Endogenous Strategic Issue Linkage in International Negotiations
by
Carlo Carraro
(University of Venice and Fondazione Eni E. Mattei)
and
Carmen Marchiori
(Fondazione Eni E. Mattei)
Abstract
This paper analyses the idea of issue linkage as a way to increase co-operation on issues where the incentives
to free-ride are particularly strong. The goal is to determine under what conditions players actually prefer to
link the negotiations on two different issues rather than negotiating on the two issues separately. If players
are asked to vote on issue linkage before starting negotiations, it is not clear at all that they would vote in
favour of issue linkage. This may indeed increase the number of cooperators on the provision of a public
good (where the incentives to free-ride are particularly strong). However, at the same time, issue linkage may
reduce the number of cooperators on the issue which is linked to the provision of a public good. There is
therefore an economic trade-off, in addition to the higher transaction costs that are likely to affect a linked
negotiation. This paper analyses these issues within a game-theoretic framework and shows under what
conditions issue linkage is players’ equilibrium strategy.
Keywords: international environmental agreements; coalition formation games; issue linkage.
Corresponding author: Carlo Carraro, Department of Economics, University of Venice, San Giobbe 873, 30121,
Venice, Italy. Fax: +39 041 2349176. E-mail: ccarraro@unive.it.
Endogenous Strategic Issue Linkage in International Negotiations
1.
Introduction
In recent years, the non-cooperative approach to coalition formation has often been adopted to analyse
various economic problems (Cf. Bloch, 1997; Konishi et al., 1997; Ray and Vohra, 1999; Yi, 1997). When
applying these theoretical results to the provision of public goods -- and in particular to global environmental
agreements -- the conclusion is often that no coalition forms at the equilibrium and that, if a non-trivial
equilibrium coalition emerges, it is formed by a small number of players (Hoel, 1991, 1992; Carraro and
Siniscalco, 1992, 1993; Barrett, 1994, 1997b; Heal, 1994). This result is induced by the presence of strong
free-riding incentives that become even stronger in the presence of leakage, i.e. non-orthogonal reaction
functions (Carraro and Siniscalco, 1993).
This is why different policy strategies have been proposed to address this problem. Transfers and issue
linkage are probably the most popular, even though negotiation rules and treaty design can also be used to
achieve equilibria in which large size coalitions form at the equilibrium (Carraro and Moriconi, 1998).
In this paper we focus on issue linkage. The basic idea behind issue linkage is to design an agreement in
which countries do not negotiate only on one issue (e.g. the environmental issue), but are forced to negotiate
on two issues, e.g. the environmental one and another interrelated (economic) issue. For example, Barrett
(1995, 1997) proposes to link environmental negotiations to negotiations on trade liberalisation, Carraro and
Siniscalco (1995, 1997), Katsoulacos (1997) propose to link environmental negotiations to negotiations on
R&D co-operation, Mohr (1995) and Mohr et al. (1998) propose to link climate negotiations to international
debt.
The idea of issue linkage was originally formulated by Folmer et al. (1993) and Cesar and De Zeeuw (1996)
to solve the problem of asymmetries among countries. The intuition is that some countries gain on a given
issue, whereas other countries gain on a second one. By linking the two issues, the agreement in which the
countries decide to co-operate on both issues may become profitable to all of them.
However, the idea of issue linkage can also be used to offset the free riding incentives that characterise many
environmental (or other public good) agreements. Suppose there is no profitability concern (either because
countries are symmetric or because a transfer scheme is implemented to make the agreement profitable to all
countries). Then, the equilibrium number of signatories of the linked agreement may be larger than the
equilibrium number of signatories of the agreement whose equilibrium number of signatories is small
because of the presence of strong incentives to free-ride (Carraro and Siniscalco, 1995). Of course this
2
depends on the relative economic weights of the two agreements and on their nature. In particular, to be
successful, an agreement concerning a (quasi) public good should be linked to an agreement concerning a
(quasi) club good.
Let us consider an example. Suppose the environmental negotiation is linked to the negotiation on R&D cooperation1, which involves an excludable positive externality (club good) and increases the joint coalition
welfare. In this way, the incentive to free-ride on the benefit of a cleaner environment (which is a public
good fully appropriable by all countries) is offset by the incentive to appropriate the benefit stemming from
the positive R&D externality (which is a club good fully appropriable only by the signatories). The latter
incentive can stabilise the joint agreement, thus also increasing its profitability because countries can reap
both the R&D co-operation and the environmental benefit (this second benefit would be lost without the
linkage).2
This idea is exploited in Carraro and Siniscalco (1995, 1997), Katsoulacos (1997), Botteon and Carraro
(1998). Carraro and Siniscalco (1995, 1997) show that issue linkage may be a powerful tool to increase the
number of signatories of an environmental agreement. For example, if developed countries on the one hand
increase their financial and technological co-operation with developing countries, and on the other hand
make this co-operation conditional on the achievement of given environmental targets, then a number of
countries is likely to be induced to join the environmental coalition, i.e. to sign a treaty in which they commit
themselves to adequate reductions in their emission growth. Katsoulacos (1997), which accounts for
information asymmetries, provides additional support to the conclusion that issue linkage can be very
effective in guaranteeing the stability of an environmental agreement. By contrast, Botteon and Carraro
(1998), where asymmetric players are also considered, show that issue linkage may be counter productive
because of potential conflicts among asymmetric players on the “optimal” members of the R&D club.
