modesty may pay!
TRANSCRIPT
The 6th MEETING ON GAME THEORY AND PRACTICE
Zaragoza, Spain 10-12 July
Modesty May Pay!*
Michael FINUS†
Department of Economics University of Hagen††
Stefan MAUS
Department of Economics University of Hagen
Abstract. Standard non-cooperative game theoretical models of international environmental
agreements (IEAs) draw a pessimistic picture of the prospective of successful cooperation.
However, in reality, there are some IEAs with high participation and some success. In order to
explain this phenomenon, this paper departs from the standard assumption of joint welfare
maximization of coalition members, implying ambitious abatement targets and strong free-
riding. Instead, it considers that countries agree on modest emission reduction targets. It is
shown that this can be an explanation for higher participation. Moreover, modesty can
increase welfare of signatories, global welfare and can lower global emissions if participation
is sufficiently raised. Thus, modesty may well pay, though the first best optimum cannot be
achieved.
JEL (or AMS) reference: C72, H41, Q20
Key Words: international environmental agreements, internal&external stability, modest
emission reduction
† Corresponding author: [email protected] †† University of Hagen, Department of Economics, Profilstr. 8, 58084 Hagen, Germany.
1
1. Introduction
Non-cooperative game theoretical analyses of international environmental agreements (IEAs)
draw a rather pessimistic picture of the prospective of successful cooperation between coun-
tries.1 A large amount of the literature came to this conclusion in a simple two-stage cartel
formation game.2 In the first stage, countries decide whether they become a member of an
IEA or remain a non-signatory. In the second stage, they decide on their emission levels.
Given the choices of the second stage, an IEA is called stable if no signatory has an incentive
to leave the agreement (internal stability) and no non-signatory has an incentive to join the
coalition (external stability). That is, decisions in the first stage must form a Nash equilibrium.
For the second stage, it is assumed that signatories choose emission levels that maximize the
aggregate welfare to the coalition. Non-signatories play as singletons and choose their emis-
sions that are optimal under autarky. That is, decisions in the second stage form a Nash equi-
librium between the coalition and the singleton players.
The equilibrium coalition depends on a number of assumptions.3 The "standard model"
assumes that countries´ welfare function comprise benefits (also sometimes called revenues)
from individual emissions and damage costs from global emissions4, signatories and non-
signatories choose their emissions simultaneously (Nash-Cournot assumption). For a large set
of welfare functions, it turns out that the equilibrium number of signatories is small and there-
fore stable coalitions improve only marginally upon the non-cooperative status quo (e.g.,
Botteon and Carraro 1997, Carraro and Siniscalco 1993 and Hoel 1992). The reason is that the
free-rider incentive increases sharply with the number of signatories and hence internal
stability is already violated for small coalitions. Clearly, the standard model helps explaining
the problems of international cooperation in the presence of environmental spillovers, but
cannot explain IEAs with high participation like the Montreal Protocol. This requires a
modification of the standard assumptions. Important modifications in the context of the cartel
1 More positive results are derived from cooperative game theory. See for instance Chander and
Tulkens (1995 and 1997). 2 Similar conclusions have been derived for repeated games by applying the concept of
renegotiation-proof strategies. See for instance Barrett (1994a, b and 1999) and Finus and Rundshagen (1998a).
3 For an overview, see Barrett (2003) and Finus (2001) and (2003). 4 An alternative specification of this model with the same qualitative results assumes countries´
strategies to be abatement from some base emission level and welfare comprises benefits from global abatement and costs from individual abatement. In the public goods literature, this has been referred to as an unweighted summation production technology with non-constant prizes.
2
game include for instance 1) Stackelberg leadership of signatories (Barrett 1994b, 1997a),
2) commitment (Botteon and Carraro 1997, Carraro and Siniscalco 1993 and Petrakis and
Xepapadeas 1996), 3) reputation effects (Jeppesen and Andersen 1998 and Hoel and
Schneider 1997) or preference for equity (Lange and Vogt 2003), 4) issue linkage (Barrett
1997b, Carraro and Siniscalco 1997)5, 5) minimum participation clauses (Carraro et al. 2003)
and 6) multiple coalitions (Carraro and Marchiori 2003 and Finus and Rundshagen 2003).
In this paper, we modify the assumption of joint welfare maximization. This seems
suggestive since casual empirical evidence tells us that coalition members neither choose their
individual emission reductions cost-efficiently nor the sum of individual emission reductions
is optimal for the coalition. On the one hand, abatement obligations of many IEAs are
specified as (cost-inefficient) uniform emission reduction quotas despite countries face
different marginal abatement costs.6 On the other hand, in many IEAs, the global abatement
level is below the optimal level.7 Both observations have been partly rationalized in models
that assume that countries bargain either on the level of an efficient uniform tax rate or on an
inefficient uniform emission reduction quota and agree on the smallest proposal (least
common denominator rule). In Endres (1997), it is shown that the bargaining outcome under
the quota regime may be superior from an ecological and economic point of view in a two-
country model (see also Eyckmans 1997). Later papers (Endres and Finus 2002 and Finus and
Rundshagen 1998b) have confirmed the superiority of the quota over the tax regime in a
repeated game framework if also stability is considered. Moreover, in an N-country model, it
is shown in Finus and Rundshagen (1998a) that modest abatement targets, as implied by the
least common denominator rule, are compensated under both regimes by a higher 5 The idea goes back to Folmer et al. (1993) who analyze issue linkage in a repeated game as for
instance also Cesar and de Zeeuw (1997) and Folmer and van Mouche (1994) do. 6 The list of examples is long and includes the Helsinki Protocol, which suggested a 30 percent
reduction of sulfur emissions from 1980 levels by 1993. Moreover, the "Protocol Concerning the Control of Emissions of Nitrogen Oxides or Their Transboundary Fluxes" signed in Sofia in 1988 called on countries to uniformly freeze their emissions at 1987 levels by 1995 and the "Protocol Concerning the Control of Emissions of Volatile Organic Compounds or Their Fluxes" signed in Geneva in 1991 required parties to reduce 1988 emissions by 30 percent by 1999.
7 For framework conventions, like the Vienna Convention preceding the Montreal Protocol on CFC-reductions, the Framework Convention on Climate Change (FCCC) preceding the Kyoto Protocol on greenhouse gases reduction or the Convention Long Range Transboundary Pollution (LRTAP) preceding the Helsinki and Oslo Protocols on sulfur reductions, this is evident because they are only declarations of intentions without abatement obligations. However, this conclusion is also supported by empirical studies on the Montreal Protocol (Murdoch and Sandler 1997b), the Helsinki Protocol (Murdoch and Sandler 1997a), the Oslo Protocol (Finus and Tjøtta 2003), or the Kyoto Protocol (Böhringer and Vogt 2004).
3
participation. The tradeoff between the level of emission reduction and participation in favor
of participation with overall positive ecological and economic effects has also been confirmed
by Barrett (2002).
In this paper, we focus exclusively on the effect of modest emission reduction on the
success of coalition formation like in Barrett (2002). We also stick to the assumption of
symmetric countries for analytical simplicity. However, unlike this and the other papers
mentioned above, we analyze the effect of modest emission reduction on participation, global
emissions and global welfare not in a repeated game but in a two-stage cartel game.
Moreover, unlike Barrett (2002), we do not restrict attention to the grand coalition in the case
of modest emission reduction and, unlike Barrett (2002) and the other papers mentioned
above, the “degree of modesty” is chosen endogenously in our model.
In what follows, we first lay out the standard model assuming joint welfare
maximization in Section 2. We call this “the classical case” with “ambitious emission
reduction”. Subsequently, we analyze what changes if countries agree on “modest emission
reduction” in Section 3. We call this “the non-classical case”. Section 4 confirms via
numerical simulations our results of section 3 for payoff functions for which we cannot derive
analytical results. Section 5 discuss equilibrium aspects of the overall game and section 6
summarizes the main findings and points to some issues of future research.
2. The Classical Case
Assume a cartel formation game with N symmetric countries that choose their membership in
the first stage and their emissions in the second stage. The game is solves by backwards
induction.
Consider as in Hoel and Schneider (1997) the following (representative) welfare
function (that we also call sometimes payoff function) of (a symmetric) country i I∈ :
[1] N
i i jj 1
B(x ) d x=
π = − ∑ with maxi i i i B '(x ) 0, B''(x )<0, d>0, x (0, x ]> ∈ .
