In this paper we consider symmetric aggregative games and investigate partial cooperation between a portion of the players that sign a cooperative agreement and the rest of the players. Existence results of partial cooperative equilibria are obtained when the players who do not sign the agreement play a Nash equilibrium game having multiple solutions. Some applications in the supermodular case are discussed.Noncooperative games, cooperation, aggregative games, supermodular games.

Lina Mallozzi

Stef Tijs

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 2 
  
Coordinating choice in partial cooperative equilibrium 
  
 
 
Lina Mallozzi  
University of Naples Federico II 
Stef Tijs  
University of Tilburg
Abstract 
In this paper we consider symmetric aggregative games and investigate partial cooperation between a portion of the 
players that sign a cooperative agreement and the rest of the players. Existence results of partial cooperative equilibria 
are obtained when the players who do not sign the agreement play a Nash equilibrium game having multiple solutions. 
Some applications in the supermodular case are discussed.
 
Citation: Lina Mallozzi and Stef Tijs, (2009) ''Coordinating choice in partial cooperative equilibrium'', Economics Bulletin, Vol. 29 no.2 pp. 
1459-1465. 
Submitted: Mar 14 2009.   Published: June 28, 2009. 
 
     1. Introduction
As in many ¯elds of Economics, the formation of coalitions (i.e. coop-
eration) is an important topic. The classical approach is in the setting of
cooperative games with the core and related solutions concepts (the stable
set, the nucleolus, the bargaining set, the kernel). A more recent approach of
coalition formation is in the setting of a mixed framework of both cooperative
and non-cooperative games. Players within coalitions cooperate but coali-
tions act noncooperatively with each other. Some examples in Economics
are cartels, R& D agreement, joint venture, environmental issues. See, for
example, Yi (1997), Ray and Vohra (1999), Finus (2001), Montet and Serra
(2003) and the references therein.
We deal with the concept of partial cooperative equilibrium. We suppose
that a portion of the players (signatories) decide to cooperate and sign a
cooperative agreement, the rest of the players (non-signatories) choose their
strategies by playing a non-cooperative game, i.e. by solving a Nash equilib-
rium problem.
The de¯nition of partial cooperative equilibrium has been presented in
Mallozzi and Tijs (2008a) in the context of symmetric potential games, i.e. a
subclass of symmetric strategic games with a potential in the sense of Mon-
derer and Shapley (1996). An existence result of partial cooperative equilibria
has been obtained for symmetric potential games by assuming the unique-
ness of the non-signatories optimal strategy for any possible decision taken
by the signatories and by using concavity-like assumptions. The uniqueness
assumption has been removed in Mallozzi and Tijs (2008b) and a de¯nition
of partial cooperative equilibrium, where the Nash equilibrium problem of
the non-signatories may have also di®erent solutions, has been presented for
non symmetric games depending on a selection choice in the set of equilibria.
In this paper we avoid the uniqueness assumption and, in order to reduce
coordination problems, suppose that the non-signatories choose among the
others only the symmetric Nash equilibria. We deal with aggregative games
(Dubey et al. 1980; Corchon 1994), i.e. games having the payo®s depending
only on individual strategies and an aggregate of all strategies. Several con-
crete situations correspond to these games, i.e. public good games (Batina
and Ihori 2005), global emission games (Finus 2001).
We present the generalized de¯nition of partial cooperative equilibrium,
then speci¯c examples are given and the existence question is investigated.2. Coordinating Choice Model
Let ¡ =< n;X;f1;:::;fn > be an n-person normal form game with player