All the above-mentioned papers assume that issue linkage has been chosen by the negotiating countries.3
Then they analyse the effectiveness of issue linkage in increasing the equilibrium number of signatories of
the joint agreement (with respect to the environmental agreement). However, the decision of linking two
economic issues should also be considered as a strategic choice that players undertake. A game therefore
describes the incentives to link two issues. And this game could also be characterised by free-riding. Hence,
even if issue linkage increases the number of signatories -- and therefore the amount of public good provided
(e.g. emission abatement) -- it may not be an equilibrium outcome. The crucial question is therefore the
following: do players have an incentive to link the negotiations on two different issues instead of negotiating
1 See Carraro-Siniscalco (1997) for a full presentation of the model.
2 A detailed proof of this result is in Carraro-Siniscalco (1997).
3 An interesting exception is the paper by Tol et al. (1999).
3
on the two issues separately? Namely, is the choice of issue linkage an equilibrium of the game in which
players decide non-cooperatively whether or not to link the negotiations on two different economic issues?
This paper answers the above questions by analysing a game in which, in the first stage, players decide
whether to link two issues on which they are trying to reach an agreement or to negotiate on two separate
agreements. If they decide not to link the two issues, in the second stage they can decide whether or not to
sign either one or both separate agreements. If they decide in favour of issue linkage, in the second stage they
decide whether or not to sign the linked agreement. Finally, in the third stage they set the value of their
policy variables.
Two cases will be considered. One in which the benefits accruing to the signatories of one of the two
separate agreements are perfectly or almost perfectly excludable (co-operators provide a club good) and one
in which the degree of excludability is low. Moreover, the choice made in the first stage of the game will be
analysed under two voting rules. One is unanimity, because the choice of issue linkage can be considered as
a negotiation rule whose determination precedes the beginning of actual negotiations and which therefore
should be taken with the consensus of all countries involved in the negotiations. A second one is majority
voting, because it may be relevant to analyse how a less restrictive voting rule increases the chances of
adopting issue linkage.
The structure of the paper is as follows. Section 2 will introduce the basic definitions and assumptions.
Section 3 will analyse the equilibrium of the three stage game under different degrees of excludability of the
club good and using two voting rules (unanimity and simple majority). Finally, Section 4 will discuss the
main conclusions of our analysis and their policy implications.
2.
Definitions and Assumptions
Assume n players face the following situation. Either they decide to link the two negotiations or they do not.
In this latter case, they then decide whether or not to participate in the first agreement, or in the second
agreement, or in both. By contrast, if the two agreements are linked, they decide whether or not to sign the
joint agreement.
The game has therefore three stages. In the first stage, the linkage game takes place, where the n players
decide simultaneously and non-cooperatively whether or not to introduce a rule that forces all players to
negotiate on a single agreement in which two issues are linked. In the second stage, the coalition game, they
decide simultaneously and non cooperatively whether or not to sign one of the available treaties (i.e. to join a
coalition c of cooperating countries). In the third stage, they play the non cooperative Nash policy game,
4
where players which signed the agreement play as a single player and divide the resulting payoff according
to a given burden-sharing rule (any of the rules derived from cooperative game theory).
A few assumptions are necessary to simplify our analysis.
A.1 (Uniqueness): The third stage game, the policy game, in which all players decide simultaneously, has a
unique Nash equilibrium for any coalition structure.
A.2 (Cooperation): Inside each coalition, players act cooperatively in order to maximise the coalitional
surplus, whereas coalitions (and singletons) compete with one another in a non cooperative way.
A.3 (Simmetry): All players are ex-ante identical, which means that each player has the same strategy space
in the second stage game.
This latter assumption allows us to adopt an equal sharing payoff division rule inside any coalition, i.e. each
player in a given coalition receives the same payoff as the other members. Furthermore, the symmetry
assumption implies that a coalition can be identified with its size c. As a consequence, the payoff received by
the players only depends on the coalition sizes and not on the identity of the coalition members.
Given the above assumptions, a per-member partition function (partition function hereon) can be defined. It
can be denoted by p(c; ), which represents the payoff of a player belonging to the size-c coalition in the
coalition structure . Let  = a (r), b(s), ... be a coalition structure formed by r size-a coalitions, s size-b
coalitions, etc.
A.4 (Single coalition): Players are proposed to sign a single agreement. Hence, those which do not sign
cannot propose a different agreement. From a game-theoretic viewpoint, this implies that only one coalition
can be formed, the remaining defecting players playing as singletons. Hence  = c, 1(n-c), where 1(n-c)
denotes the n-c singletons, and the partition (payoff) function can simply be denoted by P(c).
A.5 (Open Membership): Each player is free to join and to leave the coalition without the consensus of the
other coalition members.
This assumption enables us to adopt the usual Nash equilibrium concept to identify the equilibrium of the
coalition game. Different results could be obtained under exclusive membership or coalition unanimity
(Carraro and Moriconi, 1998).
5
Let us introduce a few definitions. Let cu* the equilibrium number of players which sign the joint agreement
(i.e. when linkage is chosen). Then Pu(cu*) is their equilibrium payoff. The remaining n-cu* players are the
free-riders of the linked agreement. Their equilibrium payoff is Qu(cu*).
If linkage is not adopted, we have two agreements. For simplicity, let us use the index “a” to identify the
agreement whose benefits are not (completely) excludable (e.g. the environmental agreement), whereas “t”
identifies the agreement with (partly) excludable benefits (e.g. the agreement on technological cooperation).
Then, let ca* be the equilibrium number of players who sign the (quasi) public good agreement, or “aagreement”, whereas ct* is the equilibrium number of signatories of the (quasi) club good agreement, or “tagreement”. Pa(ca*) is the equilibrium payoff of the former, whereas Pt(ct*) is the equilibrium payoff of the
latter. Finally, free-riders of the “a-agreement” obtains a payoff equal to Qa(ca*), whereas free-riders of the
“t-agreement” obtains Qt(ct*).
