Partial cooperation in strategic multisided decision situations
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Abstract
We consider a normalform game in which there is a single exogenously given coalition of cooperating players that can write a binding agreement on preselected actions. The actions representing other dimensions of the strategy space remain under the sovereign, individual control of the players. We consider a standard extension of the Nash equilibrium concept denoted as a partial cooperative equilibrium as well as an equilibrium concept in which the coalition of cooperators has a leadership position. Existence results are stated and we identify conditions under which the various equilibrium concepts are equivalent. We apply this framework to existing models of multimarket oligopolies and international pollution abatement. In a multimarket oligopoly, typically, a merger paradox emerges in the partial cooperative equilibrium. The paradox vanishes if the cartel attains a leadership position. For international pollution abatement treaties, cooperation by a sufficiently large group of countries results in a Pareto improvement over the standard tragedy of the commons outcome described by the Nash equilibrium.
Keywords
Noncooperative game Partial cooperation Partial cooperative equilibrium Leadership equilibrium Multimarket oligopoly International pollution abatement1 Introduction
We consider a game theoretic framework for understanding collaborative decision situations embedded in competitive environments. Often, these kinds of agreements on a specific economic issue involve a single group of decision makers who cooperate. This coalition writes a binding agreement concerning certain exogenously given jointly decided actions, while the noncollaborative decision makers remain uninvolved. Moreover, while decision makers subject to this binding agreement may cooperate on certain issues, these same players may act noncooperatively when interacting with all other players along other dimensions. Examples where these tools are applicable range from areas as diverse as environmental agreements (Barrett 1990; Carraro and Siniscalco 1993), R&D collaborations (Yi and Shin 2000; BanalEspañol et al. 2013), and financial alliances between banks (White 1996; Popov and Ongena 2011; In’t Veld and van Lelyveld 2014).
A rigorous analysis of these games with partial cooperation calls for the development of specifically tailored equilibrium concepts. Our notion of partial cooperative equilibrium extends standard Nash best response rationality to our framework and is similar to the coalition equilibrium concept of Ichiishi (1981). Here, the cooperating players write a binding agreement with regard to a specific set of actions, and, simultaneously, act noncooperatively with regard to their individualistic or “private” strategy. The noncooperators act independently from the cooperators and select a standard best response strategy to all other players’ actions. By supplementing the strategy space of the cooperators with a private strategy, this definition generalises the concept of a partial cooperative game in Chakrabarti et al. (2011). Our partial cooperative equilibrium existence result extends the existence theorems seminally stated in Mallozzi and Tijs (2008a, 2009, 2012) for more restricted environments. Our result is based on techniques seminally developed by Glicksberg (1952).
Next, we consider a leadership equilibrium concept, which postulates that the cooperating coalition has a (Stackelberg) leadership position. Hence, after a binding agreement has been signed by the cooperative players, all players—cooperators as well as noncooperators—make independent decisions with regard to all other actions.
The underlying sequential decision process in a normalform game was seminally discussed and developed in Section 3.5 of Ray (2007). This structure was first implemented to partial cooperative games by Mallozzi and Tijs (2008a) and subsequently extended in Mallozzi and Tijs (2008b, 2012) and Chakrabarti et al. (2011). Our leadership equilibrium notion builds on this work. Mallozzi and Tijs (2008a) proposed this concept for the class of symmetric potential games. In a subsequent study, Mallozzi and Tijs (2009) consider symmetric aggregative games. The situation when the cooperating agents are faced with multiple Nash equilibria when interacting with the noncooperative agents is discussed in Mallozzi and Tijs (2008b). Chakrabarti et al. (2011) extended this further to arbitrary noncooperative games.
We design an extension to the leadership equilibrium notion along several dimensions. First, we extend the strategy space for the cooperators, such that actions not subject to coalitional decisionmaking are decided simultaneously by all players—cooperators as well as noncooperators. Second, we consider a generalised aggregation of the payoffs of the cooperators to evaluate coalitional decisions. A commonly applied aggregator is the utilitarian aggregator. The utilitarian aggregator is imposed, for example, in the theory of (standard) partial cooperative games as seminally developed in Mallozzi and Tijs (2009). Alternative aggregators that can be handled by our generalised framework are the Rawlsian aggregator (Rawls 1999) and the Nashian aggregator (Nash 1950). Third, following Chakrabarti et al. (2011), we assume that in the case of multiple Nash equilibria, the cooperators act optimistically and employ a max–max strategy with regard to their payoffs. Finally, we study the existence properties of our notions in a much larger class of normalform games.
