In this paper we present a new definition related to super-balanced games interpreted as a particular type of cooperative game with transferable utility (TU). A concept of solution is defined and an existence result is determined. The fundamental aim of the paper is to extend the results previously obtained by the same author.TU cooperative games, super-balanced coalition

Massimiliano Ferrara

English

Research Papers in Economics

BADANIA OPERACYJNE I DECYZJE
Nr 2 2009
Massimiliano FERRARA*
MORE ON ONE-COMMODITY MARKET GAMES
In this paper we present a new definition related to super-balanced games interpreted as a par-
ticular type of cooperative game with transferable utility (TU). A concept of solution is defined and
an existence result is determined. The fundamental aim of the paper is to extend the results previ-
ously obtained by the same author.
Keywords: TU cooperative games, super-balanced coalition
1. Introduction
Cooperative game theory has been frequently used to model various economic
scenarios. As is well-known, economic systems have been treated as cooperative
games, with the core of an economy commonly used as the main concept of solution.
Some specific economic scenarios have been modeled within a cooperative/competitive
framework. Shapley and Shubik [9] derived the “market game” from an exchange
economy. “Production games” are used to study production problems in a competitive
environment [7]. Cost allocation problems have also been modeled as cooperative
games by several authors [1], [4] and [5]. Our approach concerns a particular market
with several agents who trade one commodity. Each bundle of the commodity sold by
an agent has a social value which includes all the relevant investments, costs and ex-
pected profit. The agent should trade a bundle of his commodity at a price which re-
flects this value. Here, we focus on the problem of the stability of prices when they
are negotiated by super-balanced coalitions. This paper develops some results previ-
ously obtained by the same author [3].
                                                     
