Tilburg University
On the sensitivity of von Neuman and Morgenstern abstract stable sets
Greenberg, J.
Publication date:
1993
Document Version
Publisher's PDF, also known as Version of record
Link to publication in Tilburg University Research Portal
Citation for published version (APA):
Greenberg, J. (1993). On the sensitivity of von Neuman and Morgenstern abstract stable sets: The stable and
the individual stable bargaining set. (Reprint Series). CentER for Economic Research.
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners
and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
• You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal
Take down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
CBM
R
8823
1993
108
for
~mic Research
On the Sensitivity of von Neumann
and Morgenstern Abstract Stable
Sets: The Stable and the
Individual Stable Bargaining Set
by
J. Greenberg
IIIIVVIIIIlllnIII~IIIIllliIIII'IlhlIIIIIIII
Reprinted from
International Journal of Game Theory,
Vol. 21, No. 1, 1992
CENTER FOR ECONOMIC RESEARCH
Board
Harry Barkema
Helmut Bester
Eric van Damme, chairman
Frank van der Duyn Schouten
Jeffrey James
Management
Eric van Damme (graduate education)
Arie Kapteyn (scientific director)
Marie-Louise Kemperman (administration)
Scientific Council
Anton Barten
Eduard Bomhoff
Willem Buiter
Jacques Drèze
Theo van de Klundert
Simon Kuipers
Jean-Jacques Laffont
Merton Miller
Stephen Nickell
Pieter Ruys
Jacques Sijben
Université Catholique de Louvain
Erasmus University Rotterdam
Yale University
Université Catholique de Louvain
Tilburg University
Groningen University
Université des Sciences Sociales de Toulouse
University of Chicago
University of Oxford
Tilburg University
Tilburg University
Residential Fellows
Lans Bovenberg
Werner Guth
Anna-Maria Lusardi
Jan Magnus
Theodore To
Karl Wárneryd
Research Coordinators
Eric van Damme
Frank van der Duyn Schouten
Arie Kapteyn
Theo van de Klundert
CentER, Erasmus University Rotterdam
University of Frankfurt
Princeton University
CentER, LSE
University of Pittsburgh
Stockholm School of Economics
for
Economic Research
On the Sensitivity of von Neumann
and Morgenstern Abstract Stable
Sets: The Stable and the
Individual Stable Bargaining Set
by
J. Greenberg
Reprinted from
International Journal of Game Theory,
Vol. 21, No. 1, 1992
International Journal of Game Theory (1992) 21: 41-55
On the Sensitivity of von Neumann and Morgenstern
Abstract Stable Sets: The Stable and the Individual
Stable I3argaining Set
J. Greenberg'
Absrracr: The purpose of this paper is twofold: First, to study the properties of the notions of
the `stable" and "individual stable" bargaining se[s (SBS and ISBS). Second, to point out the
sensitivity of the von Neumann and Morgenstern (vNácM) abstract stable set to the dominance
relation that is being employed: InS~sting that each member of the coalition be made bettet.otf
yields the SBS, whilc requiring thàt at least one member of the coalition is better ot( and all óthëis à~e not worse of( 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 ihat such games have a nonempn~ rore. The paper concludes with an open qucstion.
1
Introduction
The purpose of this papcr is twofold: First, to study the properties of the notions of
the "stable" and "indívidual stable" bargaining sets. Second, and perhaps more
im-portant, the paper points out the sensitivity of the von Neumann and Morgenstcrn
(vNRcM) ahstract stable set to the dominance relation that is being employed. Mtire
specifically, we shall distinguish between the following two dominance
relation-ships:
(i) A coalition may object to a proposed payoff only if each of its members
can be made better of(.
(ii) It suffices that at Icast one member is better off while all others are at lcast as ~~ell off for a coalition to object to a proposed payoff.
Condition (i) gives rise to the stable bargaining set (S[3S), while condition (ii) yiclds
the notion of the individual stable bargaining set (ISBS). And, rather surpritint;ly, it
' Professor Joscph Greenberg, Department o( Economics, McGill Univcrsity, 855 Sherhrnoke Strcet West, Montreal, Qurbec, Canada H3A 2T7, and C.R.t).E. Universué dr ~lontrral I tiish to thank Uov Monderer and Benyamin Shitovit7. for many stimulaung anJ rnjoyahlr discussion,, and an anonymous referee for several useful comments. This revitiiun of hlcGill Umversity working paper 4~90 was written during my pleasant and fruitful visit to Centtr, Tilburg. Fínancial support ( rom the NWO, The Netherlands, and the Natural Scirnces and
Fngineering Research Council of Canada (NSERC OGP121665) is gratefully acknowl-edged.
S2 J. Greenberg will be shown that these two dominance relations might yield two very diffcrent vN~fM abstrart stable sets. In particular, the SBS and the ISBS may have an empty intersection, even in games with transferable utilities. In fact, such is always the case in 3-person games that have an empty core.
We fully charaaerize both the SBS and the ISBS in 3-person superadditive games with transferablc utilities. It turns out that in such games both sets contain only Parcto optimal payoffs.
Vv'e also show that in ordinally convex games both the SBS and the ISBS coin-cide with the cure. Since (or the general n-person superadditive game both sets are nonempty (Grecnbcrg 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.)
Thc various notions of the bargaining set were originally introduced (for games with side paymentsZ) by Aumann and Maschler (1964). A payoff belongs to (the') bargaining set if and only if every "objection" to it can be "countered". Aumann and Maschler insisted that the objection be directed towards one other specific coal-ition. Mas-Colell (1989) modified the bargaining set in two ways. First, any coalition can mal.e objertions (see footnote 3), and second, an objection need not be directed to a particular coalition, but rather, can be countered by any other coalition. As noted by Dutta et al. (1989), while each of these bargaining sets "does test objections against counter objections, it does not similarly test the counterobjections or any further objections, and in this sense it is not consistent" (p. 94). To amend this defi-ciency, they reyuire that not only "objections", but also "counterobjections", "counter-counterobjections", etc. be "justified" or "credible", yielding the notion of the "consistent bargaining set" (CB)'. They show, however, that the CB might be empty, even for 4-person games with transferable utilities. (See section 5).
It turns out that the set of Pareto optimal payoffs in the ISBS coincides with the
CB. This is quite remarkable since the notions of the SBS and the ISBS were first
suggested within the framework of the theory of social situations (Greenberg 1990),
which is a new and integrative approach to the study of formal models in the social
and behavioral sciences. More specífically, the theory of social situations has two
main ingredients. First, it offers a unified way to represent cooperative and
nonco-operative social environments - by means of "situations". Second, it offers a unified
criterion for the recommendations, namely, that the "standard of behavior" (for the
given situation) be "stable"'. Shitovitz (Greenberg 1990, Theorem 4.5) observed that
onc`' of the stability concepts in this theory (specifically, the "optimistic stable
stand-' For extensions o( these notion to games without side payments see, e.g., Asscher (1976, 1977), I;illera (1970), Greenberg (1979), and Peleg (1963).
' In almost all applications of the "classical bargaining set", the coalition that makes the ob-jection, as well as the coalition to whom this objection is made, consist of a single
individu-al.
' See Remark 2.6.
' In contrast, "classical game [heory" offers three distinct representations of a social environ-ment, namely, games in extensive form, normal ( or strategic) (orm, and characteristic func-tion (or coalifunc-tional) form. Moreover, to each type of game, game theory offers an abun-dance of solutiun concepts whose underlying motivations differ considerably.
On the Sensitivity of von Neumann and Morgenstern Abstract Stable Sets 43
ard of behavior") can formally be associated with a vNBh1 abstract stable set'.
Us-ing this result, some o( the better known game-theoretic solution concepts", as well
as interesting new solution concepts9, were derived from the unique vNBcM abstract
stable sets that correspond to different negotiation processes (Greenberg 1990).