That is, benefits iB(x ) are strictly increasing and strictly concave in individual
emissions ix , emissions are strictly positive, there is a maximal emission level maxix in each
country, and damage costs N
jj 1
d x=∑ are linear in global emissions jx∑ with constant marginal
4
damages d. The case of non-linear damage cost functions (implying non-dominant strategies)
is analyzed in section 4.
We divide the set of countries I into the set of signatories SI and the set of non-
signatories NI , S NI I I= ∪ . Let n denote the number of signatories and assume that the n
signatories jointly maximize the aggregate welfare to their coalition whereas every of the N-n
non-signatories maximizes its own welfare. Then, a signatory Si I∈ derives its optimal
emission level from
[2] Si
N
j kj Ix k 1
max B(x ) d x∈
=
⎡ ⎤−⎢ ⎥⎣ ⎦∑ ∑ ,
which leads to the first order condition of a signatory:
[3] 'iB (x ) dn=
because of symmetry (i.e., i jx x= ∀ i,j SI∈ ). A non-signatory Ni I∈ simply maximizes [1],
which leads to the first order condition of a non-signatory:
[4] 'iB (x ) d= .
Since n 1≥ and d>0, it suffices for an interior equilibrium to require that maxix is
sufficiently large and that 'iB (x 0) dn= > . Due to constant marginal damages, countries have
a dominant strategy and hence there exists a unique emission vector for any n [1, N]∈ , n∈ .
Signatories choose Six (n) which decreases in the number of participants n, and non-signato-
ries choose Nix irrespective of n. Thus, total emissions, T S N
i ix (n) nx (n) (N n)x= + − , decrease
in the number of signatories. Therefore, a non-signatory’s payoff, N N Ti i(n) B(x ) dx (n)π = − ,
increases with the number of signatories because damages decrease in n. However, also a
signatory´ s payoff, S S Ti i(n) B(x (n)) dx (n)π = − , increases in n and hence the total payoff,
T S Ni i(n) n (n) (N n) (n)π = π + − π , increases in n.8
The equilibrium number of signatories follows from the condition of internal and
external stability:
8 In section 3, it will become apparent that the classical case is one special case of the non-classical
case and hence we prove all relations (in Lemma 1 and 2) mentioned above in this section.
5
[5] internal stability: S Ni i(n) (n 1)π ≥ π − Si I∀ ∈ and
[6] external stability: N Si i(n) (n 1)π > π + Ni I∀ ∈ .
That is, no signatory should be better off by leaving the agreement with n members to
become a non-signatory so that there are only n-1 signatories left. By the same token, no non-
signatory should have an incentive to join the coalition of n members to become a signatory in
a coalition of n+1 members. Coalitions that satisfy conditions [5] and [6] simultaneously are
called stable and we denote the equilibrium number of signatories by *n . Note that the grand
coalition (n=N) is externally stable and the singleton coalition (n=1) internally stable by defi-
nition.9
For further analysis it is helpful to define as in Hoel/Schneider (1997) stability function S N
i i i(n) (n) (n 1)Φ = π −π − , noting that internal stability implies i (n) 0Φ ≥ Si I∀ ∈ and exter-
nal stability i (n 1) 0Φ + < Ni I∀ ∈ .
For instance, consider the following two examples:10
[7] N
2i i i j
j 1
1b ax x d x2 =
⎛ ⎞π = − −⎜ ⎟⎝ ⎠
∑ , b>0, a>0, d>0
[8] N
i i jj 1
b ln x d x=
π = − ∑ b>0, d>0
which give rise to stability function i (n)Φ as drawn in Figure 1.
Figure 1: about here
For [7], the equilibrium number of signatories is *n 3= and for [8], it is *n 2= . That is, *n is the largest integer equal or smaller than i (n) 0Φ = .
9 It is evident that the definition of internal and external stability de facto corresponds to the
definition of a Nash equilibrium, except that we assume that in the case of indifference a non-signatory would accede to the cartel.
10 For welfare function [7], a sufficient condition that 'iB (x ) 0> and max
i ix (0,x ]∈ hold is b dN / a> in order to ensure ix 0> since N S S
i i ix x (n) x (N) a dN / b 0≥ ≥ = − > has to be satisfied for coalitions of any size and max
ix a≤ because 'i iB (x ) ab bx= − . For welfare function [8], we require b dN>
for ix 0> since N S Si i ix x (n) x (N) b / dN 1 0≥ ≥ = − > and max
ix b / d 1≥ − because Nix b / d 1= − (and
'iB (x ) 0> is true anyway).
6
For the further analysis it is not important whether function i (n)Φ is convex or
concave. However, it is important that our welfare function and the associated assumptions
imply i (n 2)Φ = >0 and a sufficient condition that i (n)Φ is strictly decreasing for n 2≥ is ´´́
iB (x ) 0≥ as shown in Hoel and Schneider (1997) in their appendix. For the remainder, ´´́
iB (x ) 0≥ is assumed and thus *n [2, N]∈ .
We conclude this section with a brief discussion of the implications that a stable
coalition of size *n 2≥ forms. Since Siπ and N
iπ increase in n as mentioned above and
because n=1 corresponds to the “classical Nash equilibrium” and n=N to the “classical social
optimum”, global welfare in *n will be higher than in the Nash equilibrium and lower (equal
if *n N= ) than in the social optimum. By the same token, global emission in *n will be lower
than in the Nash equilibrium and higher (equal if *n N= ) than in the social optimum since Six
decreases in n and Nix remains constant as mentioned above.
3. The Non-Classical Case
The last section reviewed the classical case where signatories conduct “ambitious” emission
reductions. Now, we consider “modest” emission reductions that we call the non-classical
case. We proceed as follows:
In a first step, we explain how we model modesty. We lay out that this includes the
classical case for the “extreme” assumption of “no modesty”. In a second step, we show how
emissions (Lemma 1) and welfare (Lemma 2) are affected by modesty and by the number of
signatories. It will become apparent that for a given number of signatories, modesty has a
“negative impact” on individual and total emissions (because they increase) as well as on
individual and total welfare (because they decrease). Consequently, modesty can only be
beneficial if it helps to stabilize larger coalitions. Therefore, in a third step, we show that for
any coalition size there is a modesty level that internally stabilizes this coalition (Proposition
1). This raises the question, whether it is desirable to stabilize larger coalitions through
modesty. This is clarified in a fourth step that looks at the impact of modesty on global
emissions (Proposition 2), welfare of signatories (Proposition 3) and global welfare (Proposi-
tion 4).
The idea of modest emission reduction is captured in a simple way by assuming that
signatories Si I∈ do not perform [2] but instead perform
7
[9] Si
N
i kj Ix k 1
max B(x ) d x∈
=
⎡ ⎤−α⎢ ⎥⎣ ⎦∑ ∑
with 0 1< α ≤ . Then, a signatory’s first order condition is given by
[10] iB '(x ) nd= α .
For non-signatories Ni I∈ , we continue to assume that every non-signatory maximizes
its own payoff (given by [1]) as in the classical case. Below (after Lemma 1), we will further
comment on α . At this stage, it suffices to note that there exists a unique solution to [10] that
we denote by Six ( , n)α , Si I∈ , 0 1< α ≤ , n∈ because of dominant strategies. As in the
classical case, we have to require for an interior solution that maxix is sufficiently large and
now that 'iB (x 0) dn= > α holds in order for ix 0> . The assumption of a uniform modesty
level α for all signatories seems obvious because of our assumption of symmetric players.
Moreover, it assures that for any given modesty level α and any given number of signatories
n, emissions of signatories are efficiently allocated.11
Let Six (t) , t∈ , be the solution to iB '(x ) td= defined in section 2. Obviously, we
have S Si ix ( , n) x ( n)α = α . This relation stresses that the classical case is a special case of the
non-classical case if 1α = . Let global emissions be given by T S N
i ix ( , n) nx ( , n) (N n)xα = α + − .
Lemma 1
a) For a given modesty level α , emissions of signatories strictly decrease in n, emissions of
non-signatories remain constant and hence global emissions decrease in the number of
signatories n (i.e., <Si
d x ( ,n ) 0dn
α , =Ni
d x 0dn
and <Td x ( ,n ) 0dn
α ) .
b) For a given number of signatories n, emissions of signatories strictly decrease in α , emis-
sions of non-signatories remain constant and hence global emissions decrease in the modesty
level α ( i.e., <Si
d x ( ,n ) 0d
αα
, =Ni
d x 0dα
and <Td x ( ,n ) 0d
αα
).
