Tilburg University
Independence of irrelevant alternatives and revealed group preferences
Peters, H.; Wakker, P.P.
Publication date:
1992
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Peters, H., & Wakker, P. P. (1992). Independence of irrelevant alternatives and revealed group preferences.
(Reprint Series). CentER for Economic Research.
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by
Hans Peters and
Peter Wakker
Reprinted from Econometrica,
Vol. 59, No. 6, 1991
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Reprint Series
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no. 90
puiiuiqniHiin~uiunuHNiHiiiyuun
Independence of Irrelevant
Alternatives and Revealed Group
CENTER FOR ECDNOMIC RESEARCH
Research Staff
Helmut Bester Eric van Damme Board
}{elmut Bester
Eric van Damme, director
Arie Kapteyn
Scientific Council Eduard Bomhoff Willem Buiter Jacques Drèze
Theo van de Klundert Simon Kuipers Jean-Jacques Laffont Merton Miller Stephen Nickell Pieter Ruys Jacques Sijben Residential Fellows Svend Albaek Pramila Krishnan Jan Magnus Eduardo Siandra Hideo Suehiro Doctoral Students Roel Beetsma Hans Bloemen Sjaak Hurkens Frank de Jong Pieter Kop Jansen
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
European University Institute San Francisco State University Tilburg University
UCLA
Kobe University
Independence of Irrelevant
Alternatives and Revealed Group
Preferences
by
Hans Peters and
Peter Wakker
Reprinted from Econometrica,
Vol. 59, No. 6, 1991
Economerrica, Vol. 59, No. 6(November, 1991), 1787-1801 INDEPENDENCE OF IRRELEVANT ALTERNATIVES
AND REVEALED GROUP PREFERENCES
BY HANS PEIERS AND PEIER WAKKER
l. 1NTRODUC['ION
IN CONSUMER DEMAND 711EORY the ooncept of revealed preference is based On tht assumption that, by choosing from budget sets, a oonsumer reveals his preferences over the available oommodiry bundles. Analogously, in bargaining game thcory the agree-ments reachcd in bargaining games may be thought to reveal the prefercnces of the bargainers as a group. In this paper we oonsider, more generally, singlc-valued choice functions dcfined on the oonvex compact subsets of the positive orthant of R". ~ese subuts are called choice situations. In bargaining game theory choice functions are called bargaining solutions and choicc situations are called bargaining gamcs. In con-sumer demand theory choice functions are called demand functions and choice situations are called budget sets. Compact convex budget sets may be regarded as "generalized" budget sets where certain oommodity bundles from the full simplices (linear budget sets) are not available. An example is the case of piecewise linesr budget sets (see Hausman (1985)); our results would remain valid under the restriction to this case as well. Works ooncerned with revcaled preference in consumer demand theory are, e.g., Richter (1971), Varian (1982), and Pollak (1990). 711e latter discusses generalized budget sets.
One purpose of this paper is to find conditions under which a choice function maximizes a real-valued function. In oonsumer demand theory such a function is called the consumer's utility function. Another purpose is to prwide a thorough study of the consequences of the well-known independence of irrelevant alternatives (IIA) condition. A third purpose is to generaliu the Nash bargaining solution.
We will first observe that a choice function maximizes a binary relation if and only if it satisfies IIA. This condition was introduced by Nash in his seminal 1950 paper on the bargaíníng problem. Next we show that the combination of Pareto optimality and IIA for a choice function in general only excludes cycles of length 1 or 2 in the revcaled binary rclation. If the dimension is 2, then also cycles of length 3 are excluded, but cycles of length at least 4 may still occur. For the latter case (i.e., n- 2), adding a weak form of oontinuity called Pareto continuiry suffices to exclude circularity of the revealed binary relation; in general, howcver, cvcn "iull" continuity does not exclude cycles. For the case of 2-dimcnsional linear budget sets, related work was done by Samuelson (1948) and Rose (1958).
The main result of ihe paper is obtained by strengthening Pareto continuiry to continuity: this condition together with Pareto optimality, and IIA for n- 2 or the (stronger) strong axiom o[ revealed prcference for n~ 2, is su(ficient for the existence of a function representing the revealcd binary relation, i.e., of a function which is maxi-mized by the choice function. We finally show that this representing function must be strongly monotonic and strictly quasiconcave and, conversely, that the existence of a represcnting function with these propertics implies the oonditions of continuiry, Parcto optimaliry, IIA, and the strong axiom of revealed preference for the choice function.
~e organization of the paper is as follows. Section 2 gives elementary definitions and oonsiders the role of IIA. Sections 3 and 4 study the (akyclicity of revealed preference without and with continuiry conditions, respectively. Section 5 is dcvotcd to the aforc-mentioned main result and briefly discusses an application to bargaining game theory. Seccion 6 shows that the results can be extended to other domains, and concludes.
17~ HM7s PETERS AND PEIER WAKKER 2 THE ROLF OF IlA
We denote by X the set ot all possible altcmatiues (for a consumer, a group o[ bargainers,...). In this paper, with the exception of Section 6, X- R;,, and a choice situation is a nonempty eonvex compact subset of X. The collection o[ all choice situations is denoted by i.
A choice junction is a map F: E-i X with F(S) E S for every S E E. Note that in this paper a choice [unction is singk-uolucd by definition. From F we derive a binary relation R on X as follows: zRy ("x is directly rcucaled prejemd to y") it there is an SE~ with x-F(S), yES.
Sometimes choice functions can be derived from binary relations. A binary relation Y on X repnesenu a choice function F i[ [or every choice situation S wc have
(2.1) (F(S)} -{x E S: x~ y for every y in S}, i.e., F uniquely maximizes a on S.
Obviously not every binary relation represents a choicc [unciion, and not every choice tunction can be represénted by a binary rclation. The following condition will charactcr-ize, within the set-up of this paper, the choice functions which can be represented by a binary relation. It was introduced in Nash (1950) for bargaining game theory, and is central in ihis paper.
Dee~NmoN 2.1: The choice function F satisfies independence of imkuant altematiuu (IIA) if for aU choice situations S and T with S e T and F(T ) E S we have F(S) - F(T). THEORSJ.t 2.2: 7ire choice junction F can be rcprrscntcd by a binary rclation y ij and
only ij Fsatirfus IIA.
PROOS- First supposc F is represented by s. Lct S, T E~ with S c T and F(T ) e S. By definition {F(T)) ~{x e T: x s y for every y e T}. So {F(T)) -{z E S: x s y tor every y E S}. From this we oonclude that F satisfies IIA.
In order to prove the converse, suppose F satisfies IIA. Define a:~ R. Then, tor
every S E~, F(S) s y for every y E S. We still have to show that F(S) uniquety
maximizcs a on S, tor evcry S e E. Suppose therc is an S E E with y E S and y s F(S),
i.e., yRF(S). Then there is a T E E with F(S) E T and y ~ F(T), so by IIA applied
twice, y- F(T rl S) - F(S). This completes the proof.
Q-F-D-[n detending the IIA-condition Nash (1950, p. 195) argues that ( two) rational individu-als, agreeing on a common choice x from T, should find the agreement to choose x from
S e T "ot lesser restrictiveness" than the agreement to choose x [rom T, and thus should
also agree to choose z from S. Theorem 2.2 and Formula ( 2.1) clarify how the presence of fewer points in S may make it "of lesser restrictiveness" to agree on the choice x [rom S: in S the players must agree on [x sy] for fewer points y. Thus Theorem 2.2 clarifies two ideas which may have bcen undcrlying Nash's intuition: firstly, that the two players should choose in acoordance with a binary "group preference" relàtion, and, secondly and more basic, the idea that the two players may be considered as one decision unit on which consistency requirements can be imposed.