These definitions enable us to introduce our last assumption:
A.6 (Additivity): Pu(c) = Pa(c) + Pt(c) ,c and Qu(c) = Qa(c) + Qt(c) ,c. Hence, the payoff that can be
obtained from linking the two agreements is equal to the sum of the payoffs of the two individual
agreements, both for co-operators in the joint agreement and for its free-riders.
Finally, under open membership, the equilibrium concept for the coalition game is the usual Nash
equilibrium. Hence, a coalition c* is an equilibrium coalition if it is profitable and stable, where profitability
and stability are defined as follows.
Profitability (Rationality): A coalition c* is profitable if each cooperating player gets a payoff larger than the
one he would get when no coalition forms. Formally:
(1)
P(c*)  P(0)
for all players in the coalition c*, 2c*n.
In the case of symmetric countries this condition is fairly trivial: it simply means that each country’s choice
must be rational and that, if a coalition is profitable for one country, it is profitable for all other ones.
Stability (Nash equilibrium): A coalition formed by c* players is stable if on the one hand there is no
incentive to free-ride, i.e.:
(2a)
Q(c*-1) - P(c*)  0
and on the other hand there is no incentive to broaden the coalition, i.e.:
6
(2b)
P(c*+1) - Q(c*) < 0
Notice that, if a coalition c* is profitable and stable, then no player has an incentive to modify its decision to
sign or not to sign the treaty. Hence, c*, 2c*n, is the outcome of a Nash equilibrium in which each
country’s strategy set is sign, not sign.
In particular, cu* identifies the size of the equilibrium coalition when issue linkage is adopted iff:
(3a)
Pa(cu*) + Pt(cu*)  Pa(0) + Pt(0)
(3b)
Pa(cu*) + Pt(cu*)  Qa(cu*-1) + Qt(cu*-1)
(3c)
Pa(cu*+1) + Pt(cu*+1) < Qa(cu*) + Qt(cu*)
From (3a) it is clear that, if the individual agreements are profitable, then the linked agreement is also
profitable. But not vice-versa. This is why, as explained in the Introduction, issue linkage has been proposed
to solve the profitability problem (Cesar and De Zeeuw, 1996).
Let us define the structure and the payoffs of the linkage game. If players decide to link the two issues and
negotiate on a joint agreement, the equilibrium payoffs are:
(4a)
Pu(cu*) = Pa(cu*) + Pt(cu*)
for a signatory of the agreement;
(4b)
Qu(cu*) = Qa(cu*) + Qt(cu*)
for a free-rider.
If instead players prefer not to link the two issues, they decide whether or not to participate in two different
agreements. In this case they obtain at the equilibrium:
(5a)
Pa(ca*) + Pt(ct*)
if they decide to cooperate on both issues;
(5b)
Pa(ca*) + Qt(ct*)
if they cooperate in the “a-agreement”, but they
free ride on the “t-agreement”;
(5c)
Qa(ca*) + Pt(ct*)
if they cooperate in the “t-agreement”, but free ride
on the other issue;
(5d)
Qa(ca*) + Qt(ct*)
if they free-ride on both issues.
Hence, without linkage, there are four “types” of countries, where the identity of countries is irrelevant
because of symmetry. The structure of the linkage game and its payoffs are summarised in Figure 1.
7
Figure 1. The structure of the game
No Linkage
Coop on both Coop on a only
Linkage
Free-ride on both
Coop on t only
Pa(ca*)+Pt(ct*)
Pa(ca*)+Qt(ct*)
Coop on the
Free-ride on the
linked agreement
linked agreement
Qa(ca*)+Pt(ct*) Qa(ca*)+Qt(ct*)
Pu(cu*)
Qu(cu*)
Under what conditions will players choose to link the two issues and negotiate on a joint agreement?
3.
The equilibrium of the game.
A first answer to the above question is provided by the following proposition.
Proposition 1: Assume ci*, i=a,t,u, 2 ci*<n are the equilibrium coalitions of the a (e.g. public good), t (e.g.
club good) and linked agreement respectively. If the cooperative payoffs Pi(ci), i=a,t,u, increase
monotonically with the coalition size, then issue linkage is players’ equilibrium choice under unanimity
voting iff:
(6)
Pa(cu*) - Qa(ca*) > Qt(ct*) - Pt(cu*)
8
Proof. From Figure 1, all countries are going to choose the right-hand tree if the payoff they get whatever
their decision concerning cooperation under issue linkage is larger than the payoff they get whatever their
decision without issue linkage.
In the presence of issue linkage, the minimum payoff is Pu(cu*) = Pa(cu*) + Pt(cu*), because Pu(cu*) < Qu(cu*) if
the payoff function Pu(cu) is monotonic. Indeed, from conditions (2a)(2b) and the monotonicity of Pu(cu) we
have:
(7a)
Qu(cu*-1)  Pu(cu*)  Pu(cu*+1) < Qu(cu*)
which also implies the monotonicity of Q(cu).
In the absence of linkage, the highest payoff that a player can obtain is the one he gets when free-riding on
both agreements. Indeed, Pa(ca*) + Pt(ct*) < Pa(ca*) + Qt(ct*), because Pt(ct*) < Qt(ct*) if the payoff function
Pt(ct) is monotonic. Indeed:
(7b)
Qt(ct*-1)  Pt(ct*)  Pt(ct*+1) < Qt(cu*)
In addition, Pa(ca*) + Qt(ct*) < Qa(ca*) + Qt(ct*), because Pa(ca*) < Qa(ca*) if the payoff function Pa(ca) is
monotonic (just replace the index “t” with the index “a” in (7b)). And finally, P a(ca*) + Pt(ct*) < Qa(ca*) +
Pt(ct*) < Qa(ca*) + Qt(ct*), because Pi(ci*) < Qi(ci*), i=a,t, if the payoff functions Pi(ci), i=a,t, are monotonic.
