we discussed in Section 4, superadditivity is a desirable property for cooperative games that may lead to the nonemptiness of the core. While it is easy to verify that the EPM game is superadditive, this still does not imply that its core is nonempty; for this, we need convexity. Unfortunately, the EPM game is not convex. To see this, consider the EPM-DQT game and let, for instance, $S=\left\{\mathit{EAU}\right\},$ and $T=\left\{\mathit{EAK}\right\}$. Then, $S\cap T=\left\{\mathit{EA}\right\},S\cup T=\left\{\mathit{EAUK}\right\}$, and

$$\begin{array}{ccc}\hfill v(S\cap T)+v(S\cup T)& =& v\left(\right\{\mathit{EA}\left\}\right)+v\left(\right\{\mathit{EAUK}\left\}\right)=1174.66+2898.78=4073.44,\hfill \\ \hfill v\left(S\right)+v\left(T\right)& =& v\left(\right\{\mathit{EAU}\left\}\right)+v\left(\right\{\mathit{EAK}\left\}\right)=2296.18+2104.53=4400.71;\hfill \end{array}$$

thus, $v(S\cap T)+v(S\cup T)<v\left(S\right)+v\left(T\right),$ which violates the convexity condition.

In addition, although DQT show that the efficiency gains increase with the size of the linking group, the efficiency gain allocation is not PMAS; for example, in the EPM-DQT game, the U.S. receives an allocation of USD 472.80 in $EU$, while it receives USD 229.53 in $EAU$. This, however, does not imply that the EPM game does not have a PMAS, as illustrated in the following example.

Example 1.

Consider the EPM-DQT Game from Section 3 and the allocation ${\left(y_i^S\right)}_{i\in S\subset N,S\ne \varnothing }$ presented in

Table 6. PMAS for EPM-DQT game from Section 3.

S	$\mathit{v}\left(\mathit{S}\right)$	${\mathit{y}}_{\mathit{E}}^{\mathit{S}}$	${\mathit{y}}_{\mathit{A}}^{\mathit{S}}$	${\mathit{y}}_{\mathit{U}}^{\mathit{S}}$	${\mathit{y}}_{\mathit{C}}^{\mathit{S}}$	${\mathit{y}}_{\mathit{K}}^{\mathit{S}}$
$\left\{\mathit{EA}\right\}$	1174.66	345.49	829.17	-	-	-
$\left\{\mathit{EU}\right\}$	719.57	365.49	-	354.08	-	-
$\left\{\mathit{EC}\right\}$	370.46	370.46	-	-	-	-
$\left\{\mathit{EK}\right\}$	836.60	672.76	-	-	-	163.84
$\left\{\mathit{AU}\right\}$	1605.23	-	1251.15	354.08	-	-
$\left\{\mathit{AC}\right\}$	876.00	-	876.00	-	-	-
$\left\{\mathit{AK}\right\}$	1169.35	-	829.17	-	-	340.18
$\left\{\mathit{UC}\right\}$	305.53	-	-	305.53	-	-
$\left\{\mathit{UK}\right\}$	506.62	-	-	354.08	-	152.54
$\left\{\mathit{CK}\right\}$	475.80	-	-	-	319.63	156.17
$\left\{\mathit{EAU}\right\}$	2296.18	672.76	1269.34	354.08	-	-
$\left\{\mathit{EAC}\right\}$	1689.91	466.54	876.00	-	347.37	-
$\left\{\mathit{EAK}\right\}$	2104.53	672.76	829.17	-	-	602.60
$\left\{\mathit{EUC}\right\}$	1026.84	672.76	-	354.08	-	-
$\left\{\mathit{EUK}\right\}$	1297.81	672.76	-	354.08	-	270.97
$\left\{\mathit{ECK}\right\}$	1204.42	672.76	-	-	319.63	212.03
$\left\{\mathit{AUC}\right\}$	1958.64	-	1269.34	354.08	335.22	-
$\left\{\mathit{AUK}\right\}$	2146.66	-	1251.15	354.08	-	541.43
$\left\{\mathit{ACK}\right\}$	1797.08	-	910.49	-	345.16	541.43
$\left\{\mathit{UCK}\right\}$	829.88		-	354.08	319.63	156.17
$\left\{\mathit{EAUC}\right\}$	2643.55	672.76	1269.34	354.08	347.37	-
$\left\{\mathit{EAUK}\right\}$	2898.78	672.76	1269.34	354.08	-	602.60
$\left\{\mathit{EACK}\right\}$	2578.80	672.76	956.07	-	347.37	602.60
$\left\{\mathit{EUCK}\right\}$	1617.44	672.76	-	354.08	319.63	270.97
$\left\{\mathit{AUCK}\right\}$	2510.01	-	1269.34	354.08	345.16	541.43
$\left\{\mathit{EAUCK}\right\}$	3253.39	680.00	1269.34	354.08	347.37	602.60

. It is easy to verify that ${\left(y_i^S\right)}_{i\in S\subset N,S\ne \varnothing }$ is PMAS.