L.et us further note that Theorem 2.2 essentially depends on the restrictive framework in this paper, in which the choice function is single-valued and has a domain which is intersection-closed. Under more genèral circumstances many other conditions [or choice functions have been [ormulated in the litcrature which in the context ot this paper are equivalent to IIA. We mention the weak axiom of revealed pre[erence ( see Samuelson
INDEPENDENCE OF IRREIEVAPr1' AL7ERNAT7VES 1~I89
nondecreasing eligibility in Wakker (1989a), the independence of~from irrelevant alter-natives of Luce (1959) and Kaneko (1980), and the V-axiom of Richtor (1971). Most of these properties were studied in the oontext of consumer demand theory. Arrow (1959) showed that 11A (called G there) is necessary and sufficient for the existence of a
transitive complete representing binary relation under the restrictive assumption that the domain of the choice function contains all finite subsets of X.
The ncxt two sections deal with the (akyclicity o[ the binary relation in Theorem 2.2. In Section 3 we consider choict functions without the (Pareto) continuity property; in Section 4 we will add Pareto mntinuity and oontinuity.
3. (A)CYCI.IC7TY OF REVEAIED PREFERENCE Wl'i7iOlIT CONT[NUITY
L.et F be a choice function and R the corresponding directly revealcd preference relation. We write xPy if therc exists sn S e i with x ~ F(S) and y E S, y~ F(S). P is called the dircct(y rcucakd strict prcjtrenct relation. For x-(xl, z2,..., z„),
y-( y„ y2, ... , y„) E X, we write x 1 y if xt ~ y; for i- 1, 2, . .. , n and x ~ y if x; ~ y; far
i- 1, 2, ... , n; x ~ y, x c y are analogous. For T e X, conv ( T ) denotes the convex hull of
T and comv(T) :- (s e X: x ~y for some y E oonv ( T )} denotes the comprehensiue
convcx hull of T. For S E F, P(S) ~- {x E S: there is no y e S with y~ z, y ~ x} denotes the Parcto optima! subset ojS. F aatisfiu Parcto optimality (PO) if F(S) E P(S) for every
SEE.
LEMMw 3.1: (i) For etxry x E X xrc hatx xRx and nor xPx. (ii) Suppose Fsatisjus PO and !!A. Let x, y E X with x f y. ?htn tht fo(lowing thrct stattments are tquiualent: (a) xRy, (b) xPy, (c) x- F(S) jor ttxry S E E with x E S and S c comv {x, y}.
PROOF. (i) xRx since F((x}) -z. (Not xPx] is obvious. ( ii) (b) -~ (a) by definition. To prove ( a) -~ (c), suppose xRy. Then x- F(T ) for some T E ï with conv (x, y} e T; so F(conv{x, y)) x by IIA, hence by PO and IIA, F(S) x for every S E E with P(S) -conv (x, y}; so by IIA again x- F(S) for every S E~ with x e S and S c comv (x, y}. We have proved (a) y(c). Suppose (c) it true; then x - F(conv (x, y}) so xPy. (c) y(b)
follows, which completes the proof. Q.E.D.
Lemma 3.1 (and the arguments in its proof) will often be used without explicit mentioning.
DEFINmoN 3.2: The choice function F satisfies the strong aziom oj reuenled prejcrence (SARP) if there does not exist a cyckx-xoPxlPx2 --. x4-IPzk -x, where k~ 0 is the length of the cycle.
The condition SARP and the following result ( fonnulated hcre in a way suited for our
context) have becn obtained by Vilte (1946) and, independently, by Houthakker (1950) for gencral contexts. Kim (1987) has shown that slight weakenings o[ the transitivity of the binary rclation do not affect the characterizing condition.
TIIEOREM 3.3: 771erc txists a tranritiue binarY rclatéon rcpresenting F if and only ij F satisfics SARP.
17~jQ FiANS PE7ERS AHD PEZER WAKKFJt
l~Mt.tw 3.4: L,et F satisfy IIA. Then thcn do not ecist cycla of kngth 1 or 2 in the
reuealed prejerence relation.
PROOr:: (.ycles of Icngth 1 are excluded by L.emma 3.1(i), which also excludes cycles of length 2: xPyPx would by 11A imply x- F(rnnv (x, y}) - y and hence zPx. Q.E.D. The last property in L.emma 3.4, nonexistence of cycles o[ Icngth 2, is known as the
Weak Aziom ojReuealed Praference (WARP); see e.g. Richter (1971). Further discussion
is postponed until the cnd of the next section.
In the sequel we shall always assume Pareto optimaliry. In consumer demand theory it is an implicit condition; in bargaining game theory it is fairly standard. In what follows, !(a, b) denotes the straight line through the points a t b in X.
L.Er.~r,~n 3.5: Let n - 2, and kt F satisfy PO and IU. T7un therc do not exist cycles oj
kngth 3.
PROOF Assume the Tollowing:
(3.1) a, b, x e X satisfy aPb and x~b.
In view of the reflexiviry of R(Lemma 3.1(i)) and the definition of P it follows from (3.1) that a~ b, a rtx, b vi x. We will show that x~ta; in some cases the additional require-ment bPx will be needed. Nonexistence of cycles of length 3 then follows immediately. In order to prove z~a, we list the tollowing cases, which essentially exhaust all possible configurations of (a, b, x}.
(3.1.a) a 7 6,
(3.1.b) bt ~ a„ bz ~ a2,
(3.1.b.1) xt c b~, x on or above I(a, b),
(3.1.b.2) xt ~ bt, x2 ~ 62, x strictly bclow I(a, b), (3.1.b.3) z E comv{a, b},
(3.1.b.4) z~ ~ at, x on or bclow !(a, b), (3.1.b.5) xZ ~a2, z strictly above l(a,b), (3.1.b.6) x~ ] a„ a2 ~ x2 ~ b2,
(3.1.b.7) zt t at, x2 c b2, x strictly above !(a, b).
Note that the case 6 7 a is excluded by oPb and PO. Also the case x] b is excluded by
x~b and PO. Further, the cases with b~ ~ a , 62 ~ aZ are analogous to (3.1.b. U-(3.1.b.7)
INDEPENDENCE' OFIRREIFVANT AITERNATIVES 1791
S~P 1: In the cases ( 3.1.a), (3.1.b.1), (3.1.b.3), and (3.1.b.4), we have z~ta.
PROOF: (3.1.a): xRa would by Lemma 3.1(ii) imply x- F(conv {z, a, b)), contradicting z~tb.
(3.1.b.1): Same proof as for case (3.1.a).
(3.1.b.3): By aPb and [.emma 3.1(ii) we have a - F(oonv {a, b, x}), so aPx, hence xEta. (3.1.b.4): C.et S :- conv {z, a, b}. If F(S) E com {a, b} then F(S) - a and hence aPx. If F(S) E conv {a, x), then F(S) ~ x since otherwise xRb; so by IIA also F(conv (a, z}) ,f x,
which by Lemma 3.1(ii) implies z1~a. This completes the proof of Step 1.
SrEr 2: Suppose 6Px. Then zXa in the cases (3.1.a), (3.1.b.1)-(3.1.b.4), and (3.1.b.7). The cases (3.I.b.S) and (3.l.b.tí) cannot oocur.
PROOF. In view oi Step 1 we still have to oonsider the cases (3.1.b.2), (3.1.b.5)-(3.1.b.7).
(3.1.b.2): Let S:- conv ( z, a, b). If F(S) E oonv [o, b} then F(S) - a by IIA, so aPx. [f F(S) E conv (x, 6}, then F(S) - b since bPz, which leads to the contradidion bPa.
(3.1.b.5), (3.I.b.tí): By Lemma 3.1(ii), bPx, and o E comv {b, x}, we would have bPa, a
contradiction.
(3.1.b.7): L.et T:- conv ( a, b, x). if F(T) E com (b, x}. then F(T )- b, which would imply bPa, a contradiction. So F(T) E conv (x, a} and F(T) ,f x since otherwise zPb. So by IIA, F(conv {x, a}) ~ x, hence zlta by Lemma 3.1(ii).
This completes the proot of Step 2, and of the lemma. Q.E.D.
The following example, which was not easy to oonstrvct, shows that for n - 2, IIA and PO are not sufficient to exclude cycles of length greater than 3.
1792 10 9 I! G3 G~
FtcuRe 1.-A choice function satisfying IIA and PO, but viotatinQ SARP. G~ ~.. .~ G~. On
G„G~,Gs,G~, the first coordinate is maximiud. On Gz,G~,G6,G~, the second coordinate is
muimiud. On Gy, xtxi is maximiud. A cycle aP6PcPd results.
