Rational choice and revealed preference without...

Soc Choice Welfare (1997) 14: 403—425

Rational choice and revealed preferencewithout binariness

Klaus Nehring

Department of Economics, University of California at Davis, Davis,CA 95616, USA

Received: 28 August 1995/Accepted: 14 February 1996

Abstract. This paper attempts to provide a unified account of the rationaliza-tion of possibly non-binary choice-functions by ‘‘Extended Preference Rela-tions’’ (relations between sets and elements). The analysis focuses on transitiveEPRs for which three choice-functional characterizations are given, two ofthem based on novel axioms. Transitive EPRs are shown to be rationalizableby sets of orderings that are ‘‘closed under compromise’’; this novel require-ment is argued to be the key to establish a canonical relationship between setsof orderings and choice-functions.

The traditional assumption of ‘‘binariness’’ on preference relations or choicefunctions is shown to be analytically unhelpful and normatively unfounded;non-binariness may arise from ‘‘unresolvedness of preference’’, a previouslyunrecognized aspect of preference incompleteness.

1. Introduction

Following Arrow’s [4] characterization of the choice behavior resulting fromthe maximization of a transitive and complete preference relation, a sizableliterature has attempted to broaden the notion of ‘‘rationalizable choice’’ toaccommodate phenomena of incompleteness and non-transitivity (for anelegant summary, see Moulin [11]). A maintained assumption of almost all ofthat literature has been the identification of ‘‘rationalizability’’ with beingderived from a binary (preference) relation over alternatives. As Sen realized inhis seminal contribution (Sen [17]), this assumption implies that if an alterna-tive x is ‘‘acceptable’’ (or ‘‘chosen’’) in each pairwise choice Mx, yN, for all y3A,then x must also be acceptable in the choice from the set MxNXA (‘‘Conditionc’’). c has been used to characterize choice-functions rationalizable by acyclicrelations (Sen [17]) and by transitive relations, i.e. partial orders (Schwartz[16]). However, once one allows for ‘‘incompleteness of preference’’ (in anintuitive, yet unformalized sense), as these contributions do, c does not seem to

be very compelling as an a priori restriction on intelligible or even on‘‘rational’’ choice behavior, as the following example shows.

Example 1. Consider an agent (named ‘‘Eta’’) who evaluates three alternativesx, y, and z based on the criteria ‘‘quark’’ and ‘‘charm’’. In terms of quark, x scores10, y scores s, and z 0. In terms of charm, x scores 0, y scores s, and z 10. ¹healternative y scores comparatively low on both criteria, i.e. 0(s(5. ¹he twocriteria determine her evaluations in additively separable manner, that is to say,alternatives are evaluated by a weighted average of scores, with weights(w, 1!w).

Quark Charmx 10 0y s sz 0 10

In view of the heterogeneity of the criteria, Eta finds it very hard to assigndeterminate weights. For simplicity, let us assume that she thus considers anyweight w between zero and one ‘‘relevant’’ or ‘‘plausible’’. ¹hus, in any pairwisechoice, all alternatives are acceptable, each being best for some relevant weightw. Condition c would imply that all alternatives are acceptable in the tripleMx, y, zN. ½et Eta (Eta being Eta and not Gamma, after all ) thinks differently.She reasons that on any assignment of weights she might employ to justify herchoice as acceptable, either x turns out to be best or z, never y. ¹hus, to Eta, y isunacceptable once both x and z are feasible.

Eta feels confirmed in her reasoning when she notices that, given herrisk-neutrality towards sources, if she were to flip a fair coin in choosing betweenx and z she would score 5 on both quark and charm, and thereby be unambigu-ously better off than if she were to choose y. ¼hile this reasoning does notdetermine her to flip a coin to make a choice, it proves to her the unambiguousinferiority of y.

