Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol37.Issue2.Mar1999:

A necessary and sufficient condition for the convexity in

oligopoly games

* Jingang Zhao

Department of Economics, Ohio State University, 1945 North High Street, Columbus, OH 43210-1172, USA

Received 23 October 1997; received in revised form 5 March 1998; accepted 30 March 1998

Abstract

This paper establishes a necessary and sufficient condition for the convexity (or supermodulari-ty) in oligopoly games. 1999 Elsevier Science B.V. All rights reserved.

Keywords: Convex games; Supermodularity; Oligopoly industry;a-core; b-core

JEL classification: C71; D43; L13

1. Introduction

Convex games are equivalent to supermodular functions, which represent the rank function of matroids in combinatorics. Such convexity (or supermodularity) arises not only in economic problems like coalition production economies, cost allocation problems, and oligopoly problems, but also in optimization problems like network

1

problems. The literature on convex games is characterized by two features. First,

2

convexity was used in the early works to establish the geometric structure of its core allocations (i.e. the extreme points of the core of a convex game are the marginal value

3

vectors of the game), then many weak versions (or extensions) of convexity followed. Because these properties and extensions follow from the original convexity, they all can

*Tel.: 11-614-292-6523; fax:11-614-292-3906; e-mail: Zhao.18@Osu.Edu

1

See (Fujishige, 1991; McLean and Sharkey, 1993) for short survey.

2

See (Edmonds, 1970; Maschler et al., 1972; Shapley, 1971).

3

˜

These include average convexity (Inarra and Usategui, 1993; Sprumont, 1990), k-convexity (Driessen, 1986), odd submodularity (Qi, 1988), permutational submodularity (Granot and Huberman, 1982), semiconvex

¨

games (Driessen and Tijs, 1985), extension to convex set functions (Rosenmuller and Weidner, 1974), and extension to games without sidepayments (Sharkey, 1981; Vilkov, 1977).

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}

F5hN,vjis a set functionv:1→R withv(5)50, which specifies a joint payoff for each

nonempty S[1_.

Definition 1. The game G 5hN, vj is convex if for any S, T[1,

v(S )1v(T )#v(S<T )1v(S>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 )2v(S )#v(T<i )2v(T ). (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

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 5hN, vj is defined by

n i i

Co(G)5

H

p[R u

O

p $v(S ) for all S[1, and

O

p 5v(N ) .

J

(3)

i[S i[N

In combinatorics, Co(G)5B(v) is called the base polyhedron of the supermodular set

functionv. Any splitp* in the core divides the total payoffv(N ) in such a way that no S

is allocated less than its own payoffv(S ), so no S alone can do better thanp*, 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 in1_{). 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 x5

1 n i i

hx , . . . ,x j[Y, let xS5hxui[Sj[YS5Pi[S Y be the production vector of S; x2S5

i i

hxui[⁄ Sj[Y2S5Pi[⁄ S Y be the production vector of the outsiders (i.e. N\S ). We shall

write xN5x, YN5Y and p 5pN for simplicity.

An oligopoly market of a homogeneous good is defined by a decreasing inverse

i i i i i i

¯ demand function p(x)5P(X )5P(Sx ) 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, x5hx , . . . ,x j[Y5Pi[NY is the production

i i

vector. This market is equivalent to a game in strategic form G 5hN, Y ,p ), where

i i i i i i i

p(x)5p(x)x 2C (x )5P(X )x 2C (x ),

is i’s profit function for each i[N.

i i

There are two ways of converting the game G 5hN, Y , p j to a coalition function

form gameG 5hN,vj, and they lead respectively to the concepts ofa- andb-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 leastv (?) (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

¯

Thea-core is defined by computing the punishment function by N /S, z2S(x ), and theS

¯

guaranteed profit function, pS(x ), which are defined respectively as the minimumS

solution function and the extreme value function in

i i

¯ ¯

pS(x )S 5 Min

O

p (x , zS 2S)5

O

p (x , zS 2S(x )),S (4)

z2S[Y2Si[S i[S

i i i i

where (x , z ) denotes a vector yS S [Y such that y5x if i[S and y 5z if i[⁄ S. Thus for

each S[1_{and x}[Y, we can write x5(x , x ) for convenience. For each S[1_{, let}

S 2S

i

˜ ˜ ˜

¯ ¯ ¯

v (S )5Max

h

p(x )ux _[Y

j

5p (x )5

O

p (x , z (x )), (5)

a S S S S S S S 2S S

˜

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 5hN,v (?) ,j (6)

a a

and any core allocation of Ga is the a-core profit allocation for the original market.

