Coalition Stability in International Environmental Matching Agreements MASTER THESIS

: In this study, we conduct an empirically calibrated simulation of an environmental matching agreement between twelve asymmetric players to determine whether the matching mechanism can produce larger stable coalitions and increase players’ abatement contributions and payoffs as compared to a standard agreement. We also analyze a version of the matching game with emissions trading. The matching model features a uniform, collective payoff maximizing matching rate by which all coalition members’ flat contributions are matched, while non -members play the standard Nash game. The simulation considers players with linear abatement benefit and quadratic cost functions, calibrated based on the STACO 3 model, and uses emissions data from the Shared Socioeconomic Pathways database. We find that the matching mechanism significantly increases the abatement and payoff levels beyond the non-cooperative Nash baseline when full agreement accedence by all players is assumed, but that such matching agreements perform worse in terms of stability than standard agreements. In our simulation, the full matching agreement without emissions trading bridges 44 percent of the global gap between the Nash baseline abatement level and the socially optimal abatement level. This, in turn, means that 37 percent of the gap between the global Nash baseline payoff and the socially optimal global payoff is bridged by this agreement. The matching agreement with emissions trading and constrained emission allowances bridges 149 percent and 76 percent of the abatement and payoff gaps, respectively. However, both the matching games with and without emissions trading fail to produce any stable coalitions, while the standard coalition formation game produces a two-player coalition that narrows the global abatement and payoff gaps by 8 and 10 percent, respectively.


Introduction
In December of 2015, the Conference of Parties (COP) to the United Nations Framework Convention on Climate Change (UNFCCC) adopted the Paris Agreement, which aims to limit this century's global temperature rise to well below 2 degrees Celsius above pre-industrial levels (UNFCCC, 2015).Recent work by the Potsdam Institute for Climate Impact Research (PIK) confirms that this goal is likely economically optimal as it strikes a balance between present mitigation costs and future climate damages (Hänsel et al., 2020).Under the agreement, 190 state parties and the European Union have so far committed themselves to so-called nationally determined contributions (NDCs), which are voluntary national pledges to reduce greenhouse gas (GHG) emissions post-2020 (UNFCCC Secretariat, 2021).However, these pledges are still largely insufficient to achieve the 2-degree goal (Climate Action Tracker, 2020).
As with many other transboundary environmental problems, GHG abatement is a matter of nonexcludable public good provision, characterized by the lack of a central policy enforcement authority.In such cases, ill-defined property rights create an incentive for freeriding, which is reinforced by the lack of credible punishment for non-cooperation.The continuing underprovision of national efforts to reduce global GHG emissions and resulting failure of international environmental agreements (IEAs) is therefore all but surprising.In fact, Barrett (1994) demonstrated that cooperative behavior in large coalitions of (symmetric) parties is inherently unlikely when the gains from cooperation are large.These features made non-cooperative game theory the primary approach for analyzing international environmental problems and devising associated policies (Folmer & von Mouche, 2000).
One mechanism that has been put forward to address this issue of collective action without the introduction of a formal enforcement authority is Guttman's (1978) matching game.Here, countries commit to subsidizing other coalition members' emission reductions by a matching factor before choosing their own flat contributions (Guttman, 1978).The mechanism has been shown to produce agreements that are Pareto-efficient and increase abatement and collective welfare beyond the Nash equilibrium outcome in the absence of an agreement (e.g.Guttmann, 1978;Guttman & Schnytzer, 1992;Rübbelke, 2006;Boadway et al., 2011;Fujita, 2013;Molina et al., 2020).
However, most studies about matching agreements so far are theoretical and many are based on the assumption of homogenous players (e.g.Guttman, 1978;Rübbelke, 2006;Boadway, Song & Tremblay, 2007;Fujita, 2013).This assumption is problematic not only because it does not reflect reality, but also because player heterogeneity can drastically change the game outcomes due to the possibility of exploiting inexpensive abatement options or payoff transfers (Mäler, 1989).Moreover, prior studies commonly consider matching agreements in the form of two-stage games where all players are assumed to accede to the agreement ab initiowith the exceptions of Fujita (2013), Buchholz et al. (2014), Wood & Jotzo (2015) and Liu (2018).A more realistic scenario would allow players to make autonomous decisions on whether to join or opt out of an agreement.
To date, a single study has investigated a numerical application of a matching mechanism in the context of IEAs: Kawamata & Horita (2013) simulated a matching game based on the Stability of Coalitions (STACO) model's estimations of six regions' damage and benefit functions and show that the matching mechanism produces significantly higher abatement and collective welfare than the singletons case without an agreement.Although this is an encouraging result, the study does not demonstrate whether the matching agreement's benefits persist when players can opt out of it.
The present study thus sets out to analyze a numerical application of a matching game between heterogeneous players with an initial coalition accedence stage.We consider twelve players with specified (linear) abatement benefit and (quadratic) abatement cost functions, calibrated based on the STACO 3 model, and use updated emissions data from the Shared Socioeconomic Pathways (SSP) database.We assume that coalition members negotiate a uniform, collective payoff maximizing matching rate by which all members' flat contributions are matched, while non-members play the standard Nash game.A comparison of the game's equilibrium abatement and payoff levels to those of the no-agreement Nash baseline and of the socially optimal grand coalition will give an indication of the matching mechanism's potential effectiveness in overcoming the general free-rider problem of IEAs.In addition to the pure matching game, we also analyze an extended version of the model including a final emissions trading stage as proposed by Boadway et al. (2011), which brings about an equalization of all players' marginal abatement costs.
The remainder of this thesis is structured as follows.Section 2 contains an overview of the most relevant game-theoretic literature on coalition stability and matching agreements.Section 3 introduces the research questions of this study.Section 4 outlines the mathematical models, as well as the data sources and calibrations underlying the simulation.Section 5 presents the simulation results and Section 6 discusses their implications.Section 7 concludes and offers recommendations for further research.Hoel (1992), Carraro & Siniscalco (1993) and Barrett (1994) were likely the first to apply stability concepts from the realm of cartel formation games to the context of IEAs in an effort to explain the widespread failure of such agreements.While Barrett (1994) analyzed a Stackelberg abatement game between symmetric players, Hoel (1992) and Carraro & Siniscalco (1993) investigated emissions games in which the symmetric players act simultaneously.In these games, environmental agreements are equivalent to coalitions in which members determine their contributions cooperatively, while the decision to join the coalition is made in a non-cooperative manner by each player.An agreement is thus considered stable when coalition members have no incentive to defect, and non-members have no incentive to join the coalition.In general, the literature studying the stability of IEA's according to these conditions seems to agree that large and effective agreements are highly unlikely by nature.