set I = f1;2;:::;ng, with the same strategy space X for each player i 2 I and
where fi: Xn 7! R is the payo® function of player i 2 I. If player i chooses
xi 2 X, then he obtains a pro¯t fi(x1;:::;xn). Each player wants to maximize
his own pro¯t. We denote by x¡i the vector (x1;:::;xi¡1;xi+1;:::;xn) 2 Xn¡1.
We deal with aggregative games, i.e. games ¡ for which there exist a function
f : X £R 7! R and a function g: Xn 7! R (called aggregator) such that for
all i and for all x = (x1;:::;xn) 2 Xn we have fi(x1;:::;xn) = f(xi;g(x)).
We denote by ¡ =< n;X;f;g > an aggregative game. In the following we
assume that ¡ =< n;X;f;g > is a symmetric aggregative game where:
a1) X is a closed real interval;
a2) f;g are continuous functions;
a3) g is a symmetric function.
By assumption a3) we have g(x) = g(x0) for all permutations x0 of the
vector x 2 Xn, then fi(xi;x¡i) = fj(x0
j;x0
¡j) so ¡ is a symmetric game. If
noncooperative behavior is assumed between the n players, the equilibrium
solution considered is the well known concept of Nash equilibrium. Let us
denote by NE the set of the Nash equilibrium vectors of the game ¡, and by
SNE the set of the symmetric Nash equilibrium vectors of the game ¡, i.e.
vectors with identical components.
We suppose now that a portion of the n players may sign a cooperative
agreement. Let Pk+1;:::;Pn be the players acting in a cooperative way and
P1;:::;Pk the players acting in a noncooperative way, for each k = 0;:::;n.
We assume that the last n ¡ k players (cooperating players or signatories)
use the same strategy, i.e. xk+1 = xk+2 = ::: = xn = y for y 2 X. This
assumption is common in some concrete situations, for example in Interna-
tional Environmental Agreement it means the available level of a certain gas
emission and countries sign the agreement by choosing the same strategy.
For simplicity we will denote (y;:::;y) 2 Xk by yk, for any y 2 X and
k = 0;:::;n. The ¯rst k players (noncooperating players or non-signatories)
with payo®s
f(xi;g(x1;:::;xk;yn¡k))
for any i = 1;:::;k do not cooperate and choose a Nash equilibrium pro¯le.
For each k = 0;:::;n and for all y 2 X, let us consider the normal form
game ¡k(y) =< k;X;f(¢;y);g(¢;y) >, i.e. the k-person game with the same
strategy space X for each player and payo® function of player i, for i =
1;:::;k, given by f(xi;g(x1;:::;xk;yn¡k) for any y 2 X. In order to de¯ne
2a concept of partial cooperative equilibrium for the game ¡, we denote by
NEk(y) (resp. SNEk(y)) the set of the (resp. symmetric) Nash equilibrium
vectors of the game ¡k(y) for any y 2 X.
Now, the uniqueness of the Nash equilibria of the non-signatories not
always occurs and their multiplicity leads to coordination failures, as inten-
sively studied by many authors (Cooper 1999). Here we suppose that the
non-signatories select in the set of Nash equilibria only the symmetric ones
in order to reduce coordination problems. Moreover, there may be still a
coordination problem if the set SNEk(y) is not single-valued for a y 2 X.
We suppose that the signatories maximize their pro¯t also in the set of the
non-signatories optimal reaction, as speci¯ed in the following.
A vector x(k)=(xN
k ;»C
n¡k)2Xn such that
»
C 2 argmax
y2X
µ
max
xk2SNEk(y)
f(y;g(xk;yn¡k))
¶
and xN
k any symmetric equilibrium in SNEk(»C), is called a partial coopera-
tive equilibrium of the game ¡ where n ¡ k players sign the agreement.
Let us note that in the context of hierarchical two-stage games, the above
de¯nition corresponds to the concept of strong hierarchical Nash equilibrium
widely studied in the literature (for example Leitmann 1978; Luo et al. 1996
and the references therein). The de¯nition of partial cooperative equilibrium
has been given by Mallozzi and Tijs (2008a) for symmetric potential games
by considering a unique symmetric Nash equilibrium for non-signatories, to-
gether with an existence result. Here we remove the uniqueness assumption