The relationship between the ISBS and the bargaining set is remarkable also
because, as Shubik (1984, p. 348) notes:
"We should stress, however, that the (bargaining set~ concerns the stability of a
sin-gle imputation, whereas the vNBtM concept concerns the stability of a set of
imputa-tions. In a certain sense, then, the bargaining set (like the core and the kernel) is not
a solution but a set of solutions - the collectivity of all possible outcomes using the
particular solution concept. In contrast, each stable set in toto is a single solution,
and the collectivity of all outcomes using this concept (the union of all stahle sets) is
generally not a stable set."
The paper concludes with the following open question. As noted above, Dutta
et al.'s example of a 4-person transferable utility game demonstrates that, in
gencr-al, the CB is empty. Hence, ISBS need not contain Pareto optimal payoffs. In
con-trast, the SBS in this example does contain Pareto optimal payoffs. Whether this is
always the case in superadditive games with transferable utilities remains an open
question. 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 (and hence than the CB): Individuals often insist on
some compensation if the "status quo" is to be changed.
Walter Bossert and Abhijit Sengupta pointed out to me that Dut[a and Ray (1989) and Sengupta and Sengupta (1992) are related works, in the sense that they, too, explore the sensitivity of solution concepts to the strict and the weak dominance relations.
For ease of exposition the paper does not (explicitly) use the terminology or
tools from the theory of social situations. Rather, it is cast completcly within
classi-cal game theory, making use of the concept of vNBcM abstrac[ stable set. 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.
' This notion was introduced by vNBcM in a few pages at the end o( the second edition tin 1947) of their classical book. They offered it purely as a mathematical extension of "the vNRht sohuion"; they neither motivated it, nor suggested an application of it. In contrast, the theory of social situations stemmed from the basic question of "rational choice".
' Such as: The core and the vNBcM solution in cooperatíve games, coalition-proof Nash equil-ibrium in normal form games and the set of subgame perfect equilibria in extensive fnrm games.
44 J. Grcenberg
2
The tilable and Individual Stable E3ar~aininK Sels
This ~ection pro~ides the (ormal definitions of the stable and the individual stable bart;aining sets, and studics some of thcir properties.
Lct ;V bc a finite nonempty sct, and let R be the set of all real numbcrs. A coalition is a nanempty subsct of N. For a coalition SCN, R' dcnutcs thc .S-dirncn-sional Luclidcan spare. If xe R'v and S is a coalition, then x' denotes the restriction ol .i on .S. I.ct S bc a coalition and Ict x', y'eRs. Thcn, xs?y` if x'?y' for all ieS: r`~y` if ,r'?y` but x'~yS: and x`~y' if x'~y' for all ieS. Recall the following dcfinition:
Dejinilion 2.1: An n-person game in characleristic function jorm (henceforth a garne) is a pair (N, v) where N is the nonempty finite set of playcrs and u is the
characteristic function which assigns to every coalition SCN, a nonempty and com-pact subset of RS , denoted u(S).
r~ gan,e (N, v) is called a Quasi transjerable utiliry (QTU) game, if it satisfies
the following ("nonlevelness") property:
For all SCN and x, yeu(S), if xcy then there exists zeu(S) such that xaz. A game (N, u) is called a transferable utilities (TU) game, and is denoted by (N, u), if for all SCN there exists a nonnegative scalar, p(S), such that u(S) is given by: L)(S)- {,YERS I~,esX'Si~(S)}.
In this paper we shall be concerned only with QTU games. For a QTU game
(N, u), let v'(S) denote the set of all S-Pareto optimal payoffs in u(S), that is,
u'(S)- {xeu(S)~ there is no yEU(S)
such that y'~x'
for all ieS}.
(The compactness of u(S), see Definition 2.1, implies that u'(S) is compact and nonempty. )
Evidcntly, there are many negotiation processes that can be employed by
players in a characterislic function form game, (N, v). Thus, many abstract systems,
describing these negotiation processes, can be associated with (N, u). (See Greenberg
1990, Chapter 6, for negotiation processes that lead to the core and the vNBM
solu-tion). This paper is concerned with the procedure where each player updates his
re-scrvation price according lo the last offer that was made to him'o. More specifically,
assumc that a payof f x is offered. Coalition S can object to x if thcre is an S-Pareto
opumal payofr y`eu(S) which makes each member strictly better off, that is,
ysy~.s h1embers of N`S continue to believe in xN". The new modified o(fer then
becomes y~(yS,xN`~S). Now, another coalition, T, may object to y, again, on the
basis that there is a T-Pareto optimal payoff zreu(T) such that each member of T is
strictly better off under zr than he is under y, that is, zrsyr. The resulting new
rnodified offer is then z~(zr yN~r) and the bargaining process continues in the
On the Sensitiviry of von Neumann and ~torgensrern Abstrart Stable Sets 45
same manner. Observe that the bargaining procedure described here is such that
modified of(ers need no longer be feasible, i.e., need not belong to r~(N).
The above procedure can be described by the following abstract system"
(D,L), where D~R';, and forx,yED,
xLy e~ 3 SCN, ySEU'(S), ysaxs, and yN~s-xN.s.
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 com-pensated for doing so. It is, nevertheless, ínteresting to study the consequences of modi(ying this requirement, and allow coalitions to object to a proposed payoff whenever at least one member is better off while all others are not worse off. De-fine, therefore, the second abstract system (D,L ~) as follows: D- R;' , and for
X, y E D,
xL~y p 3 SCN, y'sEU'(S), ys~z's, and yN~.S-xN~s
Theorem 1.1: Both (D,L) and (D,L.) admit unique vNBcM abstract stable sets, A
and A', respectively.
Proof.' Greenberg 1990, Theorem 6.5.7.
The [individual] stable bargaining set is defined as the set of all feasible payoffs
z, i.e., all payoffs xev(N), which belong to the unique vNBcM abstract stable set for
(D, L)[(D, L s)]. That is,
DeJinirion 2.3: Let (N, u) be a QTU game. The slable bargaining se~ (SBS) of (N, v)
is the set SBS(N, u)~Anu(N), and the individual srable bargaining ser (ISBS) of
(N, u) is the set ISBS(N, u)~A'nu(N).
Theorem 1.4: Both the SBS and the ISBS are nonempty for all superadditive''
games. Moreover, each of these two sets contains the core" of the game.
ProoJ: Grecnberg 1990, Theorems 6.5.4 and 6.5.6.
Rather surprisingly, the seemingly minor modification in the definition of the dominance relations might yield totally distinct solution concepts; the ISBS and the SBS may have an empty intersection, even for TU games! In particular, perhaps counter-intuitivcly, Example 3.10 shows that the SBS does not, in general, include the ISBS. Ho~~c~er, Section 4 establishes that in ordinally conver games, the ISBS coincides ~~ith thr SBS.
" f cir definitions an~ nciration of well-known gamr-throreuc notion~, a~ Hrll a. fe~r mort~a-uon, sce Grrenberg 19y0, Chaptrrs 4 and 6.
'' That ic, (N, r) is such that for all .S, TCN, Sr~ I-O, ~~r have that r~erl-til and r'e~~(T) im~ly ~er(SvT), where z'-.r' i( ~e.S, and z'-y' if ieT.
" Notr that thr core of a QTU game is the same for borh dominance rclan~~m L and L..