*  Department of Historical, Law, Economics and Social Sciences, University Mediterranea of Reggio
Calabria, Via dei Bianchi, 2, Palazzo Zani, 89127 Reggio Calabria, Italy, e-mail: massimiliano.ferrara@unirc.itM. FERRARA 30
2. Notation and definitions
Let be N = {1, 2, ..., n} the set of agents. 
n R+  denotes the positive orthant of
n-dimensional Euclidean space. If x = (x1, ..., xn), y = (y1, ..., yn) ∈ R
n we shall write
x ≥ y, if xi ≥ yi  for all i = 1, 2, ..., n, and x > y, if x ≥ y, but x ≠ y. If x ∈ 
n R+  and
} ..., , 2 , 1 { n S ⊆ , one denotes  the n-vector whose i-th component is xi if i ∈ S and 0 if
i ∉ S by xS. Finally, let S  denote the complement of S with respect to {1, 2, ..., n}.
Let G be the family of subsets of the set N = {1, 2, ..., n}.
Definition 1. G is said to be super-balanced if there exists a system of positive
weights (wS)S∈G such that
1
,
≥ ∑
∈ ∈ S i G S
S w ,  N i∈ ∀ .( 1 )
Let us consider  } ..., , { 1 n S S G =  as a collection of non-empty subsets of N = {1, 2,
..., n}. A family S is said to be balanced if there exists a system of positive weights
(wS)S∈G  such that  1 ,
, = ∈ ∀ ∑ ∈ ∈ S i G S S w N i .
Remark 1. A super-balanced family of coalitions is simply a covering of the
grand coalition N.
3. The positive core of a TU cooperative game
Starting from classical results in Game Theory, it is known that a transferable
utility (TU) cooperative game can be represented by the pair (N,a), where a: 2
N → R
is the characteristic function. The properties required for a depend on the interpreta-
tion of the game. To analyse various economic scenarios, it might be useful to con-
sider a(S) to be the maximum total revenue of the players in S, but alternatively we
can interpret a(S) as the minimum total cost (penalty) incurred by S.
For this latter case, it is natural and convenient to reverse the inequalities tradi-
tionally used in cooperative game theory. Therefore, the core C(N, a) of the game
(N, a) will be defined as
⎭
⎬
⎫
⎩
⎨
⎧
= ⊂ ∀ ≤ ∈ = ∑∑
∈S i
i i
n N a y N S S a y R y a N C ) ( , ), ( | ) , ( .( 2 )More on one-commodity market games 31
From the Bondareva–Shapley Theorem [2], [8], the core is non-empty if and only
if the game is balanced. In this paper we study the case in which there exist super-
balanced coalitions.
Definition 2. A game is said to be super-balanced, if for every super-balanced
family 
N G 2 ⊆ , and for every associated system of weights  G S S w ∈ ) (  one has
) ( ) ( N a S a w
G S
S ≥ ∑
∈
.( 3 )
4. The model
We want to consider the realistic situation of a market in which several agents
trade the same type of commodity. Each agent sells the amount of the commodity
demanded by his consumers. In principle, each consumer has his own preferred agent,
but he can change if the prices of others agents become more attractive. Therefore,
each agent should satisfy the demand for his produce which is dependent on all prices.
Each group of agents (coalition) negotiates the individual prices of its members, in
order to equilibrate the value of the good sold with the total amount of money the
consumers should pay for it.
Let us introduce the following notation into our model:
A system of prices is any 
n R p + ∈ . If  ) ..., , ( 1 n p p p = , then pi is the i-th agent’s price
for a unit of the good. The demand for the good is given by a vector function
n n R R d + + → : . For 
n R p + ∈ , the component  ) (p di  represents the demand for the good
from the i-th agent given the price vector p. The symbol  . 〈 ,.〉 denotes the scalar product
