`oRe~.~, paperi 6111111111111~IUId~In~IIINN~1 l~~1

No. 9147ON T1U; SENSITIVITY OF VON NEUMAN AND MORGENSTERN

ABSTRACf STABLE SETS: THE STABLE AND THEINDIVIDUAL STABLE BARGAINING SET r.~~-

by Joseph Greenberg

September 1991

330,~~~.2z

iSSN o924-7815

On the sensitivity of von Neum ann and Morgenstern abstract

stable sets: The stable and the individual stable bargaining set

Joseph Greenberg~

ABSTRACT

The purpose oi this paper is twofold: First, to study the properties of

the notions of the "stable` and "individual stable" bargaining sets (SBS and

ISBS). Second, to point out the sensitivity of the von Neumann and

Morgenstern (vN8M) abstract stable set to the dominance relation that is

being employed: Insisting that each member of the coalition be made better

off yields the SBS, while requiring that at least one member of the

coalition is better off and all others are not worse off yields the ISBS.

Rather surprisingly, the SBS and the ISBS may have an empty intersection.

We fully characterize both the SBS and the ISBS in 3-person games

with transferable utilities, and we also show that in ordinally convex

games these two sets coincide with the core. As a by-product we thus

derive a new proof that such games have a nonempty core.

~ I wish to thank Dov Monderer and Benyamin Shitovitz for manystimulating and enjoyable discussions, and an anonymous referee for severaluseful comments. This revisíon of McGill University w orking paper 4I90w as written during my enjoyable and fruitiul visit to CentER, Tilburg.

1

On the sensitivity of von Neumann and Morgenstern abstract

stable sets: The stable and the individual stable bargaining set

Joseph Greenberg

1 IntroductionThe purpose of this paper is twofold: First, to study the properties of

the notions of the 'stable' and 'individual stable' bargaining sets w hich are

ctosely related to Mas-Colell's (1989) except that it is required that not

only "objections', but also "counterobjections" be 'justified" or 'credible'.

Second, and perhaps more important, the paper points out the sensitivity of

the von Neumann and Morgenstern (vNBM) abstract stable set to the

dominance relation that is being employed. More specifically, we shall

distinguish between the Tollowing two dominance relationships:

(i) A coalition may object to a proposed payoff only if each oi its

members can be made better off,

(ii) lt suffice that at least one member is better ofi while all others

are at least as well off for a coalition to object to a proposed

payoff.

Condition (i) gives rise to the stable bargaining set (SBS), while

condition (ii) yields the notion of the individual stable bargaining set

(ISBS). And, rather surprisingly, it will be shown that these two dominance

relations might yield two very different vN8 M abstract stable sets. In

particular, the SBS and the ISBS may have an empty intersection, even in

2

games with transferable utilities. In fact, such is always the case in 3-

person ga mes that have an empty core.

We fully characterize both the SBS and the 1SBS in 3-person

superadditive games with transferable utilities. It turns out that, in such

games, both sets contain only Pareto optim al payoffs. In ordinally convex

games, both the SBS and the 1SBS coincide with the core. Since for the

general n-person superadditive game both sets are nonempty, (Greenberg

1990, Theorem 6.5.6), we get, as a by-product, a new proof that ordinally

convex games have a nonempty core. (See Vilkov 1977, and Greenberg 1985.)

The set of Pareto optim al payoffs in the ISBS coincides with the

'consistent bargaining set`, a solution concept that was recently suggested

by B. Dutta, 0. Ray, K. Sengupta, and R. Vohra (1989). They show, however,

that the consistent bargaining set might be empty (for 4-person ga mes with

transferable utilities. See section 6.) In contrast, the SBS in their example

does contain Pareto optimal payoffs. Whether this is atways the case in

superadditive games with transferable utilities, remains an open question.

The notions of the SBS and the ISBS were first suggested within the

framework of the theory of social situations (Greenberg 1990). But, this

paper will be cast completely within classical game theory, making use of

the concept of vN8M abstract stable set. This notion w as introduced by

vNBM in a few pages at the end of the second edition (in 1947) of their

classical book. They of fered it purely as a mathematical extension of `the

vN8M solution`; they neither motivated it, nor suggested an application ot

it. In contrast, the theory of social situations stemmed from the basíc

question of "rational choice`. Shitovitz (Greenberg 1990, Theorem 4.5)

observed that one of the concepts in this theory (specifically, an

"optimistic stable standard of behavior") can formally be associated with a

3

vNBM abstract stable set. Using this result, some of the better known game-

theoretic solution concepts~, as well as interesting new solution concepts2,

were derived from the unique vNBM abstract stable sets that correspond to

different negotiation processes (Greenberg 1990).