Further o :- (9,1), b:- (4, 8), c:- (8, 4). and d :- (1, 9). Let the transitive binary relation i on X be defincd as follows:
(i) z i for atl i ~ j, x e G;, y E G;, i, j E{l, 2, .. ., 9}. (ii) On G„ G~, Gs, G„ i is the lexicographic order.
(iii) On Gz, G3, G6, G~, á is the reversed lexicographic order (firat maximizing the second ooordinate).
(iv) On C9, i maximizes the product xtx2.
We define F as the choice function maximizing `s . It can be seen that F is well-defined, and satisfies IIA, PO, and SARP. We define s to bc equal to ~ with one exception: b s c instead of c i b. So y is not transitive. We define F as the choice function maximiiing s. 71ten also F is well-defined and satisfies PO and IIA (by 7iteorem 2.2), but F does not satisfy SARP: aPbPcPdPa,a cycle of length 4.
This section is concluded by an example showing that if n 1 Z, IIA and PO admit
cycles o[ length 3.
Ew~r~rtE 3.7: C.et n- 3 and let the choice function F: f~ X be defined as follows.
Let Y:- {x e X: x ~( 1,1,1)) and let S E i. lf S contains an interior point of Y, then let
F(S) be the unique point of Yr1 S where the product (x~ - 1Xxz - 1Xx3 - 1) is
maximized on this set; then F(S) ~ (1,1,1). If S n Y- 0, then let F(S) be the unique point of S where the product xtx~z~ is maximized on S. If S n Ys 0 and xi (resp.
z2, x~) - 1 for all x e S rl Y, then Ict F(S) be the Pareto optimal point of S n Y with
maximal third ( resp. first, second) coordinate. Then F can be seen to be a well-defined
choice function satisfying IIA and PO. The corresponding revealed preference relation
contains cycles of length 3, e.g. (2,1,1)P(1,1, 2)P(1, 2,1)P(2,1,1).
tiANS PETERS AND PETER WAKKER
1 2 ~ 0 g 10
~~2V2
4. (A)CYCLICITY OF REVEALED PREFERENCE W1TH CONTINUITY
INDEPENDENCE OFIRRELEVANT ALTERNA77VES 1793
DEFINmoN 4.1: A choice function F: E-~ X satisfies Panto continuity (PC) if for
every sequence S, SI, SZ,... in ~ with S4 -~ S and P(Sk) -~ P(S) ( where the limits are
taken with respect to the Hausdorfi metric) we have F(Sk) ~ F(S).
For n- 2 and S e E, let Dt(S) be the point of P(S) with maximal first coordinate,
and let DZ(S) be the point of P(S) with maximal second coordinate. Dt and DZ are
choice functions satisfying PO, IIA, and Pareto continuiry but not continuity (sce Def. 4.9). Note that for choice functions F satisfying PO and IIA we have F(S) - F(T ) whenever P(S)-P(Tr so, for such F, requiring Pareto continuiry instead of continuiry
seems reasonable.
The remainder of this section is devoted, firstly, to proving that the combination of PO, PC, and IIA for a choice function F implies SARP if n- 2; secondly, to showing that for n~ 2 these conditions, even with full continuiry instead of PC, do not suffice to exclude cycles. For x t y, Is(z, y) denotes the straight closed halfline through x and y with endpoint x.
L.EMntw 4.2: l,et Fsatisjy PO, !!.!, and PC. Let u, w e X wirh u f w.
(i) Ij wPu tlttn wPx jor aU z e l~,(w, u)`{w}, and wPx' jor all z' L x E!„(w, u)`{w}, (ii) ( xPu or zPw 1 jor ol! x e conv {u, w) `{u, w), and [x'Pu or x'Pw ] jor all x' ~ x with x e com (u, w} `(u, w).
PROOF: ( i) Suppose wPu. By convexiry of choice situations this immediately implies
wPx for all x e conv ( u, w} `(w}, The case remains where x e f„(w, u), z not between u
and w. Let S :- com (x, w]. If F(S) E conv {u, w) then, by IIA. F(S) - F(conv ( u, w}) - w, so wPx. 7ite case remains where F(S) ~ conv {u, w]. We will shaw that this case cannot occur. By PC the function y -~ F(conv { y, w}) is continuous on I(u, w). Its image must be connected, so there is a y e conv {x, u} such that F(conv ( y, w}) - u. This and
F(conv{u,w))-w contradict IIA. So everything concertiing x in (i) has been proved.
The result conceming z' follows from consideration of comv(x,w).
(ii) C.et x' be as in (ii) (possibly x' - x). If x' 7 u or x' ~ w, then we are done.
Otherwise, note that conv ( w, x'} U conv ( x', u} is the Pareto optimal subset of
conv ( w, z', u]. W.I.o.g. suppose F(conv ( w, z', u)) ~ conv {w, x'). By PC the function y--~
F(conv (y, z', u)) from conv {w, x') to conv {w, x') u conv {x', u) is continuous. Its image
must be connected; hence F(conv{y,x',v))-x' for some yeoonv(x',w}. This implies
x'Pu. Q.E-D.
Up to Thcorem 4.8 we make the following assumption:
(4.1) n- 2 and F satisfies PO, IIA, and PC.
We will show that P has no cycles by induction based on Lemma 3.5, which says that there are no cycles of length 3. Fix a sequence a, b,..., y, x of Iength at least 4 with
aPbP --. PyPx. We want to show: xbla. The induction hypothesis is that no cycles of
length smallcr than the length of (a,b,..., y,x) exist. This implies:
For all u and w in this sequence with uP ... Pw and not both u- a (4.2) and w-x, we have wbtu. Further, x~ta if there are v and w in the
sequence with w not the immediate successor of u and uPw.
1794 HANS PETERS AND PETFA WAKKEA
LEMMw 4.3: In tlu casu (3.1.a), (3.1.b.1), (3.1.b3), and (3.1.b.4), we haut x~ta. The remaining cases (3.1.b.2), (3.1.b.5), (3.1.b.6), and (3.1.b.7), are treated in the [ollowing lemmas.
L.Entn~w 4.4: In case (3.1.b.2r xt e b~ ~ a„ xz 1 bz 102, x srrictly below l(a, b), we hatx xRa.
PROO~ From Lemma 4.2(i) with a in the role of w and b in the role of u, it follows
that (even) aPx. Q.E.D.
I,En~t.~w 4S: Casc (3.1.b.6): x~ ~ a~ 1 b~, a2 tx2 ~ bz, cannor occur.
PROOF yPx, x] a, and L.emma 3.1(ii) imply y - F(conv {a, y, x)). So yPa in
contra-diction with (4.2). Q.E.D.
I~naaw 4.6: Case (3.1.6.7r b~ ~ al, xt e at, 62 ~ a2, b21 x2, z strictly abotx l(a, b), cannot occur.
PROOF. We eonsider all possible locations of y. I[ y~ t a~ and y on or below l(a, b), then aPy in view of L.emma 4.2(i), so from (4.2) we obtain x6ta. Since by (4.2) atso x.(tb, a contradiction with L.emma 4.2(ii) follows. If y~ ~ a„ and y on or below 1(x, a), then xPa would by Lemma 4.2(i) imply xPy which is a oontradiction. So x~ta, but as before that is also impossible. If yz ~ bz and y on or above 1(a, b), then b e comv (x, y), so yPb by Lemma 3.1(u) (since yPx), in sontradiction with (4.2). If yz e a~ and y on or above 1(x, a), then a e comv {x, y), so yPa (sinoe yPx), in oontradiction with (4.2). Also y~ a would imply the oontradiction yPa. The only possibility left is: y strictly above I(a,b), y2 c 62, y~ ~ a~. In that case, yPa or yPb by Lemma 4.2(ii), in contradiction with (4.2). Q.1~D. LEMt.tw 4.7: In case (3.1.b.5): bt t at, 621 a2 ~ x2, x strictly aboue I(a, b), wrc hatx x.ó~a.