of two vectors. If total demand is represented by a non-negative vector x, then the value
supported by the market is v(x), where  R R v
n → + :  is the value function. The triple
) , , ( v d N Γ =  is called the one-commodity market game [3]. Now we will define an
optimal system of prices 
n R p +
∗∈  to be one that satisfies the following conditions:
〉 〈
∗ ∗ ) ( , p d p  =  )) ( (
∗ p d v .( 4 )
Also, there are no 
n R p + ∈  and ∅ N S ⊆ ≠  such that
∗ < S S p p , 
∗ = S S p p ,( 5 )
) ( ) (
∗ ≥ p d p d S S ,( 5 ′)
〉 〈 ) ( , p d p S S  =  )) ( ( p d v S .( 5 ″)M. FERRARA 32
Condition (4) simply says that the total revenue equals the value of the total de-
mand  )) ( (
∗ p d v  at price p
*. From conditions (5′)–(5″), p
* is stable; no coalition of
agents can become more attractive to consumers by reducing their prices while si-
multaneously maintaining their income from sales.
Definition 3. The pseudo-core of the one-commodity market game  ) , , ( v d N Γ =  is
the set of price vectors satisfying conditions (4)–(5″). This set will be denoted by  ) (Γ PC .
The market considered here comes within the framework of imperfect competition.
In fact, we know that the agents operating in such an economic framework can decide on
the price of their commodity. But when all the agents find an agreement by which it is
possible to obtain stability of the market, then we may introduce the concept of a stable
system of prices satisfying the above conditions, (4) together with (5), (5′) and (5″). In
this case, when these conditions are satisfied the market considered is at equilibrium.
A Case study. Let us consider a Cournot–Bertrand oligopoly, where agent i sells
the good at price pi, i = 1, 2, ..., n. The demand for the i-th agent’s good di depends on
the price vector p = (p1, ..., pn), and the cost function (including a reasonable profit) is
linear with respect to demand, i.e. ci￿ di(p), where ci > 0. According to our terminol-
ogy (N,  d,  v) is a one-commodity market game, where  R R v
n → + :  is defined by
∑ ∈ =
N i i ix c x v ) ( . From condition (4) it follows that  ) ..., , (
* *
1
*
n p p p =  is a stable sys-
tem of prices if
) ( ) (
* * * p d c p d p i
N i
i i
N i
i ∑ ∑
∈ ∈
=
and
S ⊂ N, 
*
S S p p <  ⇒  ) , ( ) , (
* *
S S i
S i
i S S i
S i
i p p d c p p d p ∑ ∑
∈ ∈
< .
In particular, it follows that
∀i ∈ N, 
*
i i p p < ⇒ ). , ( ) , (
*
} {
*
} { i i i i i i i i p p d c p p d p ⋅ < ⋅
Assuming that the demand functions are strictly positive, it clearly follows that
) ..., , ( 1
*
n c c p =  is a stable system of prices.
Let us introduce the following assumptions useful in proving the central result:
(a) For every  , N i∈  di is a continuous, strictly positive and decreasing function, i.e.
) ( ) ( y d x d y x i i < ⇒ > ,
] [
n
i R C d + ∈ ,
0 ) ( > x di .More on one-commodity market games 33
(b) The value function v is continuous, non-negative and increasing, i.e.
) ( ) ( y v x v y x > ⇒ > ,
] [
n R C v + ∈ ,
0 ) ( > x v .
(c) For every 
n R x + ∈ , for every super-balanced family 
N G 2 ⊆ , and for every as-
sociated system of weights  G S S w ∈ ) ( ,
) ( ) ( x v x v w S
G S
S ≥ ∑
∈
.
(d) There exists a positive constant M, such that  M p d p d v i i ≤ ) ( / )) ( ( , for every
N i∈  and all 
n R p + ∈ .
(e) The scalar product  〉 〈 ) ( , p d p , is a non-decreasing function of p on 
n R+ .
Remark 2. What is actually restrictive is the requirement in assumption (c) that
certain inequalities should be satisfied by the associated weights. It is easy to see
that, with respect to these inequalities, only minimal systems of weights are inter-