It follows from Example 1 that the EPM-DQT game has a non-empty core. Although the efficiency gain allocation is not PMAS, it can still be a member of the core, as we show in our first result.

Theorem 1.

The efficiency gain allocation belongs to the core of the EPM game.

Proof. 

By definition, ${\sum }_{i\in N}{\delta }_{N,i}=v\left(N\right)$. This shows the efficiency part of Equation (5). We next prove the coalitional rationality part.

Consider $\left|N\right|=n$ and $\left|S\right|=m$ for some arbitrary $m<n$. Then, it follows from (3) that

$$\begin{array}{cc}\hfill \sum _{i\in S}{\delta }_{N,i}& =\frac{1}{{\Gamma }_N^2}\sum _{i\in S}\sum _{j\in N\backslash \left\{i\right\}}\left\{({\Gamma }_N-{\gamma }_i){\Gamma }_{\{i,j\}}{\Delta }_{\{i,j\}}-\frac{{\gamma }_i}{2}\sum _{k\in N\backslash \{i,j\}}{\Gamma }_{\{j,k\}}{\Delta }_{\{j,k\}}\right\}\hfill \\ \hfill & =\frac{1}{{\Gamma }_N^2}\left\{\sum _{\begin{array}{c}i,j\in S,\\ i<j\end{array}}(2{\Gamma }_N-{\Gamma }_S){\Gamma }_{\{i,j\}}{\Delta }_{\{i,j\}}+\sum _{\begin{array}{c}i\in S,\\ j\in N\backslash S\end{array}}({\Gamma }_N-{\Gamma }_S){\Gamma }_{\{i,j\}}-{\Delta }_{\{i,j\}}\right.\hfill \\ \hfill & \left.-\sum _{\begin{array}{c}i,j\in N\backslash S,\\ i<j\end{array}}{\Gamma }_S{\Gamma }_{\{i,j\}}{\Delta }_{\{i,j\}}\right\}\hfill \\ \hfill & =\frac{1}{{\Gamma }_N^2}\left\{({\Gamma }_N-{\Gamma }_S)\sum _{\begin{array}{c}i,j\in N,\\ i<j\end{array}}{\Gamma }_{\{i,j\}}{\Delta }_{\{i,j\}}+{\Gamma }_N\sum _{\begin{array}{c}i,j\in S,\\ i<j\end{array}}{\Gamma }_{\{i,j\}}{\Delta }_{\{i,j\}}-{\Gamma }_N\sum _{\begin{array}{c}i,j\in N\backslash S,\\ i<j\end{array}}{\Gamma }_{\{i,j\}}{\Delta }_{\{i,j\}}\right\}\hfill \end{array}$$

(7)

$$\begin{array}{c}=\frac{1}{{\Gamma }_N^2}\left\{({\Gamma }_N-{\Gamma }_S){\Gamma }_N{\Delta }_N+{\Gamma }_N{\Gamma }_S{\Delta }_S-{\Gamma }_N{\Gamma }_{N\backslash S}{\Delta }_{N\backslash S}\right\}\hfill \end{array}$$

(8)

$$\begin{array}{c}=\frac{1}{{\Gamma }_N}\left\{({\Gamma }_N-{\Gamma }_S)({\Delta }_N-{\Delta }_{N\backslash S})+{\Gamma }_S{\Delta }_S\right\},\hfill \end{array}$$

(9)

where (8) follows from (4), and (9) from the fact that ${\Gamma }_N={\Gamma }_S+{\Gamma }_{N\backslash S}$. It is then easy to see that

$$\begin{array}{ccc}\hfill \sum _{i\in S}{\delta }_{N,i}-v\left(S\right)& =& \frac{1}{{\Gamma }_N}\left\{({\Gamma }_N-{\Gamma }_S)({\Delta }_N-{\Delta }_{N\backslash S})+{\Gamma }_S{\Delta }_S\right\}-{\Delta }_S\hfill \\ \hfill & =& \frac{1}{{\Gamma }_N}\left\{({\Gamma }_N-{\Gamma }_S)({\Delta }_N-{\Delta }_{N\backslash S})-({\Gamma }_N-{\Gamma }_S){\Delta }_S\right\}\hfill \\ & =& \frac{1}{{\Gamma }_N}\left\{({\Gamma }_{N\backslash S}({\Delta }_N-{\Delta }_{N\backslash S}-{\Delta }_S)\right\}\ge 0\hfill \end{array}$$

because linking increases the total profit. This completes the proof. □

DQT considered two extensions of their main model: linking with endogenous cap selection and linking with banking and borrowing. We discuss our results in view of these extensions below.