We investigate two applications of our general framework. In the first application, we borrow the model of cartel formation in a multimarket Cournot oligopoly from Billand et al. (2014). Firms compete in two separate markets. Cooperators collude with regard to quantity choice in one market and not in the other. In the partial cooperative equilibrium, cooperators are worse off than in the standard competitive Cournot–Nash equilibrium due to the merger paradox (Salant et al. 1983) if the market on which cooperation occurs is sufficiently large relative to the competitive market. If the market on which firms cooperate is sufficiently small, however, it may be the case that partial cooperation leads to strictly higher payoffs to all firms compared to the standard competitive equilibrium. On the other hand, in the leadership equilibrium, cartel members as well as regular competitors are betteroff relative to the Nash equilibrium, achieving a strict Pareto improvement, thereby restoring a clear incentive to establish a cartel in such a multimarket oligopoly.
In the second application, we consider the effects of international pollution abatement treaties. Such treaties are best described as partial cooperative agreements: A single coalition of treaty countries writes a binding agreement on certain aspects of the spectrum of economic controls at the disposal of a country’s government. The 1997 Kyoto treaty, for example, only regulated emissions of carbon dioxide (“carbon production”) rather than the carbon consumption, which normally results from many other economic variables (Helm 2012; Newell et al. 2013).
Our partial cooperative approach supplements Chander and Tulkens (1997)’s model of emissions control with an additional labour input besides the usual polluting factor. Cooperators choose the amount of labour freely, but form a cooperative agreement with regard to the amount of pollutants. In this context, there is no difference between the partial cooperative equilibrium and the leadership one, but both differ from the Nash outcome. Here, only cooperators reduce the amount of pollutants and the level of reduction is greater, the larger is the number of cooperators. All nontreaty countries act as free riders in these equilibrium situations.
2 Generalised partial cooperative games
We consider normalform games in which an exante postulated group of players collaborates and writes binding agreements on a given subset of actions. All players outside this coalition of cooperators are assumed to follow their individual objectives.

Each individual \(i \in C\) controls the selection of a private action \(x_i \in X_i \), where \(X_i \ne \varnothing \) denotes i’s private action set. We let \(X = \prod \nolimits _{i \in C} X_i\) be the private action tuple set of the coalition of cooperators C.

The coalition of cooperators C selects cooperatively a collective action \(y \in Y \ne \varnothing \).^{1}

Finally, each noncooperator \(j \in N\) selects an individual action \(z_j \in Z_j \ne \varnothing \). We denote \(Z = \prod _{j \in N} Z_j\) the nonempty action tuple set of the group of noncooperators N.
In addition, following accepted conventions, for every cooperator \(i\in C\) and private action tuple \(x\in X\), we denote by \(x_{i}=(x_{1},\ldots ,x_{i1},x_{i+1},\ldots ,x_{k})\) the actions assigned to all other cooperators in C. Similarly, for every noncooperator \(j\in N\) and action tuple \(z\in Z\), we denote by \(z_{j}=(z_{1}, \ldots ,z_{j1},z_{j+1}, \ldots ,z_{n})\) the actions assigned to all other noncooperators in N.
Moreover, each cooperating player \(i \in C\) is endowed with a payoff function \(u_i :A \rightarrow {\mathbb {R}}\) and noncooperator \(j \in N\) is similarly endowed with a payoff function \(v_j :A \rightarrow {\mathbb {R}}\). We denote by \(u = (u_1, \ldots ,u_k)\) the list of payoff functions over all cooperators, and by \(v = (v_1, \ldots ,v_n)\) the list of payoff functions for all noncooperators.
Clearly, the collective payoff function U is founded on the aggregator \(\Lambda \) and the payoff functions \(u_1, \ldots ,u_k\) of its members.
Definition 2.1
A generalised partial cooperative game is a list \(\Gamma = \langle C,N,X,Y,Z,u,v, \Lambda \rangle \).
Note, here as well, that the case where Y is a singleton set reverts any generalised partial cooperative game to a standard \((k+n)\)player normalform game.