11 That is, ' S ' S
i jB (x ( ,n)) B (x ( ,n))α = α for all (0,1]α∈ , n∈ , Si, j I∈ , i j≠ . Of course, the level of emissions is not globally optimal for a given n if 1α < as we explain below, but this would also be true if we assumed that α is differentiated across signatories.
8
Proof: See Appendix 1. QED
Lemma 1a confirms our claim in section 2 that individual emissions of signatories as
well as global emissions decrease in the number of signatories, though this holds not only for
1α = but for the general case of 0 1< α ≤ . Lemma 1b implies that if we lower α from 1α = ,
emissions of signatories and global emissions increase. Thus, increasing the “degree of
modesty” implies in our setting to lower the modesty level α . From a global perspective and
for a given number of signatories, this implies that modesty has a negative impact. It seems
plausible to assume that emissions of signatories cannot exceed that of non-signatories, i.e., S Ni ix ( , n) xα ≤ , and hence a lower bound of α is 1/ nα ≥ .
We now turn to payoffs and define as in the classical case, a signatory’s welfare by S S Ti i( , n) B(x ( , n)) dx ( , n)π α = α − α , a non-signatories welfare by N N
i i( , n) B(x )π α =
Tdx ( , n)− α , recalling that T S Ni ix ( , n) nx ( , n) (N n)xα = α + − .
Lemma 2
a) For a given modesty level α , ≤ ≤1/n α 1 , welfare of signatories and non-signatories and
hence global welfare increase in the number of signatories n (i.e., >Si
d ( ,n ) 0dn
π α ,
>Ni
d ( ,n ) 0dn
π α and >Td ( ,n ) 0dn
π α ).
b) For a given number of signatories n, welfare of signatories and non-signatories and hence
global welfare increases in the modesty level α , ≤ ≤1/n α 1 , (i.e., >Si
d ( ,n ) 0d
π αα
,
>Ni
d ( ,n ) 0d
π αα
and >Td ( ,n ) 0d
π αα
).
Proof: See Appendix 2. QED
Again, Lemma 2a generalizes an observation that we made in section 2 and Lemma 2b
confirms the negative impact of modesty also for welfare. Consequently, modesty with a
modesty level 1α < can only have a positive global but also individual impact if it leads to
larger stable coalitions than in the classical case. In fact, in order to obtain an overall positive
effect, the negative effect of modesty for a given number of signatories must be “overcom-
pensated” by a sufficiently high increase in participation in stable coalitions. Therefore, in a
next step, we have a closer look how modesty affects the size of stable coalitions.
9
As in the classical case, the equilibrium number of signatories follows from the
condition of internal and external stability, which now depend not only on n but also on the
modesty level α:
[5α] internal stability: S Ni i( , n) ( , n 1)π α ≥ π α − Si I∀ ∈ and
[6α] external stability: N Si i( , n) ( , n 1)π α > π α + Ni I∀ ∈ .
From the definition, it is evident that we do not only have to ensure that S Ni ix ( , n) xα ≤
but also that S Ni ix ( , n 1) xα − ≤ holds. Hence, we restrict α further and require henceforth
1/(n 1)α ≥ − . Again, it turns out that it is helpful to use the stability function for the analysis,
which is now given by S Ni i i( , n) ( , n) ( , n 1)Φ α = π α −π α − . We say that for a given level of α a
coalition of size n is internally stable if i ( , n) 0Φ α ≥ Si I∀ ∈ , externally stable if
i ( , n 1) 0Φ α + < Ni I∀ ∈ and stable if both conditions are simultaneously satisfied.
In order to illustrate the idea of our next step, we use again our two examples introduced
in section 2 (see [7] and [8]).
Figure 2: about here
From Figure 2, it is evident that if we gradually lower the level of α from 1 1α = to 2α , 3α
and so on with 1 2 3 ....α > α > α > , the size of the stable coalition increases. This confirms our
intuition that in the context of public goods, the free-rider incentive is closely related to the
provision levels of these goods. That is, the lower α , the higher the degree of modesty, the
higher emissions of signatories for a given n which means lower abatement maxi ix x− . It is
evident that there is not only one level α that stabilizes a coalition of a particular size n. For
instance, in the first example (welfare function [7]), any level kα for which 1 k 21= α ≥ α > α
is true will stabilize a coalition of size n=3. Only if k 2α = α , it will be possible to stabilize a
coalition of size n=4 and for k 2α < α it may be possible to stabilize even larger coalitions. In
other words, the equilibrium number of signatories *n is a discontinues function of α
(because *n must be an integer value). Thus for given number of signatories n, the best
signatories can do from their perspective is to choose the highest level of α that is still suffi-
ciently low to stabilize their coalition with n members.12 This follows immediately from
12 In the first example, this implies for instance to choose k 1α = α for n=3 and k 2α = α for n=4.
10
Lemma 2 which showed that welfare of signatories increases in α for a given number of
signatories n. From Lemma 2 we also know that such an “optimal” choice of α is also in the
interest of non-signatories and from Lemma 1 we know that a similar relation holds for global
emissions. Hence, it seems obvious to concentrate in the following on such “optimal” α .
For the analysis of the properties of “optimal” α , it turns out that we can focus on
internal stability as subsequent results will confirm.13 However, as the examples above
showed, it is important that the analysis accounts for the fact that the number of signatories
must be an element in the set of natural numbers n∈ , n [2, N]∈ . Thus, we define:
[11] i1(n) : max{ [ ,1] | ( , n) 0}.
n 1α = α∈ Φ α ≥
−
to which we referred above as the optimal α that stabilizes a given coalition of size n and
which has the following properties.
Proposition 1
The function →α : {2,…,N} [0,1] defined in [11] is well defined and satisfies the inequality
[12] ≥n -1α(n+1) α(n)
n.
Proof: See Appendix 3. QED
Proposition 1 implies that coalitions of any size n can be made internally stable by
choosing the optimal modesty level (n)α . Moreover, inequality [12] gives a lower bound on
the modesty level α that is necessary to stabilize a coalition of size n+1 instead of n. Further-
more, as the proof in Appendix 3 shows, it is inequality [12] that makes the function
:{2, , N} [0,1]α →… well defined, i.e., inequality [12] guarantees that (n) 1/(n 1)α ≥ − holds.
It is certainly useful and in fact necessary to know that there is an optimal modesty level
that can stabilize a coalition of any size. However, it is even more interesting to find out
whether modesty pays at all. Therefore, we analyze in the next step the impact of an increase
in participation via modesty on global emissions (Proposition 2), welfare of signatories
(Proposition 3), and global welfare (Proposition 4).
13 This procedure seems also to be justified for two reasons. First, Lemma 2 showed that signatories´
(and also non-signatories´) welfare increases in the number of signatories. Second, in the context of public goods, the main problem of forming large stable coalitions is to prevent potential signatories from leaving but not to prevent outsiders from joining.
11
Recall that global emissions are given by T S Ni ix ( , n) nx ( , n) (N n)xα = α + − and that
S Si ix ( , n) x ( n)α = α holds. Suppose we go from an internally stable pair ( (n), n)α to an inter-
nally stable pair ( (m),m)α with m>n. Then if (m)mα is larger than (n)nα , emissions of
signatories decrease through this move and so will global emissions since there are more sig-
natories that emit less than before (and emissions of non-signatories remain constant anyway).
Thus, the problematic case is if (m)mα is smaller than (n)nα which cannot be ruled out a
priori. In order to establish T Tx ( (n), n) x ( (m),m)α > α , m>n, in this case, we have to intro-
duce a mild condition. This condition guarantees that the decrease in emissions N Si i(n m)(x x ( (m),m))− − α arising from countries that become “new signatories” is larger
than the increase in emissions of “old signatories” S Si in(x ( (m),m) x ( (n), n))α − α through the
lower modesty level.14
Proposition 2
Let m>n. Suppose that
[13] 3α(3)5
≥ .
Then, ≥T Tx (α(n),n) x ((α(m),m)) .
Proof: See Appendix 4. QED
From now on, we assume that condition [13] holds. While it is certainly desirable from
an environmental point of view to decrease global emissions via modesty, countries incentives
are determined by their individual welfare. We first have a look at signatories.
Proposition 3
Let m>n. Then, ≥S Si iπ (α(m),m) π (α(n),n) .
Proof: See Appendix 5. QED
Proposition 3 means that countries that are signatories for pair ( (n), nα ) and which are
also signatories for pair ( (m), mα ) are better off through the move from a coalition with n
14 We have not been able to find a counterexample where condition [13] is violated. It is
straightforward to check that the two examples [7] and [8] satisfy condition [13].