esting.
At this point we can introduce the following:
Theorem 1. Let  ) , , ( v d N = Γ  satisfy assumptions (a)–(e). Then,  ≠ Γ) ( PC Ø.
The proof of this theorem makes use of the following lemma.
Lemma 1. Let  ) , ( a N  be a transferable utility cooperative game with characteris-
tic function  R a
N → 2 : . Suppose that a has strictly positive values and
) ( ) ( S a w N a
G S
S ∑
∈
≤ (6)
for every super-balanced family G and associated system of weights  G S S w ∈ ) ( .
It follows that the positive core 
n R v N C v N C +
+ = I ) , ( ) , (  of the game  ) , ( v N  is
nonempty.
Proof:
⎪ ⎭
⎪
⎬
⎫
⎪ ⎩
⎪
⎨
⎧
= ⊂ ∀ ≤ ∈ = ∑∑
∈
∈ +
+
S i
N i i i
n N a y N S S a y R y a N C ) ( , ), ( ) , ( .M. FERRARA 34
As in the proof of the Bondareva–Shapley theorem regarding the balanced core,
one observes that  ) , ( a N C
+  consists of all the optimal solutions of the linear pro-
gramming problem
⎪ ⎭
⎪
⎬
⎫
⎪ ⎩
⎪
⎨
⎧
⊆ ∀ ≤ ∈ ∑∑
∈∈
+
N iS i
i
n
i N S S a y R y y L ), ( , max : ) ( ,
provided that the optimal value of (L) is  ) (N a .
Let (D) be the dual of (L), i.e.
min : ) (D
⎪ ⎭
⎪
⎬
⎫
⎪ ⎩
⎪
⎨
⎧
∈ ∀ ≥ ∈ ∑∑
∈ ∈
+
N
N
S S i
S S N i w R w S a w
2
2 , 1 , ) ( .
Since  N S S a ⊆ ∀ > , 0 ) ( , then the problem (L) has feasible solutions.
Since  ) (N a y
N i i ≤ ∑ ∈ , then it has optimal solutions and its maximum value is
bounded above by  ) (N a . In order to prove the lemma, it is sufficient to show that the
minimum value in (D) is exactly  ) (N a .
Setting  1 = N w ,  N S wS ≠ ∀ = , 0 , one obtains a feasible solution of (D). Hence, the
minimum value of (D) does not exceed  ) (N a .
Now, let  N S S w w
2 ) (
∈ =  be any feasible solution of (D). Set  } 0 | { > = S w S G . Then G
is super-balanced and  G S S w ∈ ) (  is a system of weights. From (3) one obtains
) ( ) ( ) (
2
N a S a w S a w
G S
S
S
S
N
≥ =∑ ∑
∈ ∈
.
Therefore, the minimum value of (D) is  ) (N a .
Now let us denote  } , | { N i M p R p P i
n
M ∈ ∀ ≤ ∈ = + .
For every 
n R p + ∈  consider the game  ) , ( p a N  whose characteristic function is de-
fined by  )) ( ( ) ( p d v S a s p = ,  N S ⊆ ∀  and define the correspondence (the set-valued
function) ϕ from PM to 
n R+  by
)} , ( )) ( ( | { ) ( p N i i i
n a N C p d q R q p
+
∈ + ∈ ∈ = ϕ .
Lemma 2. Let  ) , , ( v d N Γ =  satisfy assumptions (a)–(d). It follows that ϕ is
a closed correspondence from PM to itself, with non-empty, compact and convex
values.
Proof: Obviously, for  0 ) ( , > ∈ p d P p S M  and  0 ) 0 ( )) ( ( ≥ > v p d v S .More on one-commodity market games 35
If we apply Lemma 1, assumption (c) is satisfied and taking into account that
≠ ) (p ϕ Ø for every  M P p∈ , if  ) (p q ϕ ∈ , then  ) , ( )) ( ( p N i i i v N C p d q
+
∈ ∈ ,  0 ≥ q  and
N i p d v p d q i i i ∈ ∀ ≤ )), ( ( ) (  (from the definition of the core). Hence, from (d) it fol-
lows that  M P q∈ , since  ) (p ϕ  is clearly convex.
Let us now show that the correspondence ϕ is closed.
Consider the convergent sequences  ,
0 p p
k →   M
k P p ⊆ ) ( ,  ,
0 q q
k →
k p q
k k ∀ ∈ ), ( ϕ .
For every k one has
∑ ∑
∈ ∈
= ⊂ ∀ ≤
N i
k k
i
k
i
k
S
k
S i
i
k
i p d v p d q N S p d v p d q )) ( ( ) ( , )), ( ( ) ( .
From the continuity of di and v, it follows that:
= ⊂ ∀ ≤ ∑
∈
, , )) ( ( ) (
0 0 0 N S p d v p d q
S i
S i i ∑
∈N i
i i p d v p d q )) ( ( ) (
0 0 0 .
Since  , 0
0 ≥ q  it follows that  ) (
0 0 p q ϕ ∈ . Finally, the compactness of  ) (p ϕ  fol-
lows from the closedness of ϕ and the compactness of PM.
Proof of Theorem 1
From lemma 2, the correspondence 
M P
M P 2 : → ϕ  satisfies all the assumptions of
Kakutani’s fixed point theorem. Let p