Indred, in QTU games, it is possible to make one player bcrter off (all ~ithrrs as wrll ofll if
46 J. Greenberg
Remark LS: The proof of Theorem 2.2 (which enabled the definitiuns of the SBS
and the ISBS) makes use of von Neumann and Morgenstern's (1947, p. 597-6W)
result, asserting that if the dominance relation is strictly acyclic then there exists a
uniyue abstract stable set. It is because of this that 1 require a coalition S to
domí-nate by using S-Pareto optimal payoffs, while payoffs in SBS and ISBS werc not
rcyuired to bclong to u' (N). An interesting question is whether A and A',
respcc-tively, remain (the unique?) vNBrM abstract stable sets for the (possibly more
ap-pealing) systems (D,LL) and (D,LL~), wherè DáRN, and (or x, yeD,
xLLy p 3 SCN, ySEU(S),
ySaxS,
and yN`s-X,v~s
.rLL~y p 3 SCN, ytEU(S), yS~xs,
and yN`s-~~s
Remark 2.6: As was mentioned in the introduction, Dutta et al. (1989) offered the
notion of the consis[en( bargaining set, CB(N, u). These four scholars were moti-vated neither by the theory of social situations nor by vNBcM abstract stable sets, but rather, they wanted to amend the deficiency in the definition of Mas-Colell's bar-gaining set" by treating "objections" and "counterobjections" symmetrically. Ron Holtzman (private communication) observed that, as is easily verified, it turns out thal their (recursive) definition is equivalent to: CB(N,u)-1SBSnu'(N). That is,
CB(N, u) is the set of Pareto optimal payoffs that belong to the individual stable
bargaining set. Thus, as a by-product, we get a new characterization of the consis-tent bargaining set, based on vNBcM abstract stable set. One of the advantages of this characterization is that it enables to extend this notion to games with an infinite number of players. (Recall that the original definition of the CB is recursive.) This 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.
3
Three Players TU Games
Consider the 0-normalized TU game, (N, p), where N- { 1, 2, 3} and for all reN,
N(r)-0. (Recall that ~(S)?0 for all S; see Definition 2.1.) Denote:
S,-{1,2},
SZ~{2,3},
S,~{1,3},
and for all xeRN and SCN,
" Recall the following definitions: Let (N, u) be a cooperative game. A pair (S, y) is an
objec-rion tu.x'en(N) if SCN, yeR"~, y`eu'(S), y'sxs, andyN~S-xN~s. Let (S,y) be an
On the Sensitivity of von Neumann and Morgenstern Abstract Stable Sets 47
z(S)3~;Esx',
and
e(S,x)~N(S)-x(S).
The function e(.,.) is known as the excess junction.
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 optimal payoffs
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
ex-cess of each of these coalitions is less than the sum of the exex-cesses of the other two.
It follows that if the core is empty, then the SBS and the ISBS are to[ally distinct
sets. Formally,
Theorem 3. l: Let (N, N) be a 0-normalized TU three person game. Then,
xESBS(N,t~) if and only if, x is Pareto optimal, i.e., x(N)-N(N), and either
xeCore(N,N), or else, there exist j,ke{1,2, 3} such that e(S;,z)-e(Sk,x)~0 and
e(S„z)sOfor {t}-{1,2,3}`{j,k}.
Theorem 3.2: Let ( N, N) be a 0-normalized TU three person game. Then,
xeISBS(N,N) if and only if, z is Pareto optimal, i.e., x(N)-N(N), and
either
xeCore(N, p),
or
else,
e(S;,z)10
for
j- 1, 2, 3,
and
moreover,
e(S;, x) c e(Sk, x) t e(S„ x), for any choice of j, k, t, {j, k, t}-{ 1, 2, 3}.
Iri order to establish Theorem 3.1 we first need the following three Lemmas.
Lemma 3.3: Let yeR ;' be such that y(N)?N(N), and there exísts a choice of
j, k, t, {j, k, t} -{ 1, 2, 3}, that satisfies: e(S,, y)-e(S,, y), and e(S„ y)s0. Then,
yeA.
Proof.~ Otherwise, there exists zeA such that yGZ. Since e(S„y)~0, e(,V, y~)s0,
and the game is 0-normalized, it follows that e(S,, y)-e(Sk, y)~0. W.I.o.g., assume
that z blocks y using S„ i.e., z(S,)-p(S,). Then, e(S,z)~0 for all S~S,, and
e(Sk, z)10. Hence, there exists w such that zL w using Sk, and w(S)?p(.S) for all
SCN. Therefore, weA, which, together with the stability of A, contradicts ~eA.
Q.E.D.
Lemma 3.4: If xeA then there exists at least one coalition S, ~S~ -2, such that
x(S)~N(.S).
Prooj.~
Otherwise,
e(S,x)~0
(or
all
~S~-2.
Assume,
~t.l.o.g.,
that
e(S,,.r)~e(S,,x)?e(S,,x)10. Fora, Osase(S,,x), denote
y" -(.r~ f n, x2 t fe(S~. .r) -rY), x,).
Define the function g, g: (0, e(S,, x)(-~R,
ax
In parti~ular,
9(~) - e(Sz. x) -e(S,. -r) -e(S,, x) s -e(S,, x) c 0,
and
9(e(S~, ~))-e(Sz,x)-e(S,,.i)te(S,,x)?e(SZ,x)~0.
1. Grecnberg
Since g is continuous in a, there exists Q, OGQGe(S,, x), such that g((1) - 0. Define
y-y~'. Thrn, .~Ly (using coalition { 1, 2}). Since A is stable, yéA. By Lemma 3.3,
[recall that c(S~, y)-e(S,, y), and that for all a, e(S,,y")-0], we have that
y(N) ~N(,~).
Define i~-(y,, yz, x, t[Jr(N) -y(N)]). Then, xLy (using N). Since A is stable,
yéA. But, by Lemma 3.3, yeA (since y(N)-p(N), e(SZ, y)-c(S,, y), and
e(S,, f-)-0). Contradiction.
Q.E.D.
Leirrnru 3.5: If xeSBS thcn xEU'(N), i.e., x(N)-p(N).
Proof. Otherwise, xeSBS`u' (N). Assume, w.l.o.g., that e(S,, x)? e(SZ, z)? e(S,, x).
By Lemma 3.4, e(S,, x)s0. Hence, every coalition that blocks x contains player 2. Denote:
A~1-C,1ax{c(S,x)~SCN},
B-{S~c(S,.r)-M},
and
ó-[p(N)-.r(N)J~3.
Then, ,ti1~ó10. Distinguish among the (ollowing three cases:
I. {S,, S:} CB: Define y, where y'~.r' f ó, i- l, 2, 3. Clearly, xLy (using N) and hence, xeA implies yéA. But, e(S,,x)-e(S:,x)-M implies e(.S„ y) -e(S,, y), and therefore, (recall that e(S,,.r)50), by Lemma 3.3,
t~eA. Contradiction.
2. S, é B: Then, Sé B for all ~ S ~- 2. Thus, B-{N} . Recall that e(S,, x) 5 0.
Thus, player 2 belongs to all the blocking coalitions. It follows, that there
exists a payoff y,
(with yZ,xztMax{e(S,,z),0}),
such that y~z,
y(N) -N(N), and y(S)?N(S) for all SCN. Thus, yeA. But then, xLy
con-tradicts xeA.
3. S,e6 and S~éB: Then, there exists y that satisfies: y,~x,, y21xZ, y,-x,, y'11.2)-p(I, 2), and Y(2.3)?~(2, 3). Since y(N)-x(N)fti1?u(IV), we have that yeA, contradicting xLy and xeA. Q.E.D.
ProoJ oj Theorem 3.1: Lemma 3.3 yields the " if" part of the theorem. To prove the
On the Sensitivity o( von Neumann and Morgenstern Abstract Srable Sets 49
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: If xeA' and z can be blocked, then x(S)GN(S) for all ~S~ -2.
Prooj.- Otherwise, there exists SCN, ~S~ -2 such that e(S, x)s0. Assume, w.l.o.g.,
that e(S,,x)?e(SZ,x)?e(S,,x). Then, e(S,,x)s0, implying that player 2 belongs
to
all
the
blocking
coalitions.
Define
z,
where
z, - x,,
z, - x,,
and
zZ gx: t Max {e (S, x)} . Since x can be blocked, zZ 1.rZ. Thus, xL sz. But z cannot be
blocked, hence zeA', contradicting xL ~z and xeA'.