PROO~: Suppose xPa. Then xPaPb ~-- Py, and yPx. By the previous lemmas, yPx is excluded in all possible configurations except for the configuration dcscribed in this lemma, so a~ ~x„ 021 x2 ~ y2, y strictty above 1(x, a). If z is the immediate predeces-sor of y, then yPzPaPb --- Pz and zPy. Again, the only possible configuration for this is: x~ c y~, x21 y2 ~ z2, z strictly above !( y. x). Repeating this argument we find tor the final step bPcP --- PzPyPxPa and aPb: et c b„ cz ~ 62 ~ a2, o strictly above I(b, c). In particular, b~ ~ c~ ~ ---~ yt ~ xt ~ at ~ b„ an obviotu impossibiliry. Q.~D.
Lemmas 3.4 and 3.5, and Lemmas 4.3-4.7, imply the following theorem. THEOREM 4.8: For n- 2, PD, !U, aná PC imply SARP.
Samuelson (1948) and Rose (1958) essentially showcd that PO and WARP suffice to exclude cycles, for a singlc-valued choice function defined on only 2-dimensional linear
choice sítuations (i.e., budget sets of the form comv((a,0),(O,b)) where a,6 e R~). Theorem 4.8 extends this result to choice functions dcfined on nonlinear 2-dimensional
INDEPENDENCE OFIRRELEVANT ALTERNATIVES 1795
The next question is whether Theorem 4.8 will still hold if n 7 2. Gale (1960) has proviAod an example of a continuous dcmand function defined on 3-dimensional linear budgd aets which satisfies PO and WARP but not SARP. In other words, the result of Rose (1958) mentioned before does not have to hold if there are more than 2 commodi-ties. In the Appendix we will show that Gale's example can be extended to 3-dimensional nonlinear budget sets (our choice scts) as well. This can be done even with PC strengthened to full continuity:
DEF1NrnoN 4.9: A choice function F: E~ X is conrinuous if for every sequcnce
S, SI,S2,... in E with S4 --~ S(where the limit ia taken with respect to the Hausdorff
metric) we have F(Sk) -~ F(S).
The appendix shows that WARP dou not imply SARP tor dimension n~ 3, by extending the example of Gale (1960) to nonlinear choice sets. The extension to higher dimensions, also for linear budget sets, will be given in Peters and Wakker (1991). For lincar budget sets a theoretical argument has already been given in Kihlstrom, Mas-Colell, and Sonnenschcin (1976, first paragraph of page 97S).
Another interesting question is whether IIA can be atrengthened in an appealing way in order to imply SARP. For instance, for each dimension n, can one find a natural number k(n) such that requiring the exclusíon of cycles of length smaller than or equal to k(n), instead of IIA, implies SARP? For linear budget sets the answer is negative, as tollows from Shafer (1977). For our case the answer is also negative: this can be shown by extending Sha[er's 3~imensional example to nonlinear budget sets in the same way as is done in the Append'Ix with Gale's example.
S. REPRESENTA770N OF REVEAI.ED PREFERENCE
Let F be a choice function. x e X is rcuealedprcjcrrcd to y e X, notation zRy, if there exists a sequence r ~xo,zl,...,zk -y in X with xoRxlR .. - Rxk. If in this se~uence x'Px~~ 1 for some i e{0,1,..., k- 1], x is rcrxoled strictly prejcmd to y, notation xPy. By Wakker (1989b, Corollary I.2.12, (vi) and (vii), and Theorem I.2.S, (ii) and (vi)), F satisfies SARP if and only if P is the asymmetrie part of R. Note that in our case, by Lemma 3.1(ii), if z t y and xRy, then xPy.
Although it is not impossible that R is complete (i.e., xRy or yRx for all x, y e X; for instance let n - 2 and F- DI ), this will in general not be the case. For instance, if n- 2 and F is the Nash choicc function N(that is, N(S) is the point of S E.I; where the product zlz2 is maximized over S), then neither (1,2)R(2,1) nor (2,1)R(1,2)). Also, R docs not have to be "rcprescntable" by a real-valued function on X; j: X~ R rcprescnrs the binary relation Y on X if [x Y y-~ f(x) ~ f( y)] and (x } y-~ j(x)1 f( y)] for all x, y E X, whcre Y is the asymmetric part of s. For instance, if R is revealed by DI thcn R is the Icxicographic order on X which is well-known not to be representable by a real-valued function.
The main purpose of this section is to find suflicicnt conditions for F such that thc corresponding revealed preference relation R is representable by a real-valued function j. Such a tunction will be callcd a utility function (of the consumcr, or the group o[ bargainers). It will be shown that f is strongly monotonic and strictly quasi-concavc (sce abovc L.cmma 5.4). Up to Thcorem S.3 we assume:
(S.1) F is continuous and satisfics PO and SARP.
The following Icmma can be derived from Corollary 1 in Jaffray (1975) applied to the
17~ HANS PE~RS AND PETER WAKKER
LEMMw 5.1: Ij thert exists a countaóle subset A ojX such that jor al! x, yE X with xPy
there is an a E A with xPaPy, then there uists a function j: X~ R such thot [xPy ~
j(x)~j(y)1 jorallx,yEX.
ReMwRK: L.omma 5.1 is a variation on a result by Debreu (1954, Lemma II); the latter holds tor weak orders ( transitive complete binary relations). Actually, given an enumera-tion A-{a„ a~,...} of the set A, a funcenumera-tion J as in Lemma 5.1 is easily defined:
j: x--~ Ew:,p,~2-4- ~- ~e Jafíray (1975) tor further details. A set A as in L.emma S.1 can be obtained as follows:
(5.2) A:-{oEX:a-F(conv{z,y})forsomex,yEXnQ"}.
LEMMw 5.2: L.et z, y E X with xPy. Thcn then ezisu an a e A with xPaPy.
PROOF: First assumc~xPy. Choose sequcnces (xj),(yt) eX tl Q" with x' -~ x, yt ~ y, and with for all j: zt ~ x, yt ~ y, and ;xt t~yr E comv {z, y). By the continuity of F wc have F(conv {xt, ~')) ~ F(conv {x, y}) -x which implies: there is some k E N such that a:- F(conv {xk,y )) E comv {x, y}. So o E A, and zPa in view of Lemma 3.1(ii). Since y E comv{a, yk}, also aPy. So this point a has the desired properties.
Next assume xPy. Then x-xoRxr -. ~ Rz~-1Rz~ "' Rx4 ~Y with, say, j-~Px~. So by the first part of the proot we have xt - rPaPx~ tor some a E A, hence also xPoPy. Q.E.D. For an arbitrary choice function F and a real-valued function j on X, F inaximiza f if j(F(S)) ~ j(x) for cvery S e~ and x E S, x f F(S).
THEORFJd 5.3: Let F be a Parrto optimal continuota choice function. Then the jollowing two srarcments arc equiuolent:
(a) F satisfies SARP.
(b) F inaxirnizes a rcal-ualtttd junctron f at X.
PROOF: Suppose F satisfies SARP. Then F satisfies condition (5.1), so by Ltmmas 5.1 and 5.2 there is an j: X-~ R with xPy ~ f(z) ~ j(y) tor all x, y EX. Since F(S)Px for all F(S) ~ x E S and S E E, F inaximizes j. The implication (b) ~(a) is straightforward. Q.E.D.
Consequently, if the consumer's demand function, or the bargainers' solution, is continuous, Pareto optimal, and satisfies SARP, then the consumer chooses as if maximizing a utility function, and the bargainers reach a wmpromise as if maximizing a group utility function. .
Next we will show that the function j in Theorem 5.3 is strongly monotonic, i.e.,
strictly increasing in each coordinate, and strictly quasiconcaue, i.e., the set {y e X:
j(y)~j(x)) is "strictly convex." for every zEX. A set TcX is srrictly conua it
ax t(1 - a)y is an interior point of T whenever z, y E T, x t y, 0~ a c 1.
L.EMMA 5.4: L.et F be a Parcto optimaJ continuotts choice junction which maximizcs a
rcal-ualued junction j on X. 77ien j is strongfy monotonic and strictly quasiconcaue.