* be any fixed point, i.e.  ) (
∗ ∗∈ p p ϕ .
We will show that  ) (Γ ∈
∗ PC p .
It follows easily that  0 ≥
∗ p  and
〉 〈
∗ ∗ ) ( , p d p )) ( ( ) (
∗ ∗
∈
∗ = =∑ p d v p d p i N i i  ,
so (4) is satisfied.
Now let us suppose that there exists 
n R p + ∈  satisfying (5)–(5″), for some S. Since
* p p < , it follows that  , ), ( ) ( N i p d p d i i ∈ ∀ >
∗  so
〉 〈 ) ( , p d p S S () = ≥ > =
∗
∈
∗ ∗ ∑ ) ( )) ( ( ) ( p d p p d v p d v i
S i
i S S 〉 〈
∗ ∗ ) ( , p d p S S .( 7 )
From assumption (e), it follows that
〉 〈 ) ( , p d p 〉 〈 ≤
∗ ∗ ) ( , p d p .
Firstly, it follows from the monotonicity of  S i di ∈ , , see assumptions (a) and (e),
thatM. FERRARA 36
〉 〈 ) ( , p d p 〉 〈 = ) ( , p d p S S + 〉 〈 ) ( , p d p S S
〉 〈 =
∗ ) ( , p d p S S + 〉 〈 ) ( , p d p S S 〉 〈 ≥
∗ ∗ ) ( , p d p S S + 〉 〈 ) ( , p d p S S ,
and secondly one has
〉 〈
∗ ∗ ) ( , p d p = 〉 〈
∗ ∗ ) ( , p d p S S + 〉 〈
∗ ∗ ) ( , p d p S S .
From the last three relations, it follows that
≥ 〉 〈
∗ ∗ ) ( , p d p S S 〉 〈 ) ( , p d p S S ,
which contradicts (7).
5. Conclusions
A deeper understanding of the concept of pseudo-core could result from the fol-
lowing comments. Let us associate the one-commodity market game G = (N, d, v) with
the non-transferable utility (NTU) game represented by (N, V), where the coalitional
function V is defined by
} )) ( ( ) ( , | { ) ( p d v p d p R p S V S S S
n ≥ 〉 〈 ∈ = + ,
if ∅ ≠  N S ⊂  and
} )) ( ( ) ( , | { ) ( p d v p d p R p N V
n = 〉 〈 ∈ = + .
We can observe that the pseudo-core of G contains the core of (N, V). Thus, in
particular, every system of prices belonging to the core of the game (N, V) is stable.
References
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Więcej o grach rynkowych z jednym dobrem
Teoria gier kooperacyjnych jest często używana do modelowania różnych scenariuszy ekonomicz-
nych. Podstawowym pojęciem w modelowaniu tych scenariuszy jest pojęcie jądra jako rozwiązania takich
gier. Niektóre specyficzne scenariusze modelowane były w ramach teorii gier kooperacyjnych/konku-
rencyjnych. Shapley i Shubik w 1969 roku otrzymali „grę rynkową” jako wynik modelowania ekonomii
wymiany; Owen w 1975 roku wprowadził „grę produkcyjną”, która może służyć do badania problematyki
produkcji w środowisku konkurencji ekonomicznej; zagadnienie alokacji kosztów było z kolei modelowane
jako gra kooperacyjna przez różnych autorów (Bendali F. i in. [1], Gambarelli G. [4]) czy Granot D. i in.
[5]). Nasze podejście dotyczy specyficznego rynku z kilkoma agentami, którzy handlują jednym dobrem.
Każdy pakiet sprzedawanego przez agenta dobra posiada wartość społeczną, zawierającą odpowiednio
inwestycje, koszty i spodziewane zyski. Dany agent może sprzedawać pakiet dobra po cenie, która od-
zwierciedla tę wartość. W pracy, która jest rozwinięciem niektórych wyników otrzymanych wcześniej
przez Ferrarę [3], skupiamy swoją uwagę na problemie stabilności ceny negocjowanej przez super-
zrównoważone koalicje.
Słowa kluczowe: gry kooperacyjne z transferowalną użytecznością, superzrównoważona koalicja

A Cooperative study of one-commodity market games,

A generalization of Brouwer’s fixed point theorem,

Giochi Competitivi e Cooperativi,

On Balanced Sets and Cores,

On the Core and Nucleolus of Minimum Cost Spanning Tree Games,

On the Core of Linear Production Games,

Some Applications of Linear Programming Methods to the Theory of Games, Problemi Kibernetiki,

Tarification par des jeux cooperatifs avec demandes elastiques,

More on one-commodity market games

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