Q.E.D.
Lemma 3.7.~ I f xe1SBS then e(S;, z) G e(5~., x) t e(S,, .r), for any choice of j, k, t,
{j,k,t}-{1,2,3}.
Proof.- Otherwise, w.l.o.g., e(S,, x)?e(S,, x) t e(S,, z). Then, there exist y, and ,v,
that satisfy: y, ?.r, t e(S,, .r), y,?xZ t e(S,, .r), and y, t y. -~t(1, 2). Let z be equal
to y~(y,,y.,x,) if y(N)?N(N), and if y(N)Gf~(N) let Z be such that z~y, and
z(N)-N(N). Then, z(S)?p(S) for all S, implying zeA'. But xL~z, ( using either
N or { t, 2}), contradicting xEA'.
Q.E.D.
Lemma 3.8.~ ]f xe1SBS then x is Pareto optimal, i.e., z(N)-t~(N). Alternatively,
ISBS(N, N) coincides with [he consistent bargaining set, CB(N, p).
ProoJ.- Assume, in negation, that there exists xeISBS`v'(N). Define,
a, s(2) [e(S,, x) t e(S,, x) -e(SZ, x)]
aZ g(2) (e (S,, x) t e(Sz, x) - e(S,, x)] a, 3(2) [e(SZ, x) t e(S,, x) -e(S,, x)]
y~(z,ta,,xZtaZ,x,tcti!,).
By Lemma 3.7, yax. Now, if y(N)sN(N), then there exists z?y with z(N)-N(N).
It follows that z(S)?p(S) for all S, implying that zeA'. But xL~z, (using N),
contradicts xeA'.
Thus, y(N)~tt(N). Since x(N)Gtt(N), there exists z such that z(N)-N(N),
zax and zay. Since xL~z ( using N), ZEA'. That is, there exists wEA' witit
zL~w. Since z(N)-p(N), w is supported by, w.l.o.g., S, -{ I, 2}. That is, w-(w,,wZ,z,), w,?z,, wz?zz, and w(S,)-~(S,). Therefore, by Lernrna 3.tí,
w cannot bc blocked. In particular, w(S,)- w, tz,?E~(I, 3) and w'(Si)-w~tz,?tt(2.3). Hence, w,twZt2z,-N(1,2)t2Z,?p(t,3)t,u(2,3). implying that z,?(:)[l~ll.3)f~~(2,3)-~~(1,2)]-.r",fa,-Y,. But zG.t~.
Contradic-tion. Q.L.D.
PronjoJ Theorem 3.2: I-et ( N, N) be a 0-normalized TU three person game. In vicw
of Lemmas 3.6-3.8, it suffices to show that if .ret~'(~V)`Core(,N, ~i) is such Ihat
S~ J. Grcenbcrg
zeA' su~h that xL~;. Since .rEU'(;N), we ha~e that z(.S,)-~~(.S,). fur somc jejl,',3}. f3y Icmtna 3.6, Ihcrefore, c(S,z)~0 fur all ~5~-2. But e(S„ x)ce(S,, .r) t cIS„ .i) implics that thcre exists k, such that e(S,, z)~0-
Contra-di~tion. Q.E.D.
Thcurems 3.1 and 3.2 yield the following surprising result.
Coro!lurv 3.9: Let (N, p) be a 0-normalized 3-person TU garne. Then,
~t nA' - {.rED~ x is not blocked}. It follows that SBSnISBS-Core(N, N). In
par-ticular, if the core is empty, thcn SBSnISBS-O.
Proof.~ Sincc (N, p) is a TU game, 0( D) - 0' (0)". By the (external) stability of A,
A~(D`A(U)j. Similarly, by the (external) stability of A', A' ~[D`v' (D)}. Hence,
A nA' ~ (D`0 (D)] - ( D`~' (D)].
To show that the reverse inclusion also holds, consider xe 0(D) (- ~' ( D)]. By
Lemma 3.6, if xEA' then x(S)cu(S) for every coalition S that consists of two
players. By Lemma 3.4, therefore, xEA.
Q.E.D.
I end this section by studying two examples that demonstrate some of the
pos-sible relationships among the three notions: The core, the SBS, and the ISBS. In
both examples, 1 personally find the SBS to be more appealing than the 1SBS16.
Exainple 3.I0: The three perso~r majority game: There are three players. Every
coal-ition that has a ( simple) majority, that is, every coalcoal-ition 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 environmcnt is given
by (N, p), where
N- { I, 2, 3}, p(S)-2 if ~S~ ?2, and for ieN, p({r})-0.
As is well-known, Core(N,;~)-0. Using Theorems 3.1 and 3.2, it is easy to see that
sas(N, N)- {(1, 1, o). (1, o, I). to, 1, 1)},
and
ISI3.S(N,p)-{x~x(N)-u(N)-2,
and x;cl
forall reN}.
T"hus, the core of this game is empty while the individual stable bargaining set,
which coincides with the consistent bargaining set, is given by the shaded area in
Figure I. In contrast, the stable bargaining set consists of the three outcomes (the
" For a set BCD, 0(B) is the set of elements in D that are dominated by B, i.e.,4(B)~ (xeD~there exists yeB, xLy}. Similarly, 0'(B)- {xeD~there exists yeó, x L 'y}.
On the Sensitivity of von Neumann and Morgenstern Abstract Stable Sets 51 (2,0,0)
Fig. I
(0,2,0) (0,1,1) (0,0,2)
three heavy dotted points in Figure 1) that stem from two players forming a
coali-tion, and distributing the 2 doltars evenly between them.
Pxample 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
in-cludes the ISBS (implying, by Corollary 3.9 that the ISBS coincides with the core).
Example 3.11: Consider the game (N,N), where N- {1, 2, 3}, w(l, 2)-N(1,
3)-N(N)- 100, and p(S)-0 otherwise. Then,
Core - ISBS - {( I 00, 0, 0) }
and SBS - {(100 - 2a, a, a) ~ 0 ~ a s 50 }.
It is obvious that Core- {(100, 0, 0)}. The following observations verify the other
assertions.
(i) ISBS- {(100, 0, 0)}: Consider x-(a, Q, y)ev(.v) with ac 100. Assume,
w.l.o.g., that Q?y. Then, (100-y)~a, implying that y-(100-y,~1,Y)
cannot be dominated. Hence y belongs to A'. Since xLky, it follows
that
z~ISBS.
Since
Core- {(100, 0, 0)},
Thcorem
2.4
yields
that
Core -1SBS - { (100, 0, 0) } .
(ii) SBS- {(100-2a, a, a)~05a550}: Observe, first, that z-(a, ~1, y)éA
whenever Q~y. Indeed, define, áL(Qty)~2, and y-(100-ó„~,ó)EA. Then, y cannot be dominated. Hence y belongs to A. Since xcy, xéA. It follos.s that x- (100-2a, a, a)eA, since otherwise, there exists zeA such that .rLZ. But then z-(z,, zz, z,) with zz~z,, and we just saw that such z cannot belong to A.
5` 1. Greenberg
4
Ordinatlly Convex Gamest'
There are, of course, games in which the SBS and the ISBS do intersecr in fact,
there are games in which the two sets coincide. One class of such games is the set of
"convex games", where
DeJinirion 4.1: A game (N, u) is called
1. (ordinallyJ convex if it satisfies: For any xERN and any two coalitions S and
T, if xSeu(S) and xreu(T), then either xs"reu(Sv T) or x"'Teu(SnT).
2. comprehensive if for all SCN, xeu(S) implies yeu(S) for all Osysx. ln
particular, OEU(S).
The main result of this section is
Tlteorein 4.2: Let (N, u) be a comprehensive QTU convex game. Then,
A- A' -{xE R': ~ x cannot be blocked }.
In particular,
SBS (N, u) -1SBS (N, u) - Core(N, u) .