2.1 Partial cooperative equilibrium
This equilibrium notion extends Nash’s best response rationality to all decisions, namely the individual and private actions of the players as well as the collective action of the coalition of cooperators. The latter evaluates the outcomes of their selection through the aggregated payoff function U based on the aggregator \(\Lambda \).
Definition 2.2
 (i)For every cooperating player \(i \in C\), it holds thatfor every private action \(x_i \in X_i\).$$\begin{aligned} u_i (x^*,y^*,z^*) \geqslant u_i (x_i ,x_{i}^*,y^*,z^*) \end{aligned}$$(3)
 (ii)For the coalition of cooperators C, it holds thatfor every collective action \(y \in Y\), where U is given by (2).$$\begin{aligned} U (x^*,y^*,z^*) \geqslant U (x^*,y ,z^*) \end{aligned}$$(4)
 (iii)In addition, for every noncooperator \(j \in N\), it holds thatfor every individual action \(z_j \in Z_j\).$$\begin{aligned} v_j (x^*,y^*,z^*) \geqslant v_j (x^*,y^*,z_j,z_{j}^*) \end{aligned}$$(5)
Next, we investigate the existence of partial cooperative equilibria in arbitrary generalised partial cooperative games.
Theorem 2.3
 (i)
All action sets \(X_{i} \ (i \in C)\), \(Z_j \ (j \in N)\) and Y are compact and convex subsets of Euclidean spaces.
 (ii)
The payoff function \(u_{i}\) is continuous on \(X_i \times Y\), quasiconcave on \(X_i\), and concave on Y for every cooperator \(i \in C\), and the payoff function \(v_{j}\) is continuous and quasiconcave on \(Z_j\) for every noncooperator \(j\in N\),^{2}
 (iii)
The aggregator \(\Lambda \) is continuous, Paretian, and quasiconcave on \({\mathbb {R}}^k\).
For a proof of Theorem 2.3, we refer to the appendix of this paper.
2.2 Leadership equilibrium
Next, we implement a sequential structure on the decisionmaking process in generalised partial cooperative games. We consider an internal twotier hierarchical structure in which the coalition of cooperators C has a leadership position in the decisionmaking process. Thus, in the first instance, C writes a binding agreement resulting in the selection of some collective action \(y \in Y\), and, subsequently, all cooperators \(i \in C\) and all noncooperating players \(j \in N\) respond to that selection by selecting private actions \(x_i \in X_i\), respectively \(z_j \in Z\).
The leadership equilibrium concept reflects a standard backward induction logic in which the coalition of cooperators C acts as a Stackelberg leader. The coalition of cooperators C is assumed to act from an optimistic point of view in this procedure and considers only outcomes from the secondstage interaction between all agents that correspond to the best possible outcome for them collectively.
To define the leadership equilibrium concept properly, we have to introduce some hypotheses on the fundamentals.
Axiom 2.4
 (i)
For every \(i\in C\), it holds that \(X_{i}\) is a compact and convex subset of some Euclidean space.
 (ii)
For every \(j \in N\), it holds that \(Z_{j}\) is a compact and convex subset of some Euclidean space.
 (iii)
The set of collective actions Y is a compact subset of some Euclidean space.
 (iv)
For every cooperator \(i \in C\), the payoff function \(u_i :A \rightarrow {\mathbb {R}}\) is a continuous function and the section \(u_{i}(\cdot ,x_{i},y,z):X_{i}\rightarrow {\mathbb {R}}\) is quasiconcave on \(X_i\) for all \((x_{i},y,z) \in X_{i}\times Y\times Z\);
 (v)
For every noncooperator \(j \in N\), the payoff function \(v_j :a \rightarrow {\mathbb {R}}\) is a continuous function and the section \(v_{j}(x, y, \cdot ,z_{j})\) is quasiconcave on \(Z_j\) for all \((x,y,z_{j}) \in X \times Y\times Z_{j}\).
 (vi)
The aggregator \(\Lambda \) is continuous on \({\mathbb {R}}^k\).
The assumptions introduced in Axiom 2.4 are weaker than the ones imposed in Theorem 2.3.