Eta’s evaluation, have been endowed with maximal structure both to makeher reasoning maximally transparent and to make her case as difficult aspossible. Of course, a less sophisticated Eta might arrive at the same con-clusion. Note also that Eta’s rejection of y in Mx, y, zN stems from y’s low scoreon the two criteria (s(5). If s55, acceptance of y in Mx, y, zN can be justifiedby the weights (0.5, 0.5). We have chosen a multicriterion description of Eta’sdecision problem as the most general. For some readers, the cases for notaccepting y in Mx, y, zN may be stronger, if quark and charm are interpretedas individuals (and Eta’s preferences as those of benevolent dictator) or asuncertain states of nature.

Eta’s reasoning seems sound enough, certainly sound enough to denyc a valid claim to general applicability. Inspired by Eta, we shall investigate inthis paper the consequences of dropping c as a general choice-consistencycondition. In Sects. 2 and 3, we obtain a rationalization by ‘‘extended prefer-ence relations’’ (EPRs) P-F (X )]X (where F (X) denotes the set of non-empty finite subsets of some possibly infinite set X ). EPRs describe strictpreferences of sets over elements; in Eta’s preference, for instance, Mx, zNPywhile neither xPy nor zPy. Concepts of acyclicity and transitivity for EPRsare defined. Acyclic EPRs are characterized by non-empty choice-functionssatisfying the condition a that an acceptable choice remains acceptable when

404 K. Nehring

some alternatives are removed. Transitive EPRs (called ‘‘Extended PartialOrders’’, EPOs) are characterized by the additional property g that theremoval of an inferior alternative does not make other previously inferioralternatives acceptable; this is the first key result of the paper on which muchof the further analysis builds.

It is also shown that the concept of an extended preference relations unifiesthe general abstract theory of choice-functions. In particular, all rationaliza-tion is of one kind (Moulin [11], by comparison, distinguishes at least threevarieties of rationalization). Moreover, the other main results of the literature,Arrow’s and Schwartz’s, emerge as refinements of our characterization ofExtended Partial Orders.

Binariness plays essentially no role in this unification, whether conceptuallyor in terms of its deductive architecture. Thus, the stance of this paper towardsthe tradition is basically conservative. In terms of content, it goes beyondreceived theory merely in incorporating ‘‘unresolvedness’’ of preferences asa previously unrecognized dimension of incompleteness (see especiallySect. 6).

That dropping binariness may lead to a deeper understanding of abstractchoice theory is further confirmed in Sect. 4, in which a ‘‘Generalized Axiom ofRevealed Preference’’ is introduced which generalizes the classic WARP byaccommodating preference incompleteness, and which is shown to charac-terize extended partial orders.

Extended relations have made their first (and hitherto only) appearance inthe highly original and thorough work of Aizerman, Malishevski and theircollaborators (see Aizerman and Malishevski [2] (henceforth, A-M) andAizerman [1]). Section 8 contains an extensive discussion of the relation ofthis body of works to ours.

While we argue for extended relations as canonical ‘‘preference structure’’to rationalize choice-functions, A-M interpret and specify extended relations(under the name of ‘‘hyper-relations’’) rather differently, and consider ex-tended relations as only one among a number of ‘‘choice structures’’ deservingserious study. In particular, at least as prominent a role is played in their workby a ‘‘rationalization’’ of choice by sets of preference-orderings, for which theyprovide an alternative characterization of the class of choice-functions satisfy-ing a and g. In Sect. 5, we translate this result into a representation of EPOs asintersection of a set of (extended) linear orders.

In Sect. 6, we argue that for sets-of-preferences to canonically determineand properly rationalize an EPO, such sets need to be ‘‘closed under compro-mise’’. To capture this requirement, ‘‘convexity’’ of a set of preferences isdefined, and it is shown that every EPO can be rationalized by a convex set ofpreferences; it also follows from this result that every EPO can be backed upby a cardinal multi-attribute scenario as in Example 1.