PROOF: L.Ct x, y E X with z~ y, x s y. 7'hen F(conv [x, y)) - x by PO of F, so
INDEPENDENCE OF IRRELEVANT AL7ERNA77VFS I7~I7
Ncxt, for contradiction let z e X and T-~ (s E X: j(x) ~ f( z)). For oonvexity of i, let
x, x' e T with x~ x' and y ~ ax f(I - a)x' whcre 0~ a~ 1. We have to show j( y) ~ j(z). By Lemma 4.2(ii), we have yPx or yPx', so f( y)1 f(z) and y e T.
Finally, suppose that T is not strictly rnnvcx. If u~ w e T and conv[v,w] contains nh
interior point t of T, then by convexity of T all points in conv (u, w) `{u, w} are interior. So the assumption that T is not strictly convex implies the existence of v~ w e T sueh that conv ( u, w} is a subset of the boundary of T. l,et F(conv ( u, w}) - u (otherwise rnntinue the proof with F(conv (u, w)) in the role of u if F(conv ( u, w}) .~ w, or with the roles o[ u and w reversed if F(conv ( u, w)) - w). Note that j(u) ~ j(w) ~ j( z). Also,
f(x) ~ j(z) for every x in the interior of comv{u,w} since otherwise, by PO, corn(u,w}
would contain an interior point of T. L.et ul,u2,... EX be a sequence in the interior of
comv{u,w) converging to u. Then F(conv(uk,w))-w for every keN whereas F(mnv(u,w)) - u. This contradicts the continuity of F. Q.E.D.
L.EMMA 5.5: Let F be a choice feenction which maximizes a strongly monotonie and
strictly quasiconeave rcal-ualued function f on X. Then F is Parctooptimal and continuous.
PROOF: Pareto optimality of F is an immediate consequence of strong monotonicity ot j. Next suppose for contradiction that F is not continuous. Using compat:tness, subscquences, and IIA, we can arrange sequences p,pl,pZ.... and q,ql,qZ ... in X with Pl~P. 9k~9, F(conv{pR.9k))-Pk, F(conv(D,q))-9. From j(q)~j(P) and strict quasiconcavity of j it follows that ~ f~q is an interior point o[ (x : f(x) ~ f( p));
so j(3p f~9) ~ j( P) by monotonicity of j. Similarly 3q~4 t p f4 is an interior point of (x : f(x) ~ j( ~ f;q)); so bY monotonicity of f there is a q ~ 3q~4 t p~4 such that f(q) ~ f(~p t~q). Further, there is a p~ p such that ~ t yq ~ w for some w e
conv(p,q). Then j(q) ~ f(;p f~9) ~iw). By strict quasiconcavity of j: j(w) ~
min(f(P). f(9)}. We conclude that j(p) ~ j(q).
So q~ 3~~4 f p~4, p~ p, j( p) ~ f(q). Take k E N so large that pk c p and q e
oonrv{Pk,9 ). Then j(P4)~f(P)~I(4) whereas F(oonv{pk,94})-Pk. Since q"E comv{pk,q") this contradicts I.cmma 3.1(ii).
Q.E.D-Theorem 5.3 and Lemmas 5.4 and Sá lead to the following theorcm.
THEOREM S.tí: For a choice function F tRe jollowing two stattments arc equiualent:
(a) F is conrinuous and sotisfra PO and SARP.
(b) F inazimizu a strongly monotonic stricrly quasiconcaue real-ualutd junction j on X. For n- 2, Theorems 4.8 and 5.3 imply the following corollary, which further
illus-trates the meaning of Nash's IIA.
COROLLARY 5.7: Ltt n ~ 2 and Itt F be a Porcto optimal continuous choict funetlon.
Then tht jo!lowing two stattmenrs arc equiualent:
(a) F satisfees IL4.
(b) F inaximizes a rcal-ualutd function f on X.
The function f in Theorem 5.6 may fail to be continuous. This can be inferred from the straightforward adaptation of Example 1 and Remark 4 in Hurwicz and Richter (1971) to our context.
179g HI1N5 PETERS AND PEIER WAKKER
the bargainers may appeal to a choice function called (bargaining) solution in this context. This is essentially the model proposed by Nash (1950), who showed that the solutíon N defined before is the unique solution satisfying, besides some other proper-tics, IIA. The results of this paper may contribute to clarifying the role of IIA. Theorem 2.2 and Corollary 5.7 characteriu large classes of solutions with the IIA property. These solutions can be interpreted as generalizations of the Nash bargaining solution that allow tor intcraction between players, and paymcnts in more realistic quantities than von Neumann-Morgenstern utilities. Similarly, Theorem 5.6 characterius a large class of n-petson solutions with the SARP property. Further discussion was given at the end of Section 2.
6. pOMA1N F.X7FNSIONS AND CONCLUDING REMARKS
The choice of domain X~ R;t was made for convenience. Examples 3.6 and 3.7 (with any strictly quasiconcave strongly monotonic function instead of x~x2z3 for case (c) bclow) can be adapted to the cases below in a straighttorward manner. Also the example elaborated in thè Appendix can be adapted to these caus (see there). Further we have the following:
(a) If X- R", thcn all theorerns and lemmas in this paper remain true.
(b) If X~ R;. then all theorems and lemmas before L.emma SS remain true. L.cmma SS and Theorem 5.6 are no longer valid. This case is the mathernatically most deviating one since the (essential) domain is ttot open.
Details are as tollows. In the first part of the proof of L.emma 5.2 allow x~ and y~ to have uro ooordinates whenever the corresponding ooordinates of both x and y are uro, and wppress these coordinates. Take any x' - y.x f(1 - K)y (0 c ~a c 1). Then zPx'Py by Lemma 4.2. Proceed with x' instead ot z. In the last part of the proof ot Lemma 5.4 the interior of oomv((u,w)) must be replaced by the interior relative to the subspace where those coordinates are uro that are zero for both u and w. The claim about (nonrelative) interior points of T then remains true. In the proof of L.emma SS, the point q does not have to exist.
(c) If X- R4, ~ is restricted to the scts T which contain a stríctly positive point, F(T )~ 0 for all T, and the function j and the revealed and represeating binary relations are restricted to R;4, then all theorems and lemmas of this paper remain true. (Note that all intersections required in the proof of Theorem 2.2 are in F.)
Case (b) is of interest for consumer dcmand theory. Case (c) can be uscd to derive the Nash bargaining solution from our results. One of the conclusions from this paper is that the [IA condition, combined with Pareto optimality and continuity, only has strong implications in the 2-dimensional case. This case is relatively important: bargaining situations often include two parties, and if there are more than two parties intermediate coalitions should usually be allowed, which restricts the importance of n-person pure bargaining games; in consumer demand thcory, many situations can be modeled as involving only two goods by consídering composite goods. Nevertheless it is unfortunate that, in general, we obtain the n-dimensional analogue only by strengthening IIA to SARP.
INDEPENDENCE OFIRRELEVANT ALTERNATIVES 1Í99
dimensions refer to commodities which may have physical interaction, and even more in group decision making whcre dimensions refer to individuals who may have social interaction, violations of additive separability are of considerable interest. This motivated the general approach of this paper.
Department oj Mothematics, Unitxrsity oj Limóurg, P.O. Box 616, 62tX1 MD Movstricht, The Netherlands
and
Nijmegtn Institute jor Cognitinn Research and Information Technology, Uniuersity oj Nijmegen, P.O. Bar 9104, 6500 HE Nijmegen, The Netherlands
Manuscripr meiutd Augtast, 1987,~ jtnaf rreision rcceitrdJanuary, J991.
APPENDIX
In this appendix a choice fundion F is constructed for dimension n- 3, satisfying mntinuity, PO, and 11A, but not SARP. F extends a demand function, proposed try Gale (1960) in order to show that WARP does not imply SARP if there are at least three goods. L.et A be the matru
3 4 0
0 -3 1 . 4 0 -3
For all (pricc) vectors D. q E X w~ith ( dcmand vectors) Ap 7 0, Aq ~ 0 ( ct. Gale, 1960, sect. 3): (~) If pA9 eDAD and qAD e qAq, then Ap -A9 ("WARP~).
