Envelopes and Geometrical Covers of Side-Payment Games and Their Market Representations

TLDR: This paper deals with issues concerning the core as a solution concept for games in coalitional form as well as the use of these games in representing economies of a certain formal type.

Abstract: This paper deals with issues concerning the core as a solution concept for games in coalitional form as well as the use of these games in representing economies of a certain formal type. Side-payment games are imbedded in the more general class of no-side-payment games. It is shown that to a given side-payment game having an empty core one may associate two different no-side-payment games with the same nonempty core: the “envelope” and the “geometrical cover.” The discrepancy is explained in terms of market games.

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Envelopes and Geometrical Covers of Side-Payment
Games and their Market Representations
Philippe Artzner, Claude d’Aspremont, Louis-André Gérard-Varet
To cite this version:
Philippe Artzner, Claude d’Aspremont, Louis-André Gérard-Varet. Envelopes and Geometrical Covers
of Side-Payment Games and their Market Representations. Mathematics of Operations Research,
INFORMS, 1986, 11 (1), pp.19-29. �10.1287/moor.11.1.19�. �hal-00950764�

Envelopes and Geometrical Covers of Side-Payment Games and
their Market Representations
∗
Philippe Artzner
†
, Claude d’Aspremont
‡
and Louis-Andr´e G´erard-Varet
§
Received September 27, 1982; revised July 5, 1984
Abstract
This paper deals with issues concerning the core as a solution concept for games in
coalitional form as well as the use of these games in representing economies of a certain formal
type. Side-payment games are imbedded in the more general class of no-side-payment games.
It is shown that to a given side-payment game having an empty core one may associate two
different no-side payment games with the same (nonempty) core: the “envelope” and the
“geometrical cover”. The discrepancy is explained in terms of market games.
Key-words: Side-payment games, core, market games
AMS 1980 subject classification: Primary: 90D12
IAOR 1973 subject classification: Main: Games
OR/MS Index 1978 subject classification: Primary: 234 Games/group decisions/cooperative
∗
Reprinted from Mathematics of Operations Research, 11(1), 19–29, February 1986.
†
Institut de Recherches Math´ematiques Avanc´ees, Universit´e Louis Pasteur
‡
Center for Operations Research and Econometrics, Louvain-la-Neuve
§
Universit´e des Sciences Sociales de Toulouse and GREQE.
1

Introduction
The purpose of this paper is to clarify some issues concerning the core as a solution concept for
games in coalitional form, as well as the representation of these games by economies of a certain
formal type. Our main interest is in games with side-payments in coalitional form. However,
we shall imbed their study in the more general theory of games without side-payments.
It is well known that some games with side-payments may have an empty core. Following
Shapley and Shubik [9] one associates to such a game its totally balanced cover, which is a
game of the same type having a nonempty core. On the other hand, one may also look at the
original side-payment game from a geometrical point of view and consider it as some particular
(hyperplane) game without side-payments. This in turn allows one to use the notion of totally
balanced cover of a game without side-payments as introduced by Billera and Bixby [4]. Hence
one is actually lead to contrast two types of “covering” for a side-payment game: namely the
totally balanced cover of its geometrical representation as a game without side-payments – we
call it the envelope – and the geometrical no-side-payment representation of its totally balanced
side-payment cover, called the geometrical cover (§1). Although these two no-side-payment
totally balanced games associated to the same side-payment game may differ, their cores always
coincide (§2).
There is an “economic interpretation” for such a discrepancy. We consider a modification of
the notion of “direct market” of Shapley-Shubik [9] that we call “restricted direct market”. We
show that the envelope of a side-payment game is the game without side-payments associated
to a restricted market, while the geometrical cover is associated to a direct market (§3). The
essential difference b etween these two markets is whether or not a player can obtain utility from
a coalition without being a member. Finally (in §4), we compare this new type of market (which
is still a pure exchange economy) to the production representation of Billera [3]. In fact we show
that an “input net trade equivalent” of this production representation is the same (up to a trivial
identification) as the Rader net trade equivalent of the restricted direct market (see [6]).
2