Hence, all players choose issue linkage if Pu(cu*) = Pa(cu*) + Pt(cu*) > Qa(ca*) + Qt(ct*) or Pa(cu*) - Qa(ca*) >
Qt(ct*) - Pt(cu*).
Q.E.D.
How can condition (6) be interpreted? Qt(ct*) - Pt(cu*) is the loss from reducing the coalition on the “tagreement” from ct* to cu* (Proposition 2 below will show that indeed ct*  cu*). This loss can be written as
Qt(ct*) - Pt(cu*) = [Qt(ct*) - Qt(cu*)] - [Pt(cu*) - Qt(cu*)] where the first term represents a free-rider’s loss when
they get less benefits from a smaller coalition, whereas the second term represents the excess benefit of cooperation when cu* < ct*. Notice that Qt(ct*) - Pt(cu*) > 0, because it is also equal to [Qt(ct*) - Pt(ct*)] + [Pt(ct*) Pt(cu*)], where [Qt(ct*) - Pt(ct*)] > 0 and [Pt(ct*) -Pt(cu*)] > 0, if ct* > cu* and if Pt(ct) is monotonic.
Pa(cu*) - Qa(ca*) is the benefit (e.g. the environmental benefit) which a free-rider on the “a-agreement”
achieves when joining the expanded coalition. It can be also written as [Pa(cu*) - Pa(ca*)] - [Qa(ca*) - Pa(ca*)]
where the first term is the increased gain which a cooperator on the “a-agreement” achieves from expanding
the coalition, whereas the second term is a free-rider’s relative gain when a coalition ca* forms. The positivity
of Qt(ct*) - Pt(cu*) implies that (6) holds if Pa(cu*) - Qa(ca*) is also positive, i.e. if the increased gain which a
9
cooperator on the “a-agreement” achieves from expanding the coalition is larger than a free-rider’s relative
gain when a coalition ca* forms.
This is only a necessary condition. The sufficient condition says that the increased gain which a cooperator
on the “a-agreement” (e.g. a signatory of an environmental agreement) achieves from expanding the
coalition from ca* to cu*, plus the excess benefit of co-operation on the “t-agreement” when cu* < ct*, must be
larger than a free-rider’s relative gain when a coalition ca* forms plus the loss that a free-rider suffers
because of the smaller spillovers from the reduced co-operation on the “t-agreement”.
However, in which cases is condition (6) likely to be satisfied? Are the assumptions stated in Proposition 1
satisfied in the economic contexts usually analysed in the recent literature on coalition formation? For
example, Carraro and Siniscalco (1997) show that the function Pt(ct) may not be monotonic. Hence, (7b) may
not hold. In addition, Yi (1997) show that ct* = n -- i.e. the grand coalition forms -- in the case of negotiations
on R&D cooperation. Hence, Qt(ct*=n) may be smaller than Pt(ct*=n) and again (7b) may not hold. What does
condition (6) become in these cases? Finally, what are the conditions for issue linkage to be chosen under a
majority voting rule rather than under unanimity?
To answer the above questions and to discuss the above issues, a more detailed specification of the economic
model is necessary. Let us start by specifying the players’ payoff functions in two cases. In the first one, the
benefits from co-operation on the “t-agreement” are sufficiently excludable to provide the incentives for the
formation of the grand coalition on this agreement. In the second one, a coalition smaller than the grand
coalition forms on the “t-agreement”, because benefits from co-operation spill over the free riders.
Let , [0,1], be the degree of excludability of the benefits produced by “t-agreement”. If =1, then benefits
are perfectly excludable, namely they go only to cooperators. Hence, Qt(ct) = 0, ct[2,n]. If =0, the
benefits produced by cooperators are a public good and go to free-riders as well. In the case of R&D cooperation,  depends on the possibility of patenting innovations and on the duration and extension of the
patent. If 0  1, then we have a case of partial excludability. The smaller , the larger the benefits achieved
by free-riders and hence the larger the function Qt(ct) for any given ct.
Let denote by ° the value of  such that Pt(ct) = Qt(ct) when ct* = n. In words, when   °, the degree of
excludability is so high that the benefits from participating in the agreement are larger than the benefits from
free-riding for all 2ctn. Hence, assuming that the profitability condition is satisfied for all c t in the interval
[2,n], the grand coalition forms, i.e. all players prefer to sign the “t-agreement” (ct*=n).4 By contrast, when 
4 Notice that, when  > °, all coalitions ct where 2ctn satisfy the internal stability condition (2a), but not the external stability
condition (2b). In this case, all players want to join the coalition. Hence, we assume that the equilibrium is achieved when ct = n.
10
< °, only a partial coalition forms, i.e. only a subset of countries sign the agreement. The function Q t(ct) for
low ,  = ° and high  is represented in Figure 2.
Figure 2. Payoff functions for different values of .
Pt(c)
Pt(c)
Qt(c-1) for low 
Qt(c-1) for  = °
Qt(c)
Qt(c-1) for high 
c
n
Hence, in the rest of the paper we will analyse two cases:
Case A: °    , i.e. the case in which all players sign the “t agreement” (ct* = n);
Case B:    < °, i.e. a partial coalition forms (2  ct* < n)
In addition, we will restrict the analysis to the case in which the payoff function Pt(ct) is humped-shaped. As
shown in Carraro and Siniscalco (1997), this is the case when the “t-agreement” concerns R&D co-operation
and is generally the case when benefits are excludable. In the case of R&D co-operation, the intuition is as
follows. The decision to sign the R&D agreement has two positive effects for signatories: on the one hand,
production costs decrease because co-operative R&D makes more efficient technologies available; on the
other hand, market share increases because firms with lower costs have a higher market share (a standard
Cournot oligopoly is assumed). However, this latter effect becomes smaller and smaller as the coalition size
increases and goes to zero when ct = n. Hence, the benefit from belonging to the coalition ct decreases when
ct is above a given intermediate value ct°.