In order to prove this theorem, we first need to establish the following Lemma.
Lenrma 4.3: Let ( N, u) be a comprehensive QTU convex game. Thcn, F-D`~(D)- {xER'v~x cannot be blocked}
is a vN~CM abstract stable set for (D,L).
PruuJ: Clearly, 0(F)CD`F. To conclude the proof we need tu show that
U`FC,S(F). The proof of this inclusion generalizes, but closely follows that of
Pro-position 3.3 in Uuna et al. Otherwise, there exists xeD`F but xé ~(F). Thus, there
exists S that blocks x through some y, and (yS, xN~s)éF. By the comprehensiveness
u( (N, u), [herefure, the set Q, ~ 0, where
Q~- {(S,~')~y'EU'(S),
yN,s-xN`s and ys~xs}.
By considering minimal (in the set inclusion ordering) coalitions in Q,, it is easy to see that there exists (S, y)E Q, such that no subset of S can block y, and S is a maxi-mal such coalition. Since (yt, xN`~)éF, lhe comprehensiveness of (N, u) implies that SCL Q. ~ t~, N'lllre
On the Sensitivity of von Ncumann and Morgenstern Abstract Stable Sett
Qz- {ÍT. Z)IlTEPfT). ~w`IT~.f1-}'.v~tT~.SI
and zT`f;Dl
By the choicc of ~, (T, z)EQZ implies T`S~A. Let
~~
Sl
(T,
ï)EArgMaxt~,.,rEQzlh1in~,F„~~z'-Since (i~s xN`~)EF. Mint,Et~.~tï'10.
Then,
ys-ïfeu(S)
and
iteu(T).
By
the
choice
of
S,
Zs~t-~.c~télnt.u(SnT). By Lemma 3.3 in Dutta et aL (1989), it follows that
ïs"reu(~vT). Now, if there was a coalition WC~vT that blocks ï, using
cueu(W), then, by the choice of ~, W`5~0. Moreover, cu"`~;aï"`~ implies that
Min,~,,.~,~cv'~h4in,Et~.tï'. Since cu~nt~y.t~t and (N,u) is comprehensive, the
above strict inequality contradicts the choice of ï. Thus, i is not blocked by any
subset of S~ T'.
But this contradicts the maximality of ~. (Recall that T`S ~ 0, implying
~~vT~ ~ ~5~.)
Q.E.D.
ProojojTheorem 4.2: By Lemma 4.3 we have that the unique vNAM abstract
sta-ble set for (D, L) is: A- {xe R;' ~ x cannot be blocked }. Using Theorem 2.2, in
or-der to conclude the proof of the theorem, it suffices to show that A is a vNBtM
abstract stable set for (D,L~).
Indeed, since (N, u) is QTU, A n 0(A) - 0 implies that A n A' (A) - 0, that is,
0'(A)CD`A. To see that the reverse inclusion also holds, consider xeD`A. By
Lemma 4.3, there exists yEA such that xLy, implying xG~y. Hence,
~' (A) ~ D`A. Thus, 0' (A) - D`A, i.e., A is a vNBtM abstract stable set for
(D,L ~). 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).
Corollarv 4.4: The core of a comprehensive QTU convex game is nonempty.
Proof.~ Let (N, u) be a comprehensive QTU convex game. Then, since Oeu(Q) for all
QCN, the convexity of (N, v) yields that for all xsev(S), (xS, t)N`S)eu(N). By
The-orem 2.4, therefore, SBS(N, u) ~ 0. TheThe-orem 4.2 concludes the proof.
Q.E.D.
ft is noteworthy that Peleg (1986) proved that in convex games the core is also
the unique vNRtM solution.
5
An Open Question
Sy J. Greenberg
Dutta et al. showed lhat it is possible that ISBSnu'(N)-0. Specifically,
they
consider
the
4
player
TU
game
(N, N),
where
N- { 1, 2, 3, 4},
N(l, 2, 3)-N(2, 3, 4)-66, N(1, 4)-46, N(1, 2, 4)-p(l, 3, 4)-63, u(N)-80, and
N(S)-0 otherwise.
They show ( Dutta et al.
1989,
Proposition 4.1)
that
CB (N, p ) - 0.
!n contrast, the SBS in this game does contain Pareto optimal payoffs. For
example, x' ~(23, 17, 17, 23)eSBS. Indeed, { l, 4}, { l, 3, 4}, and { l, 2, 4} cannot
block x'. It remains to check for coalitions { l, 2, 3} and {2, 3, 4}. Now, if { 1, 2, 3}
blocks x' with y, then y, ~z!, implying that [yz t y,J c [66 -23J. But then, there
exists a payoff z such that z(2, 3, 4)-N(2, 3, 4)-66 and ycz. Since z(S)zN(S) for
all S, zEA. The stability of A implies, therefore, that yEA. An analogous argumen[
shows that {2, 3, 4} cannot block z' using a payoff in A. Thus, x'EA.
It remains an open question whether in superadditive TU games we have tha[
SBSC u' (N), or, at least, SBSn u' (N) ~ 0. An affirmative answer to lhis yuestion
will be particularly pleasing, since if the "real world" is to provide a guídeline, then
the notion of the SBS seems to be more appropriate than the ISBS (or the CB);
individuals otten insist on some compensation if the "status quo" is to be
changed.
References
Aumann R, Maschler M(1964) " The bargaining set for cooperative games", in Advances in
Game Theory, Uresher, Shapley, and Tucker, eds. Princeton, N.J.: Princeton University
press, 443-447.
Asscher N(1976) " An ordinal bargaining set for games without sidepayments", Mathematics of Operations Research, l, 381-389.
Asscher N(1977) 'A cardinal bargaining set for games without sidepayments", International
Journal of Game Theory, 6, 87-114.
Billera L1 (1970) "Existencr o( general bargaining sets for cooperative games without side
payments", Bullerin of the American h1athematical Society, 76, 375-379.
Uutta B, Ray U, Sengupta K, Vohra R(1989) "A consistent bargaining set", Journal of
Eco-numic Theory, 49, 93-I 12.
Uutta B, Ray D(1989) "Constrained egalitarian allocatíons", Games and Economic Behavior,
3, 403-422.
Grcenberg J(1979) `Stability when mobility is restricted by the existing coalition structure", Juurnal of Economic Theory, 21, 213-221.
Greenbrrg J(19ë5) "Corrs of convex games without side payments", htathematics of Opera-tions Krxarch, 523-525.
Grcenberg J(Iy90) The throry of social situations: An alternative game-theoretic approach, Cambridge Univrrsity Press.
C`las-Colcll A(lyb9) "An eyuivalrnce theorem for a bargaining set", Journal of Mathrwati~al [cunomirs, 18, 129-138.
Pelrg B(1963) "Bargaining sets of cooperative games without side paymenls", Israel Journal of Mathematic~, 3, 127-138.
Pelrg B(198G) "A prouf that the rure of an ordiual cunvex game is a vun Nrumann and !~lorgrnstrrn sulutiun", C,tarh. Soc. Sci. 11, 83-87.
Srngupta A, Srn~upta K(1992) "Viable proposals", mimeu.
On the Scnsitivit~ of von Ncumann and Morgenstern Ahatracl Stahlc Scts t5 Vilkov VV (1977) `Convcs games without sidepayments-, Vestnik Leningradskica
Uni~crsita-la 7, 21-2d.
von Neumann 1, Morgenstern O(1947) Theory of Games and Economic E3ehavior. Princcton
NJ: Princeton Universin~ Press. Received February 1991
Reprint Series, CentER, Tilburg University, The Netherlands:
No. 1 G. Marini and F. van der Plceg, Monetary and fiscal policy in an optimising model with capital acxumulation and finite Gves, The Economic Jowrwl, vol. 98, no. 392, 1988, PP. 772 - 786.