It seems natural to try to further substantiate the notion of an extendedpreference relation as preference over sets of alternatives by explicitlymodeling choices of such sets. This is done in Sect. 7 in a sequential setting inwhich the decision-maker faces a choice among opportunity sets in the firststage and then (in an implicit second stage) chooses a particular alternative. Inthis setting, the axioms a and g can be replaced by two axioms that express theidea that the choice-worthiness of an opportunity set is monotone withrespect to set-inclusion. One obtains in this way a cogent foundation for

Rational choice without binariness 405

the transitivity axiom g, as well as a choice-functional characterization ofindirect utility rankings of sets for extended partial orders. All (non-trivial)proofs are collected in the Appendix.

2. Rationalization of choice by extended preference relations

An agent contemplates a variety of hypothetical choice situations A takenfrom some universal set of alternatives X; he describes his choice judgements(perhaps incompletely) by assessing alternatives x as ‘‘inferior’’ (xNC (A )) or‘‘acceptable’’ (x3C (A )). These judgements are summarized by a set-valuedchoice-function C :APX that maps choice situations from a domain A tosubsets of alternatives.

Interpreting choice as judgment rather than behavior is advantageous inpreserving generality. In a normative social-choice context, for instance, oneis typically concerned with evaluating the consequences of the interactionof many agents within a given institutional setting, without necessarily pre-supposing that the agents (even collectively, even counterfactually) intend tobring about those outcomes.

Let F (X) denote the class of non-empty finite subsets of the possiblyinfinite set X. In the following, we will often ignore the distinction betweenelements x and singletons MxN. In particular, we will often write ACx forACMxN and identify X with MMxN D x3XN. Throughout, we will maintain thefollowing two assumptions.

Assumption 1 (‘‘x1’’). C (MxN)"MxN for all MxN3A .

Assumption 2 (‘‘Universal Domain’’). A"F (X ) .

We do not always require non-emptiness of the set of acceptable alterna-tives (‘‘the chosen set’’) in order to allow for the possibility that an agent mightfind himself bereft of any alternative he would willing to choose, for instancedue to the existence of a preference cycle; while perhaps ‘‘irrational’’ froma normative point of view, such cycles are not unintelligible. Nor do werequire single-valuedness; for a variety of reasons, the agent may ‘‘run out’’ ofpreference judgments before eliminating all but one alternative.

If the choice-set is non-empty, a judgment of inferiority ‘‘xNC (A )’’ hasclear behavioral meaning: ‘‘I, the decision-maker, would not choose x insituation A’’. The behavioral meaning of ‘‘acceptability’’ is less clear since thechoice among different acceptable acts may not be a matter of indifference tothe agent; acceptability may fall short of optimality, one might say. It thusseems too strong to say of an acceptable act that an agent may choose it.Rather, he merely failed to rule it out, as is more properly expressed in Englishby the subjunctive ‘‘might’’. Indeed, it is argued in Nehring [12] that ifincompleteness of preference is to be fully rationalized as committed suspen-sion of judgment (‘‘non-comparability’’), some alternatives must ultimatelyturn out to be ‘‘more acceptable than others’’, in a possibly context-dependentway. For present purposes, however, in agreement with the literature, we takebeing acceptable as being acceptable enough.

406 K. Nehring

In a trivial sense, any choice-function can be ‘‘rationalized’’ by an ‘‘ex-tended relation’’ P-F (X)]X in terms of the following condition:

(o1) APx8xNC (AXMxN) for all x, A .

The expression ‘‘APx’’ is to be read as: ‘‘The choice (sub)-situation A isstrictly preferred to the alternative ("degenerate choice (sub-)situationMxN) x’’. Since P has the role of rationalizing choice, this preference is to beunderstood canonically as ultimately based on the value of final choices(which may be indeterminate); this will be referred to as indirect utilityinterpretation of an extended relation.

In particular, any linear order ("asymmetric, transitive and connectedbinary relation) P

b-X]X has a canonical indirect utility extension to an

extended relation given by the condition ‘‘APx8yPbx for some y3A’’.