*

In contrast, theb-core is defined by computing the reaction function of S, x (zS 2S),

*

and its reaction profit function, pS(z2S), which are respectively the maximal solution

function and the extreme value function in

i i

* *

pS(z2S)5Max

O

p(x , zS 2S)5

O

p(x (zS 2S), z2S). (7)

x_S[Y_{Si[S i[S}

For each S[1_{, let}

i

ˆ ˆ ˆ

* * *

v (S )5Min

h

p (z )uz [Y

j

5p (z )5

O

p(x (z ), z ), (8)

b S 2S 2S 2S S 2S S 2S 2S

i[S

ˆ

where z2S is the minimum solution. This converts the oligopoly game to a coalition

function form game in the b-core fashion:

G 5

h

N,v (?) .

j

(9)

b b

It follows from v (N )5v (N ) andv (S )#v (S ) for S±N that Co(G)#Co(G ). 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 Ga andGb.

3. Main results

We shall focus our attention on linear industries, where the inverse demand function and cost functions become

i i i i i i i i

¯ P(X )5a2bX, and C (x )5d 1c x , x [Y 5[0, y ],

i i

where a.0, b.0; d $0 is i’s fixed cost, and c$0 is its marginal cost. The standard

i i

¯

symmetric Cournot industry is a special case in which d 50, c5c, y5 1` for all i.

Without loss of generality, we assume the negative slope of the inverse demand is 1 or

b51 (Otherwise, simply scale up or down all the parameters).

1 n 1 n 1 n

¯ ¯ ¯

Let y5(y , . . . y ), c5(c , . . . ,c ), and d5(d , . . . ,d ) denote respectively the

vectors of capacities, marginal and fixed costs, then a linear industry is defined by a

3n11

¯

vectorha, c, d, yj[R₁ . We first state two sufficient conditions on the parameterha, c,

¯

d, yj for convexity (Lemma 2 and Theorem 2) and then establish the necessary and

sufficient condition for convexity (Theorem 3).

3n11 i

¯ ¯

i

¯

The lemma is proved in Section 4. Though the condition Minhy ui[Nj$a is a strong

assumption, it includes the standard symmetric or asymmetric Cournot industries as special cases. As readers shall see, our main results are much more general than Lemma 2, which are obtained by using the following three assumptions:

Assumption 1. (Weak Synergy): For each coalition S, its marginal cost and capacity are

i i

¯ ¯

respectively cS5Minhcui[Sj and yS5oi[S y .

Assumption 2. (Capacity sufficiency): For each coalition S, its optimal choices in thea

-andb-core fashions are both bounded by its capacity.

Assumption 3. (No shut-down price): For each coalition S, its average variable cost is

always less than or equal to the price.

Assumption 1 assumes that a coalition’s most efficient technology can be costlessly adopted by all firms in S. Thus, the contributions of all nonefficient members in S are their capacities, and the contributions of the most efficient member are its capacity and its ability to reduce other member’s marginal cost. Assumption 2 assumes that a coalition always has sufficiently large capacity. Assumption 3 rules out the possibility that some coalition might be forced to shut down. These can alternatively be given as the next lemma:

Lemma 3. (a) Assumption1 holds if and only if each coalition’s production set is equal

to

S i i

˜ _¯

YS5

H

xS[R1u

O

x #

O

y

J

;

i[S i[S

(b) Assumption 2 holds if and only if

i i i

¯ ¯

a2

O

y #Min c

h

1y ui[N ;

j

i[N

(b) Assumption 3 holds if and only if

i i i

¯ ¯

Max c

h

2y ui[N

j

#a2

O

y .