 First, for every cooperator \(i \in C \), we consider the conditional payoff function \(w^y_i :X \times Z \rightarrow {\mathbb {R}}\) given by$$\begin{aligned} w^y_i (x,z) = u_i (x,y,z) . \end{aligned}$$(6)
 Second, for every noncooperating player \(j \in N\), the given action set \(Z_j\) as well as a conditional payoff function \(w^y_j :X \times Z \rightarrow {\mathbb {R}}\) is defined as follows:$$\begin{aligned} w^y_j (x,z) = v_j (x,y,z) . \end{aligned}$$(7)
Lemma 2.5
Consider a generalised partial cooperative game \(\Gamma =\langle C,N,X,Y,Z,u,v, \Lambda \rangle \) satisfying Axiom 2.4. Then, for every collective action \(y \in Y\), the set of Nash equilibria \(E_y\) of the conditional partial cooperative game \(\Gamma _y = \langle C \cup N,X \times Z,w^y \rangle \) forms a nonempty, compact subset of the (Euclidean) strategy set \(X \times Z\).
Lemma 2.5 is an immediate corollary from the more general assertion stated as Lemma A.2 in the Appendix of this paper.
The set \(\Phi ^s\) of \(U^s\)maximisers describes the best responses of the coalition of cooperators C to the other players’ actions. Indeed, \(\Phi ^s\) is the collection of collective actions that maximise the maximum payoff envelope function \(U^s\) over Y, i.e., these collective actions are coalition C’s best responses given the collective payoff function \(U^s\).
We can now define the leadership equilibrium concept for \(\Gamma \) as follows:
Definition 2.6
The existence of a leadership equilibrium in a partial cooperative game can be established under the assumed conditions.
Theorem 2.7
Let \(\Gamma =\left\langle C,N,X,Y,Z,u,v\right\rangle \) be a generalised partial cooperative game that satisfies Axiom 2.4. Then, there exists at least one leadership equilibrium in \(\Gamma \).
For a proof of Theorem 2.7, we again refer to the Appendix.
2.3 Separability
Separability guarantees that the two notions of equilibrium defined here for the class of generalised partial cooperative games coincide. Indeed, in a separable game, the collective decision of coalition C concerning the collective actions y is completely described by the maximisation of \({\tilde{U}} = \sum \nolimits _{i \in C} {\tilde{u}}_i\) and, therefore, is strictly independent of the selection of individual actions of the cooperators \(i \in C\). This implies immediately the following assertion.
Proposition 2.8
Let \(\Gamma \) be a separable generalised partial cooperative game satisfying Axiom 2.4. Then, the notion of leadership equilibrium is equivalent to that of the partial cooperative equilibrium concept in \(\Gamma \) in the sense that every partial cooperative equilibrium is a leadership equilibrium.
In a separable partial cooperative game with multidimensional collective actions in the sense that \(Y = \prod \nolimits _{i \in C} Y_i\) for appropriately chosen \(Y_i\), one can also introduce the notion of a standard Nash equilibrium. This notion is rather useful in the analysis of the effects of collective decisionmaking on certain specified actions.
Definition 2.9
An action tuple \((x',y',z')\) is a Nash equilibrium in a separable partial cooperative game \(\Gamma \) with \(Y = \prod \nolimits _{i \in C} Y_i\) if, for every cooperator \(i \in C\), the pair \((x'_i,y'_i)\) maximises \(u_i(\cdot ,x'_{i}, \cdot ,y'_{i},z')\) over \(X_i \times Y_i\), and for every noncooperator \(j \in N\), the individual action \(z'_j\) maximises \(v_j(x',y', \cdot ,z'_{j})\) over \(Z_j\).
In general, the class of Nash equilibria is different from the class of Partial Cooperative and Leadership equilibria in these separable games. This is shown through the following example, which considers a separable generalised partial cooperative game with two collaborators.
Example 2.10
We can now state the following assertion without proof. A proof is based on the insight that the determination of the collective action \(y \in Y = \prod \nolimits _{i \in C} Y_i\) is not only completely separated from the determination of the other actions \((x,z) \in X \times Z\), but that it is essentially based on the maximisation of the common payoff function \({\tilde{u}}\). Thus, the optimal collective action coincides with the individually optimal choice of \(y_i \in Y_i\) for every cooperator \(i \in C\).
Proposition 2.11
Let \(\Gamma \) be a homogeneous generalised partial cooperative game. Then, the notion of partial cooperative equilibrium is equivalent to that of Nash equilibrium in \(\Gamma \) in the sense that every partial cooperative equilibrium is a Nash equilibrium.