Let
( 9 12 16
B--A't-1~37I16 9 12 .
l 12 16 9
Let S e i Ix fixed, and kt M~(x e S : thcre is no y e S w~ith z~ - y~ for all t~ 1 and xt t y~}. For every x E M let a(x) e R; be defined by rr(z)~ -z~ for all i f 1, ir(x)~ - 0, i.e., a is the projection
on the hyperplane z~-0. 71~en u:M~~r(M) is a homeomorphism, and sr(M) is nonempty, compad, and rnmez. Further, for every z~ 0 let N(x) be the supporting hyperplane of S with normal x and such that S is txlow H(x). Then the eorrcspondence I: x--~ H(x) n S - H(x) n 1~S) for every x~ 0 is upper semicontinuous ( as an tx shown dircctly, or as a consequence of lhe Maximum 7luorem).
Finally, let the correspondenoe p:~r(M) y ~r(M) be defined by
p(z)-n~J~B(~s-t(z))~~ toreveryz.
Then ekarly p is comex-valued and upper semicontinuous, so by Kakutanï a fixed point theorcm therc exists a faed point x' ep(z').
Nact we show that such a fued point x' is unique. Suppose z' e u(z') is another fuced point. 77tenar''(z')eI(B(a-'(x')))andsr-t(z')el(B(~r-t(z'))).Sobydefinitionof l:
~Ba-t(za)~tr-t(za) c ~B.rr-t(za)~rr-t(za) and ~B~-t(x')ÍA-t(za) c ~Bir-t(xa))n-1(z').
Hence
1HOO tiANS PETERS AND PETER WAKKER
L.et F assign the point ~r"t(z') to every choice siluat'an, with z` the unique fixed point as above. Then F is a well-defined choice function. PO and !lA of F follow straighttorwardly from its definition. Ncxt we prove that F is continuous.
Let S, St, SZ, ... E f, S; y S in the Hausdorff-metric, and F(S;) - y' -~ y e S. For t:very i let
p' ~ B( y'); by construction, p~ is a normal of a suppoRing hyperplane of S; at y~. Since y' ~ y, we
have B( y' )-~ B( y) ~ p, so p' - r p. lt is straightforward to show that (z : px ~py) supporls S at y. (Incidentally, it also follows that p - B( y) ~ 0, since all entries of B are pasitive. Hence y e P(S).) So s-'( y) is the fued point of ~a, and F(S) - y folbwt.
Finally, a violation of SARP is obtained, adapting the exampk o[ Gale ( 1960, Section 5). Thc following observation will be uscd. Let S e F be such that P(S) c {z e X: pr ~ c) for some vector p~ 0 and some constant c ~ 0. 1f the point c( pAp)- rAp is an elemenl of P(S), then by construction of F it is equal to F(S). We now turn to the cxam~k.
Letz'-(1,0.001,0.001),x~-(0.6,0.001,0.3),z -(0.3,O.OOl,0.6),x~-(0.001,0.001,1),andlet pr (9.028, 16.021, 12.025), P2 (10.212, 13.209, 9.9Ifi), p~ (12.312, 12.009, 9.016), P~ -(16.021,12.025,9.028). Then each z~ is a multiple of Ap~. FuRher, we have:
ptzt ~ptr~. so
pazZ ~ Dazs. .~
F(oorn(It,r~}~ ~.tt. F(oonv (zi, z~}~ -z~,
p~z~ ~ p~z~, so F(oonv (z~, z~}~
~z~-So zt is revealed preferred to z~, i.e., (I,O.OOI,O.OOI) is revealed pre[erred to (0.001,0.001,1). By interchanging the appropriate numben one similarly shows that (0.001,0.001,1) is revealed pre-ferred to (0.001,1,0.001), arui that (0.001, L,0.001) is rweakd prepre-ferred to (1,0.001,0.001). So F violates SARP.
The above construction holds for the prevalent case in the paper where all choice situations are strictly positive. We mnclude this Appendix by tnoditying the construction for the cases (a)-(c) in Section 6. We construct G as follows. L.et c ~ 0 be a eonstant such that the set
C-~z e Rs : z~ 0, stzszs i c~
contains all the poinu needed for the construction ot Ute ~ycle above. Now for S e E, let
G(S) :- F(S fl C) if S tl C rt 0. lf S n C- PJ, then let as ~ 0 be the minimal number such that S n(C - as(1, l, t)) rt 0, and lel G(S) be the ( unique!) point in this intersection. This G is
continuous and satisfies PO and IIA, but not SARP. !t can be used in the caus ( a) and (b) in
Section 6. For the case (c) there, take, instead, the choice function C with C(S)-G(S) if S fl C rt 0, and C(S) is the unique point of S with maximal product of lhe ooordinates, otherwise.
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No. 10 F. van du Ploeg and A.J. de Zeeuw, ConIIict over arms scxumulation io market and command ooonomie; ia F. van der Pbeg and AJ. de Zceuw (eds.~ Dyrwmic PoGcy Carnes in Eowwmics, Contnbutions to Eoonomic Analysis 181, Amuer-dam: Elsevier Scienoe Puhlisfters B.V. (North-Holland), 1989, pp. 91 - 119. No. 11 J. Driffill, Macroeoonomic policy games with inoomplete informatioa: some
extension; in F. van du Ploeg and Al. de Zeeuw (eds.), Dyrwnuc PoGcy Camer in Ecoraornirs, Contnbutions to Eoonomic Analysis 181, Amuerdam: ELsevier Science PuWishus B.V. (North-Holland), 1989, pp. 289 - 322
No. 13 RJ.M. Alessie and A Kapteyn. Conwmption, savinYs and demography. in A Wenig, KF. Zimmermann (eds.~ lkmo~hic CJiartgr and F.canorr~it Developnurt, Bulin~Heidelberg Springer-Verlag, 19~89, pp. 272 - 303. No. 14 A Hoque. I.R Magnus and B. Puaran, The a~ad multi-paiod mean-sqwro
forecast error [or the fustorder autore~essive model, lounnl of F.conomcaics.
voL 39, oo. 3, 1988, pp. 327 - 34á
No. 15 R Alasie, A Kaptcyn and B. Mdenberg, The effeas of liquidity oocutnints on
oonwmption: estimation from household pand data, EuropcanEca~ornrc Review,
voL 33, no. 2~3, 1989, PP. 347 - SSS.
No. 16 A Holly and J.R Magnus, A note oa instrumental variabla and maximum líkeli-óood -estimation procedures, Ah~s d'Économie u de Swtfcrique, no. 10, April-]une, 1988, pp. 121 - 138.
No. 17 P. ten Hadceo, A Kaptryn and L Woittiez, Unemployment beae[its and the labor market, a micro~macro approach, in BA. Gustafsson and N. Anders Klevmarken (eds.), Tht PoGácal Economy oj Social Stcuriry, Contrbutions to Eoonomic Analysis 179, Amuudam: Elseviu Scienoe Publishers B.V.
(North-Holland~ 1989, pp. 143 - 164.
No. 18 T. Wansbeek and A Kapteyn, Estimation oí the errorcomponents mode! with
inoomplete paneLs, lowno! oj Economttrics, voL 41, no. 3, 1989, pp. 341 - 361.
No. 19 A Kapteyn, P. Kooreman and R Willemse, Some methoddogical i~wea in the
implemcntatioo ot wbjaxive poverty definitions, Tlu lourra! oj Humart
Riso~urccr, voL 23. no. 2, 1988, pp. 222 - 242.
No. 20 Zh. van de Klundert and F. van du Ploeg, Fiscal poliry and futite liva in
intudependent eoonomies with real and nominal wage rigidity, GàforrlF.oo~novnit
Papas,vol 41, oo. 3, 1989, pp. 459 - 489.
No. 21 J.R Magnus and B. Pesaran, The rsad multi-period mean-square torecast error
tor the Grst-ordcr autoregrestive modd with aa intertxpt, lounwJ oj
F.canometrier, voL 42, no. 2, 1989, pp. L57 - 179.