Following the associate editor's suggestion, the reader is referred to

Greenberg (1990) for the motivation behind the abstract systems that define

the SBS and the ISBS, as well as for definitions of well-known game-

theoretic concepts.

~ Such as: The core and the vNBM solution in cooperative games, coalition-

proof Nash equilibrium in normal form games and the set of subgame perfect

equilibria in extensive form games.

2 5uch as: Refinements of subgame perfect equilibria, equilibria when either

contingent threats or irrevocable commitments can be made by either an

individual or a group oi individuals.

4

2 The stable and individual stable bargaining sets

This section provides the form al definitions of the stable and the

individual stable bargaining sets, and studies some of their properties.

Let N be a finite none m pty set, and let R be the set of all real

numbers. A coalition is a nonemptu subset of N. For a coalition SCN, RS

denotes the S-dimensional Euclidean space. If xeRN and S is a coalition,

then xS denotes the restriction of x on S. Let S be a coalition and let

xS,ySeRS. Then, xS?yS if x~?yi for all ieS ; xS~yS if xS?yS but xS:yS ; and

xS~~yS if x~~yi ior all ieS. Recall the following definition:

Definition 21: An n-person qame in characteristic Iunction

Iorm (henceforth a qame~ is a pair (N,v), where N is the nonempty finite

set of players and v is the characteristic function w hich assigns to every

coalition SCN, a nonempty and compact subset of RS„ denoted v(S).

A game (N,v) is called a Ouasi transferable utility (OTU) game, if

it satisfies the following ('nonlevelness") property:

For all SCN and x,yev(S), if x~y then there exists zev(S) such that x~~z.

A game (N,v) is called a transferable utilíties ( TU) game, and is

denoted by (N,y). if for all SCN there exists a nonnegative scalar, ~(S),

such that v(S) is given by: v(S)-{xeR~S I iiEgx~sy(S)l.

In this paper we shall be concerned only with QTU games. For a OTU

game (N,v), let v"(S) denote the set of all S-Pareto optimal payoffs in v(S),

that is,

v'(S)-{xev(S)Ithere is no yev(S) such that yt~xt for all ieSF.

(The compactness of v(S), see Definition 2.1, implies that v"(S) is compact

and nonempty.)

5

Evidently, there are many negotiation processes that can be employed

by players in a characteristic function form game, (N,v). Thus, m any abstract

systems, describing these negotiation processes, can be associated with

(N,v). (See Greenberg 1990, Chapter 6, for negotiation processes that lead to

the core and the vN8M solution.) This paper is concerned with the procedure

w here each ptayer updates his reservation price according to the last offer

that was made to him.3 More specifically, assume that a payoff x is offered.

Coalition S can object to x if there is an S-Pareto optimal payoff ysev(S)

which makes each member strictly better off, that is, yS~~xS. Members of

N`S continue to believe in xN~S. The new modified offer then becomes

y-(yS,xN`S). Now, another coalition, T, may object to y, again, on the basis

that there ~s a T-Pareto optimal payoff zTev(T) such that each member of T

is strictly better off under zT than he is under y, that is, zT ~~yT. The

resulting new modified offer is then z-(zT,yN`T), and the bargaining process

continues in the same manner. Observe that the bargaining procedure

described here is such that modified offers need no longer be feasible, i.e.,

need not belong to v(N).

The above procedure can be described by the following abstract

system4 (D,L), where D~RN. , and for x,yeD,

x~y c~ 3 SCN, ySev'(S), yS~~xS, and yN`S,xN`S,

3 This procedure is reminiscent of the housing market, where every sellerevaluates his house according to the last offer he had, even though it ispossible that one potential buyer made several oifers to different sellers,so that it is impossible for all sellers to get their reservation price.

4 For definitions and notation of well-know n game-theoretic notions, aswell as for motivation, see Greenberg 1990, Chapters 4 and 6.

6

The definition of the dominance relation L is traditional: It is

customary to require that a coalition will object to a proposed payoff only

if all. of its members are made better off, since changing the 'status quo' is

costly, and individuals have to be compensated for doing so. lt is,

nevertheless, interesting to study the consequences of modifying this

requirement, and allow coalitions to object to a proposed payoff whenever

at least one member is better off whíle all others are not worse off. Define,

therefore, the second abstract system (D,L~) as follows: DáRN, and

for x,yeD,

xL~y c~ 3 SCN, ySEV"(S), yS~xS, and yN`S-xN`S.