11
Notice that, in Case A, if Pt(ct*) is humped-shaped, then Pt(ct*=n)  Pt(ct*+1). Moreover, at the equilibrium
Pt(ct*=n)  Qt(ct*-1). We also assume for simplicity that, in case A, Pt(ct*+1)  Qt(ct*=n). Hence, Qt(ct*) 
Pt(ct*). Notice that this implies the impossibility of applying Proposition 1.
As for the other payoff functions -- Pa(ca), Qa(ca), and Qt(ct) -- we assume them to be monotonically increasing
with the coalition size. This is a standard assumption in the economic literature on environmental coalition
formation (see the surveys by Barrett, 1997,2000; Carraro, 1998; Carraro and Moriconi, 1999). In particular, a
monotonic Pa(ca) means that the benefits from providing a public good (e.g. from abating emissions) increase
with the number of countries that participate in the agreement. As consequence, the following assumption will
be used:
A.7 (Incomplete monotonicity): The payoff functions Pa(ca), Qa(ca), and Qt(ct) are assumed to be
monotonically increasing in ca, and ct respectively. The payoff function Pt(ct) is assumed to be increasing in
ct for ct < ct° and monotonically decreasing in ct for ct > ct°.
The shape of the payoff functions for co-operators and free-riders is shown in Figure 3 for Case A (°<   )
and in Figure 4 for Case B (  °).
Figure 3. Shape of the payoff functions in Case A (°<   )
P,Q
P,Q
Qa(c-1)
ca*
Pa(c)
Pt(c)
n
c
Qt(c-1)
ct*=n
c
12
Figure 4. Shape of the payoff functions in Case B (  °).
P,Q
P,Q
Qa(c-1)
Pt(c)
Qt(c-1,°)
Pa(c)
Qt(c-1)
ca *
n
c
ct *
n
c
Notice that in Figures 3 and 4 we represent the case in which c a* < ct*. This reflects the implicit assumption
that the equilibrium coalition in the case of an agreement on a (quasi) public good is smaller than the
equilibrium coalition in the case of an agreement on a (quasi) club good. Indeed, where ca*  ct*, the idea of
linking the negotiation on the provision of a (quasi) public good to a different negotiation would be
meaningless.
Also notice that the monotonicity of Qa(c) and Qt(c) implies the monotonicity of Qu(c). By contrast, Pu(c) =
Pa(c) + Pt(c) can be both monotonic and humped-shaped. However, given Assumption A.7, if Pu(c) is
humped-shaped, it is monotonically increasing for cu < c°u and monotonically decreasing for cu > c°u, with
c°u  c°t .
Assume that issue linkage is effective in increasing the number of players who provide the (quasi) public
good, i.e.:
(8a)
c u* > c a *
13
and profitable
Pu(cu*)  Pu(0)
(8b)
and let us focus on the stability of the linked agreement. First we show that cu* is smaller than ct*, namely
that the equilibrium coalition emerging from the linked negotiation is always smaller than the equilibrium
coalition in the “t-agreement”.
Proposition 2: At the equilibrium, cu*  ct*, i.e. the number of players who participate in the joint agreement
is always smaller than or equal to the number of players who participate in the agreement linked to the
public good agreement.
Proof: The joint agreement is internally stable if Pu(cu*)  Qu(cu*-1), i.e. if :
(9)
Qa(cu*-1) – Pa(cu*)  Pt(cu*) – Qt(cu*-1)
For cu*  ca*, the left hand side of (9) is positive because there is an incentive to free-ride on the “aagreement” for all c > ca*. This implies that the right had side is also positive, i.e. Pt(cu*) > Qt(cu*-1).
Therefore, when countries co-operate on the “t-agreement”, there is still an incentive to enter the coalition.
Hence, cu* must be smaller than or equal to the equilibrium coalition size, i.e. cu*  ct*.
Q.E.D.
This result holds both in Case A and in Case B. The only difference is that, in case A, P t(c) – Qt(c-1) is non
negative for all c in the interval [2,n] because this is the condition which implies c t*= n. Hence, Pt(c) – Qt(c1) is obviously non negative also for c = cu*.
Notice that Proposition 2 and the preceding analysis lead to the following ordering:
(10)
c a * < c u*  c t *
and
c°t  c°u .
The payoff functions of the two separate games and of the linked game are shown in Figure 5 and 6 for Case
A and B respectively. In both Figures, the situation in which cu* > cu° is reproduced. This will be useful to
clarify the analysis of the equilibrium of the game.
14
Figure 5. Payoff functions for the linked and separate agreements in Case A and cu* > cu°.
Qu(cu-1)
Pu(cu)
Qu(cu-1)
Pu(cu)
cu* ct*=n
cu°
P, Q
Qa(ca-1)
Pa(ca)
c
Qt(ct-1)
Pt(ct)
ca *
ct°
cu°
ct*=n
c
15
Figure 6. Payoff functions for the linked and separate agreements in Case B.
Qu(c-1)
Pu (c)
Qu(c-1)
Pu(c)
cu°
P(c)
Qa(c-1)
Pa(c)
cu*
ct *
n
Qt(c-1)
Q(c-1)
Pt(c)
ca *
ct° cu°
ct*
n
16
3.1 Case A. A perfect club good.
We are now ready to determine players’ equilibrium choice in the first stage of the game (the linkage game).