i[N i

¯

Remark 1. In symmetrical case (c5c, y5y, all i ), Assumptions1 –3 become (a2c) /(n11)#y#h(a2c) / 2 [2 /(nj 21)],

where (a –c) /(n11) is a firm’s Cournot supply, and (a–c) / 2 is the monopoly supply. In

other words, a firm’s capacity is assumed to be above its Cournot equilibrium supply,

but bounded from above by 2 /(n21) of the monopoly supply. Thus, under these

assumptions, capacities are assumed to be just right, not too large and not too small.

i

cS5Min c

h

ui[S

j

(10)

denote the marginal cost of the most efficient firm in S, and

j

¯ ¯

P(0, y2S)5

S

a2

O

y

D

(11)

j[⁄ S

¯

denote the price when the outsiders chose y2Sand the coalition produces zero. Given the

¯

outside choice y2S, a coalition’s unconstrained optimal total supply is

¯

0 if P(0, y2S),c .S

¯ ₁

XS5 _{] ¯}j _¯ (12)

a2

O

y 2c if P(0, y )$c .

5

₂

S

S

D

2S S

6

j[⁄ S

i

¯

Plugging (12) intooi[Sp(x , yS 2S) at y2S5y2S, we obtain the equivalence between the

a- and b-cores:

Lemma 4. Under Assumptions1 –3, a coalition’s profits in thea- and b-core fashions

are equal and given by:

2

1 j i

] ¯

v(S )5v_a(S )5v_b(S )5

S

a2

O

y 2c

D

2

O

d . (13)

S

4 j[⁄ S i[S

For future developments, it is useful to introduce a new TU game

˜ _˜

G 5hN,vj (14)

˜

by removing the fixed costs in (13). That is, for each S in the game G, its payoff is

2

1

i j

˜ ] ¯

v(S )5v(S )1

O

d 5

S

a2

O

y 2c

D

. (15)

S

4

i[S j[⁄ S

Since (14) is obtained from (13) by adding an additive game, the convexity is

unharmed. Therefore, we only need to study the game (14). A game G 5hN, vj is

superadditive if for any S>T55_,v(S )1v(T )#v(S<T ). We first show that the oligopoly

game (14) is superadditive.

Theorem 1. Under Assumptions 1, 2 and 3, the oligopoly TU game (14 ) satisfies

˜ ˜ ˜

v(S )1v(T )#v(S<T )

for all S, T[1 _{and S}>T50.

In order to define our key conditions, we introduce the following notations and definitions given by (16–19). Let

V 5

h

(S, T, i )uS,T,N, i[N /T and cS2cS<h ji .cT2cT<h ji

j

(16)

6

supermodularity. This can alternatively be interpreted as those firm whose ability of

reducing a coalition’s marginal cost decreases as the coalition expands (i.e. cS2cS<hij is

reduced to cT2cT<hij when S is enlarged to T ). Thus, the elements inV are the potential

factors that might destroy the convexity of (14), because marginal cost enters v(S ) as negative terms. As shown in the next theorem, the game (14) is convex when there are

no such potential damaging factors (V 55).

Theorem 2. Under Assumptions 1, 2 and 3, the oligopoly TU game (14 ) of a linear

industry is convex if V 55.

i j

Remark 2. A special case of V 55 is c 5c for all i, j (all firms have identical marginal costs). Because this case allows firms to have different fixed costs and different capacities, and allows a coalition to make positive profits, it is more general than the

condition of Lemma 2. This special case can be established without using the complex notations designed for the general condition, and this is provided as Claim 2 in Section 4.

Now assume V±5. For any (S, T, i )[V, let

2 2 2 2 i j

¯ ¯

f(S, T, i )5cS2cS<h ji 2(cT2cT<h ji )12y

S

O

y 1cS2cT

D

j[T / S

j j

¯ ¯

12

F

O

y (cS2cS<h ji )2

O

y (cT2cT<h ji ) ,

G

(17)

j[⁄ S, j±i j[⁄ T, j±i

f(S, T, i )

]]]]]]]]

F(S, T, i )5 , (18)

2[cS2cS<h ji 2(cT2cT<h ji )]

v 5Min F(S, T, i )

h

u(S, T, i )[V

j

. (19)

The next lemma shows that the above Co-value is positive.

Lemma 5. If Assumptions 1 –3 hold, then v .0.