12
members to a coalition with m members, n<m, via modesty. This group of countries has been
labeled “old signatories” above (group 1).
It is evident that non-signatories for pair ( (n), nα ) that are still non-signatories for pair
( (m),mα ) will also be better off because global emissions (and hence damages) decrease by
Proposition 2 but they chose the same emission level Nix as before (and hence benefits remain
constant). This group of countries may be just labeled “non-signatories” (group 2).
Finally, there is a group of countries that are non-signatories for pair ( (n), nα ) but
signatories for pair ( (m), mα ) and which we labeled “new signatories” above (group 3). This
group faces lower damages by Proposition 2 but also lower benefits because they have to
reduce their emissions as new signatories, i.e., N Si ix x ( (m),m)≥ α . This makes predictions
difficult. Nevertheless, clear-cut conclusions can be derived at the aggregate level.
Proposition 4
Let m>n. Then ≥T Tπ (α(m),m) π (α(n),n) .
Proof: See Appendix 6. QED
Thus, modest emission reduction, associated with higher participation and lower global
emissions, increases global welfare. The intuition is as follows. First, lower global emissions
imply lower individual and aggregate environmental damages. Second, aggregate benefits
from emissions increase because the emission reduction burden is shouldered by more
countries.
We close this section with the conclusion that global emissions will attain its lowest
level (Proposition 2) and global welfare its highest level (Proposition 4) for pair ( (N), Nα ),
i.e., the modesty level that internally stabilizes the grand coalition which is externally stable
by definition. Since typically (N) 1α < (if N is sufficiently large and 1α = implies *n N< ),
global welfare will be below and global emissions will be above those in the social optimum
(which would be obtained for 1α = and n=N) at (N)α . However, modesty and in particular
level (N)α will typically improve upon the classical case without modesty, i.e., 1α = . Thus,
modesty may well pay individually and globally.
4. Quadratic Damage Cost Functions
In order to test the robustness of our results obtained for a linear damage cost function, we consider in
this section a quadratic damage cost function. This seems to be the simplest extension. Nevertheless,
we have not been able to derive analytical solutions. Of course, this is not really surprising when
13
noticing that even for the classical case simulations have been frequently used in the literature.
Therefore, we consider the following two examples:
[14] 2
N2
i i i jj 1
1b ax x d x2 =
⎛ ⎞⎛ ⎞π = − − ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
∑ , a 0, b 0, d 0> > >
and
[15] 2
N
i i jj 1
b ln x -d x=
⎛ ⎞π = ⎜ ⎟
⎝ ⎠∑ , b>0, d>0
and conduct a large number of simulations. The two examples assume the same benefit func-
tions as in our previous two examples but the linear damage cost function is replaced by a
quadratic damage cost function as for instance considered in Barrett (1994b). For [14] and
[15], we determine equilibrium emissions of signatories, Six ( , n)α , and non-signatories,
Nix ( , n)α , by solving simultaneously the first order conditions of signatories
[16] S S Ni i iB '(x ) 2 nd(nx (N n)x )= α + −
and of non-signatories
[17] N S Ni i iB '(x ) 2d(nx (N n)x )= + − .
That is, we proceed exactly as in section 3. The only difference is that there are no
dominant strategies any longer as this is evident from [16] and [17]. We then compute welfare
of signatories Si ( , n)π α and of non-signatories N
i ( , n 1)π α − as an input in stability function
S Ni i i( , n) ( , n) ( , n 1)Φ α = π α −π α − . For a given n, we can then compute optimal (n)α as
defined in [11], which determines emissions (welfare) of signatories Six ( (n), n)α
( Si ( (n), n)π α ), non-signatories N
ix ( (n), n)α ( Ni ( (n), n)π α ) and global emissions (welfare)
Tx ( (n), n)α ( T ( (n), n)π α ). It turns out - as explained in more detail in Appendix 7 - that
simulations simplify because some parameters have only a scaling effect. Thus, we can set
a=1 and d=1 in [14] and conduct a sensitivity analysis for the parameter range
( )21
1P : {(b,N) | b (k 1) N 1 ,1 k 20,2 N 50}2
= = + − ≤ ≤ ≤ ≤ . This implies that we test for every N
between 2 and 50 for 20 values of b. Similarly, we can set b=1 and d=1 in [15] and conduct a
14
sensitivity analysis in the parameter range 2P : {N | 2 N 50}= ≤ ≤ . The purpose of this analysis
is to confirm our main results of section 3 associated with Proposition 1 to 4. That is, we test
numerically the following conjecture.
Conjecture
Assume payoff functions [14] and [15].
a) There is an optimal modesty level α(n) that stabilizes a coalition with n members for every
n [ 2,N ]∈ , ∈n .
For every pair (α(n),n) and (α(m),m) , m>n,
b) global emissions are lower
c) signatories welfare is higher and
d) global welfare is higher
at m than at n.
For all parameter values in the range described above, it turns out that the conjecture is
confirmed. Thus, the positive effect of modesty carries over to a quadratic damage cost
function. This is not surprising since non-linear damage cost functions should reinforce the
positive effect of modest abatement targets. With non-dominant strategies, reduced abatement
efforts of signatories through modesty are matched by an increase of abatement efforts of
non-signatories.15 Thus, membership in a coalition becomes even more attractive, making it
easier to stabilize large coalitions via modesty.
5. Some Remarks on Equilibrium Analysis
In this section, we discuss conceptual issues related to the equilibrium analysis. One issue is
to relate the non-classical case with modesty to the classical case without modesty. An other
issue is to predict the equilibrium level of modesty and the equilibrium coalition.
For the first issue it helpful to recall that the cartel formation game assumes that
countries decide in the first stage on their participation and in the second stage on their
emission level. The game is solved by backwards induction assuming a Nash equilibrium in
each stage.
An obvious real world interpretation of this coalition formation process is as follows.
Suppose n countries negotiate on emission reductions. In the classical case, their agreement
leads to emission vector S Nx(n) (x (n), x (n))= . This agreement is stable if and only if *n n= .
15 This has a similar effect as the Stackelberg assumption mentioned in the Introduction.
15
Two remarks are in order. First, in the context of symmetric players, it seems not
controversial to interpret Sx (n) (that is technically derived from the assumption of joint
welfare maximization) as the outcome of a bargaining game because Sx (n) is not only
optimal for the entire coalition but also for each individual member, given N-n non-
signatories choose Nx (n) . Second, whenever not *n but n countries negotiate S S *x (n) x (n )≠ , leading to equilibrium emission vector *x(n) x(n )≠ , the agreement will not
come into force.
In the non-classical case, the coalition formation has to be interpreted slightly different.
Also n countries negotiate on emission reductions but under the restriction that the outcome
must be internally stable. This leads to emission vector x( (n), n)α . This agreement is stable
for any n∈ . Here, also two remarks are in order. First, again, this bargaining outcome
seems not controversial in the context of symmetric countries. Moreover, we may recall that
(n)α is the optimal modesty level for a given n, i.e., leads to the highest welfare of
signatories, provided that n is stable. Second, by the nature of the restriction, all n negotiators
will actually join the agreement.
These observations are interesting in two respects. First, the way we modeled modesty
may be seen to fulfill consistency in a wider sense. This seems not only important from a
theoretical point of view but also from an applied point of view. If n players negotiate on the
strategies of joint action, it seems natural to presume that once they find an agreement, then
these n players will actual form this agreement. In contrast, in the classical case, for any *n n≠ , the agreement on joint actions does not materialize. Second, however, modesty
implies multiple equilibria because for every n∈ , (n)α internally stabilizes n. This
problem also persists if external stability is considered additionally because typically (n)α
also externally stabilizes n.16 In contrast, without modesty, we usually have a unique
equilibrium (which is always true for the assumptions in section 2 and the functions
considered in section 4). This aspect of modesty raises the question which coalition will
emerge in equilibrium and brings us to the second issue we want to discuss in this section.
The answer to this question is not straightforward and we are aware that there may be
many approaches to tackle this question. Nevertheless, we suggest the following approach.
First, it is evident that because (n)α is optimal for coalition members (and also for non-
16 This is always true for the two examples considered in section 3 and the payoff functions
considered in section 4. It can be shown for the setting in section 3 (following a similar line of arguments as in Appendix 3) that a sufficient condition for external stability is 2 / nα ≥ .
