In this paper we investigate partial cooperation between a portion of the players and the rest of the players who do not cooperate and play a Nash game having multiple equilibria. Some properties of the partial cooperative equilibrium are studied and applied to a public goods situation.noncooperative games, cooperation, public goods games

Lina Mallozzi

Stef Tijs

English

Research Papers in Economics

AUCO Czech Economic Review 2 (2008) 21–27
Acta Universitatis Carolinae Oeconomica
Received 19 November 2007; Accepted 10 January 2008
Partial Cooperation and Non-Signatories Multiple
Decision
Lina Mallozzi∗, Stef Tijs†
Abstract In this paper we investigate partial cooperation between a portion of the players and
the rest of the players who do not cooperate and play a Nash game having multiple equilibria.
Some properties of the partial cooperative equilibrium are studied and applied to a public goods
situation.
Keywords noncooperative games, cooperation, public goods games
JEL classiﬁcation C71, C72, H41 ∗†
1. Introduction
In the last decades the problem of international pollution control has been approached
in a Game Theory setting, cooperative as well as non-cooperative. Particularly, se-
veral papers have been devoted to the coalition formation process and the stability of
the formed coalition (Ray and Vohra 1997, Yi 1997, Finus 2001 and the references
therein).
In the context of International Environmental Agreements (IEA), in a competition
between several countries, usually only a portion of the participants signs an agree-
ment. The IEA framework, together with other situations, leads to study concepts of
partial cooperation, a mixture of cooperative behavior and non-cooperative one. It
is supposed that the non-coalition members choose their strategy according to Nash
behavior and the coalition maximize the aggregate welfare of its members.
The deﬁnition of partial cooperative equilibrium has been given by Mallozzi and
Tijs (2006) for symmetric potential games together with existence results. In line with
the deﬁnition used by Barrett (1994), it has been supposed that the cooperating play-
ers choose the same strategy and the non-cooperating ones react by playing a Nash
equilibrium problem admitting a unique solution. The case where the non-cooperating
reaction set is not a singleton has been studied in Mallozzi and Tijs (2007) for sym-
metric aggregative games.
In this paper we present the notion of partial cooperative equilibrium in the non
symmetric case and an existence result. Then an application to a public goods game is
∗ University of Naples Federico II, Department of Mathematics and Applications, Via Claudio 21, 80125
Naples, Italy. Phone +390817682476, E-mail: mallozzi@unina.it.
† University of Tilburg, Center and Department of Econometrics and Operations Research, P.O.Box 90153,
5000 LE Tilburg, Netherlands. Phone +31134662348, E-mail: S.H.Tijs@uvt.nl.
AUCO Czech Economic Review, vol. 2, no. 1 21L. Mallozzi, S. Tijs
discussed. The paper is organized as followed: in Section 2 a more general deﬁnition
of partial cooperative equilibrium is given and some results are proved; in Section 3
the results are applied to a public goods game; Section 4 concludes indicating some
possible generalizations of the obtained results.
2. l-partial cooperation
LetG=hn;X; f1,..., fnibeann-personnormalformgamewithplayersetI ={1,2,...,n},
with the same strategy space X and payoff function fi: Xn 7→ R for player i ∈ I. If
player i chooses xi ∈ X, then he obtains a proﬁt fi(x1,...,xn). Each player wants to
maximize his own proﬁt. We denote by x−i the vector (x1,...,xi−1,xi+1,...,xn) ∈ Xn−1.
A noncooperative behavior between the n players is described by the well known
concept of Nash equilibrium. A Nash equilibrium is a vector h = (h1,...,hn) ∈ Xn
such that for any i ∈ I
fi(h1,...,hn) = max
y∈X
fi(h1,...,hi−1,y,hi+1,...,hn).
Let us denote by NE the set of the Nash equilibrium proﬁles of the game G. For any
j = 1,...,n we denote by yj the j-dimensional vector (y1,...,yn).
We suppose now that a group of the n players participate in an agreement, say
Pk+1,...,Pn (cooperating players or signatories), the remaining players P1,...,Pk (non-
cooperating players or non-signatories) acting in a noncooperative way for each k =
0,...,n. In this case k is called the level of non-cooperation. The game is a two-stage
game: signatories behave as a Stackelberg leader1 and announce their joint strategy.
Non-signatories are the followers and react by playing a non-cooperative game: they
choose a Nash equilibrium in the k-person subgame. The solution is given then by
using backward induction. For k = 0 all the players are signatories and maximize their
joint payoff åj fj(yn); for k = n all the players are non-signatories and we have a Nash
game for all n players.
