Focus is here on coalitional games among economic agents plagued by aggregate pollutions of diverse sorts. Defecting players presumably pollute more than others. Then, granted convex preferences and technologies, the core is proven nonempty. In fact, under natural assumptions, a specific, computable core solution comes in terms of shadow prices on the said aggregates. Such prices may, in large part, implement the cooperative treaty by clearing a competitive market for emissions.Cooperative Games; Pollution Control Adoption Costs; Distribution Effects; Employment Effects

Flåm, Sjur Didrik

English

Research Papers in Economics

WORKING PAPERS IN ECONOMICS
                                                                                         
No. 17/02
SJUR DIDRIK FLÅM
BALANCED ENVIRONMENTAL
GAMES
  Department of Economics    ________________________
 U N I V E R S I T Y  OF  B E R G E NBalanced Environmental Games
Sjur Didrik Flåm∗
Sept. 23, 2002
Abstract. Focus is here on coalitional games among economic agents
plagued by aggregate pollutions of diverse sorts. Defecting players presumably
pollute more than others. Then, granted convex preferences and technologies,
the core is proven nonempty. In fact, under natural assumptions, a speciﬁc,
computable core solution comes in terms of shadow prices on the said aggre-
gates. Such prices may, in large part, implement the cooperative treaty by
clearing a competitive market for emissions.
1. Introduction
Environmental degradation - and the prospect of climate change - has motivated
many game theoretic studies, often focused on cooperation and core solutions. This
note pursues that line of research. It adds to the results of Helm (2001), Chander and
Tulkens (1997) by allowing technological externalities, more general utility functions,
and several pollutants. More important, it treats the aggregate discharge from defect-
ing players axiomatically, presuming merely that these agents will, on the average,
pollute most.
Pollutants are ”uniformly dispersed” in global commons. So, given the total emis-
sion (say, of greenhouse gases), it does not matter for external eﬀects who contributed
how much.1 Then, granted convexity in preferences and production, the core proves
nonempty. Thus, in principle, the prospects for eﬃcient and stable cooperation may
be rather good. That point is reinforced here in two ways: First, under natural as-
sumptions, a speciﬁc core solution can be computed. Second, that solution, being
determined by shadow prices on aggregate emissions, seems implementable by trades
in competitive emission markets.
2. The Game
Accommodated here is a ﬁnite, ﬁxed set I,e a c hm e m b e ri b e i n gac o n s u m e r ,p r o d u c e r
and polluter - all at the same time. Correspondingly, the consumption and emission
bundles of these agents will come into focus next. Such bundles belong to ordered,
real vector spaces (X,≤) and (E,≤), respectively.2 Since goods and ”bads” are
∗Economics Department, Bergen University 5007 Norway; sjur.ﬂaam@econ.uib.no. Thanks for
ﬁnancial support is due Ruhrgas and Røwdes fond.
1Clearly, if dispersion is tilted towards some receiving regions, then polluters are not on equal
footing. Acid rain is an important case. See [7], [11].
2A vector space V is ordered by a binary relation ≤ if there exists a convex cone K ⊂ V such
that v ≤ v0 ⇔ v0 −v ∈ K. For example, the nonnegative orthant deﬁnes the customary order in the
ambient Euclidean space.
1Balanced Environmental Games 2
manifold, and diﬀerentiated by their availability in location or time, the spaces X