Theorem 2.2- Both (D,L) and (D,L') admit a unique vNBM abstract stable

sets, A and A", respectively.

Prooi: Greenberg 1990, Theorem 6.5.7.

The [individual] stable bargaining set is defined as the set of all

feasible payoifs x, i.e., all payoffs xev(N), which belong to the unique vNBM

abstract stable set for (D,L) [(D,L")). That is,

Definition -2.3: Let (N,v) be a OTU game. The stabfe ba~Qainin set

LSB~ of (N,v) is the set SBS(N,v)-Anv(N), and the individual stable

baryaining---set -jISBS of (N,v) is the set ISBS(N,v)-A"nv(N).S

S For the relation of the SBS and the ISBS to the concepts of the bargainingsets (Aumann and Maschler (1964), and modified by Mas-Colell (1969)), seeGreenberg (1990, Section 6.5).

7

Thearem 2.4: Both the SBS and the ISBS are nonempty for all

superadditiveb games. Moreover, each of these two sets contains the core~

of the game.

Proof: Greenberg 1990, Theorems 6.5.4 and 6.5.6.

Rather surprisingly, the seemingly minor modification in the

definitions of the dominance relations might yield totally distinct solution

concepts; the 1SBS and the SBS may have an empty intersection, even for TU

games! In particular, perhaps counter-intuitively, Example 3.10 shows that

the SBS does not, in general, inctude the tSBS. However, Section 4

establishes that in ordinally convex games, the ISBS coincides with the

SBS.

Remark --2-5: The proof of Theorem 2.2 (which enabled the definitions of

the SBS and the lSBS) makes use of von Neumann and Morgenstern's (1947, p.

597-600) result, assertrny that rf the dominance relation rs strictly acyclic

then there exists a unique abstract stable set. (t is because of this that I

require a coalition S to dominate by using S-Pareto optimal payoffs, while

payoffs in SBS and ISBS were not required to belong to v"(N). An

interesting question is whether A and A", respectively, remain (the

unique?) vN8M abstract stable sets for the (possibly more appealing)

systems (D,~L) and (D,LLfi), where D-RNt, and for x,yeD,

6 That is, (N,u) is such that for all S,TCN, SnT-ra, we have that xsev(S) and

yTev(T) imply zev(SUT), where zr-xr if ieS, and zr-yr if ieT.

7 Note that the core of a QTU game is the same for both dominancerelations L and ~~. Indeed, in OTU games, it is possible to make one playerbetter off (all others as well off) if and only ii it is possible to make all

players better oif.

8

xLLy a 3 SCN, ySev(S), yS~~xS, and yN`S-xN`S.

x. ..y ~~ 3 SCN, ySev(S), ys~xs, and yN`S-xN`S.

Remark 2.6: In a recent paper, Dutta et al. (1969) offered a new solution

concept - the consistent bargaining set, CB(N,v). These four scholars

were motivated neither by the theory of social situations nor by vN8M

abstract stable sets, but rather, they wanted to amend the deficiency in the

definition of Mas-Colell"s bargaining set by treating "objections" and

"counterobjections" symmetrically. Ron Holtzman (private communication)

observed that, as is easily verified, it turns out that their (recursive)

definition is equivalent to: CB(N,v)-ISBSnv~(N). That is, CB(N,v) is the

set of Pareto optim al payoffs that belong to the individual stable

bargaining set. Thus, as a by-product, we get a new characterization of the

consistent bargaining set, based on vNBM abstract stable set. 0 ne of the

advantages of this characterization is that it enables to extend this notion

to ga mes wtth an infinite number of players. (Recall that the original

definition oi the CB is recursive.) Thts is particularly appealing since it is

market games with an atomless space of agents that initiated Mas-Colell's

(1989) modified bargaining set. Of course, one has, then, to address the

issues of existence and uniqueness of A and A" for such games.

9

3 Three players TU gamesConsider the 0-normalized TU game, (N,y), where N-{1,2,3i and lor all

ieN, y(i)-0. (Recall that y(S)?0 for alt S; see Definition 2.1 J Oenote:

S~a{1,21, S2s{2,3i, Saa{1,31,

and for all xeRN, and SCN,

x(S)áElEgxl, and e(S,x)~}1(S)-x(S).

The function e(.,.) is known as the excess lunction.