More precisely, given the level of non-cooperation k, the signatories choose the
same strategy xk+1 = xk+2 = ... = xn = y ∈ X and the ﬁrst k players with payoffs
fi(x1,...,xk,yn−k) for any i = 1,...,k do not participate in the agreement and choose
a Nash equilibrium against the joint strategy y ∈ X. This uniform strategy choice of
signatories may appear restrictive, but it is common in IEA models where it means the
available level (for example, the percentage) of a certain gas emission. A possibility to
avoid this assumption is discussed in Section 4.
Denote by Gk(y) = hk;X; f1,..., fki the k-person game with strategy space X and
payoff function fi(x1,...,xk,yn−k) for player i, and by NEk(y) the set of the Nash
equilibrium proﬁles. By NEk we mean the correspondence mapping to y ∈ X the
1The Stackelberg assumption could imply that either non-signatories or signatories behave as Stackelberg
leaders. However, in the proposed model non-signatories are assumed to act as singletons and only Stackel-
berg leadership of the signatories (which basically act as a single player) has been assumed in the literature
so far. In IEA context it may be argued that signatories are better informed than non-signatories about emis-
sion levels in other countries since they coordinate their environmental policies within an IEA (Finus 2001,
Chapter 13).
22 AUCO Czech Economic Review, vol. 2, no. 1Partial Cooperation and Non-Signatories Multiple Decision
set NEk(y) ∈ Xk. If the game Gk(y) has a unique Nash equilibrium for any y, say
(h1(y),...,hk(y)), the signatories Pk+1,...,Pn maximize the joint proﬁt function and
solve the problem
max
y∈X
n
å
j=k+1
fj(h1(y),...,hk(y),yn−k). (1)
Deﬁnition 1. A vector x(k) = (h1(x),...,hk(x),xn−k) ∈Xn such that x solves the
problem (1) is called a partial cooperative equilibrium of the game G where n−k
players sign the agreement.
The deﬁnition of partial cooperative equilibrium has been given by Mallozzi and
Tijs (2006) for symmetric potential games by considering the unique symmetric Nash
equilibrium for non-signatories, together with an existence result. In this case the
symmetry of the Nash equilibria allows to avoid coordination problems (Cooper and
John 1988).
The uniqueness assumption of the Nash equilibrium (h1(y),...,hk(y)) of the game
Gk(y) not always occurs even in the symmetric case, as it is shown in the following
example.
Example 1. Let us consider n = 4, Xi = [0,1],i = 1,2,3,4 and the following payoffs
fi(x) = (x1+x2+x3+x4)2−2xi, i = 1,2
f3(x) = (x1+x2+x3+x4)2−16x2
3
f4(x) = (x1+x2+x3+x4)2−24x2
4.
If two of the four players cooperate P3 and P4 by choosing x3 = x4 = y, the rest of the
players choose a Nash equilibrium of the two-player game with payoffs
fi(x1,x2,y,y) = (x1+x2+2y)2−2xi, i = 1,2.
For 0 ≤ y ≤ 1/4 there are two Nash equilibria proﬁles (0,0),(1,1) and for y > 1/4
there is a unique Nash equilibrium proﬁle (1,1).
We deal now with the case where the non-signatories have multiple equilibria for a
given decision of the signatories. Let us suppose that there is a rule to choose in the set
of the non-signatories Nash equilibrium set, namely a selection of the correspondence
NEk, that is a function
l : y ∈ X 7→ (l1(y),...,lk(y)) ∈ NEk(y).
This is a way to choose a proﬁle in the set NEk(y) for any y ∈ X. Then, the signa-
tories Pk+1,...,Pn solve the problem
max
y∈X
n
å
j=k+1
fj(l1(y),...,lk(y),yn−k). (2)
AUCO Czech Economic Review, vol. 2, no. 1 23L. Mallozzi, S. Tijs
Deﬁnition 2. A vector x(k)=(l1(x),...,lk(x),xn−k)∈Xn such that x solves the prob-
lem (2) is called a l-partial cooperative equilibrium of the game G where n−k players
sign the agreement.
Under the usual assumptions, common in practical situations, X compact subsets
of Euclidean spaces and fi continuous functions on Xn for all i = 1,..,n, if there exists
an upper semi-continuous selection l of the correspondence NEk, there is a l-partial
cooperative equilibrium with level of non-cooperation k.
In the following the special case of the max-selection is investigated.