1 A no-side-payment viewpoint to games with side-payments
In this section we give the basic definitions and notation. For a set N = {1, 2, · · · , i, · · · , n} of
“players” we consider games with side-payments in coalitional form as real-valued functions v
defined on P (N), the set of all nonempty subsets of N, called “coalitions”. To simplify and unify
their economic representation we shall consider only 0-normalized games, i.e. games v such that
∀ i ∈ N, v({i}) = 0, and impose the weak monotonicity condition that: ∀ S ∈ P (N), v(S) ≥ 0.
For any game v the core of v is the set
Core(v)
def
= {u ∈ R
N
; hu; 1I
N
i = v(N), ∀ S ∈ P (N), hu, 1I
S
i ≥ v(S)}
where 1I
S
∈ {0, 1}
N
is the characteristic function
1
of S ∈ P (N). This can be interpreted as the
set of payoff vectors feasible for the grand coalition which cannot be objected against by other
coalitions.
Define for T ∈ P (N), the set B
T
of balanced families of coefficients on T by:
B
T
def
=



β ∈ R
P (T )
+
;
X
S∈P (T )
β(S)1I
S
= 1I
T



. (1.1)
To a game v, we associate, as in Shapley-Shubik [9], the game
v, called the totally balanced cover
of v which is:
v(T )
def
= sup
γ∈B
T
X
S∈P (T )
γ(S)v(S), T ∈ P (N). (1.2)
The game v is said to be balanced if and only if
v(N) = v(N), and totally balanced if and only
if v = v.
Define the sets A
def
= {u ∈ R
N
; ∀ S ∈ P (N), hu, 1I
S
i ≥ v(S)}, and B
def
= {u ∈ R
N
; ∀ S ∈
P (N ), hu, 1I
S
i ≥ v(S)}. We have A = B. The result follows from the inequality v ≥ v (see (1.2))
and from the fact that for each T ∈ P (N ) and δ ∈ B
T
such that
v(T ) =
P
S∈P (T )
δ(S)v(S),
we clearly have (see (1.1)): hu, 1I
T
i =
P
S∈P (T )
δ(S)hu, 1I
S
i. Consider now the two following
1
For any S ∈ P (N ), we have 1I
S
(i) = 1, if i ∈ S, and 1I
S
(i) = 0 otherwise. Vectors in R
N
are called “payoff
vectors” and hu, 1I
S
i =
P
i∈S
u
i
may be interpreted as the “total amount obtained by the coalition” S.
3

programs:
min
u∈A
hu, 1I
N
i and (1.3)
min
u∈B
hu, 1I
N
i. (1.4)
By looking at the dual of (1.3), one finds that each optimal solution u of (1.3) fulfills hu, 1I
N
i =
v(N). Therefore the optimal solutions of (1.4), as well as the optimal solutions of (1.3), are
exactly the elements
2
of Core(
v). Moreover, we have Core(v) 6= ∅ if and only if v is balanced,
which is the well-known Bondareva-Shapley Theorem (see [8]).
A more general class of games in coalitional form for a set N of players is the class of games
without side-payments. Let R
S
def
= {u ∈ R
N
; ∀ i /∈ S, u
i
= 0} and R
S
+
def
= {u ∈ R
S
; ∀ i ∈ S, u
i
≥
0}. A game without side-payments is a correspondence V from P (N) to R
N
such that, for each
S ∈ P (N ), V (S) is nonempty, comprehensive (i.e. V (S) = V (S)−R
S
+
) and, for e ach a
S
∈ V (S),
V (S) ∩ ({a
S
} + R
S
+
) is compact. In connection to the market representation of games without
side-payments we shall only consider games which are compactly generated
3
namely games V
such that, ∀ S ∈ P (N), V (S) = C
S
− R
S
+
, where C
S
is a nonempty compact subset of R
S
.
The Core of a game V without side-payments is the set:
Core(V )
def
= {u ∈ V (N) : ∀ S ∈ P (N), 6 ∃u
′
∈ V (S) | ∀ i ∈ S, u
′
i
> u
i
}.
Billera [3] defines (see his definition (3.1))
4
the totally balanced cover
V of a game without
side-payments V by:
V (T )
def
=
[
δ∈B
T
X
S∈P (T )
δ(S)V (S), T ∈ P (N). (1.5)
A game V is said to be totally balanced if and only if
V = V . Such a game has a nonempty core
(see [7]).
2
To each game v, one may associate the game v
∗
, called the balanced cover of v and defined by: v
∗
(N) =
v(N),
and ∀ S 6= N , v
∗
(S) = v(S). Clearly by the same argument we have Core(v
∗
) = Core(
v).
3
This terminology is due to Billera [3].
4
The notion was also discussed in Baudier [2] and originally used by Scarf in the prepublication version of
[7]. As mentioned in Billera and Bixby [4] it is stronger than the notion of Scarf [7], which makes no use of the
summation operator. Since we will be dealing exclusively with games without side-payments derived from games
with side-payments, it seems that most objections against using such an operator vanish.
4

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Envelopes and geometrical covers of side-payment games and their market representations" ?