5.1. Linking with Endogenous Cap Selection

While the main model from DQT assumes that domestic carbon caps are provided, it may happen that regulators strategically modify their caps in anticipation of linking. In addition to the non-negative effort- and risk-sharing components of the expected efficiency gain for each jurisdiction, the expected welfare impact now also contains strategic and damage effects. The strategic effect is measured by the market value of the difference between the autarky cap and the linking cap of a jurisdiction, while the damage effect is measured by the changes in aggregate emissions; each of the two can be positive or negative. While some of the results from DQT’s main model continue to hold for this setting, Proposition 2, which is the source of Equation (3) and, thus, is critical for the proof, no longer holds. Consequently, we cannot extend Theorem 1 to the case with endogenous emission caps.

5.2. Linking with Banking and Borrowing of Permits

Unlike in their main model, DQT in this extension allowed the jurisdictions to engage in some form of intertemporal trading—banking of issued permits for future compliance or borrowing of future permits for present compliance. Each jurisdiction in this case again realizes the non-negative efficiency gains consisting of effort- and risk-sharing components, which may be lower or higher than in the case with no intertemporal trading. The good news is that Proposition 2 from DQT holds in this case, so our Theorem 1 extends to the model with the banking and borrowing of permits.

6. Stability of Linked Markets for the EPM-DQT Game

While we are not able to derive more results for the EPM game, in this section, we provide additional results for the EPM-DQT game. We first calculate the Shapley value allocations for all jurisdictions in all coalition structures. We then identify core coalition structures for both the efficiency gain and the Shapley value allocation, show that the farsighted stable outcomes identified by the EPCF coincide with the core coalition structures, and calculate the level of farsightedness required for farsighted stability.

6.1. Shapley Value Allocations for EPM-DQT Game

As previously mentioned, when each jurisdiction receives the gains from the linked markets expressed by (2), the U.S. sees the lowest benefits among the five jurisdictions. At the same time, it represents the largest market, which is not captured in its gains. To address this issue, we propose that efficiency gains should be allocated in a more “fair” manner. As mentioned in Section 4, the Shapley allocation takes into account players’ marginal contributions to different coalitions and can thus be perceived as more “fair”. In this section, we calculate the Shapley allocation for each jurisdiction in different linkage structures. Our results are shown in

Table 7. Shapley allocations of gains in different partitions (in billions of 2005 USD per year).

Partition	E	A	U	C	K
$\{\mathit{UCK},\mathit{EA}\}$	587	587	253	238	339
$\{\mathit{ACK},\mathit{EU}\}$	360	781	360	435	581
$\{\mathit{AUK},\mathit{EC}\}$	185	1009	678	185	460
$\{\mathit{AUC},\mathit{EK}\}$	418	965	679	315	418
$\{\mathit{ECK},\mathit{AU}\}$	444	803	803	264	497
$\{\mathit{EUK},\mathit{AC}\}$	523	438	358	438	417
$\{\mathit{EUC},\mathit{AK}\}$	422	585	390	215	585
$\{\mathit{EAK},\mathit{UC}\}$	647	813	153	153	644
$\{\mathit{EAC},\mathit{UK}\}$	529	782	253	379	253
$\{\mathit{EAU},\mathit{CK}\}$	546	989	761	238	238
$\{\mathit{EAUC},\mathit{K}\}$	545	1088	696	314	0
$\{\mathit{EAUK},\mathit{C}\}$	617	1103	648	0	531
$\{\mathit{EACK},\mathit{U}\}$	600	938	0	388	653
$\{\mathit{EUCK},\mathit{A}\}$	544	0	354	259	461
$\{\mathit{AUCK},\mathit{E}\}$	0	1109	581	338	483
$\left\{\mathit{EAUCK}\right\}$	610	1175	591	331	547

.

We can observe several things when comparing Table 7 and Table 3. As expected, the U.S. allocations increased when we adopted the Shapley allocations: the average allocation went from 162 to 504 billions of 2005 USD per year, an increase of over 200%. The biggest contribution to this increase come from Australia, for which the average allocation went from 1,030 to 878 billions of 2005 USD per year, a decrease of 15%. The remaining three jurisdictions have seen smaller changes in their average allocations: an 8% increase for the EU, a 17% decrease for Canada, and a 4% decrease for Korea. Significant changes can be observed in the allocations within groups that link only two jurisdictions, as the Shapley allocation evenly splits the total gains in two-player coalitions. Thus, for instance, when we consider $\left\{EU\right\}$, the original gain for the EU is 574 billions of 2005 USD per year and for the U.S. is 146 billions of 2005 USD per year, while the Shapley allocation assigns 360 billions to each of them.

Similar to the scenario with the expected efficiency gain allocations, we order the structures by their preference for each of the jurisdictions. We present this in

Table 8. Jurisdictions’ preferences for different linkages and corresponding gains (in billions of 2005 USD per year) under Shapley gain allocations.