16
members) for any n∈ , the problem basically reduces to the prediction of the equilibrium
coalition size *n . This is related to the first stage of the coalition formation process. Second,
without further information, it seems plausible to assume that at the start of the game no
coalition has formed. Now, if we sticked to the conventional model, nothing would be gained.
Countries announce to participate or not to participate, no player would have an incentive to
revise its strategy because any agreement with n signatories is internally stable and - as
pointed out above - also most likely externally stable. Therefore, we suggest to introduce a
pre-stage 0.
One option is to assume that countries bargain over the coalition size in stage 0 without
knowing whether they will be signatories or non-signatories. Let n be a proposal. Given pro-
posal n, let the probability that country i is a signatory with payoff Si ( (n), n)π α be S
ip (n) and
the probability of being a non-signatory with payoff Ni ( (n), n)π α be N S
i ip (n) 1 p (n)= − . Hence,
the expected payoff to country i from proposal n is E S Si i i(n) p (n) ( (n), n)π = π α
S Ni i(1 p (n)) ( (n), n)+ − π α . Let *n denote an equilibrium proposal. Then this proposal is an
equilibrium if and only if no country has an incentive to make an other proposal, i.e., E * Ei i(n ) (n)π ≥ π for all i I∈ and all n∈ , *n n≠ . Assuming that the probability of being a
member in a coalition is equal for all players (because of symmetry), Sip (n) n / N= , and hence
the equilibrium condition reads:
[18] * *
S * * N * * S Ni i i i
n N n n N n( (n ), n ) ( (n ), n ) ( (n), n) ( (n), n)N N N N
− −π α + π α ≥ π α + π α .
It is evident that by multiplying all terms with N, [18] means that the aggregate payoff
at *n must be larger than at any other n. Hence, *n N= from Proposition 4 in section 3 for
linear damage cost functions and as confirmed for quadratic damage cost functions in
section 4.
An other option is to assume as in Finus and Rundshagen (2005) that there is a regulator
or coordinator like an international organization that can make a proposal for an equilibrium.
Given that the regulator has no enforcement power, the proposal must be self-enforcing. More
precisely, let 1 Mc (c ,..., c )= be the set of equilibrium coalitions with payoff i j(c )π .17 More-
over, let the regulator propose a mixed strategy 1 Mc cw (w ,..., w )= that is a probability
17 This is a general formulation and implies for instance that a coalition with two members 1c {i, j}=
is a different coalition than coalition 2c {i,k}= . Below, we say coalition jc is of size jn .
17
distribution over c and let 1 Mc cz (z ,..., z )= with
j
j
cc c
z 1∈
=∑ be the probability distribution of
nature. Suppose the regulator´s aim is to maximize the sum of expected payoffs to all players.
Then, *w solves this maximization task:
[19] j
*c i j
i I j 1..Mw arg max w (c )
∈ =
= π∑ ∑ s.t.: j jc i j c i j
j 1..M j 1..Mw (c ) z (c )
= =
π ≥ π∑ ∑ i I∀ ∈ .
For brevity, we refrain from a general treatment as in Finus and Rundshagen (2005).
Therefore, we demonstrate only that there is a unique solution to [19] for a plausible
assumption on the probability distribution of nature.
Ignoring the constraint for a moment, we note that Si j j i j j
i I(c ) n ( (n ), n )
∈
π = π α∑
N Ej i j j i j(N n ) ( (n ), n ) N (n )+ − π α = π in the case of symmetry (see [18]) with jn the size of
coalition jc . Hence, jc i j
i I j 1..M
w (c )∈ =
π∑ ∑ j
Ec i j
j 1..M
N w (n )=
= π∑ . Since Ei j(n )π increases in coalition
size jn as we know from above, the solution to the regulators problem is to propose the grand
coalition with probability 1. Then, every country receives Ei (N)π . Now, we check whether the
constraint can be satisfied, i.e., j
Ei c i j
j 1..M
(N) z (c )=
π ≥ π∑ for all i I∈ . In the context of symmetric
players and without further information, it seems obvious to assume for the probability of
nature an equal distribution. That is, all M equilibria are equally likely, i.e., jcz 1/ M= .
Hence,
j
j
j E Ec i j i j i j i
j 1..M j 1..M n 1..N
Nn1z (c ) (c ) (n ) (N)
M M= = =
⎛ ⎞⎜ ⎟⎝ ⎠π = π = π ≤ π∑ ∑ ∑
and therefore the constraint is satisfied.
6. Summary and Conclusion
This paper started from the observation that the pessimistic prediction of non-cooperative
game theoretical models do not always match with real world observations of IEAs. It has
been argued that one important reason for this difference is the “classical” assumption of joint
welfare maximization of coalition members when choosing their emissions in the second
stage after countries have decided on their participation in the first stage. This implies the
18
implementation of ambitious emission reduction targets among coalition members and also a
high free-rider incentive. Therefore, we analyzed the “non-classical” assumption of modest
emission reduction targets. It has been shown that modesty may well pay: a higher participa-
tion compensates for lower abatement targets so that global emissions decrease in equilib-
rium. This has also a positive effect on signatories´ and non-signatories´ welfare. Also global
welfare increases through modesty since abatement burdens are shouldered by more coun-
tries. The “optimum” is achieved for a modesty level that stabilizes the grand coalition.
Though the analytical results assumed linear damage cost functions in section 2 and 3,
they were confirmed for quadratic damage cost functions by numerical simulations in section
4. Thus, overall, our model helps to explain why participation in some IEAs is higher than
predicted by previous theory. Moreover, it provides some rationale for modest emission
reductions as frequently observed in reality: in a second-best world with no enforcement
authority and large free-rider incentives, a second-best solution may achieve more than an
ambitious first-best solution.
In section 5, we discussed some equilibrium aspects associated with modesty. First, we
argued that the way we modeled modesty seems more plausible when interpreting the second
stage of coalition formation as a bargaining game between potential coalition members about
emission reduction. The bargaining outcome is not only optimal for the every coalition and
every negotiator (as in the classical case without modesty), but the agreement actually stipu-
lates the participation of all negotiators (as this does not hold always hold for the classical
case without modesty). Second, we pointed out that the non-classical assumption (with
modesty) implies multiple equilibria whereas the classical assumption usually leads to a
unique equilibrium. Therefore, we proposed two solutions that assume a pre-stage 0 where
countries either bargain over an equilibrium or a third party (without enforcement power)
proposes an equilibrium. For our model, we predicted in both cases the grand coalition as an
equilibrium outcome.
Our assumption of symmetric players eased the exposition and allowed for analytical
solutions but seems not crucial for our main conclusions. First, since we modeled the degree
of modesty by assuming that countries consider not the sum of marginal damages of coalition
members but only a fraction α of it ( 0 1≤ α < ), cost-efficiency of emission reduction within
a coalition of asymmetric players would still be ensured with a uniform α . Hence, there is no
reason why the positive welfare effect of a modest scheme should not carry over to asymmet-
ric countries. Second, interpreting the second stage as a bargaining game over emission
reduction would suggest that in a world of asymmetric players modesty should be even more
19
important: heterogeneous interests reinforce the need for modesty if countries can only agree
on the lowest common denominator. Third, our two solutions proposed for a pre-stage 0 may
also be applied in a modified form to heterogeneous players. Most likely, the outcome will not
be the grand coalition, though a larger and more successful coalition with modesty than in the
classical case.
References
Barrett, S. (1994a): The biodiversity supergame, Environmental and Resource Economics, 4, 111-122.
Barrett, S. (1994b): Self-enforcing international environmental agreements, Oxford Economic Papers, 46, 878-894.
Barrett, S. (1997a): Heterogeneous international agreements, in: Carraro, C. (Ed.): Interna-tional environmental negotiations: strategic policy issues, Edward Elgar: Cheltenham, 9-25.
Barrett, S. (1997b): The strategy of trade sanctions in international environmental agree-ments, Resource and Energy Economics, 19, 345-361.
Barrett, S. (1999): A theory of full international cooperation, Journal of Theoretical Politics, 11, 519-541.
Barrett, S. (2002): Consensus treaties, Journal of Institutional and Theoretical Economics, 158, 529-547.
Barrett, S. (2003): Environment and statecraft: the strategy of environmental treaty-making, Oxford University Press, New York.
Böhringer, C., Vogt, C. (2004): The dismantling of a breakthrough: the Kyoto protocol as symbolic policy, European Journal of Political Economy, 20, 597-618.