Literature review
One mechanism that promises to be more effective at increasing contributions by players behaving noncooperatively was proposed by Guttman (1978).In his so-called 'matching' game, players commit to reciprocating other players' contributions according to a self-determined matching rate before choosing their own contributions.Barrett (1990) was possibly the first to suggest applying the matching mechanism to the context of GHG abatement.To our knowledge, Rübbelke's (2006) study was the first to put this suggestion into practice in a game between symmetric players, building on the standard model of public goods.Since then, a number of game theorists have examined different variations of the game theoretically, the most relevant of which are touched upon hereafter.
Much of the literature on matching schemes adopts standard public goods or externality formulations, including Guttman & Schnytzer (1992), Rübbelke (2006), Buchholz et al. (2011;2012;2014), Liu (2018;2019), and Buchholz & Liu (2020).In that context, the present study is closest to Liu (2018).In his study, Liu (2018) analyzes the profitability of matching coalitions using the aggregative game approach and examines their stability according to the internal and external stability conditions outlined by Carraro & Siniscalco (1993).The game he analyzes allows for multiple heterogeneous players and features an exogenously determined, uniform matching rate.Coalition members only match other coalition members' flat contributions, while non-members play the standard Nash game.The study finds that the matching mechanism can produce pareto-improving outcomes, but that stability conditions are hard to satisfy.In particular, larger matching rates produce more profitable but less stable coalitions and vice versa.Liu (2018) also notes that stability issues may be overcome by introducing a reputation mechanism.
More akin to the current study, other literature features models that are similar to the emissions game delineated by Finus (2001) or its abatement game analogue.In two such studies, Fujita (2013) and Wood & Jotzo (2015) examine the stability of matching coalitions with symmetric players, where coalition members not only match each other's contributions, but also non-member's abatements.In Fujita's (2013) study, coalition members negotiate a common matching rate by which all players' abatements are matched.Here, the grand coalition is found to be stable, and the agreement is thus 'self-enforcing' and efficient.In Wood & Jotzo (2015), matching rates are determined non-cooperatively by each coalition member.Since all players are assumed to have identical abatement benefit and cost functions (quadratic ones, in this case), all matching rates towards coalition members are the same, and so are the matching rates towards non-members.The study observes that the matching rates towards nonmembers increase as more players join the coalition, while the matching rates towards members approach zero.The authors conclude that the matching game produces multiple stable coalitions, including the grand coalition.
Another emissions game study that is considered groundbreaking in the matching literature and serves as an important reference for the present study is the one by Boadway et al. (2011).Here, the authors consider a game between multiple heterogeneous players with generalized benefit and cost functions that match each other's emissions reductions at different, non-cooperatively determined rates.They show that the matching mechanism leads to emissions reductions relative to the baseline and that the subgame perfect equilibrium is Pareto efficient and unique.The game is also extended to include an emissions trading stage leading to an optimal allocation of contributions among players and is then analyzed in a dynamic two-period setting where the pollution stock in one period is determined by the emissions in the previous one.Boadway et al.'s (2011) optimistic results, Molina et al. (2020) later set out to refine the model on the grounds that the exogenously determined baseline emissions in the Boadway et al. (2011) model are higher than realistic.They correct for this by having players follow Nash equilibrium emissions strategies in the absence of the matching agreement and demonstrate that the equilibrium identified by Boadway et al. (2011) still holds in the two-player case with the corrected baseline emission levels.They show that the equilibrium emissions profile is unique and locally Pareto efficient, and that both players' emissions reductions and payoffs are higher than at the baseline.However, they note that these results may not be readily extendable to the multi-player setting.

Motivated by
Some other studies focus on related models of conditional public goods provision like 'provision point mechanisms' (Bagnoli & Lipman, 1989), 'contribution mechanisms' (Varian, 1994a,b), or the so-called 'exchange-matching-Lindahl' (EML) mechanism, whose equilibria are identical to matching equilibria when their baselines are the same (Dijkstra & Nentjes, 2020).Danziger & Schnytzer (1991) and Althammer & Buchholz (1993) first demonstrated that the subgame perfect equilibria of the matching game with positive contributions are Lindahl equilibria.This outcome constitutes the central design feature of the EML game, in which players are given individual exchange rates signaling to which extent all other players would match their contribution to a public good.Players then announce how much of the public good they are willing to supply above their non-cooperative Nash baseline.In equilibrium, all exchange rates are such that the matched amount is equal to the sum of all players' individual contributions (Dijkstra & Nentjes, 2020).Dijkstra & Nentjes (2020) also analyze the game with emissions trading and find the outcome to be Pareto efficient.
As previously indicated, practical applications of the matching mechanism are still scarce in the literature.Guttman (1986) himself provides some experimental evidence that matching mechanisms may be able to induce improvements in terms of public goods provision in practice.He does so by conducting an experimental study of repeated matching games in groups of three to six people and finds that the contributions chosen by the participants are higher than at the Cournot equilibrium baseline, but rarely pareto optimal in practice.In the context of IEAs and GHG abatement, however, Kawamata & Horita's (2013) simulation study is so far the only numerical application of a matching game to our knowledge.In their study, the authors demonstrate that the matching mechanism as introduced by Boadway et al. (2011) induces higher abatement levels and collective welfare than the Nash baseline case without an agreement.While this is surely a good start, their study does not show whether matching coalitions are stable, which ultimately determines whether matching agreements can be effective in practice.

Objective and research questions
The main aim of this thesis is to conduct a simulation with an empirically calibrated model of an environmental matching agreement between multiple asymmetric players and investigate whether the matching mechanism can increase players' abatements and payoffs as compared to the game without matching, even when players have the possibility to opt out of the matching agreement.A comparison of the matching agreement's abatement level and payoff with those of the no-agreement Nash baseline and the fully cooperative outcome will give an indication of the agreement's effectiveness in overcoming potential free-rider incentives.Precisely put, the following research questions are explored: Main research question: How effective are international environmental matching agreements between heterogeneous players in overcoming the free-rider incentive when players have the option to opt out?

Sub-research questions:
-What are the stable coalition structures: • under a standard agreement (no matching) with an opt-out possibility?
• under a matching agreement with an opt-out possibility?
• under a matching agreement with an opt-out possibility and emissions trading?
-What are players' individual abatement contributions and payoffs, and what are the total abatement level and the collective payoff: • without an agreement ('all singletons')?
• given a stable agreement without matching?
• given a full matching agreement (no opt-out)?
• given a stable matching agreement with an opt-out possibility?
• given a full matching agreement with emissions trading (no opt-out)?
• given a stable matching agreement with an opt-out possibility and emissions trading?

Models and research methodology
As indicated by the research questions, the current study compares the outcomes of three alternative agreement models with the no-agreement baseline and the fully cooperative outcome.Section 4.1 outlines the standard model of coalition formation without matching components.Section 4.2.1 and 4.2.2 present the matching agreement models without emissions trading and with an emissions trading stage, respectively.To obtain the grand coalition and the full matching agreement outcomes, all three models are also analyzed without the initial coalition accedence stage, i.e. assuming that all players join the respective agreements.

Stable coalitions in standard agreements
The baseline model of coalition formation without matching considers a two-stage, non-cooperative game between  heterogeneous players that generate damaging GHG emissions.The emissions are assumed to be perfect substitutes in each player's damage function.Let  = {1, … , } denote the set of players to the game.Each player incurs player-specific benefits from its own emissions and playerspecific damages from collective emissions.Conversely, players incur player-specific costs   from their own abatement efforts   and player-specific benefits   from collective abatement of emissions ∑   ∈ .The payoff   to player  ∈  is defined as In the present study, we consider linear benefit and quadratic cost functions that monotonously increase in the positive abatement domain, resulting in the payoff function where   and   are player-specific abatement benefit and cost parameters.
In the first stage of the game, players decide whether or not to accede to an agreement and join a unique coalition .We assume that the agreement is an 'open-membership agreement', meaning that any player can freely join it without needing the permission of the coalition members (Finus et al., 2005).
Players that decide to join commit to playing abatement strategies which maximize the coalition's total payoff, while non-members remain singleton players that set their contributions non-cooperatively.In the second stage, players execute their respective abatement strategies.
Both coalition members and non-members are assumed to play a Nash game with respect to their abatement strategies.Singleton players have dominant abatement strategies due to the linear benefits of abatement and select their abatement contributions such that their individual payoffs are maximized, taking all   ,  ≠  as fixed.Any singleton player  thus selects its abatement level   such that ′ (  ) = 1.Marginal costs and marginal benefits of abatement are hence equated for each singleton player.Given the linear benefit and quadratic cost functions, we have Coalition members cooperate to maximize the sum of all coalition members' combined payoffs, taking all non-members' abatement levels as fixed.Each signatory thus sets its abatement level   such that its marginal cost of abatement is equal to the coalition's marginal benefit of abatement, i.e. .In the case of the specified linear benefit and quadratic cost functions, this means that signatories choose their contributions such that Assuming Nash strategies, the coalition that is formed will be such that members reap higher payoffs from being a member to the coalition than from staying external to it, and non-members reap higher payoffs when playing as singletons than if they joined the agreement: Internal stability:   () ≥   (\{}),  ∈ When these conditions are met, a coalition is considered internally and externally stable, as no player has an incentive to change its strategy (Carraro & Siniscalco, 1993).
With  players, there are 2  −  possible unique coalition configurations.When fewer than two players have incentives to join a coalition, we term the outcome the 'all singletons' coalition structure.
Conversely, when all players sign the agreement, the so-called 'grand coalition' is realized, which results in the globally optimal level of abatement and an equalization of all players' marginal abatement costs.