E		A		U		C		K
Partition	${\mathit{\varphi }}_{\mathit{E}}$	Partition	${\mathit{\varphi }}_{\mathit{A}}$	Partition	${\mathit{\varphi }}_{\mathit{U}}$	Partition	${\mathit{\varphi }}_C$	Partition	${\mathit{\varphi }}_{\mathit{K}}$
$\{\mathit{EAK},\mathit{UC}\}$	647	$\left\{\mathbf{EAUCK}\right\}$	1175	$\{\mathit{ECK},\mathit{AU}\}$	803	$\{\mathit{EUK},\mathit{AC}\}$	438	$\{\mathit{EACK},\mathit{U}\}$	653
$\{\mathit{EAUK},\mathit{C}\}$	617	$\{\mathit{AUCK},\mathit{E}\}$	1109	$\{\mathit{EAU},\mathit{CK}\}$	761	$\{\mathit{ACK},\mathit{EU}\}$	435	$\{\mathit{EAK},\mathit{UC}\}$	644
$\left\{\mathbf{EAUCK}\right\}$	610	$\{\mathit{EAUK},\mathit{C}\}$	1103	$\{\mathit{EAUC},\mathit{K}\}$	696	$\{\mathit{EACK},\mathit{U}\}$	388	$\{\mathit{EUC},\mathit{AK}\}$	585
$\{\mathit{EACK},\mathit{U}\}$	600	$\{\mathit{EAUC},\mathit{K}\}$	1088	$\{\mathit{AUC},\mathit{EK}\}$	679	$\{\mathit{EAC},\mathit{UK}\}$	379	$\{\mathit{ACK},\mathit{EU}\}$	581
$\{\mathit{UCK},\mathit{EA}\}$	587	$\{\mathit{AUK},\mathit{EC}\}$	1009	$\{\mathit{AUK},\mathit{EC}\}$	678	$\{\mathit{AUCK},\mathit{E}\}$	338	$\left\{\mathbf{EAUCK}\right\}$	547
$\{\mathit{EAU},\mathit{CK}\}$	546	$\{\mathit{EAU},\mathit{CK}\}$	989	$\{\mathit{EAUK},\mathit{C}\}$	648	$\left\{\mathbf{EAUCK}\right\}$	331	$\{\mathit{EAUK},\mathit{C}\}$	531
$\{\mathit{EAUC},\mathit{K}\}$	545	$\{\mathit{AUC},\mathit{EK}\}$	965	$\left\{\mathbf{EAUCK}\right\}$	591	$\{\mathit{AUC},\mathit{EK}\}$	315	$\{\mathit{ECK},\mathit{AU}\}$	497
$\{\mathit{EUCK},\mathit{A}\}$	544	$\{\mathit{EACK},\mathit{U}\}$	938	$\{\mathit{AUCK},\mathit{E}\}$	581	$\{\mathit{EAUC},\mathit{K}\}$	314	$\{\mathit{AUCK},\mathit{E}\}$	483
$\{\mathit{EAC},\mathit{UK}\}$	529	$\{\mathit{EAK},\mathit{UC}\}$	813	$\{\mathit{EUC},\mathit{AK}\}$	390	$\{\mathit{ECK},\mathit{AU}\}$	264	$\{\mathit{EUCK},\mathit{A}\}$	461
$\{\mathit{EUK},\mathit{AC}\}$	523	$\{\mathit{ECK},\mathit{AU}\}$	803	$\{\mathit{ACK},\mathit{EU}\}$	360	$\{\mathit{EUCK},\mathit{A}\}$	259	$\{\mathit{AUK},\mathit{EC}\}$	460
$\{\mathit{ECK},\mathit{AU}\}$	444	$\{\mathit{EAC},\mathit{UK}\}$	782	$\{\mathit{EUK},\mathit{AC}\}$	358	$\{\mathit{UCK},\mathit{EA}\}$	238	$\{\mathit{AUC},\mathit{EK}\}$	418
$\{\mathit{EUC},\mathit{AK}\}$	422	$\{\mathit{ACK},\mathit{EU}\}$	781	$\{\mathit{EUCK},\mathit{A}\}$	354	$\{\mathit{EAU},\mathit{CK}\}$	238	$\{\mathit{EUK},\mathit{AC}\}$	417
$\{\mathit{AUC},\mathit{EK}\}$	418	$\{\mathit{UCK},\mathit{EA}\}$	587	$\{\mathit{UCK},\mathit{EA}\}$	253	$\{\mathit{EUC},\mathit{AK}\}$	215	$\{\mathit{UCK},\mathit{EA}\}$	339
$\{\mathit{ACK},\mathit{EU}\}$	360	$\{\mathit{EUC},\mathit{AK}\}$	585	$\{\mathit{EAC},\mathit{UK}\}$	253	$\{\mathit{AUK},\mathit{EC}\}$	185	$\{\mathit{EAC},\mathit{UK}\}$	253
$\{\mathit{AUK},\mathit{EC}\}$	185	$\{\mathit{EUK},\mathit{AC}\}$	438	$\{\mathit{EAK},\mathit{UC}\}$	153	$\{\mathit{EAK},\mathit{UC}\}$	153	$\{\mathit{EAU},\mathit{CK}\}$	238
$\{\mathit{AUCK},\mathit{E}\}$	0	$\{\mathit{EUCK},\mathit{A}\}$	0	$\{\mathit{EACK},\mathit{U}\}$	0	$\{\mathit{EAUK},\mathit{C}\}$	0	$\{\mathit{EAUC},\mathit{K}\}$	0

, and observe that the ordering of structures changed for all jurisdictions. In the next subsection, we will analyze the stability of different structures and compare the stability results with respect to two different gain allocations.