The two main results of this section fully characterize the SBS and

the ISBS in 0-normalized TU three person games. Both sets contain only

Pareto optim al payofis and they both contain all of the core payoffs. But,

for a non-core payoff to belong to the SBS it is necessary and sufficient

that it be blocked by exactly two 2-players coalitions, while for a non-core

payoff to belong to the ISBS it is necessary and sufficient that it be

blocked by all three 2-players coalitions, and moreover, the excess of each

of these coalitions is less than the sum of the excesses of the other two. It

follows that il the core is empty, then the SBS and the ISBS are totally

distinct sets. Form ally,

Theorem 3.1: Let (N,u) be a 0-normalized TU three person game. Then,

xESBS(N,u) if and only if, x is Pareto optimal, i.e., x(N)-y(N), and either

xeCore(N,y), or else, there exist j,ke{1,2,3} such that e(Sj,x)-e(Sk,x)~0 and

e(St,x)s0 for {t}-{1,2,3}`{j,k}.

Theorem 3.2: Let (N,u) be a 0-normalized TU three person game. Then,

xeISBS(N,y~) if and only if, x is Pareto optimal, i.e., x(N)-y(N), and either

10

xeCore(N,y), or else, e(Sj,x)~0 for j-1,2,3, and moreover,

e(Sj,x)~e(Sk,x).e(St,x), for any choice of j,k,t, {j,k,tf-{1,2,3i.

In order to establish Theorem 3.1 we lirst need the fotlowing three

Lemmas.

Lemma 3.3: Let yeRN, be such that y(N)?u(N), and there exists a choice

of j,k,t, {j,k,t{-{1,2,3E. that satisfies: e(Sj,y)-e(Sk,y), and e(St,y)50. Then,

yEA.

Proof: Otherwise, there exists zeA such that yLZ. Since e(St,y)50, e(N,y)S0,

and the game is 0-normalized, it follows that e(Sj,y)-e(Sk,y)~0. W.l.o.g.,

assume that z blocks y using Sj, i.e., z(Sj)-u(Sj). Then, e(S,z)50 for all

S:Sk, and e(Sk,z)~0. Hence, there exists w such that zLw using Sk, and

w(S)?}~(S) for all SCN. Therefore, weA, which, together with the stability of

A, contradicts zeA.

O.E.D.

Lemma 3.4- If xeA then there exists at least one coalition S, I S I-2, such

that x(S)~~(S).

Proof: - Otherwise, e(S,x)~0 for all I S I-2. Assume, w.l.o.g., that

e(S~,x)?e(S2,x)?e(S~,x)~0. For a, Osocse(S~,x), denote

ya-(x~.a,x2.[e(S~,x)-al.x3).

Define the function g, g:(O,e(S~,x)J~R,

g(o~)- e(Sz,ya)-e(S3,ya)-e(S2,x)-e(S~ ,x)-e(53,x).2a.

In particular,

11

g(0)-e(SZ,x)-e(S~,x)-e(S~,x)~-e(S~,x)~0, and

g(e(S~,x))-e(S2,x)-e(S~,x).e(S~,x De(S2,x)~0.

Since g is continuous in a, there exists ~, O~~~e(S~,x), such that g(~)-0.

Define y-ys. Then, x[y (using coalition {1,21). Since A is stable, y~A. By

Lemma 3.3, (recall that e(S2,y)-e(S3,y), and that for all oc, e(S,y~)-0], we

have that y(N)~y~(N).

Define y-(yi.yp,x3.[~(N)-y(N)]). Then, x~ y(using N). Since A is stable,

y~A. But, by Lemma 3.3, yeA (since y(N)-y~(N), e(S2,y)-e(Sy,y), and e(S~,y~~0).

Contradiction.

Q.E.D.

Lemma 3.5: If xeSBS then xev"(N), i.e., x(N)-y(N).

Proof: Otherwise, xeSBS`v"(N). Assume, w.l.o.g., that

e(Si,x)?e(SZ,x)~e(Sy,x). By Lemma 3.4, e(Sy,x)s0. Hence, every coalition that

blocks x contains player 2. Denote:

M-Max{e(S,x) I SCNi, Ba{S I e(S,x)-M}, and S-[y(N)-x(N)]I3.

Then, M~S~O. Distinguish among the following three cases:

1. {S~,S2EC B: Define y, where yi-xi.S, i-1,2,3. Clearly, x~y (using N) and

hence, xeA implies y~A. But, e(S~,x)-e(Sy,x)-M implies e(S~,y)-e(Sp,y), and

therefore, (recall that e(S~,x)50), by Lemma 3.3, yeA. Contradiction.