in a larger class of games, namely symmetric aggregative games. In fact, if
a symmetric aggregative game ¡ =< n;X;f;g > displays separability, i.e.
for all i we have f(xi;g(x)) = B(xi) ¡ C(g(x)) for some functions B and C,
then the game turns out to be a symmetric potential game.
EXAMPLE 2.1 Let us consider n = 4, X = [0;1] and the following payo®s
fi(x1;x2;x3;x4) = ¦
4
j=1xj; i = 1;2;3;4
The game is a symmetric aggregative game with f(xi;g(x)) = g(x) = ¦4
j=1xj.
If two of the four players cooperate, the rest of the players play a two-
player noncooperative game with payo®s
f1(x1;x2;y;y) = f2(x1;x2;y;y) = x1x2y
2
for any y 2 X, that admits two symmetric equilibria SNE2(y) = f(0;0);(1;1)g
for y 6= 0 and in¯nite symmetric equilibria SNE2(0) = f(x;x); x 2 [0;1]g
3for y = 0. The signatory problem then is to maximize the function y2 on
[0;1] and a partial cooperative equilibrium is x(2) = (1;1;1;1):
EXAMPLE 2.2 Let us consider n = 4, X = [0;1] and the following payo®s
fi(x) = xi + x
2
i=2 ¡ (x1 + x2 + x3 + x4)
2=2; i = 1;2;3;4
The game is a symmetric aggregative game with g(x) = (x1+x2+x3+x4)2=2
and f(xi;g(x)) = xi + x2
i=2 ¡ g(x).
If two of the four players cooperate, the rest of the players choose a Nash
equilibrium of the two-player game with payo®s
fi(x1;x2;y;y) = xi + x
2
i=2 ¡ (x1 + x2 + 2y)
2=2; i = 1;2:
For y > 1=2 the game admits the only symmetric Nash equilibrium (0;0);
for y = 1=2 the set of equilibria is NE2(1=2) = f(t;0);(0;t);t 2 [0;1]g; for
y < 1=2 the set NE2(y) = f(1;0);(1 ¡ 2y;1 ¡ 2y);(0;1)g. In this case, for
all y · 1=2, turns out that the symmetric Nash equilibrium is SNE2(y) =
f(1 ¡ 2y;1 ¡ 2y)g ½ NEk(y). The signatories problem will be to maximize
the function y +y2=2¡(2¡4y +2y)2=2 if y 2 [0;1=2], y +y2=2¡(2y)2=2 if
y 2]1=2;1] and x(2) = (0;0;1=2;1=2) is the partial cooperative equilibrium.
In order to have an existence result, we need some properties of the cor-
respondence NEk mapping y 2 X into the set of the Nash equilibria NEk(y)
and also of the correspondence SNEk mapping y 2 X into the set of the sym-
metric Nash equilibria SNEk(y). First of all let us recall some de¯nitions.
Recall that a correspondence T from X to Y (X;Y topological spaces) is
(sequentially) closed at x 2 X if, for any sequence (xm) of X converging to
x 2 X and any sequence (ym) of Y converging to y 2 Y such that ym 2 T(xm)
for all m 2 N, we have y 2 T(x) (Aubin and Frankowska 1990). T is (se-
quentially) closed on X if it is closed at x, for every x 2 X.
PROPOSITION 2.1. Let ¡ =< n;X;f;g > be a symmetric aggregative game
s.t. SNEk(y) 6= ;, for any y 2 X; under assumptions a1), a2) and a3), the
correspondences NEk and SNEk are closed on X.
Proof. Let y 2 X and (ym) a sequence in X such that ym ! y, and let
(´1m;:::;´km) a converging sequence of Xk, (´1m;:::;´km) ! (´1;:::;´k) such
that (´1m;:::;´km) 2 NEk(ym) for all m 2 N. We know that
f(´im;g(´1m;:::;´km;(ym)n¡k)) ¸ f(xi;g(´1m;::;xi;::;´km;(ym)n¡k))
for any xi 2 X and any i = 1;:::;k. By using assumption a2), we have
f(´i;g(´1;:::;´k;yn¡k)) ¸ f(xi;g(´1;::;xi;::;´k;yn¡k))
4i.e. (´1;:::;´k) 2 NEk(y).
Let us de¯ne the correspondence DIAGk mapping to y the set f(x1;:::;xk) 2
Xk; s:t: x1 = x2 = ::: = xkg. The correspondence SNEk mapping to any
y 2 X the set of symmetric Nash equilibria is the intersection correspondence
between the correspondence NEk and the correspondence DIAGk, both closed
on X. So that, SNEk is closed on X (Border 1989).
By assumptions a1), a2) and by using Berge's theorem, the marginal
function max
xk2SNEk(y)
f(y;g(xk;yn¡k)) is upper semicontinuous on X.
PROPOSITION 2.2 Let ¡ =< n;X;f;g > be a symmetric aggregative game such
that SNEk(y) 6= ;, for any y 2 X; under assumptions a1), a2) and a3), there
exists a partial cooperative equilibrium x(k)=(xN
k ;»C
n¡k)2Xn.
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Coordinating choice in partial cooperative equilibrium

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