190 J
. Zhao Mathematical Social Sciences 37 1999 189 –204
4
be understood as providing necessary conditions for convexity. Second, few studies established convexity, and these works can be understood as providing sufficient
conditions for convexity. In contrast to the existing literature, this article establishes a necessary and sufficient
condition for convexity in oligopoly games, based on the primitive parameters in an oligopoly industry. Our result also makes a step forward in solving the problem of ‘‘how
to split joint profits among firms engaged in a merger.’’ When firms engaged in a merger are asymmetric, the most difficult and most
important issue on the negotiation table is how to share the benefits of the merger. For example, when all n firms are merging into a monopolist, the issue becomes whether or
not to agree on a specific split of monopoly profit among the n members. For such a
5
merger i.e. the monopoly merger to take place, the associated core has to be nonempty otherwise, the blocking coalitions will not join the merger. When the associated
transferable utility game is convex, the issue of splitting profits can be perfectly answered by using either Shapley value Shapley, 1953 or nucleolus Schmeidler,
1969, because they are both in the core.
The rest of the paper is organized as follows. Section 2 provides the definitions and notations, Section 3 presents the main results and examples, Section 4 establishes the
proofs, Section 5 discusses some extensions and provides some concluding remarks.
2. Definitions and notations
Let 1 denote the collection of all subsets of N. A game in coalition function form F 5
hN, vj is a set function v:1 →
R with v550, which specifies a joint payoff for each nonempty S [1.
Definition 1. The game G 5
hN, vj is convex if for any S, T [1, v
S 1 vT vS T 1 vS T . 1
Equivalently, the set function v is supermodular if 1 holds Submodular if 1 is reversed. The following lemma provides an equivalent definition for the convexity.
Lemma 1. The game 1 is convex if and only if for any S ,T ,N and i [N\T i
.e. i [N, i [
⁄ T ,
v S i 2 vS vT i 2 vT .
2 The lemma see Ichiishi, 1981 or Moulin, 1988 is used in proving our main result.
From 2, a convex game can be interpreted as exhibiting increasing returns to scale in
4
For example, the minimum costs in a class of scheduling problems Curiel et al., 1989 and in a class of network problems Granot and Hojati, 1990 are submodular i.e. the games defined by these functions are
convex.
5
See Zhao, 1998 for more discussions.
J . Zhao Mathematical Social Sciences 37 1999 189 –204
191
coalition sizes, since the marginal contribution of a firm i increases as the coalition expands.
Definition 2. The core allocation for a TU game G 5
hN, vj is defined by
n i
i
CoG 5 p [ R
u
O
p vS for all S [ 1, and
O
p 5 vN . 3
H J
i [S i [N
In combinatorics, CoG 5Bv is called the base polyhedron of the supermodular set function v. Any split p in the core divides the total payoff vN in such a way that no S
is allocated less than its own payoff vS , so no S alone can do better than p , therefore the grand coalition is stable in the sense that no S has incentive to break up the
agreement p .
Throughout the paper superscripts in small letters denote individual firms, and subscripts in capital letters denote coalitions or nonempty subset in 1. For each S [1,
S
let
uSu denote the number of firms in S, and R denote the uSu-dimensional Euclidean
space whose coordinates have as superscripts the members in S. For any x 5
1 n
i i
hx , . . . ,x j[Y, let x 5hx ui[Sj[Y 5P Y be the production vector of S; x
5
S S
i [S 2S
i i
hx ui[⁄Sj[Y 5P Y be the production vector of the outsiders i.e. N\S . We shall
2S i [
⁄ S
write x 5x, Y 5Y and p 5p for simplicity.
N N
N
An oligopoly market of a homogeneous good is defined by a decreasing inverse
i i
i i
i i
¯ demand function px5PX 5PSx and a cost functions C x , x [Y 5[0, y ] for
i 1
n i
¯ each i [N, where y .0 is i’s capacity, x 5
hx , . . . ,x j[Y5P Y is the production
i [N i
i
vector. This market is equivalent to a game in strategic form G 5 hN, Y , p , where
i i
i i
i i
i
p x 5 pxx 2 C x 5 PXx 2 C x , is i’s profit function for each i [N.
i i
There are two ways of converting the game G 5 hN, Y , p j to a coalition function
form game G 5 hN, vj, and they lead respectively to the concepts of a- and b-cores see
Aumann, 1959. In the a -core approach, each S can guarantee its members at least a payoff of v ? regardless of the actions of N S; while in the b -core approach, each S
a
can not be prevented from getting at least v ? i.e. for each outside choice z , S could
b 2S
react by choosing x z so as to have a joint payoff at least v S .
S 2S
b
¯ The a -core is defined by computing the punishment function by N S, z
x , and the
2S S
¯ guaranteed profit function, p x , which are defined respectively as the minimum
S S
solution function and the extreme value function in
i i
¯ ¯
p x 5 Min
O
p x , z 5
O
p x , z x ,
4
S S
S 2S
S 2S
S z
[ Y
2S 2S
i [S i [S
i i
i i
where x , z denotes a vector y [Y such that y 5x if i [S and y 5z if i [ ⁄ S. Thus for
S S
each S [1 and x [Y, we can write x 5x , x for convenience. For each S [1, let
S 2S
i
˜ ˜
˜ ¯
¯ ¯
v S 5 Max p x
ux [ Y 5 p x 5
O
p x , z x ,
5
h j
a S
S S
S S
S S
2S S
i [S
192 J
. Zhao Mathematical Social Sciences 37 1999 189 –204
˜ where x is the maximum solution. This converts the oligopoly game to a coalition
function form game in the a -core fashion as follows: G 5 N, v ? ,
6
h j
a a
and any core allocation of G is the a -core profit allocation for the original market.
a
In contrast, the b -core is defined by computing the reaction function of S, x z ,
S 2S
and its reaction profit function, p z , which are respectively the maximal solution
S 2S
function and the extreme value function in
i i
p z 5Max
O
p x , z 5
O
p x z , z
. 7
S 2S
S 2S
S 2S
2S x [Y
S S
i [S i [S
For each S [1, let
i
ˆ ˆ
ˆ v
S 5 Min p z uz
[ Y
5 p z
5
O
p x z , z
, 8
h j
b S
2S 2S
2S S
2S S
2S 2S
i [S
ˆ where z
is the minimum solution. This converts the oligopoly game to a coalition
2S
function form game in the b -core fashion: G 5 N, v ? .
9
h j
b b
It follows from v N 5v N and v S v S for S ±N that CoG CoG . Note
a b
a b
b a
that deriving the TU games 6 and 9 only requires the continuity of demand and cost functions, because all firms have finite capacities.
We are now ready to study the convexity of G and G .
a b
3. Main results