and E could have large dimensions.3
Agent i ∈ I contemplates consuming a commodity vector xi ∈ X and emitting a
vector ei ∈ E of pollutants. Thereby he obtains real-valued payoﬀ (or transferable
utility) πi(xi,e I) where eI :=
P
i∈I ei denotes the total emission. In autarchy i would
face the technological constraint xi ≤ fi(ei,e I).
As customary, we take fi : E×E → X and πi : X×E → R∪{−∞} to be concave
functions, both increasing in the ﬁrst argument and decreasing in the second. Thus,
each player is directly and adversely aﬀected by the total emission eI.4 The extreme
payoﬀ πi(xi,e I)=−∞ serves here as a ﬁctitious, but convenient ”death penalty.”
By indicating violation of underlying constraints this simple device saves us repeated
and explicit mention of evident restrictions (such as nonnegativity or capacity limits).
Our concern is with the prospects of cooperation. Can the grand coalition form?
Can it secure eﬃciency and split the potential gains to satisfy every party? More
precisely, for a suitable characteristic function, i st h ec o r ee m p t y ?
While addressing that issue, Chander and Tulkens (1997) deﬁned the γ-worth of
coalition S ⊆ I by considering a noncooperative game against S.S p e c i ﬁcally, using
shorthand expressions eS :=
P
i∈S ei and e−S :=
P
i∈IÂS ei, then, in that game,
• S acts as one player with objective
P
i∈S πi(xi,e S+e−S) and constraints
P
i∈S xi ≤ P
i∈S fi(ei,e S + e−S),
P
i∈S ei ≤ eS;
• each outsider i ∈ IÂS plays with similar objective πi(xi,e i + e−i) and constraint
xi ≤ fi(ei,e i + e−i).
Deﬁnition (The γ-characteristic function). The worth vγ(S) of coalition S is the
Nash equilibrium payoﬀ it obtains in the described game against S. 2
Clearly, this deﬁnition is somewhat demanding. To see how, let ˜ e =( ˜ e)i∈I denote a
complete list of emissions and deﬁne, for any coalition S, its best reply (correspon-
dence) BS(˜ e) to comprise each emission proﬁle (ei)i∈S that together with a suitable
consumption pattern (xi)i∈S would solve the problem:
maximize
X
i∈S
πi(xi,e S +˜ e−S) s.t.
X
i∈S
xi ≤
X
i∈S
fi(ei,e S +˜ e−S),
X
i∈S
ei ≤ eS (1)
Then, e is a Nash equilibrium (in the game against S) iﬀ ei ∈ B{i}(e) for all i ∈ IÂS
and (ei)i∈S ∈ BS(e).
One may reasonably ask whether vγ(S) is well deﬁn e di nt h i sm a n n e r . T h a ti s :
does equilibrium exists in all these games? Is it always unique? Does the deﬁnition
invariably provide a unique value vγ(S)?5
3For simplicity one may think of one consumption good and one pollutant. Then X and E are
one-dimensional with the usual order.
4Put diﬀerently: everybody falls victim to spillovers produced by the others. Such spillovers are
discharged in global commons.
5An equilibrium exists if each pair (xi,e i) must belong to a nonempty compact convex set Ki ⊂Balanced Environmental Games 3
We shall circumvent these diﬃculties via a more axiomatic approach. For the
statement recall that, given a collection C of coalitions, a corresponding real-valued
mapping S ∈ C 7→ δS ≥ 0 is declared a balanced collection of weights iﬀ for
each i ∈ I we have
P
{δS : i ∈ S ∈ C} =1 . For simplicity we write henceforth
Ci := {S ∈ C : i ∈ S}. Thus,
P
S∈Ci δS =1for all i. Using this notion we now make
ak e y
Hypothesis (about anticipation, best reply, and free riding)
(i) (Foreseeable emissions and best reply) Any coalition S,i fi tw e r et of o r m ,w o u l d
face a foreseeable total emission ˜ e−S from the outsiders. In response, S would be
worth the optimal value v(S) of problem (1).
(ii) (External agents are excessive free-riders) For any balanced collection of weights