This paper deals with issues concerning the core as a solution concept for games in coalitional form as well as the use of these games in representing economies of a certain formal type. 

Q2. What is the definition of a game without side-payments?

A game without side-payments is a correspondence V from P (N) to RN such that, for each S ∈ P (N), V (S) is nonempty, comprehensive (i.e. V (S) = V (S)−RS+) and, for each a S ∈ V (S), V (S) ∩ ({aS} + RS+) is compact. 

Q3. What is the reason for using an equality constraint for feasibility?

Since VE(S) ∩ R S + is convex, contained in H+(vE)(S) and contains all vectors U(1IS)1I{i} = vE(S)1I{i}, the authors have:VE = H+(vE). (3.3)Define now for τ ∈ [0, 1]N the set Bτ def = {β ∈ R P (N) + ; ∑ S∈P (N) β(S)1IS = τ}. Following Shapley and Shubik [9] associate to a game with side-payments v an n-agent, m-commodity simple market, called the direct market generated by v, which the authors denote Dv:Dv = {(Z i, wi, U)i ∈ N}, (3.4)8The justification for using an equality constraint for feasibility comes from the assumption that the utility functions are monotone increasing. 

Q4. What is the net trade equivalent of PV?

The net trade equivalent Eq[PV ] generates a game without side-payments, analogously to(3.2) and (4.1), as follows:VEq[PV ](S) ={u ∈ RS : ∀ i ∈ S, ui ≤ U i ∗(z i ∗), z i ∗ ∈ Z i ∗,∑i∈Szi∗ = 0}. 

Q5. What is the core of a game without side-payments?

The Core of a game V without side-payments is the set:Core(V ) def = {u ∈ V (N) : ∀S ∈ P (N), 6 ∃u′ ∈ V (S) | ∀ i ∈ S, u′i > ui}. 

Q6. What is the net trade equivalent of PH+?

In particular, given a restricted direct market Rv, one obtains the n-agent net trade Rader equivalent:Req[Rv] = {(T i, U i∗), i ∈ N},where T i, i’s set of admissible net trades, is simply T i = [0, 1]n − 1I{i} and U i ∗ is defined by letting, for ti ∈ T i:U i∗(t i) = U i(ti + 1I{i}). 

Q7. What is the definition of restricted direct market?

The authors call restricted direct market associated to the game v the n-agent, n-good simple market denoted Rv and defined as:Rv = {(Z i, wi, U i); i ∈ N} (3.7)where, for every i ∈ N , Zi = [0, 1]N , wi = 1I{i},U i(τ) = sup γ∈Bτ ∑S∈P (N) S∋iγ(S)v(S) . 

Q8. What is the net trade vector of the last n goods?

Therefore the authors see that the net trade vector t i = (t i 1, t i 2, · · · , t i 2n) is of the form (0, t i), where ti ∈ ([0, 1]N − 1I{i}) is the net trade vector of the last n goods, and the feasibility constraint is simply ∑i∈N t i = 0 (in [6] all goods are traded). 

Q9. What is the simplest way to get a representation of H+(v)?

Letting x S = v(S)1IS/|S| if x S = 0 and xS = v(S)xS/〈xS , 1IS〉 otherwise, it is easy to see that x ≤ x def = ∑ S∈P (T ) γ(S)x S . 

Q10. Why is the equality constraint less justified in some cases?

Hence in some cases the equality constraint for feasibility (which is needed for their results) seems to be less justified from an economic viewpoint. 

Q11. What is the simplest way to prove that the inclusion of VRv is equal to H+?

To prove the inclusion VRv ⊂ H+(v), it is sufficient to prove, for each decomposition1IT = ∑ i∈T τ i, τ i ∈ [0, 1]n, and each decomposition τ i = ∑ S⊂T γ i(S)1IS , that the vector u of all ui = ∑ 

Q12. what is the decomposition defining i?

The decomposition defining τ i shows that:U i(τ i) ≥ ∑S⊂T S∋iγi(S)v(S) = ∑S⊂T S∋iv(S)>0γi(S)v(S)while:xi = ∑S⊂T S∋iγ(S)xSi = ∑S⊂T S∋iv(S)>0γ(S)xSi = ∑S⊂T S∋iv(S)>0γi(S)v(S).