The matching game
In the matching game, the basic assumptions about the set of players , as well as the nature of their abatement benefit and cost functions are retained (as outlined in the previous section).The matching game analyzed in this study is similar to the one presented in Fujita (2013), as it considers a uniform, cooperatively determined matching rate.However, it differs in that we allow players to be heterogeneous in terms of their abatement benefit and cost functions, and that non-signatories' contributions are not matched.
The matching mechanism itself consists of two stages.In the first stage, the players participating in the matching game negotiate a common matching rate  by which every such player  must match the unconditional 'flat' abatements of every one of its peers  ≠ .For instance, a matching rate of 0.5 would oblige the matching players to make additional abatements equivalent to half of the sum of all other matching players' flat contributions.As it is customary in the literature on coalition formation games, we assume joint payoff maximization by the cooperating players.The matching rate is thus selected such that it maximizes the collective payoff of all matching players.In the second stage, all players set their unconditional abatement contributions   non-cooperatively, considering the matching rate announced in the previous stage.Player 's total abatement commitment   is thus

The matching game with an opt-out possibility
To be able to determine whether matching agreements can effectively curb the general freeriding problem of IEAs, they must be examined in a context of voluntary participation.We therefore define the following three-stage game to determine the stable coalition structures, as well as the concomitant abatement and payoff levels under a matching agreement with an opt-out possibility: 1. Coalition accedence: Players non-cooperatively decide to join the matching agreement or to remain singleton players.As in the standard game of coalition formation, the matching agreement is considered to be an 'open-membership agreement' that can freely be joined by any player.

Negotiation of the matching rate:
Coalition members cooperatively set a common matching rate by which every coalition member must match the other coalition members' flat abatements.Non-members do not match other players' contributions.

Choice of flat abatements:
All players choose their flat contributions non-cooperatively.
The remainder of this section reports the analytical solution of the game.
As in the standard model of coalition formation in Section 4.1,  = {1, … , } denotes the set of players to the game.With linear benefits of collective abatement   =   ∑   ∈ and quadratic costs of its own abatement   = 1 2     2 , player 's payoff is defined as where   are the individual players' total abatements, and   and   are player-specific benefit and cost parameters, respectively.
Let  ⊆  denote the set of signatories to the matching agreement.The number of signatories in a unique coalition  is denoted by .Non-signatories do not commit to any matching, hence for  ∉ ,  = 0.In extended form, the payoff to player  ∈  or  ∉  is given by where   is player 's flat abatement and  is the matching rate set by the coalition which determines each coalition member 's matching contribution offered on the flat abatements of the other coalition members  ≠ .For coalition member  ∈ , its total contribution is defined as   =   +  ∑   ≠ with  ∈ .For singleton player  ∉ , its total contribution is equivalent to its flat abatement,   =   .We solve the game through backward induction.

Stage 3: Choice of flat abatements
At this stage, all players have made their choices on whether to join the agreement or opt out, and the coalition members have negotiated the matching rate.All players now choose their flat contributions non-cooperatively.As all singleton players have dominant strategies, their flat abatements are equivalent to the singleton contributions in the standard game of coalition formation (Eq. 3): We now derive the equilibrium flat contributions for the coalition members  ∈ , assuming an interior solution.Coalition member 's payoff function is defined by Eq. 8 or the upper side of Eq. 9. Since all non-signatories have dominant strategies, their abatement contributions do not show in the coalition members' first order conditions: Solving Eq. 11 for   yields Introducing  = (1 + ( − 1)), we have and In equilibrium, the coalition members will choose their flat contributions such that all members' payoff functions are maximized simultaneously.We can also show that the sum of all coalition members' equilibrium flat contributions is equivalent to the sum of their would-be singleton abatements.Since we can use Eq. 14 to show that which simplifies to This result implies that the existence of a stable matching coalition inevitably increases the global abatement level beyond that of the no-agreement baseline, provided that the matching rate is positive.Precisely put, the sum of all coalition members' matching contributions is equivalent to the additional global abatement above the baseline induced by the matching agreement.

Stage 2: Negotiation of the matching rate
In this stage, all players that acceded to the matching agreement in stage 1 negotiate a common matching rate that maximizes the coalition's total payoff, taking the stage 3 equilibrium abatements as given.Disregarding non-members' abatements, the coalition's collective payoff function can be expressed as follows: Using   =      (Eq.14) from stage 3, we have With the partial derivative of  with respect to  being we maximize   by setting the following function equal to zero: which is equivalent to Rearranging for , we have for every particular coalition .The equilibrium matching rate will always be such that  ∈ (0,1).

Stage 1: Coalition accedence
In this stage, all players non-cooperatively decide whether to join the agreement and commit to the matching game or to opt out and play the Nash baseline game.Anticipating the equilibrium outcomes of the subsequent stages, each player evaluates whether it could increase its payoff by staying external to the agreement.If fewer than two players accede to the agreement, the second stage is skipped and the standard game of coalition formation without matching is recovered.
The condition for player  to accede to the matching agreement is   () ≥   (\{}),  ∈  (internal stability condition, Eq. 5).It will choose to remain a singleton player if   () ≥   ( ∪ {}),  ∉  (external stability condition, Eq. 6).As signatory, player  anticipates the subgame perfect Nash equilibrium of the matching game with   =      (Eq.14).As non-signatory, it plays the Nash baseline game with   =   =     (Eq.10).The solution of this stage identifies stable coalitions  which satisfy both stability conditions.While Fujita ( 2013) manages to solve this stage analytically for a matching game between symmetric players, analytical solutions are difficult to obtain for games with heterogeneous players.We therefore resort to examining the coalition stability of this agreement numerically by means of a simulation (Section 5.2).

The matching game with an opt-out possibility and emissions trading
We now extend the basic matching game as presented in the previous section by an additional emissions trading stage.Boadway et al. (2011) include such a stage in their study and find that the combination of the matching mechanism and the emissions trading scheme brings about socially optimal abatement levels, as the emissions trading scheme equalizes the participating players' marginal abatement costs.The game with emissions trading consists of the following consecutive stages:

Coalition accedence: Players non-cooperatively decide to join the (open-membership)
matching agreement or to remain singleton players.

Negotiation of the matching rate:
Coalition members cooperatively set a common matching rate by which every coalition member must match the other coalition members' flat abatements.Non-members do not match other players' contributions.

Choice of flat abatements:
All players choose their flat contributions non-cooperatively.

Emissions quota trading:
Coalition members may trade their emissions quotas at an equilibrium price.Non-members do not participate in the carbon market.
We now characterize the solution of the matching game with emissions trading.
As before,  = {1, … , } denotes the set of players to the game.With linear benefits of abatement   =   ∑   ∈ and quadratic costs of abatement   = 1 2   (  −   ) 2 , player 's payoff is now defined as where   are individual players' abatements, and   and   are player-specific benefit and cost parameters, respectively.  is the quantity of emissions quotas purchased at the equilibrium price  by player .
⊆  denotes the set of signatories to the matching agreement.Non-signatories do not commit to any matching and do not participate in the carbon market, hence for  ∉ ,  = 0 and   = 0.In extended form, the payoff to player  ∈  or  ∉  is given by where   is player 's flat abatement and  is the matching rate set by the coalition which determines coalition member 's matching contribution offered on the flat abatements of the other coalition members  ≠ .For coalition member  ∈ , its total abatement commitment is defined as   =   +  ∑   ≠ with  ∈ .For singleton player  ∉ , its total contribution is equivalent to its flat abatement,   =   .
In the game with emissions trading, coalition members are free to meet their total abatement commitments   ≥ 0 with self-produced abatements   or by purchasing emissions quotas   from other coalition members at an equilibrium price : Emissions quotas sold to other coalition members are expressed as   < 0. Every coalition member may purchase at most as many quotas as it commits to units of abatement   , and may sell at most as many quotas as it produces units of abatement   .We solve the game through backward induction.