2. S~aB: Then, SaB for all I SI -2. Thus, 8-{N{. Recall that e(S3,x)50. Thus,

player 2 belongs to all the blocking coalitions. It follows, that there exists

a payof f y. (with y2~xp.Max(e(S~,x),OI), such that y~~x, y(N)-y(N), and

y(S)?y~(S) for all SCN. Thus, yeA. But then, xLy contradicts xeA.

3. S~eB and Szv B: Then, there exists y that satisfies: y~~x~, yy~xy, y~-x~,

y(1,2)-u(1,2), and y(2,3)?}~(2,3). Since y(N)-x(N).M?}1(N), we have that yeA,

12

contradicting xLy and xeA.

Q.E.D.

Proof of Theorem 3.1: Lemma 3.3 yields the "if' part oi the theorem. To

prove the "only if' part, it suffices, in view of Lemmas 3.4 and 3.5., to

show that if x is Pareto optim at and satisfies: e(Sj,x)~e(Sk,x)~0 and

e(St,x)s0 for {tf-(1,2,3(`{j,k), then xéSBS(N,y). Indeed, since SjnSk:ra, there

exists zev"(Sj) such that z~~xi for ieSj, e(Sj,z)-0~e(Sk,z). Thus, e(S,z)50

for all S, implying that zeA. Since x[ z, we conclude that xa A, hence

xé SBS(N,}~).

O.E.D.

We now turn to the proof of Theorem 3.2. In order to establish this

theorem, we first need the following three Lemmas.

Lemma-3.6: II xeA' and x can be blocked, then x(S)~y(S) for all I S I-2.

Proof: Otherwise, there exists SCN, I S I-2 such that e(S,x)50. Assume,

w.l.o.g., that e(S~,x)?e(S2,x)?e(S~,x). Then, e(S~,x)50, implying that player 2

belongs to all the blocking coalitions. Define z, where z~-x~, z~-x3, and

z2-xp.Max{(e(S,x)). Since x can be blocked, zp~xz. Thus, xLwz. But z cannot

be blocked, hence zeA", contradicting x~~z and xeA".

O.E.D.

Lemma 3.7: If xeISBS then e(Sj,x)~e(Sk,x).e(St,x), for any choice of j,k,t,

{j,k,t)-(1,2,3).

Proof: Otherwise, w.l.o.g., e(S~,x)?e(Sq,x).e(S~,x). Then, there exist y~ and

y2 that satisfy: y~?x~.e(5~,x), yp?xp.e(S2,x), and y~.y2-y(1,2). Let z be equal

to y-(y~,yy,x3) if y(N)?y(N), and if y(N)~y(N) let z be such that z~~y, and

z(N)-y(N). Then, z(S)?}~(S) for all S, implying zeA". But x~.z, (using either N

or {1,2f). contradicting xeA'.

O.E.D.

Lemma 3.9: If xe1SBS then x is Pareto optimal, i.e., x(N)-y(N).

Alternatively, ISBS(N,y) coincides with the consistent bargaining set,

CB(N,y).