The above v-value turns out to be critical in the convexity of oligopoly games:

Theorem 3. Under Assumptions1, 2 and 3, the oligopoly TU gameG 5Ga b in a linear industry with V±5is convex if and only if a#v.

In other words, in an industry withV±5and with no firm too small nor too large, the

oligopoly TU game is convex if and only if the intercept of the inverse demand function

6

is less than or equal to thev-value of (19). Since marginal costs and capacities are both

asymmetric, the condition ‘‘a#v’’ fully characterizes the convexity of linear industries.

Remark 3. In Theorem3, the valuev can be understood as an index or measurement for scale economies of (or increasing returns to) coalition size. If there is not enough

increasing returns to coalition size,vwill fall below the intercept a, therefore the game

will be nonconvex. If there are enough increasing returns to coalition size, v will be above the intercept a, therefore the game will be convex. The case ofV 55can also be interpreted this way, because the minimum over an empty set is defined as1`and thus satisfies a#v.

Theorem 3 is illustrated in the following example, which are computed by using the

¯

dataha, c, d, yj. It first shows if ‘‘a#v’’ fails to hold, the game is not convex. Then it

shows that the game becomes convex when the condition ‘‘a#v’’ holds.

Example. Consider a general linear industry with three firms. The inverse demand

function and the cost functions are respectively:

1 2 3 1 1 1 1 2 2 2 2

P572(x 1x 1x ); C (x )54x , x [[0, 1.3]; C (x )52.25x , x

3 3 3 3

[[0, 1.3]; and C (x )52.25x , x [[0, 1.3].

Letv(1, 2)5v_a(1, 2)5v_b(1, 2) denote the profit for S5h1, 2jas defined in (15). Using

similar notations, the profits for each coalition are:v(1)50.04,v(2)5v(3)51.1556,v(1,

2)5v(1, 3)5v(2, 3)52.9756, v(1, 2, 3)55.6406. For S5h1, 2j, T5h1, 3j,

v(S )1v(T )55.9512.v(S<T )1v(S>T )55.6806,

the game is therefore not convex. This is so because a57.v 56.6907, which violates

the condition ‘‘a#v.’’

Now let the intercept be decreased to a56.65, and all other parameters remain the

same (this will not change the value v), the payoffs become: v(1)50.0006, v(2)5

v(3)50.81,v(1, 2)5v(1, 3)5v(2, 3)52.4026,v(1, 2, 3)54.84. For any S and T, it can

be verified that v(S )1v(T )#v(S<T )1v(S>T ). The game becomes convex now,

because ‘‘a#v’’ is satisfied: a56.65,v 56.6907.

4. Proofs

i

¯

Proof of Lemma 2. It follows from Minhy ui[Nj$a that

j

¯ ¯

P(0, y2S)5

S

a2

O

y

D

#0,

j[⁄ S

for any coalition S[1_{. That is, any nonempty complementary coalition N\S can drive}

˜

the price down to zero or a negative level. By (12), (14) and (15), v(S )50 for all S±_N.

˜

Proof of Lemma 3. Part (a) is proved by replacing Y by Y in the profit-maximizationS S

problems of (5) and (7). For Part (b), notice that a coalition’s optimal unconstrained

total supply (in both the a- and b-fashions) is

1 j

¯ _{] ¯}

XS5₂

S

a2

O

y 2cS

D

,

j[⁄ S

i _¯ i

¯ ¯

where cS5Minhcui[Sj. Its capacity is sufficiently large if and only if XS#yS5oi[S y ,

that is,

1 j i

]₂

S

a2

O

y¯ 2cS

D

#

O

y ,¯

j[⁄ S i[S

which is equivalent to

i i

¯ ¯

a2

O

y #cS1

O

y .

i[N i[S

The above expression holds for all S if and only if

i i i

¯ ¯

a2

O

y #Min c

h

1y ui[N .

j

i[N

For Part (b), notice that a coalition does not shut down production if and only if price is always above the average variable cost, which equals its marginal cost. This is equivalent to

j i i

¯ ¯ ¯ ¯

P(0, y2S)5

S

a2

O

y

D

$c or aS 2

O

y $cS2

O

y .

j[⁄ S i[N i[S

The above expression holds for all S if and only if

i i i

¯ ¯

a2

O

y $Max c

h

2y ui[N . Q.E.D.