Stage 4: Emissions quota trading
When emissions quota trading is assumed, the last stage allows coalition members to trade their abatement quotas at an equilibrium price .Non-signatories do not participate in the carbon market.This ensures dominant strategies for all  ∉  and thus prevents carbon leakage.
At this stage, all coalition accedence choices have been made and the matching rate  as well as the flat abatements   have been determined.Signatories' total abatements   =   +  ∑   ≠ are hence fixed and determine the emission quotas that each of them can trade.Assuming that all players are price-takers and assuming an interior solution, any signatory's demand   for emission quotas at price  minimizes its cost As all abatement commitments   are fixed, the benefit function need not be considered.Differentiating with respect to   yields The demand for emission quotas by player  ∈  thus satisfies With   =   +   (Eq.26), every coalition member's self-produced abatements are such that As the equilibrium price satisfies  =     , the coalition members' marginal abatement costs are equalized, which ensures an efficient allocation of abatements within the coalition.In equilibrium, the For simplification, we will henceforth refer to the constant as   .We then have

Stage 3: Choice of flat abatements
When choosing their flat abatements   , the coalition members take the matching rate defined in the previous stage as given.Players also anticipate the equilibrium price of quotas in the subsequent stage.We will first characterize the unconstrained solution of this stage, where   ∈ ℝ.
Using   =   −   (Eq.26) and   =    (Eq.30) from stage 4, we have the following expression of coalition member 's payoff: In the unconstrained equilibrium, the coalition members choose their flat contributions such that all members' payoff functions are maximized simultaneously.As Boadway et al. (2011) note in their study, the equilibrium abatement levels   are socially optimal in this case and the coalition's payoff   is equivalent to the payoff realized under full cooperation by the coalition members.However, when players are heterogeneous, this equilibrium results in negative total abatement commitments by some coalition members.Helm (2003), Carbone et al. (2009) and Yu et al. (2017) analyze this phenomenon commonly referred to as 'hot air' in climate policy, whereby the prospect of generating a payoff from selling emissions quotas induces some players to increase their demand for emission allowances beyond the allowance level they would demand in the case without emissions trading, and sometimes beyond their BAU emission levels.In the unconstrained equilibrium of this game, the negative total commitments allow the respective players to cash in on the trading opportunity by selling an amount of emissions quotas   that exceeds their amount of self-produced abatements   .
To prevent players from setting negative abatement commitments and producing 'hot air', we solve this stage while enforcing the binding exogenous constraint   ≥ 0 in the simulation.To find the solution of this constrained optimization problem which is closest to the optimal payoff levels in the unconstrained problem, we minimize the sum of the squared differences between the optimal payoffs   * and the feasible payoffs    in the constrained region:

Stage 2: Negotiation of the matching rate
At this stage, all players that acceded to the matching agreement in the first stage negotiate a common matching rate  that maximizes the coalition's collective payoff, taking the equilibrium abatements and the outcome of the carbon market as given.Non-signatories do not commit to matching other players' contributions and proceed directly to stage 3.As the latter have dominant strategies due to constant marginal benefits of abatement, we need not consider their contributions when solving this stage.
Due to the equalization of the coalition members' marginal abatement costs by the carbon market, the matching rate that maximizes any coalition's payoff is always unity.We resort to numerical methods to draw this conclusion, which is consistent with the findings in other matching literature.For instance, Fujita (2013) who analyzes a matching game between multiple symmetric players with a common matching rate finds the equilibrium matching rate of the game to be unity.Boadway et al. (2011) also find that the product of all players' individual, non-cooperatively determined matching rates in their game is equal to one in equilibrium.Some illustrative examples of the game's collective payoff outcome for different coalitions and matching rates are provided in Appendix A.

Stage 1: Coalition accedence
As in the other models analyzed in Sections 4.1 and 4.2.1, in stage 1 of this game, all players noncooperatively decide whether to join the agreement and commit to the matching game or to opt out and play the Nash baseline game.Anticipating the equilibrium outcomes of the subsequent stages, each player evaluates whether it could increase its payoff by staying external to the agreement.If less than two players accede to the agreement, the second stage is skipped and the standard game of coalition formation without matching is recovered.
The condition for player  to accede to the matching agreement is   () ≥   (\{}),  ∈  (internal stability condition, Eq. 5).It will choose to remain a singleton player if   () ≥   ( ∪ {}),  ∉  (external stability condition, Eq. 6).The solution of this stage identifies stable coalitions  that satisfy both stability conditions.As for the matching game without emissions trading, we solve this stage numerically since analytical results are difficult to obtain when considering asymmetric players (Section 5.3).

Data and simulation model calibration
To generate numerical results of the different agreements introduced in the Sections 4.1 to 4.2.2, as well as the non-cooperative Nash baseline and the grand coalition outcomes, we calibrate a general simulation model consisting of a predefined set of players' abatement benefit and cost estimates.The latter are used as input for individual simulations of each of the agreements analyzed.As the data underlying the different simulations are the same, their results are directly comparable and can be used to draw conclusions about the agreements' potential to overcome the general free-rider problem of IEAs.

The Stability of Coalitions (STACO) model
The simulation data are calibrated based on the third version of the Stability of Coalitions (STACO) model developed by the Environmental Economics and Natural Resources Group of Wageningen University.STACO 3 is a dynamic, multi-region model which serves to investigate the formation and stability of international climate agreements in the time frame from 2010 to 2110 (Dellink et al., 2015).It considers a two-stage, non-cooperative game of coalition accedence and GHG abatement (as described in Section 4.1) between twelve world regions with linear abatement benefits and cubic abatement costs.Detailed specifications on STACO 3 can be found in its latest technical manual by Dellink et al. (2015).
Much of the STACO 3 model's underlying data and calibrations are based on the fifth version of the Massachusetts Technology Institutes' (MIT) Economic Projection and Policy Analysis (EPPA5) model, which is a multi-region, multi-sector computable general equilibrium (CGE) model of the world economy and its effects on the global climate (Chen et al., 2017).Information about the EPPA5 model is provided in its technical manual by Chen et al. (2017).

General simulation model set-up
The current study's general simulation model is a static representation of the dynamics that are inherent in the STACO 3 model, meaning that players' payoff functions and abatement strategies are generalized to be uniform across the time horizon of the agreement.This simplification was made to increase the model's conceptual fit with the one-shot matching game as it is presented in the literature and decrease its emphasis on the complexity of the growth-climate interactions.The procedure that was followed to calibrate the static model is described in Section 4.3.4 and Section 4.3.5 of this thesis.The simulations themselves were programmed in the general-purpose programming language Python 3.8.
In the simulations, players make a decision on their accedence to the analyzed agreement before 2020, and then set their abatement strategies for the time period from 2020 up to and including 2100.While the duration of the agreement and players' abatement strategies are limited to 81 years, the time horizon for the benefits of abatement extends beyond the agreement term to reflect the long-term effects of climate change.The general model assumes constant abatement paths over the duration of the agreement, meaning that players decide on a fixed abatement to make every year.The games are thus formulated with annual costs and benefits as the payoff space.The model follows the twelveregion structure of the STACO 3 model, which comprises the players presented in Table 1 and Figure 1.