3 Two applications
In this section, we investigate two applications of generalised partial cooperative games that clearly delineate the various equilibrium concepts. Both of these applications consider a generalised partial cooperative game formulation that allows the development of a standard Nash equilibrium. Hence, the concept of a standard Nash equilibrium is well defined and can be computed. Thus, a complete comparison of these Nash equilibria with the two main partial cooperative equilibrium concepts—partial cooperative equilibrium and leadership equilibrium—can be developed.
3.1 Cartels in multimarket oligopolies
We first consider a cartel in a multimarket oligopoly. Firms are fully competitive in one market, while a subset of firms forms a cartel in the second market only. We show that the cartel benefits in comparison with the standard Nash equilibrium outcome if it has a clear leadership position: An improvement only occurs in the leadership equilibrium, not in the partial cooperative equilibrium.
We limit our analysis to the case of three firms indexed by \(i=1,2,3\) that compete in two related markets A and B.^{5} We assume that these firms engage in quantitysetting, Cournotian competition. We denote by \(q_{i}\) and \(Q_{i}\) the quantities sold by firm i on markets A and B, respectively. Furthermore, p and P represent the market prices emerging in market A and B, respectively.
3.2 Formulation as a generalised partial cooperative game
We set up a generalised partial cooperative game that describes cartel formation in this multimarket Cournot oligopoly. We consider the formation of the cartel \(C = \{ 1,2 \}\) only in market B, while full competition is retained in market A. Here, firm 3 remains an independent producer.^{6}
In this multimarket framework, \(y = (Q_1,Q_2)\) forms the collective action of the cartel \(C = \{ 1,2 \}\), while quantities \((q_1,q_2)\) set in the Amarket constitute private actions for the two cartel members \(i=1,2\). Firm 3 acts completely independently in both markets and set all output levels \((q_3,Q_3)\) as individual actions.
The generalised partial cooperative game is completed with the selection of the utilitarian aggregator \(\Lambda ^u (\pi _1, \pi _2 ) = \pi _1 +\pi _2\) to direct the collective decisions of the cartel \(C = \{ 1,2 \}\).^{7}
3.2.1 Equilibrium analysis
In this setting, we discuss the Nash equilibrium, the partial cooperative equilibrium, and the leadership equilibrium in this model of cartel formation in a multimarket oligopoly.
3.3 Partial cooperative and leadership equilibria
Next, we investigate the partial cooperative and leadership equilibria in this generalised partial cooperative game.
Proposition 3.1
 A Partial cooperative equilibrium exists and is unique if \(\tfrac{\alpha }{5}< \beta < 5 \alpha \) and is given by$$\begin{aligned} q_1^{PE} = q_2^{PE} = \frac{20 \alpha  2 \beta }{98}; \quad q_3^{PE} = \frac{21 \alpha  7 \beta }{98};\\ Q_1^{PE} = Q_2^{PE} = \frac{15 \beta  3 \alpha }{98} ;\quad Q_3^{PE} = \frac{25 \beta  5 \alpha }{98}. \end{aligned}$$
 A unique leadership equilibrium exists and is given by$$\begin{aligned} q^{LE}_{1}&=q^{LE}_{2} = \frac{205\alpha 37\beta }{968}; \quad q^{LE}_{3} = \frac{197\alpha 45\beta }{968}; \\ Q^{LE}_{1}&=Q^{LE}_{2}= \frac{193\beta 49\alpha }{968}; \quad Q^{LE}_{3} = \frac{19\beta 3\alpha }{88}. \end{aligned}$$
We can illustrate this point more clearly by presenting the equilibrium outcome for specific values of the market parameters. For example, if \(\alpha =5\) and \(\beta =3\), we arrive at \(\pi _1^{PE} = \pi _2^{PE} = 1.9080 > \pi ^{NE} = 1.9028\) as well as \(\pi _3^{PE} = 1.9298 > \pi ^{NE} = 1.9028\). Here, a leadership role improves the profit position for the cartel, although the independent producer firm 3 still outperforms its competitors. To be precise, we compute that \(\pi ^{LE}_1 = \pi ^{LE}_2 = 1.9101> \pi _1^{PE} = \pi _2^{PE} > \pi ^{NE}\), while \(\pi _3^{PE}> \pi _3^{LE} = 1.9170 > \pi ^{NE}\).