j

i[N

Proof of Lemma 4. The claim is proved by observing that a coalition’s final supply (in

both the a- and b-core fashions) is equal to

1 j

] ¯

XS5₂

S

a2

O

y 2cS

D

. Q.E.D.

j[⁄ S

˜ ˜ ˜

Proof of Theorem 1. We need to show v(S )1v(T )#v(S<T ) for S>T55. Let

˜ ˜ ˜

g / 45v(S<T )2(v(S )1v(T )),

i

we need to show g$0. We first prove g$0 in the special case of c5c for all i. By (15),

2 2 2

¯ ¯ ¯

g(u)5(u 2y2S<T) 2(u 2y2S) 2(u 2y2T) ,

j

¯ ¯

where u 5a2c, and y2S5oj[⁄ S y . Note that g is a ‘‘U’’ shaped function whose

i 2 2

¯ ¯

¯ ¯ ¯

isu*5y. Without loss of generality, assume yS#y . Under Assumptions 1, 2 and 3, weT

have

j

¯ ¯

0,XS5

S

a2

O

y 2c

D

Y

2#y ,S

j[⁄ S

which is equivalent to

¯ ¯ ¯ ¯

y2yS#u #y1y .S

Since g is concave, and

2 2

¯ ¯ ¯ ¯ ¯

g(y2y )S 5(y )T 2(yT2y )S .0,

2 2 2

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

g(y1y )S 5(2yS1y )T 2(2y )S 2(yS1y )T 5y [2yS T2y ]S .0,

¯ ¯ ¯ ¯

we have g(u).0 for allu[[y2y , yS 1y ]. Thus,S

˜ ˜ ˜

v(S )1v(T )#v(S<T )

holds for S>T55in industries with identical marginal cost.

Now consider nonidentical marginal costs. Note first that the above inequality holds if

j

all firms in S<T have the same lowest marginal cost c5cS<T5Minhcuj[S<Tj. Since

˜ ˜

v(S ) is a decreasing function of the marginal costs, the right-hand side v(S<T ) remains

unchanged and the left-hand side terms both decrease when a member j’s marginal cost

j

is raised from c5cS<T to the true value c . By repeating this for all members, we see

˜ ˜ ˜

that v(S )1v(T )#v(S<T ) holds. This competes the proof. Q.E.D

Before we prove Theorems 2 and 3, we need to establish Claims 1 and 3. Claim 2 provides a direct proof for the special case of Theorem 2 (with identical costs) without using the notations designed for Theorem 3.

¯ ¯

Claim 1. For any coalitions S,T, S±_{T, let X and X be the optimal choices of S and T} S T

respectively. Then

¯ ¯

(a) XT50⇒XS50;

¯ ¯ ¯

(b) XS.0⇒XT.XS.0;

j 1

¯ ¯ ] _¯

(c) XT5XS12 (oj[T \Sy 1cS2c );T

˜ _˜

(d) The new gameG5hN, vj given by (14) is monotonic.

Part (a) says if the optimal choice of a larger coalition T is to shut down production, then so does the subcoalition S. Part (b) says if a coalition S produces a positive amount of outputs and this coalition is enlarged, then the enlarged coalition shall also produce a positive (larger) amount.

¯

Proof of Claim 1. For Part (a), XT50 requires

j

¯ ¯

It follows from S,T and cS$c thatT

j j

¯ ¯ ¯

P(0, y2S)5

S

a2

O

y

D S

# a2

O

y

D

#cT#c .S j[⁄ S j[⁄ T

¯

Thus XS50.

Part (b) follows directly from (c) and cS$c . Part (c) can be obtained by rearrangingT

¯

the formula for X .T

˜ ˜

The game (14) is monotonic if v(S )#v(T ) for any S,T. Part (d) follows directly from

˜

Parts (a), (b), and the formula for v(S ) given by (15). Q.E.D

i j

Claim 2. Under Assumptions 1–3, (14) is convex if c 5c for all i±_j. 2

1

2 j

¯

˜ ] ¯

Proof of Claim 2. By (12) and (15),v(S )5X 5

S

a2o y 2c

D

for any S[1. We

S ₄ S

j[⁄ S

˜ _˜

shall show that the inequality (2) holds in the game G5hN, v). That is, we shall show

that for any coalitions S,T,N, and firm i[N\T,

˜ ˜ ˜ ˜

v(S<i )2v(S )#v(T<i )2v(T ).