GDP, population and emissions data
As the baseline data underlying the STACO 3 model dates back to the first half of the past decade, the current study uses more recent GDP, population and business as usual (BAU) GHG emissions projection estimates from a different source.The data were retrieved from the second version of the Shared Socioeconomic Pathways (SSP) database, which contains data on five possible trajectories of global socioeconomic and climatic development in the 21 st century.The different scenarios each consist of a narrative of future development which is substantiated with quantitative data from economic, demographic and integrated assessment models (IAMs) (van Vuuren et al., 2017).The SSPs were jointly developed by research teams from a variety of organizations in an effort to facilitate climate change analysis.The public database is administered by the International Institute for Applied Systems Analysis (IIASA) and may be accessed via the following link.
Among the five SSPs, SSP2 describes a "middle-of-the-road" scenario in which "social, economic, and technological trends do not shift markedly from historical patterns" (Riahi et al., 2017).The current study uses data from the SSP2 scenario to avoid incorporating assumptions about drastic future socioeconomic change in the model.A more detailed description of the different SSP narratives is provided in O' Neill et al. (2017).
As each SSP was implemented by multiple IAM models, so-called 'marker' scenarios were selected from the different quantitative interpretations as being representative of the respective SSP storylines.We use the baseline emissions data from the MESSAGE-GLOBIOM model, which was selected as the SSP2 marker model.Similarly, from the three models of economic development that were available, we use the SSP2 data from the marker model by the Organization for Economic Cooperation and Development (OECD) (see Dellink et al., 2017).The population projections are based on KC and Lutz (2017).Detailed information on the SSP2 marker interpretation can be found in Fricko et al. (2016).
While the GDP and population data were readily available on a country basis, the GHG emissions projections were aggregated in a five-region structure in the SSP database.To obtain data on individual countries' BAU emissions projections which could later be used to construct the STACO aggregation, regional projections were therefore scaled by a country-specific GDP factor. Figure 2 contains a graphical representation of the resulting emissions projections for the STACO regions over the 2020 to 2100 time horizon.Graphical representations of the regions' GDP and population projections can be found in Appendix B.

Figure 1: Regional aggregation
The emissions projections include data on all GHG of the Kyoto basket, expressed in CO2 equivalents in terms of their global warming potential, which include carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O), sulfur hexafluoride (SF6) and the so-called F-gases (hydrofluorocarbons and perfluorocarbons).

Calibration of the benefits of abatement
The general simulation model's benefit estimates are largely based on the STACO 3 model's climate and benefits calibrations.Regional damages of emissions and benefits of abatement are linked to global BAU-emissions via a linearized climate module, which translates emissions into GHG concentrations, temperature change and ultimately into global damages.Abatement benefits are defined as forgone damages, i.e. the difference in damages between emissions after collective abatement and the global BAU emissions baseline.Regional benefits are calculated by scaling global benefits by a regional benefit share parameter and the region's respective yearly GDP projection.

Figure 2: BAU emissions paths per region
The linearity of the emissions-damages interaction is given by the combination of the climate dynamics' functional forms.Specifically, temperature change is commonly presented as a logarithmic function of concentrations (via an intermediate radiative forcing step) and damages as a quadratic function of temperature change.In combination, these functional forms yield a quasi-linear function for concentrations and damages, which is why the STACO model uses a linear approximation for these nonlinear dynamics (Dellink, 2004;Dellink et al., 2015).The linearity of the benefits functions constitutes a convenient feature of the model as it ensures dominant strategies for all players of the standard game of coalition formation and for the singleton players in the matching games.
In the model, atmospheric GHG concentrations   (ppmv CO2-e) are thus a linear function of concentrations in the previous period and global emissions after abatement   = ∑   (Gt CO2-e): is a calibration parameter that can be interpreted as the decay of GHG concentrations over time, and   represents a transfer coefficient from emissions to concentrations.Table 2 lists the different parameter values of the model, which were estimated using ordinary least squares regression (OLS) to fit the baseline data of the STACO model (Dellink et al., 2015).The starting value  0 = 500 ppm is equivalent to the atmospheric concentration of GHG in 2019 as reported in the annual greenhouse gas index (AGGI) by the US National Oceanic and Atmospheric Administration's (NOAA) Global Monitoring Laboratory (NOAA, 2020).
Concentrations   are then converted into temperature change ∆  (°C as compared to 1900) via the following linearized function: is a calibration parameter which can be interpreted as heat dissipation and   is a transfer coefficient from concentrations to temperature change.∆ 0 is the atmospheric temperature change in 2019 compared to 1900, which the NOAA reported to be 1.03°C (WMO, 2020).
is a scale parameter of global damages as used in STACO 3 and based on Tol (2009) whose value is included in Table 2.  ̅  are the regional GDP values of the respective years.The different regional benefit share parameters are presented in Table 3.
Table 3: Regional shares of benefits (Dellink et al., 2015) Regional benefits are then described as the difference between region-specific damages with and without abatement, discounted over time.As the abatements of any particular year produce a stream of benefits over the abatement year as well as the subsequent years, presumably to infinity, the benefits function is a concave expression of discounted benefits accruing to past and current global abatement that extend beyond the agreement term: The discount rates   are set according to the Ramsey rule, featuring a 1.5 percent pure rate of time preference  and an elasticity of marginal utility with respect to consumption  of 1, both obtained from the UK Treasury's Green Book (2020).The growth of consumption per capita ̇  is calibrated based on the different regions' GDP and population growth projections.Rather than retaining time-specific discount rates, the model uses static discount rates based on average regional growth rates.The resulting discount rates are included in Table 3.
To derive a static benefits function, we assume constant abatement paths for the duration of the agreement and no abatement after 2100.Setting the global abatement path to one unit (Mt CO2-e) of abatement per year yields regional net present values of the different regional benefit streams, which were then converted into equivalent annual annuities (EEAs) to be uniform across the abatement horizon.The static regional benefits function can then be expressed as follows, with   as an annual benefit: where  = ∑   and   is a region-specific benefit parameter and the slope of each region's marginal abatement benefit curve.An overview of the regional benefit parameters is included in Table 3.As no data was available on BAU emissions or GDP beyond 2100, benefit estimates after this year are based on the assumption of constant global emissions and regional GDP, using the values from the last available data year.Rather than accounting for an infinite stream of benefits, the model includes benefits until the year 2200, given that later benefits become marginal due to discounting.

Calibration of the costs of abatement
The model's abatement cost estimates are based on the abatement cost curves used in the STACO 3 model, which, in turn, are based on Morris et al. (2008) and the EPPA5 model (Dellink et al., 2015).The STACO model's cost estimates are described by a cubic function with two regional cost parameters   and   , as well as a region-specific technological progress parameter   : To reduce the dynamics of the model to a static relationship between abatements and costs, we make the simplifying assumption that the marginal abatement cost curves for the different regions will remain the same over the time horizon 2020 to 2100.We thus assume no technological progress over time, i.e.

21
= 0.When assuming indefinite constant abatement paths for all regions with equal abatements   in every year, the annuity of the cost is given by the following simplified cost function: The different STACO regions' cubic cost functions were approximated by quadratic ones via OLS in the domain [0,   ̅], where   ̅ denotes each region's mean BAU-emissions over the time horizon 2020 to 2100.
Abatement cost estimates in the quadratic model are thus closest to their commensurate estimates from the cubic model when regions' yearly abatements are at most as high as their mean BAU-emissions between 2020 and 2100.Figure 3 provides a graphical example of the cubic and the approximated quadratic abatement cost curves for one of the regions.