2.1 Max-selection
Let us consider a partially ordered set X that is a set X on which there is a binary
relation ￿ that is reﬂexive, antisymmetric and transitive. Recall that the function
fi(yi,x−i) has increasing differences in (yi,x−i) on X ×Xn−1 for all i if for all yi ∈ X
and x0
−i,x00
−i ∈ Xn−1 with x0
−i ≺ x00
−i, the function fi(yi,x00
−i)− fi(yi,x0
−i) is increasing in
yi (Vives 1999). If f is a differentiable function on Rn, then f has increasing differen-
ces on Rn if and only if ¶ f/¶xi is increasing in xj for all distinct i and j and all x. If
f is a twice differentiable function on Rn, then f has increasing differences on Rn if
and only if ¶2 f/¶xi¶xj ≥ 0, for all distinct i and j.
Fix now a level of non-cooperation k in the game G = hn;X; f1,..., fni.
Proposition 1. Let X be a closed real interval and f1,..., fn continuous functions on
Xn. If fi(yi,x−i) has increasing differences in (yi,x−i) on X ×Xn−1 for all i, then the
set NEk(yn−k) 6= / 0 for any y and a greatest and a least equilibrium point exist.
Proof. The game Gk(y) turns out to be a supermodular game (Vives 1999) and by
Topkis’ theorem the set of the Nash equilibrium proﬁles is a nonempty complete lattice
with respect to the natural partial ordering ￿ in Rk. ￿
By using Proposition 1 we consider the following max-selection
l(y) = max{h = (h1,...,hk) : h ∈ NEk(y)}.
Proposition 2. Let X be a closed real interval and f1,..., fn continuous functions on
Xn. If fi(yi,x−i) has increasing differences in (yi,x−i) on X ×Xn−1 for all i, for each
k = 2,...,n, the function l is upper semi-continuous on X.
Proof. Since f1,..., fn are continuous functions on X closed real interval, the Nash
equilibrium correspondence is sequentially closed at y ∈ X (Aubin and Frankowska,
1990), i.e. for any sequence (ym) of X converging to y ∈ X and any sequence (hm) of
Xk converging to h ∈ Xk such that hm ∈ NEk(ym) for all m ∈ N, we have h ∈ NEk(y).
Let us consider y ∈ X and a sequence (ym) converging to y. For any m ∈ N , the
sequence l(ym) ∈ NEk(ym) is bounded. So, there exists a subsequence of l(ym) con-
verging to l = limsupml(ym). Since NEk is closed, l ∈ NEk(y) and by deﬁnition of l
we have l ≤ l(y), i.e. limsupml(ym) ≤ l(y), i.e. l is upper semi-continuous at y. ￿
24 AUCO Czech Economic Review, vol. 2, no. 1Partial Cooperation and Non-Signatories Multiple Decision
Let us note that in Example 1, for k = 2, the max-selection is the constant function
mapping to any y ∈ X the Nash equilibrium pair (1,1).
As a consequence of Proposition 1 and Proposition 2 we have the following exis-
tence theorem.
Theorem 1. Let X be a closed real interval and f1,..., fn continuous functions on
Xn. If fi(yi,x−i) has increasing differences in (yi,x−i) on X × Xn−1 for all i, for
each k = 2,...,n−1, there exists at least a l-partial cooperative equilibrium x(k)=
(l1(x),...,lk(x),xn−k) ∈Xn of the game G.
Remark 1. It is easy to ﬁnd not upper semicontinuous selections. In Example 1, the
min-selection
l(y) = min{h = (h1,...,hk) : h ∈ NEk(y)}
is a selection of the Nash equilibrium correspondence that is not u.s.c.: for 0≤y≤1/4
we have l(y)=(0,0) and for 1/4<y≤1 we have l(y)=(1,1). The two cooperating
players P3 and P4 have to jointly maximize the function (f3 + f4)(1,1,y,y) = 2(2+
2y)2−16y2−24y2 for y > 1/4, (f3+ f4)(0,0,y,y) = 2(2y)2−16y2−24y2 for 0 ≤ y ≤
1/4, that is not u.s.c. at y = 1/4 and a l-partial cooperative equilibrium does not exist.
3. Public goods game
Let us consider a situation where n agents interact to consume a good having a “public”
character, for example roadways (Mas-Colell et al. 1995, Ray and Vohra 1997, Yi
1997, Batina and Ihori 2005). There are n identical consumers of one public good.
Each consumer can provide xi ∈ X (X real closed interval) units of the public good at
cost Ci(xi) and enjoys a beneﬁt depending on the total amount G of the public good
Bi(G). The usual assumptions in the twice differentiable case are the following for
each i:
C0
i ≥ 0,B0
i ≥ 0,C00
i ≥ 0,B00
i ≤ 0
The consumer i’s utility is fi(xi,G) = Bi(G)−Ci(xi). The quantity G represents the
public consumption. The simplest version of the public goods game assumes a linear
technology constraint, which transforms one unit of private good into one unit of public
good, G = å
n
i=1xi.