Botteon, M., Carraio, C. (1997): Burden-sharing and coalition stability in environmental negotiations with asymmetric countries, in: Carraro, C. (Ed.): International environmental negotiations: strategic policy issues, Edward Elgar: Cheltenham, 26-55.
Carraro, C., Marchiori, C. (2003): Stable coalitions, in: Carraro, C. (Ed.): The endogenous formation of economic coalitions, Edward Elgar: Cheltenham, 156-198.
Carraro, C., Marchiori, C., Oreffice, S., (2003): Endogenous minimum participation in inter-national environmental treaties, Working Paper 113, Fondazione Eni Enrico Mattei, Milano.
Carraro, C., Siniscalco, D. (1993): Strategies for the international protection of the envi-ronment. Journal of Public Economics, 52, 309-328.
Carraro C., Siniscalco, D. (1997): R&D cooperation and the stability of international environmental agreements, in: Carraro, C. (Ed.): International environmental negotiations: strategic policy issues, Edward Elgar: Cheltenham, 71-96.
20
Cesar, H., de Zeeuw, A. (1997) : Issue linkage in global environmental problems, in: Xepapadeas, A. (Ed.): Economic policy for the environment and natural resources: techniques for the management and control of pollution, Edward Elgar: Cheltenham, 158-173.
Chander, P., Tulkens, H. (1995): A Core-theoretic solution for the design of cooperative agreements on transfrontier pollution, International Tax and Public Finance, 2, 279-293.
Chander, P., Tulkens, H. (1997): The core of an economy with multilateral environmental externalities, International Journal of Game Theory, 26, 379-401.
Endres, A. (1997): Negotiating a climate convention – the role of prices and quantities, International Review of Law and Economics, 17, 147-156.
Endres, A., Finus, M. (2002) : Quotas may beat taxes in a global emission game, Journal of International Tax and Public Finance, 9, 687-707.
Eyckmans, J. (1997): Nash implementation of a proportional solution to international pollution control problems, Journal of Environmental Economics and Management, 33, 314-330.
Finus, M. (2001): Game theory and international environmental cooperation, Edward Elgar: Cheltenham, 2001.
Finus, M. (2003): Stability and design of international environmental agreements: the case of global and transboundary pollution, in: Folmer, H., Tietenberg, T. (Eds.): International yearbook of environmental and resource economics 2003/4, Edward Elgar: Cheltenham, 82-158.
Finus, M., Rundshagen, B. (1998a): Toward a positive theory of coalition formation and endogenous instrumental choice in global pollution control, Public Choice, 96, 145-186.
Finus, M., Rundshagen, B. (1998b): Renegotiation-proof equilibria in a global emission game when players are impatient, Journal of Environmental and Resource Economics, 12, 275-306.
Finus, M., Rundshagen, B. (2003): Endogenous coalition formation in global pollution control: a partition function approach, in: Carraro, C. (Ed.): The endogenous formation of economic coalitions, Edward Elgar: Cheltenham, 199-243.
Finus, M., Rundshagen, B. (2005): Participation in international environmental agreements: the role of timing and regulation, Working Paper 45.2005, Fondazione Eni Enrico Mattei, Milano, Forthcoming in: Natural Resource Modeling.
Finus, M., Tjøtta, S. (2003): The Oslo protocol on sulfur reduction: the great leap forward? Journal of Public Economics, 87, 2031-2048.
Folmer, H., van Mouche, P., Ragland, S. (1993): Interconnected games and international environmental problems. Environmental and Resource Economics, 3, 313-335.
21
Folmer, H., van Mouche, P. (1994): Interconnected games and international environmental problems II, Annals of Operations Research, 54, 97-117.
Hoel, M. (1992): International environment conventions: the case of uniform reductions of emissions, Environmental and Resource Economics, 2, 141-159.
Hoel, M., Schneider, K. (1997): Incentives to participate in an international environmental agreement, Environmental and Resource Economics, 9, 153-170.
Jeppesen, T., Andersen, P. (1998): Commitment and fairness in environmental games, in: Hanley, N., Folmer, H. (Eds.): Game theory and the environment. Edward Elgar: Cheltenham, 65-83.
Lange, A., Vogt C. (2003): Cooperation in international environmental negotiations due to preference for equity, Journal of Public Economics, 87, 2049-2067.
Murdoch, J.C., Sandler, T. (1997a): Voluntary cutbacks and pretreaty behavior: The Helsinki protocol and sulfur emissions, Public Finance Review, 25, 139-162.
Murdoch, J.C., Sandler, T. (1997b): The voluntary provision of a pure public good: the case of reduced CFC emissions and the Montreal protocol, Journal of Public Economics, 63, 331-349.
Petrakis, E., Xepapadeas, A. (1996): Environmental consciousness and moral hazard in international agreements to protect the environment, Journal of Public Economics, 60, 95-110.
22
Appendix 1
Differentiation of the first order condition of a signatory SiB '(x ( , n)) ndα = α with respect to n
on both sides gives S Si i
dB ''(x ( , n)) x ( , n) ddn
α α = α and with respect to α gives
S Si i
dB ''(x ( , n)) x ( , n) ndd
α α =α
. Rearranging terms gives
Si S
i
d dx ( , n) 0dn B''(x ( , n))
αα = <
α and S
i Si
d ndx ( , n) 0d B''(x ( , n))
α = <α α
since B'' 0.< QED
Appendix 2
We take the derivatives with respect to n and α :
S S S S S Ni i i i i i
S N Si i i
S N Si i i
d d d( , n) B'(x ( , n)) x ( , n) dx ( , n) nd x ( , n) dxdn dn dn
d( nd nd) x ( , n) d(x x ( , n))dn
d( 1)nd x ( , n) d(x x ( , n)) 0dn
π α = α α − α − α +
= α − α + − α
= α − α + − α ≥
since SiB '(x ( , n)) ndα = α by equation [10], 1α ≤ , S
id x ( , n) 0
dnα < and N S
i ix x ( , n)≥ α .
N S S Ni i i i
S N Si i i
d d( , n) dx ( , n) nd x ( , n) dxdn dn
dnd x ( , n) d(x x ( , n)) 0dn
π α = − α − α +
= − α + − α ≥
since Si
d x ( , n) 0dn
α < and N Si ix x ( , n)≥ α .
S S S Si i i i
Si
Si
d d d( , n) B'(x ( , n)) x ( , n) nd x ( , n)d d d
d( nd nd) x ( , n)d
d( 1)nd x ( , n) 0,d
π α = α α − αα α α
= α − αα
= α − α ≥α
since SiB '(x ( , n)) ndα = α by equation [10], 1α ≤ and S
id x ( , n) 0
dα <
α.
23
Finally,
N Si i
d d( , n) nd x ( , n) 0d d
π α = − α ≥α α
since Si
d x ( , n) 0d
α <α
. QED
Appendix 3
In Appendices 3 and 4, we will use the fact that Six (t) is convex in t. This is shown by taking
the second derivate with respect to t of the first order condition of a signatory SiB '(x (t)) td= .
Rearranging terms, we find S S
S i ii S
i
B '''(x (t))(x ) '(t)(x ) ''(t)B''(x (t))
= −
which is positive (and hence Six (t) convex) since B ''' 0≥ and B '' 0< by assumption (see
section 2), and Si(x ) ' 0.>
We begin now with the proof of Proposition 3. From Hoel/Schneider (1997) we know
that (2)α is well defined and that (2)α =1. Note that if inequality [12] holds, then this implies
1 2 n 2 1(n) (2) .2 3 n 1 n 1
−α ≥ α =
− −… We prove that ( , n) 0Φ α ≥ , 1
n 1α ≥
−, implies
n 1( , n 1) 0n−
Φ α + ≥ . This implies inequality [12], and hence the well-definedness of
:{2, , N} [0,1]α →… follows by induction.