6.2. Core Coalition Structures for EPM-DQT Game

As we discussed in Section 5, the efficiency gain allocation belongs to the core of the EPM game. Does this mean that the grand coalition will always form? Not necessarily. Our next result identifies some other structures that might emerge as core coalition structures.

Proposition 1.

In the EPM-DQT game,

• Under the expected efficiency gain allocation, $\left\{\mathit{EAUCK}\right\}$, $\{\mathit{EAUK},\mathit{C}\}$, and $\{\mathit{EAU},\mathit{CK}\}$ are core coalition structures;
• Under the Shapley value allocation, $\left\{\mathit{EAUCK}\right\}$, $\{\mathit{EAUC},\mathit{K}\}$, $\{\mathit{EAUK},\mathit{C}\}$, $\{\mathit{EACK},\mathit{U}\}$, $\{\mathit{AUCK},\mathit{E}\}$, and $\{\mathit{EAU},\mathit{CK}\}$ are core coalition structures.

Proof. 

For each partition that is not a core structure under either of the two allocations, we show one of the possible blocking coalitions that prohibit its stability in

Table 9. EPM-DQT game: blocking coalitions for non-core coalition structures under the efficiency gain allocation ($\Pi$) and the Shapley value allocation ($\varphi$).

Structure	Blocking Coalition for $\mathbf{\Pi }$	Blocking Coalition for $\mathit{\varphi }$	Structure	Blocking Coalition for $\mathbf{\Pi }$	Blocking Coalition for $\mathit{\varphi }$
$\{\mathit{UCK},\mathit{EA}\}$	$\left\{\mathit{EAUCK}\right\}$	$\left\{\mathit{EAUK}\right\}$	$\{\mathit{EAK},\mathit{UC}\}$	$\left\{\mathit{EAU}\right\}$	$\left\{\mathit{AUC}\right\}$
$\{\mathit{AUK},\mathit{EC}\}$	$\left\{\mathit{EAUCK}\right\}$	$\left\{\mathit{EAUC}\right\}$	$\{\mathit{EAC},\mathit{UK}\}$	$\left\{\mathit{EAU}\right\}$	$\left\{\mathit{EAU}\right\}$
$\{\mathit{AUC},\mathit{EK}\}$	$\left\{\mathit{EAUCK}\right\}$	$\left\{\mathit{EAU}\right\}$	$\{\mathit{EAUC},\mathit{K}\}$	$\left\{\mathit{CK}\right\}$	n.a.
$\{\mathit{ACK},\mathit{EU}\}$	$\left\{\mathit{AU}\right\}$	$\left\{\mathit{EAU}\right\}$	$\{\mathit{EACK},\mathit{U}\}$	$\left\{\mathit{AU}\right\}$	n.a.
$\{\mathit{EUK},\mathit{AC}\}$	$\left\{\mathit{AU}\right\}$	$\left\{\mathit{EA}\right\}$	$\{\mathit{EUCK},\mathit{A}\}$	$\left\{\mathit{AC}\right\}$	$\left\{\mathit{EAUC}\right\}$
$\{\mathit{EUC},\mathit{AK}\}$	$\left\{\mathit{AU}\right\}$	$\left\{\mathit{EA}\right\}$	$\{\mathit{AUCK},\mathit{E}\}$	$\left\{\mathit{EK}\right\}$	n.a.
$\{\mathit{ECK},\mathit{AU}\}$	$\left\{\mathit{EAU}\right\}$	$\left\{\mathit{EACK}\right\}$

. For the core coalition structures, there are no blocking coalitions, so we write n.a. □

Proposition 1 shows that, if we allocate gains according to the expected efficiency gains generated by the five jurisdictions, we can have three core coalition structures and they all contain the linkage between the EU, the U.S. (the two jurisdictions with the largest markets), and Australia (the jurisdiction with the lowest autarky permit price). Once we distribute the gains using the Shapley value—i.e., in a more “fair” way—we can see that the set of core coalition structures grows; moreover, all the structures containing linkages of four jurisdictions are stable, except $\{EUCK,A\}$. The instability of $\{EUCK,A\}$ is not surprising, as this structure leaves Australia, being the jurisdiction with the lowest autarky permit price outside the linkage. As a result, the total gains generated are only 1617 billions of 2005 USD per year, which is about one half of the gains that can be realized when all five jurisdictions are linked.