The strategic form game GPG = hn;X;B1,..,Bn,C1,..,Cni is called public goods
game. Public goods games are examples of aggregative games (Corchon, 1994), i.e.
the payoffs depend only on individual strategies and an aggregate of all strategies
å
n
i=1xi. In this case g(x1,...,xn) = å
n
i=1xi is called aggregator function; usually g is
a continuous increasing function of (x1,...,xn).
In order to apply Theorem 1, if we require for the game GPG the increasing diffe-
rences property, we would have ¶ f2
i /¶xi¶xj = B00
i ≥ 0 against the usual assumption
B00
i ≤ 0. So that in the game GPG we would have linear beneﬁt function Bi for each i
and the utility functions separable in all variables.
A way out to have increasing differences property in a public goods game is to deal
with non-linear technology constraint. We assume G =
￿
å
n
i=1xi
￿2. In this case we
AUCO Czech Economic Review, vol. 2, no. 1 25L. Mallozzi, S. Tijs
can combine multiplicity and supermodularity in a special class of public good games,
namely public goods games with linear beneﬁt function Bi for each i.
Let GPG be a public goods game with G =
￿
å
n
i=1xi
￿2, Bi linear for each i and
C00
i ≥ 0 for each i. The game satisﬁes all assumptions of the Theorem 1 and we have
the following existence result.
Proposition 3. For any level of non-cooperation k there exists at least a l-partial
cooperative equilibrium x(k)= (l1(x),...,lk(x),xn−k) ∈Xn of the game GPG.
Thegame inExample 1isapublicgoods gamewithG=
￿
å
n
i=1xi
￿2, Bi(t)=t,i=1,..4,
C1(t) =C2(t) = 2t,C3(t) = 16t2,C4(t) = 24t2. The two cooperating players P3 and P4
jointly maximize the function
(f3+ f4)(l1(y),l2(y),y,y) = 2(2+2y)2−16y2−24y2
being (l1(y),l2(y)) = (1,1) for any y. The l-partial cooperative equilibrium is then
(1,1,1/4,1/4).
For any y ∈ X and any i > k, since B0
i ≥ 0 we have
max
h∈NEk(y)
Bi
￿ k
å
j=1
hj +(n−k)y
￿2−Ci(y) = Bi
￿ k
å
j=1
l j(y)+(n−k)y
￿2−Ci(y),
where l is the max-selection of the Nash equilibrium correspondence. So that the
max-selection l is the best choice from the signatory point of view.
4. Conclusion
We investigated a partial cooperation approach between a group of the players who
cooperatebychoosingthesamestrategy; therestoftheplayersplayaNashequilibrium
game. In the case where the set of the possible Nash equilibria for the non-cooperating
players is not a singleton, we proposed a deﬁnition of partial cooperative equilibrium
depending on a selection choice in the set of equilibria. We presented some results in
the max-selection case. It could be interesting to develop further the study of l-partial
cooperation by using different selection choices besides the max-selection, in order to
have existence of the l-partial cooperative equilibrium and also properties useful in
concrete applications as the public goods game.
In the proposed model the cooperative aspect of the partial cooperative equilibrium
concept has been formalized by mean of the joint (common) strategy of the signatories.
This model ﬁts in the IEA framework where the signatory players are those countries
signing the agreement and choosing the joint emission level (Finus 2001). It is possible
to give a more general deﬁnition of partial cooperative equilibrium by assuming that
the signatories choice is a vector yn−k =(yk+1,...,yn) having different components and
maximize again their aggregate welfare
max
yn−k∈Xn−k
n
å
j=k+1
fj(h1(yn−k),...,hk(yn−k),yn−k),
26 AUCO Czech Economic Review, vol. 2, no. 1Partial Cooperation and Non-Signatories Multiple Decision
where the vector (h1(yn−k),...,hk(yn−k)) is the Nash equilibrium chosen by the non-
signatories. The deﬁnition of l-partial cooperative equilibrium and existence results
could be given in this case similarly to those proved in the paper.
A deeper formalization of the behavior of the cooperating group could be given in
terms of other possible solution concepts used in a Cooperative Game Theory setting,
for example the Shapley value. Further problems arise in these more general cases, for
example the redistribution of utility between cooperating players and the stability of
the coalition. This will be developed in a future paper.
Acknowledgment The authors thank the referees for their interesting comments.
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AUCO Czech Economic Review, vol. 2, no. 1 27

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