˜

It follows from the above formula for v(S ), Part (c) of Claim 1, and symmetrical marginal costs that

2 2

¯ ¯ ¯ ¯ ¯ ¯

˜ ˜

v(S<i )2v(S )5X 2X 5(X 1X )(X 2X )

S<h ji S S<h ji S S<h ji S

1 i 1 i

¯ ] ¯ ] ¯

F

G

5 2XS1₂(y 1cS2cS<h ji ) ₂(y 1cS2cS<h ji )

1 _¯ 1 i i

]

S

]¯

D

¯

5₂ 2XS1₂y y .

Similarly,

1 1

2 2 i i

¯ ¯ ¯

˜ ˜ ]

S

]¯

D

¯

v(T_<i )2v(T )5X 2X 5 2X 1 y y .

T<h ji T ₂ T ₂

¯ ¯

The above two expressions and the fact that XT$X lead toS

˜ ˜ ˜ ˜

v(S<i )2v(S )#v(T<i )2v(T ).

Thus the TU game (14) is convex. Q.E.D

Proof of Theorem 2. It follows from Assumptions 2 and 3, (12) and (15) that

2

1

2 j

¯

˜ ] ¯

v(S )5X 5

S

a2

O

y 2c

D

S ₄ S

j[⁄ S

for any coalition S[1_{. Similarly as in the proof of Claim 2, we shall show that for any}

coalitions S,T,N, and any firm i[N\T,

˜ ˜ ˜ ˜

This will lead to the convexity of (14). As already shown in the proof of Claim 2, we have

1 _¯ 1 i i

˜ ˜ ]

F

] ¯

G

¯

v(S<i )2v(S )5 2X 1 (y 1c 2c ) (y 1c 2c )

S S S<h ji S S<h ji

2 2

1 _¯ 1 i i

˜ ˜ ]

F

] ¯

G

¯

v(T<i )2v(T )5 2X 1 (y 1c 2c ) (y 1c 2c ).

T T T<h ji T T<h ji

2 2

It follows from V 55that (S, T, i )[⁄V. By Part (i) of Claim 3, we have

cS2cS<h ji 5cT2cT<h ji .

¯ ¯

The previous three expressions and the fact that XT$X lead toS

˜ ˜ ˜ ˜

v(S<i )2v(S )#v(T<i )2v(T ).

This finishes the proof. Q.E.D

j

Claim 3. Let cS5Minhc uj[Sjdenote the marginal cost of the most efficient firm in a

coalition S. Then for any coalitions S,T,N, and any firm i[N\T, the following three

inequalities hold:

(i) cS2cS<hij$cT2cT<hij$0;

2 2 2 2

(ii) cS2cS<hij2(cT2cT<hij)$(cT1cT<hij)[cS2cS<hij2(cT2cT<hij)];

2 2 2 2

(iii) cS2cS<hij2(cT2cT<hij)$0.

Proof of Claim 3. Part (iii) follows from (i) and (ii). We first prove part (i). If S55,

j 7

then cS5Minhc uj[5j5 1`, and Part (i) holds. Thus, we only need to prove it for

i

S±5. Given this, only the following three possible cases can happen. Case 1. c,cT$

i i i

c . Case 2. cS T#cS%c . Case 3. cT#c,c . In case 1, cS S<hij5cT<hij5c , thus (i) follows

from cT#c ; In case 2, cS S5cS<hij, and cT5cT<hij, thus cS2cS<hij5cT2cT<hij50, and

i

(i) holds; In case 3, cT2cT<hij50, and cS2cS<hij5cS2c.0, thus cS2cS<hij.cT2

cT<hij50.

2 2 2 2

Part (ii) follows from the equation cS2cS<h ji 2(cT2cT<h ji )5(cS1cS<h ji )(cS2

cS<h ji )2(cT1cT<h ji )(cT2cT<h ji ) and the facts that cS1cS<hij$cT1cT<hij, and cS2

cS<hij$0. Q.E.D

Proof of Lemma 5. There is nothing to prove ifV 55. Now suppose

V 5

h

(S, T, i )uS,T,N, i[N /T, and cS2cS<h ji .c cT S<h ji

j

±5.