No. 2 F. van der Plceg, International policy coordination in interdependent monetary
economies, Jourrtdoj Iruemationo! Ecorromics, vol. 2S, 1988, pp. 1- 23.
No. 3 A.P. Barten, The history of Dutch macroec:onomic modelling (1936-1986), in W. Driehuis, M.M.G. Fase and H. den Hartog (eds.), Challenges jorMacroeconomic
Moclelling, Contributions to Economic Analysis 178, Amsterdam: North-Holland,
1988. PP. 39 - 88.
No. 4 F. van der Plceg, Disposable income, unemployment, in~ation and state spending in a dynamic political-economic model, PubGc Choice, vol. 60, 1989, pp. 211 - 239.
No. S Th. ten Raa and F. van der Plceg, A statistical approach to the problem o[ negatives in input-output analysis, Economic Modelling, vol. 6, no. 1, 1989, pp. 2 - 19.
No. 6 E. van Damme, Renegotiation-proof equilibria in repeated prisoners' dilemma, Jouma! oj Econantic Theory, voL 47, no. 1, 1989, pP. 206 - 217.
No. 7 C. Mulder and F. van der Ploeg, Trade unions, investment and employment in a small open economy: a Dutch perspective, in J. Muysken and C. de Neubourg (eds.), Unen:ployment in Europe, London: The Macmillan Press Ltd, 1989, pp. 200 - 229.
No. 8 Th. van de Klundert and F. van der Ploeg, Wage rigidity and capital mobility in an optimizing model of a small open economy, De Econontist, vol. 137, nr. 1, 1989, PP. 47 - 7S.
No. 9 G. Dhaene and A.P. Barten, When it all began: the 1936 Tinbergen model revisited, Economic Modelling, vo4 6, no. 2, 1989, pp. 203 - 219.
No. 10 F. van der Ploeg and AJ. de Zeeuw, Conllict over arms accumulation in markct and command economies, in F. van der Ploeg and A.J. de Zeeuw (eds.), Dytwntic
Policy Camer in Economicr, Contributions to Economic Analysis 181,
Amster-dam: Elsevier Science Publishers B.V. (North-Holland), 1989, pp. 91 - 119. No. 11 J. Driffill, Macroeconomic po6cy games with incomplete information: some
extensions, in F. van der Ploeg and AJ. de Zeeuw (eds.), Dynamic Policy Ga~nes in Economicr, Contributions to Economic Analysis 181, Amsterdam: Elsevier Science Publishers B. V. (North-HoUand), 1989, pp. 289 - 322.
No. 13 RJ.M. Alessie and A. Kapteyn, Consumption, savings and demography, in A. Wenig, (CF. Zimmermann (eds.), Demogmphic Change atuf Economic Development, Berlin~Heidelberg: Springer-Verlag, 1989, pp. 272 - 305.
No. 14 A. Hoque, J.R. Magnus and B. Pesaran, The exact multi-period mean-square forecast error for the first-order autoregressive model, Jounwl oj Ecwtornetncs, vol. 39, no. 3, 1988, pp. 327 - 346.
No. 15 R. Alrssie, A. Kapteyn and B. Melenberg, The elfects of liyuidity constraints on consumption: estimation from household panel data, European Economic Review, vol. 33, no. 2~3, 1989, pp. 547 - SSS.
No. 16 A. Holly and J.R. Magnus, A note on instrumental variables and maximum likeli-hood estimation procedures, Annaler d'Éconanie et de Statistiyue, no. 10, April-June, 1988, pP. 121 - 138.
No. 17 P. ten Hacken, A. Kapteyn and 1. Woittiez, Unemployment benefits and the labor market, a micro~maao approach, in B.A. Gustafsson and N. Anders Klevmarken (eds.), T!u Political Economy oj Sociul Security, Contributions to Economic Analysis 179, Amsterdam: Elsevier Sc:ience Publishers B.V. (North-Holland), 1989, pp. l43 - 164.
No. 18 T. Wansbeek and A. Kapteyn, Estimation of the error-components model with incomplete panels, Journal oj Econontetncs, vol. 41, no. 3, 1989, pp. 341 - 361. No. 19 A. Kapteyn, P. Kooreman and R. Willemse, Some methodological issues in the
implementation of subjective poverty defmitions, The loutra! of Human
Resources, vol. 23, no. 2, 1988, pp. 222 - 242.
No. 20 Th. van de Klundert and F. van der Plceg, Fisca! policy and tinite lives in interdependent economies with real and nominal wage rigidity, Oxjord Economic
Papers, vol. 41, no. 3, 1989, pp. 459 - 489.
No. 21 J.R. Magnus and B. Pesaran, The exact multi-period mean-square forecast error for the first-order autoregtessive model with an intercept, Joumal oj
Economenics, vol. 42, no. 2, 1989, pp. 1S7 - 179.
No. 22 F. van der Ploeg, Two essays on political economy: (i) The political ernnomy of overvaluation, The Economic Joumal, vol. 99, no. 397, 1989, pp. 8S0 - 855; (ii) Election outcomes and the stockmarket, European Joumal of Political Economy, vol. S, no. 1, 1989, pp. 21 - 30.
No. 23 1.R. Magnus and A.D. Woodland, On the maximum likelihood estimation of multivariate regression models containing serially correlated error components,
Imemational Economic Review, vol. 29, no. 4, 1988, pp. 707 - 725.
No. 24 AJJ. Talman and Y. Yamamoto, A simplicial algorithm for stationary point problems on polytopes, Mathematics oj Operations Reseanh, vol. 14, no. 3, 1989, pp. 383 - 399.
No. 25 E. van Damme, Stable equilibria and forward induction, Journal oj Economic
No. 26 A.P. Barten and LJ. Bettendurf, Price formation of fish: An application of an inverse demand system, Ewopean Econontic Review, voL 33, no. 8, 1989, pp. 1509 - 1S2S.
No. 27 G. Noldeke and E. van Damme, Signalling in a dynamic labour market, Review
oj Economic Studies, vol. 57 (1), no. 189, 1990, pp. 1- 23.
No. 28 P. Kop Jansen and Th. ten Raa, The choice of model in the construction of input-output cuefficients matrices, Interstationa! Economic Review, vol. 31, no. I,
1990, PP. 213 - 227.
No.29 F. van der Ploeg and A.J. de Zeeuw, Perfect equilibrium ín a model of wmpetitive acros aocumulation, ]ntemationa! Economic Review, vol. 31, no. 1, 1990, pp. 131 - 146.
No. 30 J.R. Magnus and A.D. Woociland, Separability and aggregation, Economica, vol. 57, no. 226, 1990, pp. 239 - 247.
No. 31 F. van der Ploeg, International interdependence and policy coordination in economies with real and nominal wage rigidity, Creek Ecunomic Review, vol. 10,
no. 1, June 1988, pp. 1- 48.
No. 32 E. van Damme, Signaling and forward induction in a market entry context,
Operationr Researrh Proceedings 1989, Berlin-HeidelbCrg: Springer-Verlag, 1990,
pp. 45 - S9.
No. 33 A.P. Barten, Toward a levels version of the Rotterdam and related demand systems, Contributions to Operations Rrseurch wui Economics, Cambridge: M1T Press, 1989, pp. 441 - 465.
No.34 F. van der Ploeg, International coordination of monetary policies under alternative exchange-rate regimes, in F. van der Plueg (ed.), Advanced Lectwes in Qtuuuitative Economics, London-Orlando: Academic Press Ltd., 1990, pp. 91 - 121.
No. 3S Th. van de Klundert, On socioeconomic causes of 'wait unempluyment', Ewopean
Economic Review, vol. 34, no. S, 1990, pp. ]011 - 1022.