Additional observations can be noted if we calculate the benefits from participation in the grand coalition (that is, the gains that different coalitions can produce on their own compared to the total allocation that their members receive as members of the grand coalition); we show these values in

Table 10. Coalition values compared to sums of allocations and benefits from participation in the grand coalition for core coalition structures (in billions of 2005 USD per year).

S	$\mathit{v}\left(\mathit{S}\right)$	${\sum }_{\mathit{i}\in \mathit{S}}{\mathbf{\Pi }}_{\mathit{i}}^{\mathit{S}}$	${\mathbf{\Delta }}_{\mathit{S}}$	${\sum }_{\mathit{i}\in \mathit{S}}{\mathit{\varphi }}_{\mathit{i}}^{\mathit{S}}$	${\mathbf{\Delta }}_{\mathit{S}}$
$\left\{\mathit{EAU}\right\}$	2296	2348	52	2375	79
$\left\{\mathit{EAUC}\right\}$	2644	2694	50	2706	62
$\left\{\mathit{EAUK}\right\}$	2899	2908	9	2923	24
$\left\{\mathit{EACK}\right\}$	2579	3007	428	2663	84
$\left\{\mathit{AUCK}\right\}$	2510	2630	120	2643	133

, and denote the total benefits obtained by members of coalition S from participation in the grand coalition by ${\Delta }_S$.

As can be seen from Table 10, the benefits are the highest for $\left\{\mathit{EACK}\right\}$ and $\left\{\mathit{AUCK}\right\}$, while they are the lowest for $\left\{\mathit{EAUK}\right\}$. This finding perhaps indicates that it is easier to establish and sustain linkage of the EU, Australia, the U.S. and South Korea than the linkage of Australia, the U.S., Canada, and South Korea. In addition, the difference in the total allocations that members of $\left\{\mathit{EAUK}\right\}$ receive as members of the grand coalition and by acting on their own is rather small (0.3% under the efficiency gain allocation and 0.83% under the Shapley value allocation); thus, even a small change in the uncertainty parameters can cause the grand coalition to become unstable. This should also not come as a surprise, as Canada represents the smallest market, 2.7% of the total market, and 4.6% of the U.S. emissions. We will discuss this issue further in Section 6.3.

6.3. Farsightedly Stable Structures for EPM-DQT Game

As mentioned earlier, the core coalition structures are also stable in the farsighted sense. We further want to investigate if there are additional structures that might emerge as stable when the jurisdictions are farsighted, that is, if they consider how the others may react to their moves, and what levels of farsightedness are required to establish the stability of different structures. Our findings are summarized in the following result.

Proposition 2.

In the EPM-DQT game, when the discount factor is high enough, the set of core coalition structures and the set of farsighted stable outcomes are equivalent under both the expected efficiency gain allocation and the Shapley value allocation.

Thus, core coalition structures and farsighted stable structures under both allocations coincide. However, the way in which we arrived at the stability/instability of specific structures is rather different. For instance, consider structure $\{\mathit{EAK},\mathit{UC}\}$, which is not a core coalition structure and is not farsightedly stable. In Table 9, we show that a blocking coalition for the Shapley value allocation is $\left\{\mathit{AUC}\right\}$ (similar logic holds for efficiency gain allocation), and this defection is beneficial for defecting jurisdictions. However, the resulting structure, $\{\mathit{AUC},\mathit{EK}\}$, is not a core coalition structure, so this argument may be inconsistent. On the other hand, if we consider the farsighted stability of, say, the grand coalition, Australia alone defects from $\{\mathit{EAK},\mathit{UC}\}$ to $\{\mathit{EK},\mathit{UC},\mathit{A}\}$, and this move leads to a one-period reduction in Australia’s allocation. This initial move, however, serves only as a trigger to induce the remaining jurisdictions to join Australia in the grand coalition, which is a farsighted stable structure. Thus, the reasoning behind the stability of specific structures follows significantly different logic, based on the stability concepts that we adopt.

Note that, starting with an environment with no linkage, it is not clear which of these stable outcomes might be more likely to emerge. To gain more insights, we will analyze some additional stability aspects of the above-mentioned structures; namely, the number of jurisdictions involved in performing non-profitable moves according to the underlying PCF and the required discount factors. To show how we evaluate the defection steps and discount rates for farsighted stability, we start with an illustrative example.

Example 2.