In Case A the situation is simpler, because, if players negotiate only on the t-agreement, at the equilibrium all
countries would like to sign it (ct* = n). Hence, if players disagree on linkage, either they co-operate on both
the “a-agreement“ and the “t-agreement”, or they free-ride only on the first one. Their payoff is therefore
Pa(ca*) + Pt(ct*) or Qa(ca*) + Pt(ct*), where Pa(ca*) + Pt(ct*)  Qa(ca*) + Pt(ct*) because the monotonicity of
Pa(ca) and (2a)(2b) implies Pa(ca*)  Qa(ca*). As a consequence, for issue linkage to be the equilibrium
strategy under unanimity voting, the best payoff without linkage must be smaller than the worst payoff when
issue linkage is adopted. This leads to the following Proposition:
Proposition 3: Assume °<   , i.e. ct*= n. If either one of these conditions is satisfied:
-
Pu(cu) is monotonic in the interval [2,n];
-
cu* < cu°;
-
cu*  cu° , cu° < n, and Pt(cu*) - Qt(cu*) is smaller than Qa(cu*) - Pa(cu*) > 0,
then players adopt issue linkage under unanimity voting iff:
(11)
Pa(cu*) – Qa(ca*) > Pt(ct* = n) – Pt(cu*)
The condition becomes:
(12)
Qa(cu*) – Qa(ca*) > Pt(ct* = n) – Qt(cu*)
if either:
-
cu* > cu° , cu° < n, and Pt(cu*) - Qt(cu*) is positive and larger than Qa(cu*) - Pa(cu*); or
-
cu* = n > cu°.
Proof: If Pu(cu) is monotonic or Pu(cu) is humped-shaped with cu* < cu°, then at the equilibrium Pu(cu*+1) 
Pu(cu*), which implies Pu(cu*) = Pa(cu*) + Pt(cu*) < Qu(cu*) = Qa(cu*) + Qt(cu*) because of (2a)(2b). Hence, all
players vote for issue linkage if Pu(cu*) = Pa(cu*) + Pt(cu*) -- the worst payoff they can get under issue linkage
-- is larger than Qa(ca*) + Pt(ct*) -- the largest payoff they get without linkage. Hence, (11) must hold.
If Pu(cu) is humped-shaped with cu*  cu° , cu° < n, then Qu(cu*) may be smaller than Pu(cu*). If not, (11) holds
again. Qu(cu*) is smaller than Pu(cu*) iff Pt(cu*) - Qt(cu*) > Qa(cu*) - Pa(cu*). Notice that Qa(cu) - Pa(cu) > 0 at c=
cu*, because ca* < cu*. Hence, a necessary condition for Qu(cu*) < Pu(cu*) is Pt(cu*) - Qt(cu*) > 0, which holds
because cu* < ct*. As a consequence, if Pt(cu*) - Qt(cu*) > Qa(cu*) - Pa(cu*) > 0, all players vote in favour of
issue linkage when Qu(cu*) = Qa(cu*) + Qt(cu*) -- the worst payoff they can get under issue linkage -- is larger
than Qa(ca*) + Pt(ct*) -- the largest payoff they get without linkage. Hence, (12) must hold.
17
Finally, when cu* = n, there is no incentive to defect for any cun. Hence, Pu(cu*) > Pt(cu*+1) > Qu(cu*). As a
consequence, Qu(cu*) = Qa(cu*) + Qt(cu*) must be larger than Qa(ca*) + Pt(ct*), i.e. (12) must hold.
Q.E.D.
How can conditions (11) and (12) be interpreted? Pa(cu*) – Qa(ca*) -- the left hand side of (11) -- is the same
as the left hand side of (6). Hence, it is the gain or loss that a free-rider on the “a-agreement” achieves from
joining the expanded coalition. It can be also written as [Pa(cu*) - Pa(ca*)] - [Qa(ca*) - Pa(ca*)], where the first
term is the increased gain that a cooperator on the “a-agreement” achieves from expanding the coalition,
whereas the second term is a free-rider’s relative gain when a coalition ca* forms. Pt(ct*) – Pt(cu*) is the
possible gain or loss that goes to a cooperator in the “t-agreement” when the coalition size moves from ct* to
cu*. Hence, (11) says that the gain (loss) that a free-rider on the “a-agreement” achieves from joining the
expanded coalition must be larger (smaller) than the gain (loss) that goes to a cooperator in the “tagreement” when the coalition size moves from ct* to cu*.
Condition (12) has a different interpretation. Qa(cu*) – Qa(ca*) is the gain that goes to a free-rider when
more players co-operate on the provision of a (quasi) public good. Pt(ct*) – Qt(cu*) = Pt(ct*) – Pt(cu*) +
Pt(cu*) – Qt(cu*) is the possible gain or loss that goes to a cooperator in the “t-agreement” when the coalition
size moves from ct* to cu* plus the excess benefits of co-operation when cu* < ct* (re-call that Pt(c) > Qt(c) for
all c < ct* = n, because the agreement concerns a perfect club good). Hence, issue linkage is chosen by all
players if the gain that goes to a free-rider when more players co-operate on the provision of a (quasi)
public good is larger than the excess benefits of co-operation when cu* < ct* plus the gain (loss) that goes to
a cooperator in the “t-agreement” when the coalition size moves from ct* to cu*.
Finally, let us analyse the equilibrium conditions under majority voting. If the group of players that would
co-operate on both agreements even in the absence of linkage is larger than ½n, then linkage is chosen if:
(13a)
Pa(cu*) - Pa(ca*) > Pt(ct* = n) - Pt(cu*)
which is less stringent than (11) because Pa(ca*) < Qa(ca*) or
(13b)
Qa(cu*) - Pa(ca*) > Pt(ct*) - Qt(cu* = n)
which is less stringent than (12) because again Pa(ca*) < Qa(ca*). Otherwise, either (11) or (12) becomes
necessary.
18
Figure 7. Equilibrium conditions under majority voting (Case A).