No. 36 R.J.M. Alessie, A. Kapteyn, J.B. van Lochem and T.J. Wansbeek, Individual e(fects in utility consistent models of demand, in J. Hartog, G. Ridder and J. Theeuwes (eds.), Pane! Data and Labor Market Studles, Amsterdam: Elsevier
Science Publishers B.V. (North-Holland), 1990, pp. 253 - 278.
No.37 F. ven der Ploeg, Capital accumulation, inflation and lung-run conflict in interna[iunal objCCtives, O.tford Economic Pupers, vol. 42, no. 3, 1990, pp. SOl -525.
Nu. 38 Th. Nijman and F. Pelm, Parametcr identiGcation in ARMA Prueesses in the presence o( regular but incomplete sampling, Jounw( ojTirne Series Arwtysis, vol.
1 l, no. 3, 199U, pp. 239 - 248.
No. 39 Th. van de Klundert, Wage differentials and employment in a two-sector model
No. 40 Th. Nijman and M.FJ. Steel, Exclusion restrictions in instrumental variables
eyuations, Econometnc Review.r, vol. 9, no. 1, 1990, pp. 37 - 55.
No. 4l A. van Soest, 1. Woittiez and A. Kapteyn, Labor supply, income taxes, and hours restrictiuns in the Netherlands,JoumalojHunuui Resources, vul. 25, no. 3, 1990, pp. 517 - 558.
No. 42 Th.C.MJ. van de Klundert and A.B.T.M. van Schaik, Unemployment persistence and loss of productive capacity: a Keynesian approach, Journal oj blacro-economics, vol. 12, no. 3, 1990, pp. 363 - 380.
No. 43 Th. Nijman and M. Verbeek, Estimation of time-dependent parameters in Gnear models using cross-sections, panels, or twth, Jounutf ojEconometrics, vol. 4G, no. 3, 1990, pp. 333 - 346.
No. 44 E. van Damme, R. Selten and E. Winter, Alternating bid bargaining with a smallest money uni[, Games and Economic Behavior, vol. 2, no. 2, 1990, pp. 188 - 201.
No. 45 C. Dang, The D,-triangulation of R' for simplicial algorithms for computing solutions of nonlinear equations, Mathematics ojOpemtions Resea~cit, vol. 16, no. 1, 1991, pp. 148 - 161.
No. 46 Th. Nijman and F. Palm, Predictive accuracy gain from disaggregate sampling in ARIMA models, JoumalojBusiness 6c Economic Statisticr, vol. 8, no. 4, 1990, pp. 405 - 415.
No. 47 J.R. Magnus, On certain moments relating to ratios of quadratic forms in normal
variables: further results, Sankhya~ The Indinn Jouma[ ojStatisticr, vol. 52, series B, part. I, 1990, pp. 1- 13.
No. 48 M.FJ. Steel, A Bayesian analysis of simultaneous equation models by combining recursive analytical and numerical approaches, Journaloj Econometrics, vol. 48, no. 1~2, 1991, pp. 83 - 117.
No. 49 F. van der Ploeg and C. Withagen, Pollution rnntrol and the ramsey problem,
Environmental and Resource Economicr, vol. 1, no. 2, 1991, pp. 215 - 236.
No.50 F. van der Plceg, Money and capital in interdependent economies with overlapping generations, Economica, vol. 58, no. 230, 1991, pp. 233 - 25tí.
No. 51 A. Kapteyn and A. de Zeeuw, Changing incentives for economic research in the Netherlands, European Economic Review, vol. 35, no. 2~3, 1991, pp. G03 - 611.
No. 52 C.G. de Vries, On the relation between GARCH and stable processes, Joumal oj Econometrics, vol. 48, no. 3, 1991, pp. 313 - 324.
No. S4 W. van Grcenendaal and A. de Zeeuw, Control, coordination and conLict on international commodity markets, Econonric Modelling, vol. 8, no. 1, 1991, pp. 90 - 101.
No. SS F. ven der Ploeg and AJ. Markink, Dynamic poliry in linear models with rational expcctations of future events: A computer package, Cumputer Science in
Economict arul Munagement, vol. 4, no. 3, 1991, pp. 17S - 199.
No.56 H.A. Keuzenkamp and F. van der Ploeg, Savings, investment, government finance, and the current acxount: 71te Dutch experience, in G. Alogoskoufis, L. Papademos and R. Portes (eds.), Euernal Constrainu on Macroeconomic Policy:
The Europeun Erperiencr, Cambridge: Cambridge University Press, 1991, pp. 219
- 263.
No. S7 Th. Nijman, M. Verbeek and A. van Soest, The efticiency of rotating-panel
designs in an analysis-of-variance model, Joumal oj Econometnc.r, vol. 49, no. 3, 1991, pp. 373 - 399.
No. S8 M.FJ. Steel and 1.-F. Richard, Bayesian multivariate exogeneity analysis - an
application to a UK money demand equation, Jouma! of Eca[ornetricr, vol. 49,
no. 1~2, 1991, PP. 239 - 274.
No. S9 Th. Nijman and F. Palm, Generalized least syuares estimation of linear mode4s
containing rational future expectations, Intenuuional Econoinic Revirw, vol. 32, no. 2, 1991, PP. 383 - 389.
No. 60 E. van Damme, Equilibrium selection in 2 x 2 games, Revista Espanula de Ecorwmia, voL 8, no. 1, 1991, pp. 37 - S2.
No. 61 E. Bennett and E. van Damme, Demand commitment bargaining: the case of apex games, in R. Selten (ed.), Came Equilibrium ModeLs !II - Strategic
Burgaining, Berlin: Springer-Verlag, 1991, pp. 118 - 140.
No. 62 W. Guth and E. van Damme, Gorby games - a game theoretic analysis of disarmament campaigns and the defense efficiency - hypothesis -, in R. Avenhaus, H. Karkar and M. Rudnianski (eds.), Drjense Decisiun Muking
-Anulytica! Supporr und Crisis Marwgement, Berlin: Springer-Verlag, 1991, pp. 21S
- 240.
No. 63 A. Roell, Dual-capacity trading and the quality o( the market, Jounut! oj Finaiuial Intennediatiun, vol. l, no. 2, 1990, pp. lOS - 124.
No. 64 Y. Dai, G. van der Laan, AJ.1. Tnlman and Y. Yamamoto, A simpticial algori[hm for the nonlinear stationary point problem on an unbuunded polyhedron, Sia~n Juumul of Optimimtion, vol. 1, no. 2, 1991, pp. 1S 1- 1GS. No.65 M. McAleer and C.R. McKenzie, Keynesian and new classical models of
unemploymcnt revuited, The Economic Jountal, vol. 101, no. 406, 1991, pp. 3S9 - 381.
No.67 J.R. Magnus and B. Pesaran, The bias of forecasts from a Cust-order autoregression, Econometnc Theory, vol. 7, no. 2, 1991, pp. 222 - 235.
No. 68 F. van der Ploeg, Macroeconomic poGcy coordination issues during the various phases of economic and monetary integration in Europe, European Economy
-The Economicr oj EMU, Commission of the European Communities, special
edition no. 1, 1991, pp. 136 - 1G4.
No. 69 H. Keuzenkamp, A precursor to Muth: Tinbergen's 1932 model of rational expectations, The Economic Joutrwl, vol. 101, no. 408, 1991, pp. 1245 - 1253. No. 70 L. Zou, The target-incentive system vs. the price-incentive system under adverse
selection and the ratchet effect,loutnal ojPublic Economics, vol. 46, no. 1, 1991, pP. S l - 89.
No.71
E. Bomhoff, Between price reform and privatization: Eastern Europe in
transition, Finanznwrkt und Porfjolio Management, vol. 5, no. 3, 1991, pp. 241
-2s 1.
No. 72 E. Bomhoff, Stability of velocity in the major industrial countries: a Kalman filter approach, Intemationa! Monetary Fund StajjPapers, vol. 38, no. 3, 1991, pp. 626 - 642.