Thus, we want to show that n 1( , n 1) 0.n−
Φ α + ≥ It suffices to show that
n 1( , n 1) ( , n)n−
Φ α + ≥ Φ α since ( , n) 0.Φ α ≥ Now,
24
S S Ni i i
N S Ni i i
S S Ni i iN S Ni i i
Si
n 1 ( , n 1) ( , n)n
n 1 n 1B(x ( , n 1)) d((n 1)x ( , n 1) (N n 1)x )n n
n 1 B(x ) d((n 1)x ( , n) (N n)x )n
B(x ( , n)) d(nx ( , n) (N n)x )
B(x ) d((n 1)x ( , n 1) (N n 1)x )n 1B(x ( , n 1)
n
−Φ α + −Φ α
− −= α + − + α + + − −
−− + − α + −
− α + α + −
+ − − α − + − +−
= α + Si
S Si i
S S Si i i
n 1) ndx ( , n 1)n
B(x ( , n)) ndx ( , n)
n 1 n 1 ndx ( , n) dx ( , n 1) (n 1)dx ( , n 1).n n
−− α +
⎡ ⎤− α − α⎣ ⎦− −
+ α − α + − − α −
Since S S S Si i i i
n 1 n 1x ( , n) x ( n) x ( (n 1)) x ( , n 1)n n− −
α = α = α − = α − , this simplifies:
S Si i
S Si i
S Si i
n 1 n 1B(x ( , n 1)) ndx ( , n 1)n n
B(x ( , n)) ndx ( , n)
n 1d x ( , n 1) x ( , n 1) .n
− −α + − α +
⎡ ⎤− α − α⎣ ⎦−⎡ ⎤+ α − − α +⎢ ⎥⎣ ⎦
Let S S S Si i i i
n 1 n 1: B(x ( , n 1)) ndx ( , n 1) B(x ( , n)) ndx ( , n)n n− − ⎡ ⎤Λ = α + − α + − α − α⎣ ⎦ and
S Si i
n 1: d x ( , n 1) x ( , n 1) .n−⎡ ⎤Ω = α − − α +⎢ ⎥⎣ ⎦
We show that
Si
Si
n 1[*] (x ) '( , n 1) ( 1)nd d ,n n n
n 1 n 1[**] (x ) '( , n 1) d.n n
− α α⎛ ⎞Λ ≥ − α + α − −⎜ ⎟⎝ ⎠
− −Ω ≤ − α + α
Let f (x) : B(x) ndx= − . Then, f is concave since B is concave. Furthermore, we know
that f is maximized for Six (n) . Finally, since
2n 1 n 1(n 1) n n nn n n− − α
α + = α = α − ≤ α ≤ , we
have S S Si i i
n 1x (n) x ( , n) x ( , n 1)n−
≤ α ≤ α + . Then
25
S Si i
S S Si i i
S S Si i i
S Si i
n 1f (x ( , n 1)) f (x ( , n))n
d n 1 n 1f (x ( , n 1)) x ( , n 1) x ( , n)dx n n
n 1 n 1B'(x ( , n 1)) nd x ( , n 1) x ( ,n)n n
n 1 n 1(n 1)d nd x ( , n 1) x ( , n)n n
−Λ = α + − α
− −⎡ ⎤≥ α + α + − α⎢ ⎥⎣ ⎦− −⎛ ⎞ ⎡ ⎤= α + − α + − α⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦
− −⎛ ⎞ ⎡ ⎤= α + − α + − α⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦
since Si
n 1 n 1B'(x ( , n 1)) (n 1)dn n− −
α + = α + by equation [10]. We continue by considering the
last term in last row above. Since Six is convex ,we obtain
S S S Si i i i
S Si i
Si
Si
n 1 n 1x ( , n 1) x ( , n) x ( n 1) x ( n)n n
x ( n ) x ( n)n
(x ) '( n ) n nn n
(x ) '( n ) .n n
− −α + − α = α + − α
α= α − − α
α α⎡ ⎤≤ α − α − −α⎢ ⎥⎣ ⎦α α
= − α −
This establishes inequality [*] since n 1 (n 1)d nd ( 1)nd d 0.n n− α
α + − = α − − <
Again, because Six is convex, we obtain
S S S Si i i i
Si
Si
n 1 n 1x ( , n 1) x ( , n 1) x ( (n 1)) x ( (n 1) )n n
n 1 n 1(x ) '( (n 1) ) (n 1) (n 1)n n
n 1(x ) '( n ) .n n
− +α − − α + = α − − α −
+ +⎡ ⎤≤ α − α − −α −⎢ ⎥⎣ ⎦α −
= − α − α
This establishes inequality [**].
Now, 0Λ+Ω ≥ is equivalent to n 1( , n 1) ( , n) 0n−
Φ α + −Φ α ≥ . Hence, we are done with
the proof if we can show that Ω ≥ −Λ . From inequalities [*] and [**], we have Ω ≥ −Λ if we
can show:
[***] S Si i
n 1 n 1 n 1(x ) '( , n 1) (x ) '( , n 1) ( 1)nd d .n n n n n− − − α α⎛ ⎞− α + α ≥ α + α − −⎜ ⎟
⎝ ⎠
26
Dividing by Si
n 1(x ) '( , n 1)dn n− α
− α + >0, yields :
2 2 2
2
2
αn -1 (1-α)nn
n - n n -αn +αα(n -1) n
n n 1α .n 1 n +1 n -1
+≥
⇔⇔ ≥
⇔ ≥ =−
≥
The last inequality is satisfied since 1n 1
α ≥−
. Hence, Ω ≥ −Λ . QED
Appendix 4
We refer to Appendix 3 for a proof that Six (t) is convex in t which we will use also here.
It suffices to show T Tx ( (n), n) x ( (n 1), n 1)α ≥ α + + for all n {2, , N 1}∈ −… . Now,
T T
S N S Ni i i i
N S S Si i i i
x ( (n), n) x ( (n 1),n 1)nx ( (n), n) (N n)x (n 1)x ( (n 1),n 1) (N n 1)x
x x ( (n 1),n 1) n x ( (n 1),n 1) x ( (n), n) .
α ≥ α + +
⇔ α + − ≥ + α + + + − −
⎡ ⎤⇔ − α + + ≥ α + + − α⎣ ⎦
Letting S Si i: x (1) x ( (n 1)(n 1))Λ = − α + + and S S
i i: n x ( (n 1)(n 1)) x ( (n)n)⎡ ⎤Ω = α + + − α⎣ ⎦ , we
have to show that .Λ ≥Ω Since Six is convex and 1 (n 1)(n 1)≤ α + + , we have:
[*] Si(x ) '( (n 1)(n 1))[1 (n 1)(n 1)] 0.Λ ≥ α + + −α + + ≥
If (n)n (n 1)(n 1)α ≤ α + + , then 0Ω ≤ and we are done. If (n)n (n 1)(n 1)α > α + + , then,
as Six is convex,
[**] [ ]Sin(x ) '( (n 1)(n 1)) (n 1)(n 1) (n)n .Ω ≤ α + + α + + −α
Because of inequalities [*] and [**], it suffices to show holds
[ ]S Si i(x ) '( (n 1)(n 1))[1 (n 1)(n 1)] n(x ) '( (n 1)(n 1)) (n 1)(n 1) (n)nα + + −α + + ≥ α + + α + + −α
in order to establish that .Λ ≥Ω Dividing by Si(x ) '( (n 1)(n 1)) 0− α + + > , we have:
[***] 2
2 2
(n 1)(n 1) 1 (n)n (n 1)(n 1)n(n 1)(n 1) (n)n 1.
α + + − ≥ α −α + +
⇔ α + + −α ≥
Because of inequality [12]
27
2 2 2 2
2
2
n 1(n 1)(n 1) (n)n (n) (n 1) (n)nn
(n)(n)n (n 1) (n)nn
(n)(n)n (n)n
n n 1(n) .n
−α + + −α ≥ α + −α
α⎛ ⎞= α − + −α⎜ ⎟⎝ ⎠
α= α −α −
⎡ ⎤− −= α ⎢ ⎥
⎣ ⎦
Hence if 2n n 1(n) 1
n⎡ ⎤− −
α ≥⎢ ⎥⎣ ⎦
, i.e., 2
n(n)n n 1
α ≥− −
, we are done. From inequality [12]
and condition [13], we have n-1
k=3
k -1 2 6α(n) α(3) = α(3) .k n -1 5(n -1)
≥ ≥∏ It can be easily
checked that 2
6 n5(n 1) n n 1
≥− − −
if n 3≥ .
In case n=2, we also show that inequality [***] holds. Because of condition [13], we
have 3(3)5
α ≥ . Then,
2 2 3(n 1)(n 1) (n)n (3)9 (2)4 9 4 1.5
α + + −α = α −α ≥ − >
Thus, for all possible values of n, Λ ≥ Ω holds. QED
Appendix 5
It suffices to show S Si i( (n 1), n 1) ( (n), n)π α + + ≥ π α for all n {2, , N 1}∈ −… .