Cooperators on both issues
> 50%
Condition (13a) or (13b)
 50%
All countries
Condition (11) or (12)
3.2 Case B. An imperfect club good
Let us now consider the second case, in which the economic issue linked to the (quasi) public good issue is
an imperfect club good. This implies that the benefits from co-operation on the “t-agreement” which spill
over free-riders are strong enough to induce some players not to join the coalition. Hence, when players
negotiate on the “t-agreement” only, the equilibrium coalition ct* is not the grand coalition, i.e. ct* < n.
In this context, it is still important to adopt issue linkage as a strategy to increase the coalition size on the “aagreement” because ca* < ct*. Hence, issue linkage helps players to achieve a coalition cu* both in the “aagreement” and in the “t-agreement”, where cu*> ca*, even if cu* < ct*. This induces a trade-off in the stage of
the game where players decide whether or not to adopt issue linkage. Indeed, the benefits of a larger
coalition on the “a-agreement” must be traded off with the loss of a smaller coalition in the “t-agreement”.
The situation is complicated by the fact that in the case of the t-agreement players may prefer a smaller
coalition (a smaller club) than a large one.
19
The first step to determine the equilibrium of the game is the analysis of the payoffs of the four types of
players that emerge in the second stage of the game. We need to compare:
- Pa(ca*) + Pt(ct*), the payoff of a cooperator in both separate agreements;
- Pa(ca*) + Qt(ct*), the payoff of a player who cooperates in the “a-agreement” but free-rides on the other one;
- Qa(ca*) + Pt(ct*), the payoff of a player who cooperates in the “t-agreement” but free-rides on the other one;
- Qa(ca*) + Qt(ct*), the payoff of a free-rider on both separate agreement,
without using the monotonocity of Pi(c), i=u,t, as in the case of Proposition 1. First, notice that Pa(ca*) +
Pt(ct*) < Qa(ca*) + Pt(ct*) and Pa(ca*) + Qt(ct*) < Qa(ca*) + Qt(ct*) because the monotonicity of Pa(c) implies
Pa(ca*) < Qa(ca*). Hence, the largest payoff in the case of two separate agreements is the one in which a
player free-rides on both agreements iff:
(14)
Pt(ct*) < Qt(ct*)
In the rest of this paper we will use (14), which says that a free-rider on the “t-agreement” achieves a larger
payoff than a cooperator in the same agreement. This is reasonable if the degree of appropriability of the
benefits from co-operation in the “t-agreement” is sufficiently low. We assume that this is the case for  < °.
Then, the conditions for issue linkage to be an equilibrium strategy of the three-stage game under unanimity
voting in the first stage are described by the following Proposition:
Proposition 4: Assume 0   < °, i.e. ct*< n, and Pt(ct*)< Qt(ct*). If either one of these conditions is satisfied:
-
Pu(cu) is monotonic in the interval [2,n];
-
cu* < cu°;
-
cu*  cu° , cu° < n, and Pt(cu*) - Qt(cu*) is smaller than Qa(cu*) - Pa(cu*) > 0,
then players adopt issue linkage under unanimity voting iff condition (6) of Proposition 1 holds, i.e.
(6)
[Pa(cu*) - Qa(ca*)] > [Qt(ct*) - Pt(cu*)]
The condition becomes:
(15)
Qa(cu*) – Qa(ca*) > Qt(ct*) – Qt(cu*)
if cu* > cu° , cu° < n, and Pt(cu*) - Qt(cu*) is positive and larger than Qa(cu*) - Pa(cu*).
Proof: If Pu(cu) is monotonic or Pu(cu) is humped-shaped with cu* < cu°, then at the equilibrium Pu(cu*+1) 
Pu(cu*), which implies Pu(cu*) = Pa(cu*) + Pt(cu*) < Qu(cu*) = Qa(cu*) + Qt(cu*) because of (2a)(2b). Hence, all
20
players vote for issue linkage if Pu(cu*) = Pa(cu*) + Pt(cu*) -- the worst payoff they can get under issue linkage
-- is larger than Qa(ca*) + Qt(ct*) -- the largest payoff they get without linkage. Hence, (6) must hold.
If Pu(cu) is humped-shaped with cu*  cu° , cu° < n, then Qu(cu*) may be smaller than Pu(cu*). If not, (6) holds
again. Qu(cu*) is smaller than Pu(cu*) iff Pt(cu*) - Qt(cu*) > Qa(cu*) - Pa(cu*). Notice that Qa(cu) - Pa(cu) > 0 at c=
cu*, because ca* < cu*. Hence, a necessary condition for Qu(cu*) < Pu(cu*) is Pt(cu*) - Qt(cu*) > 0, which holds
for cu* < ct*. As a consequence, if Pt(cu*) - Qt(cu*) > Qa(cu*) - Pa(cu*) > 0, all players vote in favour of issue
linkage when Qu(cu*) = Qa(cu*) + Qt(cu*) -- the worst payoff they can get under issue linkage -- is larger than
Qa(ca*) + Qt(ct*) -- the largest payoff they get without linkage. Hence, (15) must hold.
Q.E.D.
The interpretation of this proposition is easy. Again we have two conditions for issue linkage to be chosen by
all players in the first stage of the game. The first one coincides with (6) and has already been analysed when
discussing Proposition 1. The second condition – the inequality (15) -- is new and says that the benefits
enjoyed by a free rider on the “a-agreement” when the coalition size increases must be larger than the loss
suffered by a free-rider on the “t-agreement” when the signatories of the t-agreement” decrease from ct* to
cu* (recall that benefits from co-operation spill over free-riders also in the case of the “t-agreement”).