No. 73 E. Bomhoff, Currency convertibility: when and how? A contribution [o the Bulgarian debate, Kredit und Kapital, vol. 24, no. 3, 1991, pp. 412 - 431.
No.74 H. Keuzenkamp and F. van der Ploeg, Perceived constraints for Dutch unemployment policy, in C. de Neubourg ( ed.), The Arr oj Full Employment
-Unemployment Policy in Open Economies, Contributions to Economic Analysis
203, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1991, pp. 7
- 37.
No. 75 H. Peters and E. van Damme, Characterizing the Nash and Raiffa bargaining
solutions by disagreement point axions, Mathematicc ojOpemtions Research, vol. 16, no. 3, 1991, pp. 447 - 461.
No. 76 PJ. Deschamps, On the estimated variances of regression coefGcients in
misspecified error components modeLs, Econometric Theory, vol. 7, no. 3, 1991,
pp. 369 - 384.
No. 77 A. de Zeeuw, Note on 'Nash and Stackelberg solutions in a differential game
model of capitalism', loutnal oj Economic Dynamics and Control, vol. 16, no. 1,
1992, pp. I39 - 145.
No. 78 J.R. Magnus, On the fundamental bordered matrix of linear estimation, in F. van der Ploeg (ed.), Advanced Lectutrs in Quanti[ative Economics, London-Orlando: Academic Press Ltd., 1990, pp. 583 - 604.
No. 79 F. van der Ploeg and A. de Zeeuw, A differential game of international pollution control, Systems and Control Letters, vol. 17, no. 6, 1991, pp. 409 - 414. No. BO Ttt. Nijman and M. Verbeek, The optimal choice of controls and
No. 81 M. Verbeek and Th. Nijman, Can cohort data be treated as gcnuine panel data?,
Empirica! Ecunomict, vol. 17, no. 1, 1992, pp. 9- 23.
No. 82 E. van Damme and W. Guth, Equilibrium selection in the Spence signaling game, in R. Selten (ed.), Garne EquiG'brium MoclrCr II - Metlwds, MuruLs, uru! Murkrts, Berlin: Springer-Verlag, 1991, pp. 263 - 288.
No. 83 R.P. Gilles and P.H.M. Ruys, Characterization of economic agents in arbitrary communication structures, Nieuw Archiej voor Wukutule, vol. 8, no. 3, 1990, pp. 325 - 345.
No. 84 A. de Zeeuw and F. van der Ploeg, Difference games and policy evaluation: a
conceptual framtwork, Oxjord Economic Pupers, vol. 43, no. 4, 1991, pp. 612
-636.
No. 85 E. van Damme, Fair division under asymmetric information, in R. SCl[en (ed.),
Ratiorta! Interaction - Essays in Nonor ojJoiut C. Harsanyi, Berlin~Heidelberg:
Springer-Verlag, 1992, pp. 121 - 144.
No. 86 F. de Jong, A. Kemna and T. Klcek, A contribution to event study methodology with an application to the Dutch stock market, Jouma! oj Burtking and Firtance, vol. 16, no. 1, 1992, pp. 11 - 36.
No. 87 A.P. Barten, The estimation of mixed demand systems, in R. Bewley and T. Van Hoa (eds.), Cunrributionr to Consumer Denuuul arui Econometrics, Essuys in
Norwui ojHenri Theil, Basingstoke: The Macmillan Press Ltd., 1992, pp. 31 - 57.
No. 88 T. Wansbeek and A. Kapteyn, Simple estimators for dynamic panel data models with errors in variables, in R. Bewley and T. Van Hoa (eds.), Contributions to
Conrumer Denuind and Econometrics, Essays in Nonour oJ Nrnri Titeil,
Basingstoke: The Macmillan Press Ltd., 1992, pp. 238 - 251.
No. 89 S. Chib, J. Osicwalski and M. Steel, Posterior in(erence on the degrees of freedom parameter in multivariate-t regression models, Economics Lrtters, vol. 37, no. 4, 1991, pp. 391 - 397.
No. 90 H. Peters and P. Wakker, Independence of irrelevant altCrnatives and revealed group preferences, Econometrica, vol. 59, no. 6, 1991, pp. 1787 - 1801. No. 91 G. Alogoskoufis and F. van der Ploeg, On budgetary policies, growth, and
external deficits in an interdependent world, Jounw! oj the Jupttrtese arul
Intrnwtionul Econurnies, vol. 5, no. 4, 1991, pp. 305 - 324.
No. 92 R.P. Gilles, G. Owen and R. van den Brink, Games with permission structures: The conjunctive approach, Internatiorwl Jounuel uJ Gume Theory, vol. 20, no. 3,
1992. PP. 277 - 293.
No. 93 1A.M. Potters, 1.1. Curiel and S.H. Tijs, Travelingsalesman games, Afurhentaticu!
Prugrununing, vol. 53, no. 2, 1992, pP. 199 - 21 l.
No. 94 A.P. Jurg, M.J.M. Jansen, 1.A.M. Potters and S.H. Tijs, A symmetrization for (initC two-perwn games, Zeitscluijt fur Oprrutiau Reseurch - Afrthudc utul Mu~leLs
No. 9S A van den Nouweland, P. Borm and S. Tijs, Allocatíon rules for hypergraph
communication situations, International lounui! of Garne Theory, vol. 20, no. 3,
1992, PP. 2S5 - 268.
No. 96 EJ. Bomho(f, Munetary refurm in Eastern Europe, Europeart Ecorwmic Review, vol. 36, no. 2~3, 1992, pp. 454 - 458.
No. 97 F. van der Plceg and A. de Zeeuw, In[ernational aspects of pollution control,
Enviro~unrnta! and Resource Economict, vol. 2, no. 2, 1992, pp. 117 - 139.
No. 98 P.E.M. Borm and S.H. Tijs, Strategic claim games corresponding to an NTU-game, Garnes and Economic Behavior, vol. 4, no. 1, 1992, pp. S8 - 71.
No. 99 A. van Soest and P. Kooreman, Coherency of the ind'trect translog demand
system with binding nonnegativity cons[raints, louma( oj Econometrics, vol. 44,
no. 3, 1990, pp. 391 - 400.
No. 100 Th. ten Raa and E.N. Wol(f, Secondary products and the measurement of productivity growth, Regiortal Science and Urban Economicr, vol. 21, no. 4, 1991, pp. S81 - 615.
No. 101 P. Kooreman and A. Kapteyn, On the empirical implementation of some game theoretic models of household labor supply, The Journal ojHtunan Resources, voL 2S, no. 4, 1990, pp. 584 - 598.
No. I02 H. Bester, Bertrand equilibrium in a differentiated duopoly, Intentational
Econoinic Review, vot. 33, no. 2, 1992, pp. 433 - 448.
No. 103 JA.M. Potters and S.H. Tijs, The nucleolus of a matrix game and other nucleoli,
Mathematics oj Opert~tioru Research, vol. 17, no. 1, 1992, pp. 1G4 - 174.
No. 104 A. Kapteyn, P. Kooreman and A. van Soest, Quantity rationing and concavity in a ~exible household labor supply model, Review oJEconomics and Statisrics, vol. 72, no. 1, 1990, pp. SS - 62.
No. lOS A. Kapteyn and P. Kooreman, Household labor supply: What kind of data can
tell us how many decision makers there are?, European Economic Review, vol. 36,
no. 2~3, 1992, pp. 3G5 - 371.
No. 106 Th. van de Klundert and S. Smulders, Reconstructing growth theory: A survey,
De Economist, vol. 140, no. 2, 1992, pp. 177 - 203.
No. 107 N. Rankin, Imperfect competition, expectations and the multiple effects of monetary growth, The Economicloumal, vol. 102, no. 413, 1992, pp. 743 - 753. No. 108 J. Greenberg, On the sensitivity of von Neumann and Morgenstern abstract stable sets: The stable and the individual stable bargaining set, Intemational