In general, if preferences are incomplete, extended relations cannot alwaysbe reduced to binary relations, even on an indirect utility interpretation. AsEta’s case shows, an agent may be ‘‘sure’’ to prefer some alternative in A overx without having resolved which to prefer. Such an interpretation of non-binariness as unresolvedness of preference is further supported in Sects. 6and 7.

In the sequel, we will denote the unique P that satisfies o1

for a given Cby P

C, and conversely C

Pas the unique C that satisfies o

1for given P with

the property ‘‘APx8ACxPx , for all x , A’’ ; this property is implied by theconditions of irreflexivity and monotonicity defined just below. P

Cwill be

referred to as the extended preference relation revealed by C whose standardinterpretation is as a collection of choice judgments. Readers who stronglyassociate ‘‘revelation’’ of preference to revelation by observable choice behav-ior may find the term ‘‘implied’’ preference more accurate. Our terminologyagrees with the tradition, however, in contrasting ‘‘revealed preference’’ to anotion of preference as a psychological state such as need or desire.

We shall admit P as an independent entity not necessarily defined in termsof choice by o

1as P

C. This makes room for interpretations of the theory that

take preference rather than choice as basic, as well as for alternative linkagesbetween preference and choice such as o

2below. That the notion of preference

enjoys some conceptual independence from the notion of choice is confirmedby the fact that the relevant properties of EPRs can be interpreted intuitivelywithout being defined in terms of choice.

Note that PC

defined by o1

satisfies automatically ‘‘Irreflexivity’’:

(IRR) APxN [ACxO0 and ACxPx] for all x , A .

This condition reflects the fact that, ultimately, a particular alternative needsto be selected; it is then only meaningful to reject the choice of an act x in favorof other alternatives.

To warrant the intended indirect-utility interpretation, it seems indispens-able to require that P satisfy ‘‘Monotonicity’’:

(MON) APx&A-BNBPx for all x, A, B .

Requiring Monotonicity of PC

is a move beyond tautology, as it istantamount to imposing a, the mother of all choice-consistency conditions.

(a) x3ACC (A)&A-BNxNC (B ) for all x, A, B .


For the record, we note

Fact 1. Pc satisfies MON if and only if C satisfies a.

An extended relation satisfying Monotonicity and Irreflexivity is called‘‘(strict) Extended Preference Relation’’ (EPR); since irreflexivity is maintainedthroughout, the ‘‘strictness’’ adjective will normally be omitted.

It seems appropriate to view the class of EPRs as the most general formalrepresentation of the notion of context-independent preference and to viewfurther conditions such as transitivity or binariness as non-definitional nor-mative or structural properties of preferences: this seems legitimate if ‘‘prefer-ence’’ is understood broadly in subjective rather than objective terms, that is,as ‘‘x if favored over y’’ rather than ‘‘x is better than y’’.

For EPRs, the rationalization of a choice-function can equivalently bedefined by the following condition:

(o2) x3ACC (A)8x3A & &B-ACx :BPx for all x, A .

Fact 2. For any pair SC; PT such that P is an EPR, o18o

2.

Some readers may feel drawn to generalize the notion of rationalizationin terms of EPRs to rationalization in terms of preference relations over setsQ-F (X)]F (X) according to

(o3) x3ACC (A)8 [x3A&[&B, D-A : B-ACD& DUx&BQD]].

However, a moment’s reflection shows that such a move yields nothingnew. For if one imposes on Q the natural monotonicity condition‘‘BQ D&DUxNB Q x ’’, the satisfaction of o

3by C and Q is equivalent to

the satisfaction of o2

by C and QWF (X)]X .

Remark. For infinite X, we have assumed the choice-function to be definedon finite sets only. This is a less significant restriction that it may appear atfirst sight. For if one imposes on top of a the following regularity condition of‘‘finitariness’’ u, then C is determined uniquely by its restriction to F (X) .

(u) [x3A&[x3BNx3C (B) ∀B3F(A)]]Nx3C (A) for all x, A .