and any agent i it holds that
X
S∈CÂCi
δSeS ≤
X
S∈Ci
δS˜ e−S. 2 (2)
We shall assume, of course, that the optimal value v(S) in (1) be ﬁnite. That value
need, however, not be attained. Clearly, our deﬁnition of worth ignores some problems
by tacitly presuming informational symmetry. Also, it simpliﬁes many conﬂictual
issues by coaching cooperation merely in monetary terms [12].
Inequality (2) is crucial but hard to interpret: Suppose each agent i, whenever
he belongs to a coalition S, takes part to the degree δS. Thereby he falls victim to
proportional emission
P
S∈Ci δS˜ e−S from defecting outsiders. (2) says the latter item
exceeds the aggregate discharge produced by the coalitions to which i does not belong.
Broadly speaking, if coalition memberships are balanced, each agent will experience
that defectors pollute more than do other contracting parties. To justify (2) we con-
sider next an important case:
Proposition 1. (Excessive free-riding) Suppose there exists a particular emission
pattern ¯ e =( ¯ ei)i∈I such that
eS ≤ ¯ eS and ¯ e−S ≤ ˜ e−S for every coalition S. (3)
More generally, suppose that for every balanced collection of weights δS,S ∈ C, and
agent i there exists a particular emission pattern ¯ e =( ¯ ei)i∈I such that
X
S∈CÂCi
δSeS ≤
X
S∈CÂCi
δS¯ eS and
X
S∈Ci
δS¯ e−S ≤
X
S∈Ci
δS˜ e−S. (4)
Then (2) holds.
X × E, and all functions are continuous, properly deﬁned on the ensuing sets. Moreover, given
Euclidean spaces, equilibrium will be unique if the marginal payoﬀs mi := ∂πi/∂(xi,e i) constitute
a strictly monotone map m =( mi) on the product set of all Ki.Balanced Environmental Games 4
Proof. Fix any balanced collection of weights δS,S∈ C, and consider some agent i.
Let ¯ e be an emission pattern that satisﬁes (4). Then
X
S∈CÂCi
δSeS ≤
X
S∈CÂCi
δS¯ eS =
X
S∈C
δS
X
j∈S
¯ ej −
X
S∈Ci
δS¯ eS =
X
j
X
S∈Cj
δS¯ ej −
X
S∈Ci
δS¯ eS
=¯ eI −
X
S∈Ci
δS¯ eS =
X
S∈Ci
δS¯ eI −
X
S∈Ci
δS¯ eS =
X
S∈Ci
δS¯ e−S ≤
X
S∈Ci
δS˜ e−S. 2
One may think of the emission pattern ¯ e =( ¯ ei)i∈I in (3) as one that would emerge
under total lack of cooperation. For example, it could stem from a Nash equilibrium
in the noncooperative game having atomistic player set I. Free-riding would then
explain eS ≤ ¯ eS. In addition, the tragedy of the commons would entail ¯ e−S ≤ ˜ e−S.
Chander and Tulkens (1997) bring out such results formally. They used a version
of ratio equilibrium (Kaneko 1977) to ﬁnd a core solution. We rather follow Helm
(2001)6 in proving that the game is balanced:
Theorem 1. (A balanced game) The coalitional game has a nonempty core.
Proof. Pick any balanced collection of weights δS,S∈ C. By the Bondareva-Shapley
theorem [10] it suﬃces to verify that v(I) ≥
P
S∈C δSv(S). Fix any number ε > 0 and
let εS := ε/
P
S∈C δS. For any coalition S ∈ C let (xS
i ,e S
i )i∈S solve problem (1) up to
εS-optimality, and denote by ˜ e−S the aggregate emission produced by the outsiders.
Deﬁne for each agent i ap a r t i c u l a rc h o i c e(xi,e i): =
P
S∈Ci δS(xS
i ,e S
i ). The plan so
constructed is feasible. Indeed, since production is everywhere decreasing in total
emissions, (4) entails
X
i∈I
xi =
X
i∈I
X
S∈Ci
δSx
S
i =
X
S∈C
δS
X
i∈S
x
S
i ≤
X
S∈C
δS
X
i∈S
fi(e
S
i ,e
S
S +˜ e−S)
=
X
i∈I
X
S∈Ci
δSfi(e
S
i ,e
S
S +˜ e−S) ≤
X
i∈I
fi(ei,
X
S∈Ci
δSe
S
S +
X
S∈Ci
δS˜ e−S)
≤
X
i∈I
fi(ei,
X
S∈Ci
δSe
S
S +
X
S∈CÂCi
δSe
S
S)=
X
i∈I
fi(ei,e I).
Also note that (4) implies
eI =
X
i∈I
ei =
X
i∈I
X
S∈Ci
δSe
S
i =
X
S∈C
δSe
S
S =
X
S∈Ci
δSe
S
S +
X
S∈CÂCi
δSe
S
S
≤
X
S∈Ci
δSe
S
S +
X
S∈Ci
δS˜ e−S =
X
S∈Ci
δS(e
S
S +˜ e−S).