For any given (S, T, i )[V,

7

2 2 2 2 i j

¯ ¯

f(S, T, i )5cS2cS<h ji 2(cT2cT<h ji )12y

S

O

y 1cS2cT

D

j[T / S

j j

¯ ¯

12

F

O

y (cS2cS<h ji )2

O

y (cT2cT<h ji )

G

j[⁄ S, j±i j[⁄ T, j±i

2 2 2 2 i j

¯ ¯

5cS2cS<h ji 2(ct2cT<h ji )12y

S

O

y 1cS2cT

D

j[T / S

j j

¯ ¯

12

HF

O

1

O

G

y (cS2cS<h ji )2

O

y (cT2cT<h ji )

J

j[⁄ T, j±i j[T / S j[⁄ T, j±i

2 2 2 2 j

¯

5cS2cS<h ji 2(cT2cT<h ji )12

O

y [cS2cS<h ji 2(cT2cT<h ji )]

j[⁄ T, j±i

j i i

¯ ¯ ¯

12

O

y [y 1cS2cS<h ji ]12y (cS2C ).T

j[T / S

Clearly,

i

¯y 1cS2cS<h ji $cS2cS<h ji 2(cT2cT<h ji ); and

cS2cT5cS2cS<h ji 1cS<h ji 2cT$cS2cS<h ji 1cT<h ji 2cT

5cS2cS<h ji 2(cT2cT<h ji ).

It follows from the above two expressions and Part (ii) of Claim 3 that

f(S, T, i )$(cT1cT<h ji )[cS2cS<h ji 2(cT2cT<h ji )]

j j

¯ ¯

12

O

y [cS2cS<h ji 2(cT2cT<h ji )]12

O

y [cS2cS<h ji

j[⁄ T, j±i j[T / S

i

¯

2(cT2cT<h ji )]12y [cS2cS<h ji 2(cT2cT<h ji )]

j

¯

5[cT1cT<h ji 12

O

y ][cS2cS<h ji 2(cT2cT<h ji )].

j[⁄ S

This leads to

f(S, T, i ) j

]]]]]]]] ¯

F(S, T, i )5_{2[c 2c 2(c 2c )]}$

F

cT1cT<h ji 12

O

y

G

Y

2

S S<h ji T T<h ji j[⁄ S

j

¯

5(cT1cT<h ji ) / 21

O

y .

j[⁄ S

j

¯

v 5Min F(S, T, i )

h

u(S, T, i )[V $

j

Min (c

H

T1cT<h ji ) / 21

O

y u(S, T, i )[V

J

j[⁄ S

.0. Q.E.D

The above results and claims lead directly to a proof for the main theorem.

Proof of Theorem 3. Let (S, T, i )[V, then cS2cS<hij.cT2cT<hij. Substituting

2

1

2 j

¯

˜ ] ¯

v(S )5X 5

S

a2

O

y 2c

D

S ₄ S

j[⁄ S

˜ ˜ ˜ ˜

into the expressions for v(S<i )-v(S )-(v(T<i )-v(T )), we have

2 j

˜ ˜ ˜ ˜ ¯

4[v(S<i )2v(S )2(v(T<i )2v(T ))]5

HS

a2

O

y 2c

D

S<h ji j[⁄ S, j±i

2 2 2

j j j

¯ ¯ ¯

2

S

a2

O

y 2cS

D J HS

2 a2

O

y 2cT<h ji

D S

2 a2

O

y 2cT

D J

j[⁄ S j[⁄ T, j±i j[⁄ T

5h2a[cS2cS<h ji 2(cT2cT<h ji )]2f(S, T, i )j

52[cS2cS<h ji 2(cT2cT<h ji )] ah 2F(S, T, i ) ,j

where f(S, T, i ) and F(S, T, i ) are given in (17) and (18). It follows from

cS2cS<h ji .cT2cT<h ji

and the above expression that

˜ ˜ ˜ ˜

[v(S_<i )2v(S )2(v(T_<i )2v(T ))]#0⇔a#F(S, T, i )⇔a#v

5Min F(S, T, i )

h

u(S, T, i )[V

j

.