Case 1: (n 1)(n 1) (n)nα + + ≤ α
Then, S Si ix ( (n 1), n 1) x ( (n), n)α + + ≥ α and hence S S
i iB(x ( (n 1), n 1)) B(x ( (n), n)).α + + ≥ α Since
global emissions fall by Proposition 4, we can conclude: S Si i
Si
Si
( (n 1), n 1) B(x ( (n 1), n 1)) dQ( (n 1), n 1)
B(x ( (n), n) dQ( (n), n)
( (n), n).
π α + + = α + + − α + +
≥ α − α
= π α
Case 2: (n 1)(n 1) (n)nα + + > α
Let f (x) : B(x) (n 1)dx= − + . Then, f is concave and is maximized for Six (n 1).+ Since
S S Si i ix (n 1) x ( (n 1), n 1) x ( (n), n)+ ≤ α + + < α , we obtain
28
S Si if (x ( (n 1), n 1)) f (x ( (n), n)).α + + > α
Hence,
S Si i
S S Ni i i
S S Ni i i
S Si i
S Si i
( (n 1), n 1) ( (n), n)
B(x ( (n 1), n 1)) d (n 1)x ( (n 1), n 1) (N n 1)x
B(x ( (n), n)) d nx ( (n), n) (N n)x
B(x ( (n 1), n 1)) (n 1)dx ( (n 1), n 1)
B(x ( (n), n)) (n 1)dx ( (n),
π α + + −π α
⎡ ⎤= α + + − + α + + + − −⎣ ⎦⎡ ⎤− α + α + −⎣ ⎦
= α + + − + α + +
− α − + αN Si i
S Si i
n)
d(x x ( (n), n))
f (x ( (n 1), n 1)) f (x ( (n), n))0.
⎡ ⎤⎣ ⎦+ − α
≥ α + + − α>
Thus, in both cases, we have shown that S Si i( (n 1), n 1) ( (n), n)π α + + ≥ π α . QED
Appendix 6
It suffices to show that global welfare increases from ( (n), n)α to ( (n 1), n 1).α + + Suppose
first that we hold global emission constant when moving from ( (n), n)α to ( (n 1), n 1).α + +
Consequently, signatories have to choose an emission level Six such that
S N S Ni i i i(n 1)x (N n 1)x nx ( , n) (N n)x ,+ + − − = α + − i.e.,
SN S N Sii i i i
n 1x x x ( , n) x x ( , n).n 1 n 1
> = α + > α+ +
We show that if signatories emit Six in a coalition of size n+1, then global welfare W is
greater than global welfare W( (n), n)α at the agreement ( (n), n)α . Now,
S Ni i
S Ni i
W (n 1)B(x ) (N n 1)B(x ) NdQ( (n), n)
W( (n),n) = nB(x ( (n),n))+(N-n)B(x ) NdQ( (n), n)
= + + − − − α
α α − α
as global emissions do not change. However,
S S Ni i i
S S Ni i i
W W( (n), n) (n 1)B(x ) nB(x ( (n), n)) B(x )n 1(n 1) B(x ) B(x ( (n), n)) B(x )
n 1 n 10
− α = + − α −
⎡ ⎤= + − α −⎢ ⎥+ +⎣ ⎦≥
29
since B is concave and S S Ni i i
n 1x x ( ,n) xn 1 n 1
= α ++ +
. Hence, W W( (n), n)≥ α . Thus, if we
show that W( (n 1), n 1) Wα + + ≥ , we are done with the proof.
Let f (x) : B(x) (n 1)dx= − + . Then,
[*]
( )( )
( )
S S Ni i i
N S Ni i i
S S Ni i i
SNii
W( (n 1), n 1) W
(n 1) B(x ( (n 1), n 1)) d (n 1)x ( (n 1), n 1) (N n 1)x
(N n 1) B(x ) d (n 1)x ( (n 1), n 1) (N n 1)x
(n 1) B(x ) d (n 1)x (N n 1)x
(N n 1) B(x ) d (n 1)x (N n
α + + −
⎡ ⎤= + α + + − + α + + + − −⎣ ⎦
⎡ ⎤+ − − − + α + + + − −⎣ ⎦
⎡ ⎤− + − + + − −⎢ ⎥⎣ ⎦
− − − − + + − −( )( )
( )
( )
Ni
SSii
SSii
1)x
(n 1) f (x ( (n 1), n 1) f (x )
(N n 1) Q( (n), n) Q( (n 1), n 1)
(n 1) f (x ( (n 1), n 1) f (x ) ,
⎡ ⎤⎢ ⎥⎣ ⎦
= + α + + −
+ − − α − α + +
≥ + α + + −
as global emissions decrease. For the same reason and because Six was chosen such that
global emissions remain constant, we must have S Si ix ( (n 1), n 1) xα + + ≤ . Because f is concave
and maximized for S Si ix (n 1) x ( (n 1), n 1)+ ≤ α + + , this implies
SSiif (x ( (n 1), n 1) f (x ) 0.α + + − ≥
Hence, by inequality [*], W( (n 1), n 1) Wα + + ≥ . QED
Appendix 7
From the first order conditions [16] and [17] of signatories and non-signatories, respectively,
we derive for [14]:
Ni 2
2nd( n 1)a abxb 2d( n N n)
α − +=
+ α + − and S N
i ix nx ( n 1)a= α − α − .
It is evident that Nix 0≥ since n 1α > (because 1/(n 1)α ≥ − ). Furthermore, it can be checked
that N maxi ix a x≤ ≤ (since we need max
ia x≤ for ´iB 0≥ ) and S N N
i i ix nx ( n 1)a x .= α − α − ≤ How-
ever, it can also be checked that Six 0≥ if and only if 1(N n)d(1 n) b
2− −α ≥ − . Hence, we
assume the last condition to hold to ensure an interior solution.
For [15], we find:
30
Ni
bx 12d( N n)=
+ −α
and N
S ii
xxn
=α
.
Hence, Nix 0> , S
ix 0> , and S Ni ix x≤ since n 1α > .
Inserting these solutions into the stability function S Ni i i( , n) ( , n) ( , n 1)Φ α = π α − π α − , global
emissions T S Ni ix ( , n) nx ( , n) (N n)x ( , n)α = α − − α , welfare of signatories S
i ( , n)π α and global
welfare T S Ni i( , n) n ( , n) (N n) ( , n)π α = π α − − π α , it turns out for [14] that parameter a 0> and for
[15] that parameter b 0> is only a multiplicative factor in all these functions. Thus, these parameters
do not affect the solution of i ( , n) 0Φ α = , the minimum of Tx ( , n)α as well as the maximum of
Si ( , n)π α and T ( , n)π α for a given n or α . Consequently, we normalize a to 1 in [14] and b to 1 in
[15].
Moreover, we can normalize d to 1 for [14] since solutions depend only on the ratio of b/d (i.e.,
it suffices to vary b for d=1 in the simulations).
For [16], we also normalize d to 1. In this case, when inserting optimal emissions, it turns out
that
1. d drops in the stability function
2. d is only a multiplicative factor in global emissions
3. d is just constant term that is subtracted in the welfare of signatories and
4. d is just a constant term that is subtracted in global welfare.
Therefore, d does not affect the solution of i ( , n) 0Φ α = , the minimum of Tx ( , n)α as well as
the maximum of Si ( , n)π α and T ( , n)π α for a given n or α .
Hence, we test the conjecture in section 4 for combinations of values of b and N for
payoff function [14] and for different values of N for payoff [15]. This gives immediately
parameter range 2P : {N | 2 N 50}= ≤ ≤ for [15]. For [14], we notice that an interior solution
requires 1(N n)d(1 n) b2
− −α ≥ − as stated above. Let f ( , n) : (N n)d(1 n)α = − −α . Since we
have a priori no information on α , we have to make sure that 1f ( , n) b2
α ≥ − holds for all
combinations of α and n. Hence, we set 1α = and recall that d=1 from above. Thus, we
31
minimize f (n) : (N n)(1 n)= − − which is true for N 1n2+
= . Inserting this into f (n) , a
sufficient condition for an interior solution is 21b (N 1)2
≥ − which is satisfied if
( )21
1P : {(b,N) | b (k 1) N 1 ,1 k 20,2 N 50}2
= = + − ≤ ≤ ≤ ≤ .
Figure 1: Stability Function i (n)Φ for Two Examples
a) Welfare Function [7] b) Welfare Function [8]
Figure 2: Stability Function i ( ,n)Φ α for Different Levels of α for Two Examples
a) Welfare Function [7] b) Welfare Function [8]