3.4 International pollution abatement
Our second application concerns an international environmental protection situation in which certain countries write an international pollution abatement treaty, while other countries do not participate in such a collective control of their emissions. Our analysis builds on the seminal contribution by Chander and Tulkens (1997) who, for the purpose of their analysis, developed the notion of the \(\gamma \)core for partition function form games. We remark here that their notion of partial agreement equilibrium corresponds to our notion of partial cooperative equilibrium.
We amend their analysis by supplementing the cooperators’ strategy space with an additional dimension along which binding agreements and cooperation are not possible. Therefore, we include an input factor—denoted as labour—that treaty countries choose independently, similarly to noncooperators.
There are two types of countries, namely those that cooperate on writing an international emissions control treaty—denoted by \(C = \{ 1, \ldots ,k \}\) with \(k \geqslant 2\)—and those that act independently—denoted by \(N = \{ 1, \ldots ,n \}\) with \(n \geqslant 1\). All countries produce two nontradable goods, 1 and 2, only. Treaty country \(i \in C\) produces and consumes quantities \(q_{1i}\) and \(q_{2i}\), while nontreaty country \(j \in N\) produces and consumes \(q_{1j}\) and \(q_{2j}\), respectively.
Good 1 is “clean” in the sense that its production leaves no emissions footprint. Good 2 is “dirty” in the sense that its production requires an input that leaves an emissions footprint. For all countries, the production technology is the same. For generic country \(h \in C \cup N\), the production functions for the two goods are given by \(q _{1h}=\sqrt{L_{1h}}\) and \(q_{2h}=\sqrt{d_{h} L_{2h}}\), where \(L_{1h} \geqslant 0\) and \(L_{2h} \geqslant 0\) are, respectively, the amounts of labour expended in the production of goods 1 and 2, and \(d_{h} \geqslant 0\) is the “dirty input” in the production of good 2.
We further assume that every country \(h \in C \cup N\) is endowed with the same total labour input equal to \({\overline{L}}\). Thus, \(L_{1h}+L_{2h}\le {\overline{L}}\). Throughout, we assume that \({\overline{L}} >1\).
In terms of the notation employed, \(y = ( d_1 , \ldots ,d_k )\) is the multidimensional collective action of all treaty countries in C, representing the treaty written between the countries in C to control emissions. Throughout, we postulate that the coalition of treaty countries C is guided by utilitarian principles and uses the utilitarian aggregator \(\Lambda ^u = \sum \nolimits _{i \in C} u_i\) to guide its collective decisions.
3.4.1 Equilibrium analysis
We can approach this model as a normalform strategic game as well as a generalised partial cooperative game. We first determine its Nash equilibrium that we use as a benchmark. This represents a situation in which all countries in C act independently and do not write a treaty to control their emissions. This results into a standard tragedy of the commons problem which is given by the maximisation of the objective function (16) subject to the constraints listed in (17)–(20).
Indeed, in the Nash equilibrium for each country \(h \in C \cup N\), we derive that \(L^{NE}_{1h} =1\) and \(L_{2h}^{NE} = d_{h}^{NE} = {\overline{L}}1\), resulting into \(u^{NE}_h = {\overline{L}}  \tfrac{1}{2} (k+n) ({\overline{L}} 1)\).^{10}
3.5 Partial cooperative and leadership equilibria
First, note that the formulated generalised partial cooperative game is separable and that Proposition 2.8 now implies that the partial cooperative equilibria are equal to the leadership equilibria in this model. We, therefore, limit our discussion to the partial cooperative equilibria only.
In the case of cooperation, the coalition of treaty countries C signs a binding agreement \(y = ( d_1, \ldots ,d_k )\) and each country \(h \in C \cup N\) chooses independently the amount of labour that it allocates to the production of each of the two goods \((L_{1h},L_{2h})\). Moreover, each nontreaty country \(j \in N\) makes a decision on the dirty input \(d_{j}\).