Consider the farsighted stability of $\{\mathit{EAU},\mathit{CK}\}$ under the Shapley value allocation, and suppose that the current status quo is the grand coalition. The U.S. is the only jurisdiction that prefers $\{\mathit{EAU},\mathit{CK}\}$ to the grand coalition, so the PCF requires that the U.S. performs the move, $\left\{\mathit{EAUCK}\right\}{\rightharpoonup }_{\left\{\mathit{U}\right\}}\{\mathit{EACK},\mathit{U}\}$; while Australia prefers $\{\mathit{EAU},\mathit{CK}\}$ to $\{\mathit{EACK},\mathit{U}\}$, the EU does not and would not want to initiate a sequence of moves from $\{\mathit{EACK},\mathit{U}\}$ that eventually leads to $\{\mathit{EAU},\mathit{CK}\}$; the same is true for Canada and South Korea. If Australia joins the U.S., it would lead to $\left\{\mathit{AU}\right\}$, which is the linkage most preferred by the U.S., from which it would not want to move. As a result, the PCF requires that Australia moves alone, $\{\mathit{EACK},\mathit{U}\}{\rightharpoonup }_{\left\{\mathit{A}\right\}}\{\mathit{ECK},\mathit{A},\mathit{U}\}$. From this structure, the EU, Australia, and the U.S. jointly form $\left\{\mathit{EAU}\right\}$. Thus, the sequence of moves is

$$\left\{\mathit{EAUCK}\right\}{\rightharpoonup }_{\left\{\mathit{U}\right\}}\{\mathit{EACK},\mathit{U}\}{\rightharpoonup }_{\left\{\mathit{A}\right\}}\{\mathit{ECK},\mathit{A},\mathit{U}\}{\rightharpoonup }_{\{\mathit{E},\mathit{A},\mathit{U}\}}\{\mathit{EAU},\mathit{CK}\}.$$

(10)

As the moves must be profitable for all the defecting jurisdictions, we have to evaluate the lowest value of the discount factors for which this is true. For the U.S., we compare the expected discounted profit from staying in the grand coalition with the profit from the sequence of moves described by (10):

$$\begin{array}{c}\hfill \frac{591}{1-{\delta }_U}<591+{\delta }_U·0+{\delta }_U^2·0+{\delta }_U^3·\frac{761}{1-{\delta }_U}⟹{\delta }_U>0.88.\end{array}$$

Similarly, for Australia we have

$$\begin{array}{c}\hfill \frac{938}{1-{\delta }_A}<938+{\delta }_A·0+{\delta }_A^2·\frac{989}{1-{\delta }_A}⟹{\delta }_A>0.95.\end{array}$$

The EU only participates in the last move, which is profitable for the EU, so it performs the move regardless of the discount factor. Thus, in this case, only the U.S. and Australia perform non-profitable moves, and the required discount factors are ${\tilde{\delta }}_U=0.88$ and ${\tilde{\delta }}_A=0.95$.

In Appendix A, in the proof of Proposition 2, we show all the PCFs and required discount factors. In

Table 11. Jurisdictions with non-profitable moves and required minimum discount factors for farsighted stability of different structures, and corresponding total gains from linkage (in billions of 2005 USD per year).

Allocation	Stable Structure	E	A	U	C	K	${\mathit{\delta }}_{\mathit{E}}$	${\mathit{\delta }}_{\mathit{A}}$	${\mathit{\delta }}_{\mathit{U}}$	${\mathit{\delta }}_{\mathit{C}}$	${\mathit{\delta }}_{\mathit{K}}$	Total Gains
$\Pi$	$\left\{\mathit{EAUCK}\right\}$	✓	✓	✓		✓	0.83	0.96	0.52		0.51	3253
$\Pi$	$\{\mathit{EAUK},\mathit{C}\}$	✓	✓	✓		✓	0.92	0.95	0.77		0.37	2899
$\Pi$	$\{\mathit{EAU},\mathit{CK}\}$			✓					0.91			2772
$\varphi$	$\left\{\mathit{EAUCK}\right\}$		✓					0.94				3253
$\varphi$	$\{\mathit{EAUC},\mathit{K}\}$		✓	✓				0.91	0.95			2644
$\varphi$	$\{\mathit{EAUK},\mathit{C}\}$	✓	✓	✓		✓	0.44	0.97	0.95		0.58	2899
$\varphi$	$\{\mathit{EACK},\mathit{U}\}$	✓	✓		✓	✓	0.65	0.65		0.77	0.88	2579
$\varphi$	$\{\mathit{AUCK},\mathit{E}\}$		✓		✓			0.995		0.99		2510
$\varphi$	$\{\mathit{EAU},\mathit{CK}\}$		✓	✓				0.95	0.90			2772

, we summarize the main insights from the proof.

Let us first consider the efficiency gain allocation. As we can see from Table 11, the stability of $\left\{\mathit{EAUCK}\right\}$ and $\{\mathit{EAUK},\mathit{C}\}$ requires coordination and non-profitable defections of all the jurisdictions but Canada, and the required highest values of the lowest bounds on the discount rates (in this case, for Australia) are 0.96 and 0.95, respectively. On the other hand, for the stability of $\{\mathit{EAU},\mathit{CK}\}$, we only require that the U.S. perform non-profitable steps, and the required lowest bounds on the discount rate is 0.91, which may be closer to some practical discount rates. Recall also that the U.S. is seeing the lowest allocations among the five jurisdictions in different structures, although it contributes the largest market, and that $\{\mathit{EAU},\mathit{CK}\}$ is the most preferred structure for the U.S. Thus, if we use the efficiency gain allocation, it seems that $\{\mathit{EAU},\mathit{CK}\}$ is the most likely to emerge as stable, although this leaves 15% of efficiency gains unrealized.