Figure 3: Quadratic approximation of a cubic abatement cost function
The approximated quadratic cost functions can generally be expressed as follows, with   as an annual cost: where   is a region-specific cost parameter and the slope of each region's linear marginal abatement cost curve. 1An overview of the different cost parameters and mean BAU-emissions levels is provided in Table 4.A graphical overview of the regional marginal abatement cost curves is presented in Figure 4. We include a graphical representation of the original STACO 3 marginal abatement cost curves in Appendix C. 1 We abstain from adding a linear term to the quadratic cost function as the regressions yield negative coefficients for this term, resulting in negative marginal abatement costs for low abatement levels.Although these could be interpreted as local ancillary benefits, we choose to mimic the interpretation of the cubic model, which monotonously increases in the domain {0, ∞}.

Results
The current section reports the simulation results.To demonstrate how the different agreement types perform in terms of abatement and payoff levels, we calculate so-called 'closing-the-gap indices' (CGIs) for each of them.Here, the gap is the difference between the non-cooperative Nash baseline ('allsingletons') and the socially optimal ('grand coalition') outcomes, and the index expresses to what extent each agreement can bridge the divide between the respective global abatement and payoff levels.We report the different agreements' CGIs in Table 11.
Broadly speaking, the globally optimal abatement level covers 53.2 percent of the mean global yearly BAU emissions, as opposed to the all-singletons abatement level of 6.5 percent (Table 9).While in the non-cooperative outcome, abatement contributions strongly depend on players' marginal abatement benefits, the fully cooperative outcome equalizes the players' marginal abatement costs, thus inducing those players with the smallest cost parameters to make the largest contributions (Table 9).Although the socially optimal outcome leads to the highest possible collective payoff, seven out of the twelve players therefore experience negative payoffs in the grand coalition (Table 10).Additionally, the grand coalition requires some players to make abatement contributions that exceed their BAU emissions over most or all of the agreement term, thus making negative emission technologies necessary to achieve the abatement targets.
We now turn to examine the numerical results of the stable standard agreement, as well as the two different matching agreements with and without emissions trading in more detail.Table 5 repeats the different players' benefit and cost parameters for convenience.Table 9 contains the yearly abatement values for the different agreement types and players.Table 10 reports the annual payoffs for the different agreement types and players.Finally, Table 7 presents the players' marginal abatement costs under the different agreements.

The standard agreement
With twelve players, there are 2 12 − 12 = 4,084 possible unique coalitions per game, including the allsingletons outcome.In our simulated standard game of coalition formation (Section 4.1), the only coalition which satisfies both stability conditions (Eq. 5 & 6) is the one between the USA and EUR.Both of these players have significantly higher marginal benefits of abatement than the other players, as well as relatively low marginal abatement costs (see Table 5), making it lucrative to maximize their joint payoffs for each of them.With relatively lower marginal benefits of abatement and higher marginal abatement costs, all other players have incentives to freeride when the {USA, EUR} coalition is formed.
In the standard agreement, both coalition members nearly double their abatements as compared to the singleton outcome.While the USA make yearly contributions equivalent to 56.6 percent of their mean yearly BAU emissions in the agreement term, EUR's yearly contribution is equivalent to 32.5 percent of its mean yearly BAU emissions.All singletons' contributions lie in the range of 0.15 to 6.8 percent of the respective players' yearly BAU emissions (Table 9).
The repartition of annual payoffs is generally aligned with the relative magnitude of players' marginal abatement benefits in this agreement; players with higher marginal benefits reap relatively higher payoffs.However, while the USA and EUR experience marginal payoff improvements of 8.5 and 30.3 percent from the all-singletons case, the main beneficiaries of the agreement are the non-signatories with payoff increases of 59.6 to 65.6 percent each.
In terms of global abatement, the standard agreement leads to total yearly abatement levels which are equivalent to 10.4 percent of the world's mean annual BAU emissions (as compared to 6.5 percent in the all-singletons case and 53.2 percent under the grand coalition) (Table 9).It thus manages to narrow the gap between Nash baseline levels and socially optimal levels by 8.3 percent.Similarly, the global payoff gap is narrowed by 10.1 percent under this agreement (Table 11).

The matching agreement without emissions trading
Our simulation of the matching game without emissions trading (Section 4.2.1)does not produce any coalitions that satisfy both stability conditions (Eq. 5 & 6).When players have the option to freely join and opt out of the matching agreement, all of them choose to remain singleton players and the global abatement and payoff levels are the Nash baseline levels.We therefore note CGIs of zero for the matching agreement with an opt-out possibility (Table 11).
The full matching agreement that assumes coalition accedence by all players provides more promising results.Here, 26.95 percent of the global mean annual BAU emissions are abated, narrowing the gap between the Nash baseline and socially optimal abatement levels by 43.75 percent.Similarly, the global payoff gap is narrowed by 37.09 percent (Table 11).
In the full matching agreement, the matching rate that maximizes the coalition's collective payoff is  ≈ 0.285.Coalition members have no control over the size of their matching contributions since the latter are determined by the matching rate and the respective other players' flat contributions.To adjust their total abatement commitments so as to maximize their individual payoffs, players with relatively low marginal benefits and high marginal costs of abatement are therefore forced to set negative flat abatements.However, as the USA and EUR have incentives to set large positive flat contributions, all but these two players' matching contributions are positive, resulting in positive total contributions by all players.The different players' flat, matching and total contributions in the full matching agreement are reported in Table 6.As predicted by Eq. 17, the sum of all players' flat abatements is equivalent to the global Nash baseline abatement level in the absence of an agreement, meaning that the global abatement level under any matching agreement inevitably surpasses the Nash baseline abatement level.
Under the full matching agreement, players' total abatement commitments each increase by 313.4 percent compared to the singleton abatement levels, leading to substantial payoff increases of 125.1 to 313.0 percent for all players but the USA.The latter experience a payoff decrease of 30.9 percent, giving them a clear preference for the all-singletons outcome or the standard agreement.In fact, the matching agreement demands that the USA make yearly total abatement contributions in excess of their annual BAU emissions in the agreement term, implying that they must employ negative emission technologies to fulfill their commitments.