Proposition 3.2
 (a)\(1 < {\overline{L}} \leqslant k^2 :\) For each treaty country, \(i \in C :L^{PE}_{1i} = {\overline{L}}\) and \(L^{PE}_{2i} = d^{PE}_i = 0\). For each independent country \(i \in N :L^{PE}_{1j} =1\) and \(L^{PE}_{2j} = d^{PE}_j = {\overline{L}}1\). The total pollution in this equilibrium is determined as \(\Delta ^{PE} = n ({\overline{L}}1)\) and the resulting utility levels asFor all \(1 < {\overline{L}} \leqslant k^2\), the partial cooperative equilibrium is a strict Pareto improvement over the Nash equilibrium.$$\begin{aligned} u^{PE}_i= & {} \sqrt{{\overline{L}}}  \tfrac{n}{2} ({\overline{L}}1) \\ u^{PE}_j= & {} \left( 1  \tfrac{n}{2} \right) {\overline{L}} + \tfrac{n}{2}. \end{aligned}$$
 (b)
\({\overline{L}} > k^2 :\) For each treaty country, \(i \in C :L^{PE}_{1i} = k^2\), \(L^{PE}_{2i} = {\overline{L}} k^2\) and \(d^{PE}_i =\tfrac{{\overline{L}}k^2}{k^2}\). For each independent country \(i \in N :L^{PE}_{1j} =1\) and \(L^{PE}_{2j} = d^{PE}_j = {\overline{L}}1\).
The total pollution in this equilibrium is determined asand the resulting utility levels as$$\begin{aligned} \Delta ^{PE} = \left( n + \tfrac{1}{k} \right) {\overline{L}}  (k+n), \end{aligned}$$For any \({\overline{L}} > k^2\), the partial cooperative equilibrium is a strict Pareto improvement over the Nash equilibrium.$$\begin{aligned} u^{PE}_i= & {} \frac{{\overline{L}} k^2}{k} + \tfrac{1}{2} (3k+n)  \tfrac{1}{2} \left( n + \tfrac{1}{k} \right) {\overline{L}} \\ u^{PE}_j= & {} {\overline{L}}  \tfrac{1}{2} \left( n + \tfrac{1}{k} \right) {\overline{L}} + \tfrac{1}{2} (k+n). \end{aligned}$$
The conclusion from Proposition 3.2 is that treaty countries as well as independent countries are betteroff under pollution abatement treaties than under global noncollaboration as described in the Nash equilibrium. Although, in Nash equilibrium, all countries achieve the same utility levels, under partial cooperation, the independent countries have a higher utility than the cooperators, since they act as free riders, enjoying the environmental pollution abatement imposed through the treaty. As expected, total pollution always increases with the number of independent countries. An increase in the number of treaty countries, on the other hand, decreases total pollution.
4 Concluding remarks
In this paper, we introduce two equilibrium notions that are applicable to situations of partial cooperation. We establish their existence properties in a general and widely applicable setting.
First, as our concept of partial cooperative equilibrium builds on the standard notion of Nash equilibrium, it is neutral to the assumption of observability of players’ strategies, including the existence and composition of the coalition of cooperators.
Second, even though our results are derived in the context of a single group of cooperators, our results on the partial cooperative equilibrium concept are generalisable to a setting with an arbitrary number of exclusive coalitions of cooperators. For a partial cooperative equilibrium to exist in such more general settings, the conditions on the aggregator function identified in Theorem 2.3 must apply to all coalitions of cooperators.
In the context of leadership in partial cooperation, we present our results under the assumption that the cooperators have a leadership role. A similar existence result can be derived if, instead, one of the noncooperators acts as the Stackelberg leader. To see that, it is sufficient to recognise that once the first mover takes action, the players that are moving next are de facto in a situation of partial cooperation, where the existence of a fixed point is established in Theorem 2.3.
Finally, we point out the role of complete information in our set up. Here, as in the standard Nash Equilibrium analysis, we take complete information to implicitly imply common knowledge of payoffs. In the context of leadership equilibrium, we make an implicit requirement that the collective action is observable by all players, too.
4.1 Possible extensions of our analysis
Despite its robustness, we acknowledge that our analysis is restrictive in the sense that cooperation between the coalition members is exante postulated and binding. In this respect, our work is complementary to the large literature on the endogenous formation of agreements based on the work of Ichiishi (1981), Bloch (1997), and Yi (1997).^{12} Whereas the focus of this literature is on the existence and stability of a stable coalition structure given a payoff allocation, our contribution is on the equilibrium derivation of payoffs in a strategic environment taking the coalition structure as given. While, in these contexts, the existence of equilibria is not an issue, our contribution guides future research on the entire class of games where equilibrium existence is guaranteed by deriving very general conditions on the payoff functions. Clearly, further effort is needed in the direction of studying the general properties of the simultaneous game of coalition formation and partial cooperation.