Let us now look at the Shapley value allocation. First, we can observe that the stability of $\{\mathit{AUCK},\mathit{E}\}$ requires that Australia and Canada almost do not discount future gains at all, so this structure is rather unlikely to emerge as stable. $\{\mathit{EACK},\mathit{U}\}$ requires that all jurisdictions except Canada perform non-profitable steps, and that Australia uses a discount factor of at least 0.97, which causes this structure to be less likely to occur. $\{\mathit{EAUC},\mathit{K}\}$ and $\{\mathit{EAU},\mathit{CK}\}$ both require that only Australia and the U.S. perform non-profitable moves, and both require the lowest discount factor of 0.95—$\{\mathit{EAUC},\mathit{K}\}$ for the U.S., and $\{\mathit{EAU},\mathit{CK}\}$ for Australia. The grand coalition only requires that Australia performs non-profitable moves, with the lowest discount factor of 0.94, while $\{\mathit{EACK},\mathit{U}\}$ requires the coordination of all jurisdictions except the U.S., with the lowest discount factor of 0.88. Thus, if we assume that coordination among four jurisdictions is easy, then $\{\mathit{EACK},\mathit{U}\}$ might appear as a likely candidate for stability. However, as the example of California, Quebec, and Ontario indicates, even with three jurisdictions, that the coordination may not be easy. In addition, while $\{\mathit{EACK},\mathit{U}\}$ takes advantage of the negative correlation between Canada and South Korea, it leaves the U.S., the jurisdiction with the largest market, outside of the linkage and leaves 21% of the potential gains on the table. Hence, we feel that the stability of the grand coalition, which only requires non-profitable moves by Australia, may be more likely to emerge. Note that Australia ranks the grand coalition second highest among all the possible outcomes, and is only dominated by $\{\mathit{AUCK},\mathit{E}\}$. However, as previously noted, the stability of $\{\mathit{AUCK},\mathit{E}\}$ would require almost no discounting of future payoffs and would leave 23% of the potential gains on the table, while the grand coalition maximizes the total gains obtained from the market linkage. As Australia’s participation is critical due to its low cost of emissions, the other jurisdictions might be likely to follows its lead and join the grand coalition.

If we combine the two discussions, we can conclude that, without any changes in gain allocations, the linkage between the EU, Australia, and the U.S. is most likely to occur, but it leaves 15% of the potential gains on the table. On the other hand, if the gains are apportioned according to the Shapley value allocation, it is more likely that all five jurisdictions will link their markets. Thus, it appears that, if the international community wants to maximize the global benefits, it should agree to the Shapley value, as this would cause the grand coalition to be more likely to emerge as a stable structure.

7. Concluding Remarks

The linkage of CAT markets can potentially help achieve the goals of the Paris agreement. One of the problems with market linkage typically lies in the participants’ asymmetry, which leads to different benefits for the participants and, thus, different preferences for linkage partners. These preferences can often be inconsistent, as the partner most favored by one jurisdiction might prefer to join another jurisdiction in a linkage group. We can also observe that, in practice, dynamics exist in the market linkages, as jurisdictions join and leave the linkage groups.

In this paper, we use the model from DQT to show that the all-inclusive linking of carbon permit markets is stable, although it may not be the most preferred outcome for all jurisdictions. This result holds regardless of the level of asymmetry among jurisdictions. This is an encouraging result, as it implies that we could observe formation of an all-inclusive permit trading market without additional enforcement mechanisms, which usually require a third party for implementation. However, as illustrated in our numerical example, this is not, in general, the only stable outcome, and may not always be realized.

We further study the stability of all possible CAT market linkages among five jurisdictions—Australia, Canada, the EU, South Korea, and the U.S. We show that the most likely stable outcome has two linked markets, with Australia, the EU, and the U.S. in one market, and Canada and South Korea in the other. The benefit of this approach is that we do not need to devise a specific implementation mechanism. Each jurisdiction receives its own gains from the linked market, and their allocations satisfy both myopic and farsighted stability constraints. However, this configuration leaves about 15% of efficiency gains unrealized. To mitigate this, we propose to allocate the efficiency gains among the linking parties via the Shapley value, which would cause the linking of all five jurisdictions to be the most likely option for stability, but would require the development of an implementation mechanism among the five jurisdictions.

For practical implementation, perhaps the global community can agree on a common agency or organization, such as the U.N., which could encourage the jurisdictions to either pick the all-inclusive market among all the potentially stable linkages, or to assist in distribution of benefits according to the Shapley value allocation, which would encourage formation of an all-inclusive market that would lead to higher efficiency gains.

We hope that our result is of use to jurisdictions and policy makers and lead to the establishment of more successful stable linkages between emission permit trading markets in the future. It can also provide helpful insights to global companies when deciding where to locate their manufacturing facilities, or from which countries to source from.