No. 22 F. van du Pkxg, T~vo essays on poGtical eoonomy: (i) The poGtical economy of
overvaluation, ?7u Econornic lownal, vol 99, no. 397, 1989, pp. 850 - 855; (ii) Electioa outcomes and the stoclcmarku, Ewopeon Jowno! ojPolitical F.conomy,
voL S, oo. 1, 1989, pp. 21 - 30.
No. 23 J.R Magnus and AD. Woodland, On the maaimttm lilteLliood estimation o[ multivariate regression models mntaining serially oorrelated aror componeats,
Intunoriona! Economic RevFew, voL 29, no. 4. 1988, pp. 707 - 725.
No. 24 AJJ. Talman and Y. Yamamoto, A simplicial algorithm for atationary point
problenu on polytopes, lllathtmatirs oj OperotionrRae~arich, voL 14, no. 3, 1989,
pp. 383 - 399.
No. 25 E. van Damme, Stable equílibris and forward induction, lountal oj Economk
No. 26 A.P. Bartea and LJ. Bettendort, Price tormatioa of fish: An application of an invuse dunand system, Fauopeen Eco~nanic Rmcw, voL 33, no. 8, 1989, pp. 1509
- 1525.
No. 27 G. Noldeke and E. van Damme, Signalling in a dywtnic labour marlcet, Revlcw ojEconomic swdicr, voL s7 (1~ tto. 189, 1990, pp. t- 23.
No. 28 P. Kop Jansen and Th. ten Raa, The choix ot modd ia the construdion of input-output ooe[Gcsents tnatrioa, lntanatiavta! F.cnnomic Revicw, voL 31, no. 1, 1990, pp. 213 - 227.
No.29 F. van du Pbeg artd AJ. de Zeeuw. Pufect equílíbrium in a mode! of aompetit'rve arau aocumulatioo. Jnrsrnational F.tnnornit Revitw. voL 31, tw. 1,
1990, pp. 131 - 146.
No. 30 J.IL Magnus and A.D. Woodland, SeparabíL'ry and aggregstioq F.carioinica,va1. 57, tw. 226, 1990, pp. 239 - 247.
No. 31 F. van du Ploeg, Intunational intudependenoe and poliry coordination in eoonomies with real and nominal ~vaje rigidity. Cn:rAt F.oawrrtic Revitw, voL 10, tto. 1, June 1988, pp. 1- 4g.
No. 32 E van Datnme. Signaling and torward indtxtioo in a mulcet entry context, Opantions Research Proceidirt,gs 1 A89, Bulin-Heiddbug: Springu-Vuhg, 1990, pp. 45 - 59.
No. 33 A.P. Barte0. Toward a levels version of the Rotterdam and rdated dunand syuetns, Coruributionr w Optiationc Researrk anQ F.awro~rnict, Cambridge: MIT Press, 1989. pp. 441 - 465.
No.34 F. van du Ploeg, Incunational ooord'uution of monetary policies undu alternative exchange-nte regimes, in F. van du Pbeg (ed.). AdNanced L.ecnurcr in Quanrita~ive Ecawmiu. L.ondon-0rtutdo: Acxdemic Press L.td., 1990. pp. 91 - 121.
No. 3S 1à. van de Klundut, On sociooconomic causes o[ tiwait unemploymutt', Europuyi
F.coiwrnic Rcvicw, voL 34, tto. S, 1990, pp. 1011 - 1022
No. 36 RJ.M. Alessie. A Kapteyn, J.B. van Locitaa and TJ. Wansbeek, Individual effects in utiliry mnsistent models of demand, in J. Hartog, G. Riddu and J. 7iueuwes (eds.). Ponel Unra anQ Labor Mwket Srudius, Amstudam: ELseviu
Scier~ce Publishu: B.V. (North-Holland), 1990, pp. 253 - 278.
No.37 F. van du Ploeg, Gpital aocumulatioq inflatioa and bng-run oonflicx in intunational objeaives, (hjouá Econornic Papers, voL 42, no. 3, 1990, pp. 501
-525.
No. 38 Th. Nijman and F. Palm, Parametu identit'icatioa in ARMA Processes in the
preunce of regulu but incomplete sampling loumal ojT'une SaierMolysi.r, voL 11, tw. 3, 1990, pp. 239 - 248.
No. 39 Th. van de Klundert, Wage di[tuentials and anpbyment in a two-sec.ttor model
No. 40 Th. Nijman and M.FJ. Stcel, F~dusion reurictions in instrumental variables equations. Econo~rut~e Reviews, voL 9, no. 1, 1990, pp. 37 - SS.
No. 4l A. van Soest, L Woitciez and A. Kapteyn, Labor suPPly, income taxes, and houn resuictions in the Netherlands,loumal ojHunwn !lrsaucu, wL 25, no. 3, 1990, pp. S 17 - SSB.
No. 42 Th.C.MJ. van de Klundert and A.B.T.M. van Schaik, Unemployment persistence and loss of produdive capaàt~r. a Keynesian approach, Joumal oj Macro-economict, vo1 12, no. 3, 1990, pp. 363 - 380.
No. 43 Th. Nijman and M. Verbeek, Fstimatioo of time-dependent parameters in linear models using aoss-sections, panelt, or botb,Jouma! ojEco~nornetrict, voL 46, no. 3, 1990, pp. 333 - 346.
No. 44 E. van Damme, R Selten and E. Winter, Alternating bid bargaining with a smallest money unit. Can~a and Etonornic Bdiarior, voL 2, no. 2, 1990, PP. 188 - 201.
No. 43 C. Dang, The D,-uiangulation of Y for simplicial algorithnu for computing
solutions of nonlinear eqwtions, Madurnatirs ojOpemtionr Research, voL 16, no. 1, 1991, pp. 148 - 161.
No. 46 Th. Nijman and F. PaLn, Predidive aocuracy gain Gom disaggegate sampling in ARIMA models,loumaJ ojBusinest dc Economic Sratirticr, voL 8, no. 4, 1990, pp. 403 - 413.
No. 47 J.R Magnu; On oertain momenb relating to ratios of quadratic forms in normal variables: further results, Sankhro: 77~e lndianlouma! ofStatistiu, voL 52, suies B, paR. 1, 1990, pp. 1- 13.
No. 48 M.FJ. Steel, A Bayesian analysis o[ simultaneotu equation modeb by combining recunive analytial and numerical approaches, Jounw! oj Ecawmearcr, voL 48, no. 1~2, 1991, pp. g3 - 117.
No. 49 F. van der Ploeg and C. Withagen, Pollution control and the ramsey problem, Envirwvrunta[ and Rrsowu F.conomio, vol 1, no. 2, 1991, pp. 215 - 236. No. SO F. van det Ploeg, Monry and ppital in interdependent eoonomia with
ovulaPP~B Benerationa, F.cortarnioo. vol Sg, no. 230, 1991, pp. 233 - 23tí. No. S1 A. Kapteyn and A. de T,eeuw. Changing inoentives for eoonomic research in the
Netherlands, Eumpcan F.ooutornic Review. vo1 35, no. 2~3, 1991, pp. 603 - 611. No. 52 C.G. de Vries, On the rdation betweea GARCH and stable proxsxs,louina!
oj Economcdicr, voL 48, no. 3, 1991, pp. 313 - 324.
No. 54 W. vao Groenendaal and A. de Zeeuw, Control, coordination and contlict on iaternational oommodiry mulcets, F,cortontic ModcLing, voL 8, ao. 1. 1991, pp. 90 - 101.
No. 55 F. van du Ploeg and A.J. Multink, Dynamic poliry in linear models with rational
ezpectations of tuture evencs: A oomputer package, Computcr Scienu Lt
Econaruu ana Managaria~r, voL 4, tw. 3, 1991, pp. 175 - 199.
Na 56 HA Keuzenkamp and F. van du Ploeg, Savings, investment, government
f'inance, and the current sooount: The Dutch eatperience, in G. AlogoskouGs, L Papademaa and R Portes (eds.~ Frucmal Conrnauur ort Macrotconomic Policy: ?7~e 6tu,opedn E~aien~s, Cantbridge: Cambridge Universiry Press, 1991, pp. 219
-
?b3-No. 57 Th. Nymaa, M. Verbeelc and A. van Soest, Tlte efficienry of rotating-panel
desigtu in an anatysis-ot-varisttoe model, Joumal oj F.cawmeaics. voL 49, no. 3, 1991. pp. 373 - 399.