Thus the TU game (14) is convex if and only if ‘‘a#v.’’ Q.E.D

5. Extensions and concluding remarks

We have established a necessary and sufficient condition for the convexity of oligopolistic TU games arising from linear industries. The condition is characterized by

the v-value which measures increasing returns to coalition size. The game is convex if

and only if ‘‘v’’ is above the intercept ‘‘a’’ of the inverse demand (i.e. there are enough

increasing returns to coalition size).

Our necessary and sufficient characterization is obtained by using the precise and simple formulae for each v(S ), which are made possible by Assumptions 1–3 in linear markets. It remains as a challenge to find necessary and sufficient condition for supermodularity in more general markets because v(S ) might become very complicated. One possible topic is to relax the interior solution assumption to see if convexity also exists with boundary conditions. Another possible topic is to extend the main result to

2

some special nonlinear cost structure like C(x)5d1cx .

Acknowledgements

´

I would like to thank Tatsuro Ichiishi, Mamoru Kaneko, Herve Moulin, two anonymous referees, and workshop participants at the 95 Midwest mathematical economics and trade theory meeting at U. Minnesota and 95 Southeast economic theory and trade conference at SMU for valuable comments and suggestions. I would also like to thank the Ohio State University for the seed grant support. All errors, of course, are my own.

References

Aumann, R., 1959. Acceptable points in general cooperative n-person games. In: Tucker and Luce (Eds.), Contributions to the Theory of Games IV. Annals of Mathematics Studies, vol. 40. Princeton University Press, Princeton.

Curiel, I., Pederzoli, O., Tijs, S., 1989. Sequencing games. European Journal Operational Res. 40, 344–351. Driessen, T., 1986. Solution concepts of the k-convex n-person games. International Journal of Game Theory

15, 201–229.

Driessen, T., Tijs, S., 1985. Thet-value, the core and semiconvex games. International Journal of Game Theory 14, 229–248.

Edmonds, J., 1970. Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., ¨

Schonheim, J. (Eds.), Proceedings of the Calgary International Conference on Combinatorial Structures and their Applications. Gordon and Breach, New York.

Fujishige, S., 1991. Submodular functions and optimization theory. Annals of Discrete Mathematics, No. 47, North Holland, Amsterdam.

Granot, D., Hojati, M., 1990. On cost allocation in communication networks. Networks 20, 209–229. Granot, D., Huberman, G., 1982. The relationship between convex games and minimal cost spanning tree

games: a case for permutationally convex games. SIAM Journal Algebra Discrete Math. 3, 288–292. Ichiishi, T., 1981. Super-modularity: Applications to convex games and to the greedy algorithm for LP. Journal

of Economic Theory 25, 283–286. ˜

Inarra, E., Usategui, J., 1993. The Shapley value and average convex game. International Journal of Game Theory 22, 13–29.

Maschler, M., Peleg, B., Shapley, L.S., 1972. The Kernel and bargaining set for convex games. International Journal of Game Theory 1, 73–93.

McLean, R., Sharkey, W., 1993. Review of Fujishige’ Submodular functions and optimization theory. Bulletin of the American Mathematical Society 29, 98–104.

Moulin, H., 1988. Axioms of Cooperative Decision Making. Cambridge University Press, Cambridge. Qi, L., 1988. Odd submodular functions, Dilworth functions and discrete convex functions. Mathematics of

Operations Research 13, 435–446. ¨

Schmeidler, D., 1969. The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics 17, 1163–1170.

Shapley, L., 1953. A value for N-person games. In: Kuhn and Tucker (Eds.), Contributions to the Theory of Games II, Annals of Mathematics Studies, Vol. 28. Princeton University Press, Princeton.

Shapley, L., 1971. Core of convex games. International Journal of Game Theory 1, 11–26.

Sharkey, W., 1981. Convex games without side-payments. International Journal of Game Theory 11, 101–106. Sprumont, Y., 1990. Population monotonic allocation schemes for cooperative games with transferable utility.

Games and Economic Behavior 2, 378–394.

Vilkov, V., 1977. Convex games without side payments (in Russian). Vestnik Leningradskiva Universitata 7, 21–24.