The matching agreement with emissions trading
As in the simulation of the matching game without emissions trading, our simulation of the matching game with emissions trading (Section 4.2.2) does not produce any stable coalitions.We thus note CGIs of zero for the game that allows players to opt out of the agreement (Table 11) and focus our analysis on the full matching agreement with emissions trading.
As declared in Section 4.2.2, the model of the matching game with emissions trading that we analyze prescribes that players commit to non-negative total abatements.This is equivalent to demanding that players trade at most as many emissions quotas as they abate units from their BAU emissions baselines.In the unconstrained game with emissions trading, five out of all twelve players take advantage of the trading opportunity by committing to negative total contributions and selling the resulting additional emission quotas.The results of this unconstrained matching game with emissions trading are reported in Appendix D. As Boadway et al. (2011) observe in their study, the combination of the matching mechanism and the emissions trading scheme brings about socially optimal abatement levels and the socially optimal global payoff in the unconstrained game.In the constrained version of the game, the players that would set negative total commitments in the unconstrained game are forced to forgo the benefits of doing so.Consequently, the constrained game results in higher abatement levels by all players and lower payoffs for most of them (Table 8 and Appendix D).
While the unconstrained matching game with emissions trading would achieve abatement and payoff CGIs of 100 percent, the binding exogenous constraint of non-negative total abatement commitments induces a global abatement level that exceeds the optimal abatement level and therewith reduces the global payoff to a less-than-optimal level.Specifically, 76.21 percent of the mean global BAU emissions are abated in the full constrained matching agreement with emissions trading (Table 9), bridging the gap between the Nash baseline and socially optimal abatement levels by 149.25 percent.The global payoff gap, however, is only narrowed by 76.20 percent (Table 11).
As predicted by Boadway et al. (2011), the emissions trading scheme equalizes the marginal costs of abatement across all players in this agreement, which are then equal to the equilibrium carbon price (see Table 7).As explained by Yu et al. (2016), the forgone benefits due to the exogenous constraint on abatement commitments translate into an increased shadow price for emission allowances.Hence, the equilibrium price in the constrained game is higher than that in the unconstrained game, which is the same as players' marginal abatement costs in the grand coalition without matching.In our simulation, we obtain carbon prices of 207.75 and 297.81 USD per ton of CO2-e for the unconstrained and constrained agreements, respectively (Appendix D and Table 7).The equalization of the marginal abatement costs by the emissions trading scheme causes the coalition's payoff-maximizing matching rate for any coalition to be  = 1 (Eq.46).In terms of flat abatements and matching contributions, we observe a similar pattern as in the full matching agreement without emissions trading.In the full agreement with emissions trading, the USA, EUR, CHN and ROW set large positive flat contributions, while the remaining players reduce their total commitments by setting negative flat abatements.This results in negative matching contributions for the four players with positive flat contributions and the reverse for the remaining players.In terms of total abatement contributions, the players that would have incentives to set the largest negative total contributions in the unconstrained game make do with commitments of zero and sell all of their self-produced abatements (OHI, ROE, BRA) (Table 8).The remaining players set total abatements which are between 733.8 and 5956.9 percent higher than their Nash baseline abatements.
In equilibrium, the carbon market clears when the aggregate of emissions quotas traded is zero.It follows that the sum of total abatement commitments must be equally high as the sum of self-produced abatements by all players (Eq.31).In our simulation, all emissions quotas sold are bought up by the USA, EUR and ROW.CHN, which, similarly to these players, also commits to high total contributions, fulfills the entirety of its commitment with self-produced abatements thanks to its low marginal abatement costs (Table 8).It is therewith the player with the largest absolute amount of self-produced abatements, equivalent to 119.5 percent of its mean annual BAU emissions.Besides CHN, five other coalition members self-produce amounts of abatement that exceed their mean annual BAU emissions (USA, JPN, OHI, RUS, MES), implying the use of negative emissions technologies in this agreement.
The full matching agreement with emissions trading increases all but two players' payoffs compared to the Nash baseline payoffs.While the USA's payoff is decreased by 79.4 percent, resulting in the smallest positive payoff in the coalition, CHN's payoff is reduced by 617 percent relative to the baseline, causing this player to incur a large negative payoff (Table 10).Both of these players thus have clear preferences for the all-singletons outcome.

Discussion
The aim of this study was to provide numerical evidence of the matching mechanism's potential effectiveness at increasing players' contributions and payoffs in IEAs.While prior research has substantiated that different versions of the matching game can produce pareto-efficient outcomes given full participation by all players (e.g.Guttman, 1978;Guttman & Schnytzer, 1992;Rübbelke, 2006;Boadway et al., 2011), only three theoretical studies have investigated how matching agreements fare with regard to coalition stability thus far (Fujita, 2013;Wood & Jotzo, 2015;Liu, 2018).Given that enforcing participation is generally difficult in IEAs, coalition stability determines in large part how effective such agreements can be in practice.
The merit of this study as compared to the studies by Fujita (2013), Wood & Jotzo (2015) and Liu ( 2018) is three-fold.First, while these studies' approaches are purely theoretical, we offer numerical results to provide further insight into the practical implications of the matching games analyzed.In particular, the simulations of the different matching agreements we conducted manage to convey a sense of magnitude of the matching mechanism's effect on abatements and payoffs relative to the noncooperative Nash baseline and the desired socially optimal outcome.Second, we allow for player heterogeneity (as does Liu ( 2018)), while simultaneously considering cooperatively determined matching rates (as in Fujita ( 2013)), rather than exogenously set ones (as in Liu (2018)).These game characteristics, which had not been analyzed in combination before, can be argued to be more representative of a real-world setting in which players are heterogeneous and agreements are designed to reach a collective goal.Moreover, we assume that coalition members only match each other's contributions (contrary to Fujita (2013) and Wood & Jotzo (2015)), which precludes the informational difficulty of having to match all non-signatories' contributions in addition to the coalition members'.Third, we extend the matching game to include an emissions trading stage, demonstrating how emissions trading can leverage the effect of the matching mechanism to reach significantly higher abatement and payoff levels.
Our results are generally consistent with the findings of other studies.As predicted by Molina et al. (2020), players' equilibrium (self-produced) abatement contributions are higher than at the Nash baseline when the full matching agreements are formed.Our results also confirm Boadway et al.'s (2011) prediction that the equilibrium abatement levels of the full matching agreement with emissions trading are socially optimal, provided that players are not constrained in their choice of total contributions.As observed by Liu (2018), however, coalition stability seems to be difficult to achieve in the type of matching agreement we investigated.In fact, both matching agreements that we simulated perform worse in terms of stability than the standard agreement.We can therefore conclude that the application of a matching mechanism featuring a uniform, cooperatively determined matching rate likely cannot fully resolve the general freerider problem of IEAs in settings where players are heterogeneous.In contrast, Fujita (2013) suggests that the matching mechanism produces stable and efficient outcomes when symmetric players are considered.It remains to be shown whether other types of matching games such as the one analyzed by Boadway et al. (2011) and Molina et al. (2020), where players set their own matching rates non-cooperatively, are better suited to address stability issues in IEAs between asymmetric players.
Despite having failed to produce stable coalitions, our simulation results may still provide a cause for optimism.As described in Section 5, the grand coalition in the standard game of coalition formation demands that the majority of players make contributions of sizes that cause them to incur negative payoffs.In this coalition, all but the USA and EUR have incentives to renege on their required contributions, making it highly unlikely that such an outcome materializes in practice.In contrast, the full matching agreements bring about payoffs which present much fewer players with incentives to defect.In the full matching agreement without emissions trading, none of the players incur negative payoffs, and the USA, EUR and CHN are the only ones which have incentives to leave the full coalition.
In the full matching agreement with emissions trading, the only player incurring a negative payoff is CHN, and only three other players (USA, EUR, ROW) would prefer to leave the coalition when the full agreement is formed.These results arguably make it more likely that factors unaccounted for in our analysis, such as reputational risks, sanctioning mechanisms or opportunities for arrangements like payoff transfers, may be able to induce all or most players to adhere to the contributions required for full-agreement outcomes in practice.
Although the data underlying our simulations were calibrated based on recent empirical data from widely recognized climate modelling sources, the absolute results of the different agreement simulations are likely far from representative of how these games would play out in practice and should therefore be interpreted with caution.Since the objective of this thesis was to demonstrate how the matching games and the standard game of coalition formation compare in terms of their ability to close the gap between the Nash baseline and the socially optimal abatement and payoff levels, we did not aspire to accurately reflect the complexities underlying real climate policy negotiations.
The most notable simplification we made in configurating the simulation model is the reduction of the dynamic regional abatement benefits and costs over the agreement term and beyond to static representations thereof.This was done to increase the model's conceptual fit with the one-shot matching game as it was analyzed analytically, but it implies that the abatement strategies and payoffs calculated in our simulations by no means reflect the outcomes one would obtain when analyzing the games in a more realistic, dynamic setting.Moreover, the functional forms we adopted for players' abatement benefit and cost curves are at best crude reflections of the real relationships they represent.While the linearity of the benefits functions used facilitated our analysis as it ensured dominant strategies for all outsiders to the matching game, regions' real abatement benefits curves are likely to be more complex, which would give rise to potential issues of carbon leakage by non-signatories.It is also likely that the costs of negative emissions technologies are comparable across different regions, implying that the simulation results become all the more inaccurate when players commit to abatement contributions that exceed their BAU emissions baselines.Some further limitations of our calibration approach which make our results less representative of real climate policy negotiations include the omittance of technological progress over time, the assumption of zero abatement after the agreement term, the assumption of constant emissions and GDP after the agreement term that we made due to a lack of data, the application of constant rather than time-specific discount rates, the use of potentially outdated abatement cost estimates, and the implausible regional aggregation of players.
More importantly yet, the theoretical model of the matching mechanism analyzed also has its limitations.First, the model assumes that all players dispose of perfect information about their own and the other players' abatement benefit and cost functions.This requires certainty over the different players' future economic growth and GHG emissions pathways, the relationship between global emissions and future climate damages incurred by the respective players, as well as the cost of available abatement technologies.It goes without saying that such certainty is unattainable in practice.Second, the model presupposes that all players which choose to join a matching coalition can commit to the matching contributions required by the agreement.When players can defect on their conditional abatements, the freeriding problem reemerges and coalition members may have to resort to sanctioning mechanisms to achieve the agreement goal.Moreover, when punishment is costly, a socalled second-order freeriding problem arises where players prefer others to do the punishing (Molina et al., 2020).Third, any process that involves negotiation is vulnerable to the exploitation of power asymmetries.The matching model assumes that the matching rate negotiated by the coalition members maximizes the coalition's collective payoff.In practice, it likely cannot be guaranteed that this is the case when coalition members have conflicting interests and unequal powers in the negotiation process.Lastly, Liu (2019) expresses the concern that "matching mechanisms may be too sophisticated for practical implementation".Even if there was a way to resolve the issues of incomplete information, incredible commitment and power asymmetries, it could still be too challenging for players to understand and anticipate all other players' equilibrium contributions and set their optimal strategies accordingly.While our simulation dealt with a manageable number of players, IEAs like the Paris Agreement involve significantly larger numbers of countries, making the implementation of a matching mechanism considerably more complex.
Nonetheless, the matching agreements also provide clear advantages over other types of agreements.Most importantly perhaps, the players joining a matching agreement are only required to commit to conditional contributions, as opposed to making unconditional pledges like under the Kyoto Protocol or the Paris Agreement.This allows the players to retain more of their sovereignty and reduces the threat of incredible commitment (Molina et al., 2020).If, for instance, the players which announced large positive flat abatements reneged on their commitments, the other coalition members could punish the defectors by reducing their conditional abatements accordingly, leaving the whole coalition with lower total abatement levels.This type of punishment does not impose additional costs on the punishers, as it suffices that the latter follow their equilibrium abatement strategies.Another advantage of this type of agreement, as Molina et al. (2020) observe, is that it does not rely on any direct transfers between players.While many design proposals for potential IEAs involve players subsidizing each other's contributions via monetary or in-kind transfers, the implicit subsidization mechanism of matching agreements facilitates matters because it is more likely to be implemented by players between which diplomatic relations are tense.The type of matching mechanism studied by Boadway et al. (2011) and Molina et al. (2020), in which players set their own matching rates towards each of the other coalition members non-cooperatively, provides even greater degrees of player sovereignty and robustness to commitment issues and power abuse in the sense that it circumvents the negotiation stage and allows players to influence the size of their matching contributions.However, this comes at the expense of an increased complexity of the matching game, which makes it more challenging to analyze analytically and to implement in practice.