6Helm considers single commodities xi and ei. He uses quasi-linear utility: πi(xi,e I)=xi−di(eI)
and production technology free from externalities: fi(ei,e I)=fi(ei).Balanced Environmental Games 5
Consequently, πi(xi,e I) ≥ πi(xi,
P
S∈Ci δS(eS
S +˜ e−S)) for each i.S o ,v(I) ≥
X
i∈I
πi(xi,e I) ≥
X
i∈I
πi(
X
S∈Ci
δSx
S
i ,
X
S∈Ci
δS(e
S
S +˜ e−S)) ≥
X
i∈I
X
S∈Ci
δSπi(x
S
i ,e
S
S +˜ e−S)
=
X
S∈C
δS
X
i∈S
πi(x
S
i ,e
S
S +˜ e−S) ≥
X
S∈C
δS [v(S) − εS]=
X
S∈C
δSv(S) − ε.
Since ε > 0 was arbitrary the desired conclusion follows. 2
3. A Reduced Game with Emission Rights and Trade
Suppose here that an overall emission proﬁle (ˇ ei)i∈I has been agreed upon. More
precisely, suppose total emission has been ﬁxed, by contract, at ˇ eI together with a
consistent allocation of property rights ˇ ei,i∈ I,
P
i∈I ˇ ei = ˇ eI. For example, ˇ eI could
be the total emission that solves problem (1) for S = I, and (ˇ ei)i∈I might result from
some principle of equity (or grandfathering).
In this simpliﬁed setting, where everybody holds emission rights and regards ag-
gregate emissions as given, coalition S could achieve worth
ˇ v(S): =s u p
(
X
i∈S
πi(xi, ˇ eI):
X
i∈S
xi ≤
X
i∈S
fi(ei,ˇ eI),
X
i∈S
ei ≤ ˇ eS
)
.
Then, how can a core solution be found by a decentralized procedure? To that end
suppose consumption and emission bundles could be purchased at ﬁxed nonnegative
prices x∗ and e∗. By prices we understand real-valued, linear mappings X Ä x 7→
hx∗,xi,EÄ e 7→ he∗,ei such that 0 ≤ x ⇒ 0 ≤ hx∗,xi, and 0 ≤ e ⇒ 0 ≤ he∗,ei.
While regarding total emission ˇ eI and own endowment ˇ ei as given, let
ui := sup
xi,ei
{πi(xi, ˇ eI)+hx
∗,f i(ei,ˇ eI) − xii − he
∗,e ii} + he
∗, ˇ eii
denote the highest proﬁta g e n ti could aim at under price-taking behavior. Note that
the supremal term in ui does not depend on the distribution of property rights. With
ui as just deﬁned, we declare x∗ ≥ 0,e ∗ ≥ 0 ap a i ro fshadow prices if
ˇ v(I) ≥
X
i∈I
ui.
Theorem 2. (Core solutions deﬁned by shadow prices) Suppose x∗,e ∗ are shadow
prices. Then, for established property rights ˇ ei,i ∈ I, ˇ eI =
P
i∈I ˇ ei, the payment
scheme ui,i ∈ I, belongs to the core of the game that has characteristic function
S 7→ ˇ v(S).
Proof. Since
ˇ v(S) ≤ sup
X
i∈S
{πi(xi, ˇ eI)+hx
∗,f i(ei,ˇ eI) − xii − he
∗,e ii + he
∗,ˇ eii} =
X
i∈S
ui,Balanced Environmental Games 6
no coalition S ⊆ I should reasonably block the proposed scheme of payments. In
particular, ˇ v(I) ≤
P
i∈I ui. So, invoking the preceding assumption, we get ˇ v(I)= P
i∈I ui whence Pareto eﬃciency also obtains. This proves that the core is nonempty.
2
Given our convexity assumptions, existence of shadow prices (i.e.of so-called La-
grange multipliers) is ensured under standard qualiﬁcations. Also, when the agents
are many and minor, these assumptions become relatively less important; see [2].
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A core of an economy with multilateral environmental externalities,

A Finnish-Soviet acid rain game: noncooperative equilibria, cost eﬃciency and sulphur agreements,

Equilibrium binding agreements,

External economies and cores.

On balanced sets and cores,

On the existence of a cooperative solution for a coalitional game with externalities,

Sharing nonconvex cost,

The acid rain game,

The concept of regime eﬀectiveness,

The core of an economy with a common resource pool: A partition function form approach,

The ratio equilibria and the core of the voting game G(N,V) in a public goods economy,

Balanced Environmental Games

Balanced Environmental Games

Abstract

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