What are the conditions under majority voting? Using (14) and assuming that at the equilibrium free-riders
get a relative larger benefit in the case of the “a-agreement” than in the case of the “t-agreement”5, namely:
(16)
Qt(ct*) - Pt(ct*) < Qa(ca*) - Pa(ca*)
then the following ranking holds:
(17)
Pa(ca*) + Pt(ct*) < Pa(ca*) + Qt(ct*) < Qa(ca*) + Pt(ct*) < Qa(ca*) + Qt(ct*)
This implies that, if the group of cooperators in both separate agreement is larger than ½n, the issue linkage
is chosen under majority voting iff either:
(18a)
Pa(cu*) – Pa(ca*)  Pt(ct*) – Pt(cu*)
or
(18b)
Qa(cu*) – Pa(ca*)  Pt(ct*) – Qt(cu*)
If two groups of players are necessary to achieve majority, namely cooperators on both separate agreements
and cooperators on the “a-agreement” only, then issue linkage is adopted under majority voting iff either:
5 This assumption is obviously satisfied when Qt(ct*) < Pt(ct*) (as in Case A, but this situation may occur in Case B
too) because the monotonicity of P a(c) implies Qa(ca*) - Pa(ca*) > 0.
21
(19a)
Pa(cu*) – Pa(ca*)  Qt(ct*) – Pt(cu*)
or
(19b)
Qa(cu*) – Pa(ca*)  Qt(ct*) – Qt(cu*)
If three groups of players are necessary to achieve majority, namely cooperators on both separate agreements
and cooperators on either one of the two separate agreements, then issue linkage is adopted under majority
voting iff either:
(20a)
Pa(cu*) – Qa(ca*)  Pt(ct*) – Pt(cu*)
or
(20b)
Qa(cu*) – Qa(ca*)  Pt(ct*) – Qt(cu*)
Finally if the four groups of players are necessary to achieve majority, then the conditions are the same as
under unanimity voting.
Figure 8 shows the different conditions that are necessary for issue linkage to be adopted in the first stage of
the game in the case of majority voting.
22
Figure 8. Equilibrium conditions under majority voting (Case B).
Cooperators on both separate agreements
 50%
> 50%
Conditions (18a) or (18b)
Cooperators on both agr. and on the “a-agr.” only
 50%
> 50%
Conditions (19a) or (19b)
Cooperators on at least one agr.
> 50%
 50%
Conditions (20a) or (20b)
All players
Conditions (6) or (15)
Unanimity Voting
23
4. Conclusions
The previous section have identified four conditions under which all players of the game prefer to negotiate
on two linked issues rather than on the two issues separately. In order to simplify the message which can be
derived from Proposition 3 and 4, let us assume that free-riders on the linked agreement are better off than
cooperators [Pu(cu*) < Qu(cu*)].This is the most frequent case in coalition theory. Then issue linkage is the
equilibrium strategy under unanimity voting iff:
(6)
[Pa(cu*) - Qa(ca*)] > [Qt(ct*) - Pt(cu*)]
in the case of an imperfect club good (ct* < n), or
(11)
Pa(cu*) – Qa(ca*) > Pt(ct* = n) – Pt(cu*)
in the case of a perfect club good (ct* = n).
What policy message can be derived from these inequalities? First a necessary condition. From (6) (and
(11)), a free-rider on the public good agreement who enters the coalition on the linked agreement must get a
higher payoff [Pa(cu*) > Qa(ca*)]. This is a pre-requisite without which issue linkage is not chosen. Hence,
public good (e.g. environmental) benefits provided by a larger coalition must be perceived as sufficiently
large.
Then a sufficient condition. A free-rider on the public good agreement who enters the coalition on the linked
agreement must not only increase his payoff, but this positive change must be larger than the loss a player
may suffer because the club good coalition becomes smaller (this is particularly clear in condition (11) but it
is also true in (6)).
This highlights the trade-off that players face when deciding whether to adopt issue linkage or not. Consider
again the example of an environmental negotiation linked to a negotiation on R&D co-operation. On the one
hand, players would like to reap the benefits provided from a larger environmental coalition. On the other
hand, they know that though issue linkage increases the environmental cooperators, it also decreases the
participants in the R&D co-operation agreement. Hence, environmental benefits could be offset by
technological losses.
A similar argument holds when free-riders on the linked agreement are worse off than cooperators [Pu(cu*) >
Qu(cu*)]. In this case the conditions for issue linkage to be adopted under unanimity voting are:
24
(15)
Qa(cu*) – Qa(ca*) > Qt(ct*) – Qt(cu*)
in the case of an imperfect club good (ct* < n), or
(12)
Qa(cu*) – Qa(ca*) > Pt(ct* = n) – Qt(cu*)
in the case of a perfect club good (ct* = n).
No necessary condition to be stressed, because the monotonicity of Qa(ca) implies Qa(cu*) > Qa(ca*).
However, the sufficient condition says that the gain Qa(cu*) – Qa(ca*) that a free-rider achieves when freeriding on a larger public good agreement must be larger than the loss a player may suffer because the club
good coalition becomes smaller.
As a consequence, when proposing or advocating issue linkage, policymakers must be very careful in
assessing two crucial elements. First, the relative changes of the coalition sizes cu* –ca* and ct* –cu*. The
larger cu* –ca* and the smaller ct* – cu*, the larger the likelihood that conditions (6) (or (11)) and (15) (or (12))
be satisfied. Second, the relative changes of the players’ payoffs. The larger the increased benefits induced
by a larger co-operation on the public good issue, the larger the likelihood that issue linkage be adopted.
Similarly, the smaller the loss from a reduced co-operation on the “t-agreement”, the larger the likelihood
that issue linkage be adopted.
Notice that these conditions neglect the likely existence of larger transaction costs when negotiating on two
linked issues. But their introduction is trivial. They would simply be added to the right-hand side of
conditions (6), (11), (15) and (12).
25
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