Proof: Assume, in negation, that there exists xeISBS`v"(N). Define,

~~~('Iz)[e(Si,x),e(s3,x)-e(Sz,x)]~c2~('Iz)[e(St,x).e(SZ,x)-e(S3,x)]aaa('Iz)[e(S2,x).e(Sa,x)-e(S~,x))

ys(x~.a~,xz.a2.Xy.aa).

By Lemma 3.7, y~~x. Now, if y(N)5}~(N), then there exists z?y with z(N)-y(N).

It follows that z(S)?y~(S) for all S, implying that zeA". But xL~z. (using N),

contradicts xeA".

Thus, y(N)~y(N). Since x(N)~y~(N), there exists z such that z(N)-y(N),

z~~x and z~~y. Since x[~z (using N), zeA". That is, there exists weA" with

zL~w. Since z(N)-}~(N), w is supported by, w.t.o.g., S~:{1,2f. That is,

w-(wi.wZ,zs), wi?zi. wz?z2. and w(Si)-}~(S~). Therefore, by Claim 3.6, w

cannot be blocked. In particular, w(S~)-w~.z3?~(1,3) and w(SZ)-w2.za?y(2,3).

Hence, wi.wp.2z~-y(1,2).2z3?y(1,3).}~(2,3), implying that

za~('Iy)(}~(1,3).y(2,3)-}t(1,2))-X~.oty.yy. BUt z~~y. Contradiction.

O.E.D.

14

Proof of Theore m 3.2: Let (N,y) be a 0-normalized TU three person ga me.

In view of Lemmas 3.6-3.6, it suffices to show that if xev"(N)`Core(N,}~) is

such that e(S,x)~0 for all I S I-2, and moreover, e(Sj,x)~e(Sk,x).e(St,x), 1or any

choice of j,k,t, {j,k,t}-{1,2,3[. then xe1SBS(N,}~). lndeed, otherwise, there

exists zeA" such that xt~z. Since xev"(N), we have that z(Sj)-y(Sj), for

some je{ 1,2,3}. By Lemma 3.6, therefore, e(S,z)50 for all I S I-2. But

e(Sj,x)~e(Sk,x).e(St,x) implies that there exists k, such that e(Sk,z)~0.

Contradiction.

Q.E.D.

Theorems 3.1 and 3.2 yield the following surprising result.

CorollarU 3.9: Let (N,}~) be a 0-normalized 3-person TU game. Then,

AnA":{xeD I x is not blocked}. It follows that SBSnI58S-Core(N,y). In

particular, if the core is empty, then SBSnlSBS-ra.

Proof: Since (N,}~) is a TU game, ~(D)-~"(~)-8 By the (external) stability

of A, AD[D`r,(D)]. Similarly, by the (external) stability of A",

A"~[D`~"(D)]. Hence, AnA"D(D`~(D)]-[D`~"(D)l.

To show that the reverse inclusion also holds, consider

xe~(D)[-~"(D)l. By Lemma 3.6, if xEA" then x(S)~y(S) for every coalition S

that consists ot two players. By Lemma 3.4, therefore, x~A.

Q.E.D.

8 For a set BCD, ~(B) is the set oi elements in D that are dominated by B,i.e., ~(B)-{xeD I there exists yeB, xty{.

15

I end this section by studying two examples that demonstrate some of

the possible relationships among the three notions: The core, the SBS, and

the ISBS. In both examples, ! personally find the SBS to be more appealing

than the ISBS.9

ExamDle 3.10: The three person majority game: There are three players.

Every coalition that has a(simple) majority, that is, every coalition

consisting of two or more players, can distribute among its members 2

dollars. The utilities of the three players are linear with money. The TU

game that describes this social environment is given by (N,}~), where

N-{1,2,3}, }1(S)-2 ii IS~?2, and for ieN, y({i})-0.

As is well-known, Care(N,}i)-~. Using Theorems 3.1 and 3.2, it is easy to

see that

SBS(N,yl)-{(1 ,1 ,0),(1 ,0,1),(0,1,1)},

and

ISBS(N,y~)-{x I x(N)-y~(N)-2, and xi~l for all iENI.

Thus, the core of this game is empty while the individual stable bargaining

set, w hich coincides with the consistent bargaining set, is given by the

shaded area in Figure l. In contrast, the stable bargaining set consists of

the three outcomes (the three heavy dotted points in Figure 1) that stem

from two players forming a coalition, and distributing the 2 dollars evenly

between them.

9 Recall that the ISBS is closety related to, and by Lem ma 3.8, in 3-persongames coincides with the consistent bargaining set, CB(N,u).

1G

Example 3.10 should not leave the impression that in general the SBS

"is a smaller set" than the ISBS. Indeed, in the following example the SBS

strictly includes the ISBS (implying, by Corollary 3.9 that the ISBS

coincides with the core).

ExamDle 3.11: Consider the game (N,}~), where }~(1,2)-y(1,3)-y(N)-100, and

y(S)-~ otherwise. Then,

Core-ISBS-{(100,0,0)} and SBS-{(100-2a,ot,ot) I OSo~50E.

It is obvious that Core-{(100,0,0)}. The following observations verify