In the context of the two specific applications developed here, an indication of coalition stability can be derived from the comparison between the partial equilibrium payoffs and those in the Nash Equilibrium. If one takes Nash equilibrium to be resulting when cooperation breaks down, then conditions under which partial (leadership) equilibrium payoffs for the cooperators are higher are suggestive of the range of conditions necessary to sustain cooperative agreements.
The two applications discussed also underline both the significance and the relevance of the equilibrium notions based on partial cooperation. In particular, in our first application, we embed a standard framework of multimarket oligopolistic competition into our analysis of partial cooperation. By doing so, we generalise the analysis in existing studies—e.g., Bernheim and Whinston (1990) and more recently Choi and Gerlach (2013)—that analyse the profitability of cartels in multimarket context. In this literature, cartels are of limited size and occur in the absence of outside competitive pressures. Our analysis suggests that, when firms meet on two markets, cartels are more likely to be stable on the relatively smaller market, or where the cartel can take a leadership position relative to noncartel members. To the best of our knowledge, we are the first to point out the advantages that cartels attain in multimarket oligopolies in the context of sequential decisionmaking.
Our second application concerning environmental agreements draws out the role of separability in the analysis. As discussed in the introduction, partial cooperation in this context has been studied since the 1990s. The main conclusion of our analysis—that payoffs for the signatories are equal under partial cooperative equilibrium and under leadership equilibrium—points to the fact that arguments that a success of partial cooperation is dependent on taking a firstmover advantage are not always justifiable.
Footnotes
 1.
This generalises Mallozzi and Tijs (2008a) and Chakrabarti et al. (2011), in which the set of collective actions, Y, is modelled as a kdimensional vector space with \(Y = \prod \nolimits _{i \in C} Y_i\). In this case, \(y_i \in Y_i\) denotes the individual player’s \(i \in C\) strategic (sub)action as part of the collective action \(y \in Y\). In many applications, it is additionally assumed that, for every cooperator \(i \in C\), the action set \(Y_i\) is, in fact, some subset of a Euclidean space.
 2.
Let A be convex. A function \(f :A \rightarrow {\mathbb {R}}\) is concave if for all \(\lambda \in (0,1)\) and \(a^{\prime },a^{\prime \prime }\in A :f(\lambda \cdot a^{\prime }+(1\lambda )\cdot a^{\prime \prime }) \geqslant \lambda \cdot f(a^{\prime }) + (1\lambda ) \cdot f (a^{\prime \prime })\), and f is quasiconcave if for all \(\lambda \in (0,1)\) and \(a^{\prime },a^{\prime \prime }\in A :f(\lambda \cdot a^{\prime }+(1\lambda )\cdot a^{\prime \prime })\geqslant \min \{f(a^{\prime }),f(a^{\prime \prime })\}\).
 3.
We remark that this optimistic outlook can be replaced by other approaches such as a minimax logic or even a fully pessimistic outlook.
 4.
 5.
Throughout, we use the notation of Billand et al. (2014).
 6.
We remark that our analysis can be extended to an arbitrary number of competitors and cartel members. This generalization, however, leads to rather cumbersome expressions that add rather little to the conclusions from our analysis. The case for any number of competitors is available upon request from the authors.
 7.
We mention here that replacing the utilitarians aggregator by the Rawlsian aggregator \(\Lambda ^r = \min \{ \pi _1,\pi _2 \}\) would not affect the analysis of this particular application. In particular, this is due to the symmetry of the described situation.
 8.
Detailed derivations are available from the authors upon request.
 9.
To bring to the fore the mechanism of partial cooperation, we take countries to be completely homogeneous and goods to be nontradable. As a consequence, all countries remain in a state of autarky.
 10.
Obviously, \(u^{NE} < 0\) if the total number of countries \(k+n \geqslant 3 \ (\geqslant 4)\) and the total incountry labour input \({\overline{L}}>3 \ (>2)\). In that case, the negative impact of pollution overtakes the positive utility from the consumption of the two goods.
 11.
The computations and derivations are rather tedious and, therefore, relegated to the appendix of this paper.
 12.
For an overview of this literature, we refer to Finus and Rundshagen (2009).
 13.
We refer to Negishi (1963, Proposition1) for a concise proof of this result.
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