No. SB M.FJ. Steel aad ]:F. Ridtard, Bayesian multivuiate exogeneiry analysis - an
applintion to a UK moary demand equatioqlounw! of F.oatonuaia. voL 49,
no. 1~2, 1991. pp. 239 - 274.
No. 59 Th Ngmaa and F. Palm, Generalized kast squues estimatioa o[ lineu mode4
oontaining rational future ezpedations. lruanationd Econoniic Reritw, vo1 32, tto. 2, 1991, pp- 383 - 3g9.
No. 60 E. van Damme, Equilibrium selection in 2 z 2 games, Revitm Espano~a dt
Economio, voL 8, tw. 1. 1991. pp. 3T - 52.
Na 61 E Bennetc and E. vaa Damme, Demand commitment bugaining: the ase of
apex games, ia R Selten (ed~ Caint Fquilibiuun ModeLr !11 - Saruegic Baig~ciiwtg, Berlitt: Springu-Vulag, 1991, pp. 118 - 140.
No. 62 W. G3ch and E. van Damme, Gorby games - a game theoretic saalyais of
disarmament campaigns and the detense e[ficiency - hypothesis -, in R
Avenhaus, H. Kukar u~d M. Rudniattski ( eds.), Dejenu Deci.cion Making -Ma~yrical Suppat and Crisir Maurgernent, Betlin: Springu-Vulag, 1991, pp. 215
- 240.
No. 63 A RoeU, Dualcapaciry trading and the qualiry of the muttet, Joumá oj Financial lnrcrnudiatiort,vo1 1, no. 2. 1990, pp. 105 - 124.
No. 64 Y. Dai, G. van der Laan, AJJ. Talman and Y. Yamamoto, A simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron, Sianr Jowna! of Opántizaáat, vol 1, no. 2, 1991, pp. t51 - 165.
No.65 M. McAleu and C.R McKenzie, Keynesian and new classical models of unemployment revisited, T7u F.conon~ic Joumal, voL lO l, no. 406, 1991, pp. 359 - 381.
No.67 J.R Magnus and B. Pesaran, The bias of fora~xts [rom a first-order
autoregession, Econaneaic Tiuory, voL 7, no. 2, 1991, pp. 222 - 235.
No. tí8 F. van der Ploeg, Macroeconomic policy caordinatioa iasues during the variotu
phases of eoonomic and monetary integration in Europe, F.uropean F,corwray
-The Econon~ics oj EMU, Commission of the European Coaimunities, ~pecial
edition ao. 1, 1991, pp. 136 - 164.
No. 69 H. Keuzenkamp, A prccursor to Muth: Tinbergen's 1932 model o[ rational
expectations, Tlu Economie Jouiml, vo1 101, oo. 408, 1991, pp. 1245 - 1253. No. 70 L. Zou, Ttte target-incentive system vs. the prioe-incentive system under sdverse
selection and the ratchet effect,Journni ojPublic Ec.awrnict, voL 46, oo. 1, 1991, PP- S 1- 89.
No.71 E. Bomhoff, Bctween price reform and privatizadon: Eastern Europe in
transition, Fimiumanl~ und Porrfolio Manogrmenr, voL S, no. 3, 1991, pp. 241
-ul.
No. 72 E. Bomhoft; Stability of velocity in the major industrial oountries: a Kalman ~ter approach, Inrematiorwl Monuary Fwtd StaJjPaprn, vol 38, no. 3, 1991, pp. 626 - 642.
No. 73 E. BomhofL Ctirrenry mnvertib~7ity: w~hen and how? A oontrbution to the Bulgarian debate, Kndit und 1Capital, voL 24, no. 3, 1991, pp. 412 - 431.
No.74 H. Keuunkamp and F. van der Ploeg, Peroeived oonstraints for Dutch unemployment poliry, in G de Nwbourg (ed.), T7re Art oJ Ful1 Employrnent
-Unemplayntatt PoJicy in Open Econorrties, Contrbutions to Eoonomic Analysis 203, Amsterdam: Elsevier Science Publishers B.V. (North-Holland). 1991, pp. 7 - 37.
No. 75 H. Peten and E. van Damme, Charaderizing the Nash and itaiffa bargainin`
wlutions by disagreement point auions, Madtanatics ojOp~aadons Raseonch, vd. 16, no. 3, 1991, pp. 447 - 461.
No.76 PJ. Deschamps, On the estimated varianoa of regreasion ooetticients in misspecified error components modcls, Economeaic Tbeory. voL 7, no. 3, 1991, pp. 369 - 384.
No. 77 A. de Zetuw, Note on Nash and Stadtclberg solutions in a ditferential pme
model of capitalism',loturtnl ojEconontie Dyrwmies and Cawá, voL 16, no. 1,
1992, pp. 139 - I45.
No. 78 J.R Magnus, On the fundamental borderod matrut ot linear estimation, in F. van
der Plceg (ed.), Adrnnced Lectwrs in Quantitotive Ecorwntics, London-0rlaado:
Academic Press L.td., 1990. PP. 583 - Ci04.
No. 79 F. van der Ploeg and A de Zeeuw, A differential game of international pollution mntrol, Sysanr and Conar! Leaen, voL 17, no. 6, 1991, pp. 409 - 414. No. 80 Th. Nijman and M. Verbeek, The optimal choice of controh and
No. 81 M. Vubeek aad Tà Ny`maa, Can oobort data be veatod as genuiae paael data7, F~np~u'd Ecaion~. voL 17, m. 1. 1992, pR 9'
23-No. 82 E. vaa Damme aad W. G~th, EquíLbrium adoction ia the Spenoe aignalinB 8~~, ia R Seltea (ed.~ Gmne F.quiliàiwn dlodilr D- Xethod;lNorolt. ondA(arketr. gulio; Springer-Vulag ~991. pp. 263 -
288-No. 83 RP. Gilks and P.H.M Ruys, Charactuir~tioo ot ernoomic agents ia arbitrary oommunication structure; N~suw Arrhisf rnw Wirkwdt, voL 8, ao. 3, 1990. pp.
325 - 345.
No. 84 A. de Zeeuw and F. van du Ploeg, Diffuenoe gama aad poliryevaluation: a
conxptual fram~worlc. GOrJaid Ecorioout PRPm, voL 43, oo. 4, 1991, pp.612
636.
-No. 85 E. vaa Daaune, Fair divLsioa uader acymmetric informatioo. ia R Sdten (ed~ Rotiona! Intaouion - Esralu in Hawr oJJolui C Narwnyi, Bulia~Heidelbug: SprinBu-Vulag. 1992, pp. 121 - 144.
No. 86 F. de Jong, A.1Ceeu~a and T. Kloek, A ooav~butioa to event study methodology with aa application to tAe Dutch stocJc market, Jownd of Bo~uFinB and F'uwnca, voL 16, ao. 1, 1992, pp. 11- 36.
No. 87 A.P. Barten, The estimatioa ot mixed dunand aystems, in R Bewlry aad T. Van Hoa (eds.~ Canbibwions w Conrunur Qemand and Eca~o~rnedic; Fssa~a w Nawur ojNuui TiuiJ. Basingstoke- The Maan~tan Pras Ltd., 1992. pp. 31- S7. No. 88 T. Wansbeelc and A. Kapteyn, Simple utimators tor dynamic panel data models with urors in variabla, ia R Bewlry and T. Vaa Hoa (eds.~ Conaibukoru ro Conruma Drim~~d ar~Q F.eonantdiet. Eva~r in Honour oj Haui Theif,
Basingstoke: Tbe Macmillaa Press Ltd.. 1992. PP- 238 - 251.
No. 89 S. Ctub, J. Osiewalski and M. Stecl, Postuior infuenoe oa the degreu ot freedom parametu ia multivariste-t regres:ioa models. F.c~wwrnicJ Lats~. voL 37. ao.1, 1991. pp. 391 - 397.