the other assertions.

(i) ISBS-{(100,0,0)}: Consider x-(oc,~.ó)ev(N) with a~100. Assume, w.l.o.g.,

that ~?ó. Then, (100-ó)~a, implying that y-(100-ó:~,?f) cannot be dominated.

Hence y belongs to A". Since xL~y, it follows that xe ISBS. Since

Core-{(100,0,0)}, Theorem 2.4 yields that Core-ISBS-{(100,0,0)}.

(ii) SBS-{(100-2o~,a,oc)IO5oc5S0}: Observe, first, that x-(a,~,ó)eA whenever

~~ó. Indeed, define, 6-(~.ó)~2, and y-(100-5,{3,S)eA. Then, y cannot be

dominated. Hence y belongs to A. Since xLy, xeA. It follows that x-(100-

2o~,a,a)eA, since otherwise, there exists zeA such that xLZ. But then

z-(a,{3,71), with ~~ó, and we just saw that such z cannot belong to A.

As in the previous example, 1 find the SBS for this game more

appealing than the ISBS, despite the fact that the SBS is larger. It seems to

me that players 2 and 3 are not totally 'helpless" vis a vis player 1; they

could, for example, collude.

17

4 Ordinally Convex games10

There are, of course, games in which the SBS and the ISBS do

intersect; in fact, there are games in which the two sets coincide. One

class of such games is the set oi "convex games", where

Definition--4.1- A game (N.v) is called

1. (ordinally) convex if it satisfies: For any xeRN and any two

coaliUons S and T, if xsev(S) and xTev(T), then either xSUTEV(SUT) or

xSnTEV(SnT).

2. comp~ehensiye if for all SCN, xev(S) implies yev(S) for all Osysx.

ln particular, Oev(S).

The main result of this section is

Theorem 4.2: Let (N,v) be a comprehensive, OTU convex game. Then,

A-A"-{xeR,Nlx cannot be blockedl.

In particular,SBS(N,v)-ISBS(N,v)-Core(N,v).

In order to prove this theorem, we first need to establish the

following Lemma.

Lemma 4.3: Let ( N,v) be a comprehensive, OTU and convex game. Then,

F-D`0(D)-{xeRNlx cannot be blockedl

is a vNBM abstract stable set for (D,~).

10 This section makes extensive use of the results in Section 3.3 in Dutta

et al. (1989).

18

Proof: Clearly, ~(F)CD`F. To conclude the proof we need to show that

D`FC~(F). The proof of this inclusion generalizes, but closely follows that

of Proposition 3.3 in Dutta et. al. Otherwise, there exists xeD`F but xe(,(F),

Thus, there exists S that blocks x through some y, and (yS,xN~S)eF. By the

comprehensiveness of (N,v), therefore, the set O~z~, where

0~-{(S,y) ~ ySEV"(S). yN`S-xN`S and yS~~xsl.

By considering minimal (in the set inclusion ordering) coalitions in Q~, it

is easy to see that there exists (S`,y~eQ~ such that no subset of S can block

y, and S is a maximal such coalition. Since (yS,xN~S)e F, the

comprehensiveness of (N,v) implies that set Qpz~, where

02-1(T,z) I zTev(T), zN`(T`S]-y~N`(T`5), and zT`S~~yT`SE.

By the choice of S, (T,z)eQ2 implies T`Szra. Let

(T,z7EArgMax{(T,z)e02}Min{ieT`SEz~~

Since (yS,xN~S)é F, Min{ieT`Sf z~~0.

Then, y5-z5ev(S) and z~ev(T). By the choice of S,

z'S"nT-ySnT~lnt.v(SnT). By Lemma 3.3 in Dutta et al. (1989), it follows that

z~SuTEV(SuT). Now, if there was a coalition WCSuT that blocks z, using

t,iev(W), then, by the choice of S, W`Sz~. Moreover, c,iW~S~~z~~~ implies

that MiniEtil`Sc.~i~MiniET`Sz~. Since ~SnT,ySnT and (N,v) is

comprehensive, the above strict inequality contradicts the choice of z. Thus,

z is not blocked by any subset of SuT.

But this contradicts the maximality of S. (Recall that T`Szg,

implying I SuT I~ I S I.)

O.E.D.

Proof of Theorem 4.2: By Lemma 4.3 we have that the unique vNBM

abstract stable set for (D,~) is: A-{xeR,Nl x cannot be blocked}, Using

Theorem 2.2, in order to conclude the proof of the theorem, it suffices to

show that A is a vNBM abstract stable set for (D,L~).

Indeed, since (N,v) is QTU, Anp(A)-~ implies that AnpM(A)-rd, that

is, 0"(A)CD`A. To see that the reverse inclusion also holds, consider

xeD`A. Cty Lemma 4.3, there exists yeA such that x.~y, implying xi~y. Hence,

0"(A)~D`A. Thus, ~"(A)-D`A, i.e., A is a vNBM abstract stable set for

(D.~~).

Q.E.D.

A by-product of Theorem 4.2 provides a new proof for the

nonemptiness of the core of convex games (see Vilkov 1977, and Greenberg