Conclusion and recommendations
This thesis investigated the effect of matching mechanisms on the coalition stability and the abatement contributions and payoffs in IEAs between heterogeneous players.It did so to explore whether matching mechanisms can help overcome the freeriding problem that standard environmental agreements commonly struggle with.The type of matching mechanism we analyzed features a uniform, cooperatively determined matching rate by which all coalition members match each other's flat contributions, while the non-signatories play the standard Nash game.We also extended the pure matching game by an emissions trading stage to show how the emissions trading scheme leverages the matching mechanism's effect on players' abatement contributions and payoffs.As opposed to most other studies on matching agreements, we not only developed theoretical models of the games, but also analyzed them numerically by means of simulations with an empirically calibrated model.The simulations consider twelve asymmetric regional players with linear abatement benefit functions and quadratic abatement cost curves, calibrated on the basis of the STACO 3 model, and are based on emissions data from the SSP database.
Lamentably, we find that both types of matching agreements we analyzed, with and without emissions trading, perform worse in terms of stability than the standard agreement.While the latter produces a stable coalition of two players, the matching agreements fail to produce any coalitions for which the participating players have no incentives to opt out and the non-signatories have no incentive to join.As such, the introduction of a matching mechanism in IEAs between heterogeneous players thus does not represent a solution to the common freeriding problem in international climate negotiations.However, compared to the grand coalition in the standard game of coalition formation, the individual payoffs under the full matching agreements are such that a significantly smaller number of the coalition members have incentives to freeride or even incur negative payoffs.This is encouraging as it means that additional factors like reputational risks, sanctioning mechanisms or payoff transfers may be able to countervail the freeriding incentives such that collectively optimal outcomes are achieved.
In terms of the full agreements' ability to bridge the divide between the global Nash baseline abatement and payoff levels and those of the socially optimal grand coalition outcome, the matching agreement with emissions trading fares better than the pure matching agreement we analyzed.When the players are constrained in their emissions allowance choices, the agreement with emissions trading narrows the global abatement and payoff gaps by 149.25 and 76.20 percent, respectively.The full matching agreement without emissions trading, in contrast, only manages to narrow the abatement and payoff gaps by 43.75 and 37.09 percent.Nonetheless, this is still superior to the results of the stable two-player coalition produced by the standard game of coalition formation, which bridges the global abatement and payoff gaps by 8.31 and 10.13 percent (Table 11).Hence, matching agreements as analyzed in this study may be able to offer a better alternative to standard agreements, provided that their coalition stability can be partially improved by introducing additional incentives for cooperation.We leave it up to future research to determine whether this is achievable and conclude by offering some further recommendations.
In this study, we assume that all agreements are 'open-membership' agreements, meaning that they can freely be joined by any player and external stability cannot be enforced by the coalition members.Likewise, players do not experience negative repercussions from opting out of an agreement.Hence, we only consider coalitions to be stable which meet both the internal and external stability conditions.However, it may be reasonable to assume other agreement types with membership rules that do allow exogenous access restrictions (Finus et al., 2005).Conversely, sanctioning and reputation mechanisms may be able to increase the internal stability of coalitions (Liu, 2018).For such cases, it could be interesting to investigate the internal and external stability of matching coalitions separately, as the agreements are likely to perform better when accounting for such considerations.
As noted earlier, it may be that other types of matching games are better suited to address the stability issues of IEAs.In particular, matching mechanisms with player-specific, non-cooperatively determined matching rates as analyzed by Boadway et al. (2011) and Molina et al. (2020) may be able to produce larger coalitions by allowing coalition members to freely determine their matching contributions.A numerical simulation of such a matching agreement, as Kawamata & Horita (2013) conducted it but including an initial coalition accedence stage, could potentially generate valuable insights in this regard.
Finally, the most evident next step to obtaining numerical evidence of matching agreements that are more representative of a potential real-life matching agreement would be to examine them in a dynamic multi-period model.Boadway et al. (2011) make a first move in this direction by analyzing their theoretical matching model in a two-period setting where the GHG concentrations in one period are determined by the emissions in the previous one.In reality, we can expect that not only the players' benefits of global abatement are dynamically distributed across the agreement term and beyond, depending on the accumulation of GHG in the atmosphere and the discount rates used, but that different rates of technological progress also change players' abatement cost curves over time.These dynamics would have a decisive influence on players' optimal abatement strategies and are considerations that this thesis could not take into account.

APPENDIX A: Demonstration of collective payoff maximizing matching rates in the matching game with emissions trading
The coalitions whose payoffs are analyzed hereunder were arbitrarily selected.We herewith intend to demonstrate that the coalition payoff maximizing matching rate in the matching game with emissions trading is always unity.

Figure
Figure 4: Regional marginal abatements cost curves

Table 2
are a linearized function of temperature change ∆  , which are converted into regional damage shares   using share parameters   based onFinus et al. (2006):

Table 4 :
Regional cost parameter values and mean baseline emissions

Table 5 :
Regional benefit and cost parameters

Table 9 :
Total yearly abatements per region