1985).

Corollafy- 4.4: The core of a comprehensive, QTU, and convex game is

nonempty.

Proof- Let (N,v) be a comprehensive, QTU, and convex game. Then, since

Oev(Q) for all OCN, the convexity oi (N,v) yields that for all xsev(S),

(xS,ON~S)EV(N). By Theorem 2.4, therefore, SBS(N,v)zr~. Theorem 4.2

concludes the prooi.

O.E.D.

lt is noteworthy that Peleg (1986) proved that in convex games the

core ~s also the unique vNBM solution.

zo

5 An open question

By Theorem 2.4, both the SBS and the ISBS are nonempty in all

superadditive QTU games. An interesting question is w hether, in such

games, these sets contain only, or at least some, Pareto optimal payoffs.

Dutta et al. showed that it is possible that ISBSnv"(N)-ró.

Specifically, they consider the 4 player TU game (N,u), where N-{1,2,3,4E,

y(1,2,3)-}~(2,3,4)-66, y(1,4)-46, }~(1,2,4)-}~(1,3,4)-63, }~(N)-80, and y~(S)-0

otherwise. They show (Dutta et al. 1989, Proposition 4.1) that

CB(N,u)-1SBSnv"(N)-r~.

In contrast, the SBS in this ga m e does contain Pareto optimal payoffs.

For example, x"-(23,17,17,23)eSBS. Indeed, {l,q}, {1,3,4} and {1,2,4} cannot

block x". It remains to check for coalitions i1,2,3f and {2,3,4}. Now,if

{1,2,3} blocks x" with y, then y~~x"~, implying that [y2.yyl~[66-23]. But

then, there exists a payoff z such that z(2,3,4)-}~(2,3,4)-66 and y~z. Since

z(S)?y~(S) for all S, zeA. The stability of A implies, therefore, that yeA. An

analogous argument shows that {2,3,41 cannot block x' using a payoff in A.

Thus, x"eA.

It remains an open question whether in superadditive TU games we

have that SBSCV"(N), or, at least, SBSnv"(N)x~. An affirmative answer to

this question will be particularly pleasing, since if the "real world` is to

provide a guideline, then the notion of the SBS seems to be more

appropriate than the ISBS (or the CB); individuals often insist on some

compensation if the "status quo" is to be changed.

zl

ReferencesAumann, R. and M. Maschler ( 1964), 'The bargaining set for cooperative

games,` in Advances in Game TheorU. Dresher, Shapley, and Tucker,

eds. Princeton, N.J: Princeton University press, 443-7.

Dutta, B., Ray, D., Sengupta, K., and Vohra, R. (1989), "A consistent

bargaining set," Journal- of Economic Theo~, 49, 93-112.

Greenberg, J. (1985), `Cores of convex games without side payments,"

Mathematics oi OPerations Research, 523-525.

Greenberg, J. (1990), The theory of social situations: An alternative game-

theoretic approach, Cambridge University Press.

Mas-Colell, A. ( 1989), 'An equivalence theorem for a bargaining set",

Journal o( . Mathematical Economic~ 1 B, 1 29-1 38.

Peleg, B. ( 1966),"A proof that the core oi an ordinal convex game is a von

Neumann and Morgenstern solution" Math. Soc. Sci. 11, 83-87.

Vilkov, V.V. ( 1977), "Convex games without sidepayments", Vestnik

Leningradskiva Uníversitata 7, 21-24.

von Neumann, J. and Morgenstern, 0. (1947), Theory of Games and

Economic Behavior. Princeton NJ: Princeton University Press.

Figure 1

Discussion Paper Series, CentER, Tilburg University, The Netherlands:

(For previous papers please

No. Author(s)

9031 A. Weber

9032 J. Osiewalski andM. Steel

9033 C. R. Wichers

9034 C. de Vries

9035 M.R. Baye,D.W. Jansen and Q. Li

9036 J. Driffill

903~ F. van der Ploeg

9038 A. Robson

903y A. Robson

9040 M.R. Baye, G. Tianand J. Zhou

9o41 M. Burnovsky andI. Zang

9042 P.J. Deschamps

consult previous discussion

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Robust Bayesian Inference in EllipticalRegression Models

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Aggregation and the "Random Objective"Justification for Disturbances in CompleteDemand Systems

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9063 S. Chib, J. Osiewalskiand M. Steel

9064 M. Verbeek andTh. Nijman

9065 F. van der Ploegand A. de Zeeuw

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9071 J. Eichberger,H. Haller nnd F. Milne

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wi i~in iu u n ~w i u~~~ u Mi~wiu