If C is in addition non-empty-valued onF (X ) and satisfies an appropriatecontinuity-condition, C can be shown to be non-empty-valued on compactsets (see Nehring [13]).

To illustrate the range of the developed theory, and to further motivate theinterest in set-valued choice-functions, we conclude this section by discussingbriefly a non-standard application to the context of norms. In that context, analternative describes an agent’s possible behavior, and a choice-function repres-ents the set of alternatives compatible with a given ensemble of (ethical, legal)norms in a given situation. In general, that set will depend on the situation(choice-set) in a non-trivial way; legal norms of self-defense, for instance, makethe legitimacy of particular means of self-defense dependent on the non-availability of sufficiently effective but less harmful alternative means.

Norms are particularly interesting an example since they typically lead toset-valued choice-functions; indeed, on most ethical views, set-valuedness ishighly esteemed, as a single-valued choice-function would leave the agentwithout any room for discretionary choice.

408 K. Nehring

It seems unreasonable to assume that norm-based choice-functions shouldalways satisfy standard consistency conditions; however, these sometimesapply naturally if, one might say, the underlying norms have ‘‘consequen-tialist’’ structure. One example of such consequentialist norms is the aboveconcerning legitimate means of self-defense; another, richer and more charged,is the following of ‘‘affirmative action’’. Here, a choice situation A is a setof candidates x3X applying for one position. The relevant norm is desertbased on qualification. The resulting choice-function selects a subset of ‘‘com-parably qualified’’ candidates, and the successful candidate is then selectedbased on some criterion of compensation, as, for instance, a candidate’smembership in a ‘‘disadvantaged’’ group.

Two features of this example seem particularly interesting. First, thetwo-stage selection process seems to be reasonably realistic as a description ofsome people’s view of appropriate affirmative action. Secondly, the selectivityof the desert-based stage-one selection is to a significant extent a politicaldecision; in particular, advocates of an active affirmative action policy wouldpresumably argue for a wide definition of what is to count as ‘‘comparablyqualified’’ (see Aizerman—Malishevski [3] for a mathematical analysis oftwo-stage choice procedures).

3. Consistency of choice and of preference

In this section, we will present characterizations of the following choice-consistency conditions in terms of properties of EPRs.

(b ) Mx, yN-C (Mx, yN)WAN [x3C (A)8 y3C (A)] for all x, y, A .

(c ) [x3C (Mx, yN) ∀y3A]Nx3C (AXMxN) for all x, A .

(g) [x3ACC (A) & y3ACC (A )]NxNC (ACy ) for all x, y, A .

(s) [x3C (A)& y3C (A)]Nx"y for all x, y, A .

(u) C (A)"0 for all A .

These conditions are very standard, or minor modifications of standardconditions. While not customary, it seems appropriate to list single-valued-ness and non-emptiness of the choice-function among them, and thus tofurther enrich Sen’s [17] alphabet soup. On the one hand, they correspond tosignificant and interpretable consistency properties of EPRs; moreover, theyare closely related to standard choice-consistency conditions, as the implica-tions ‘‘sNb’’ and ‘‘g&u

1Nu’’ show (note that the latter implication

hinges on the maintained assumption u1

of non-emptiness of choice fromsingletons).

Most important, perhaps, is condition g. g says that unchosen acts are ‘‘nocompetition’’; their elimination should therefore not make other previouslyunchosen alternatives acceptable. On a universal domain, g is equivalent tothe condition g@ just below, its customary version. We have chosen to modifythe latter primarily in order to obtain a closer correspondence to the sub-sequently defined transitivity condition on EPRs TRA. g@ seems to have


appeared first in Chernoff [6], employed first in a significant way in Jamisonand Lau [10] and fully recognized in importance by Aizerman andMalishevski [2].

(g@) C (A)-B-ANC (B )-C (A ) for all A, B .

The conditions b and c are slight weakenings of Sen’s [17] conditions withthe same name. In somewhat non-standard terminology, choice-functionssatisfying c will be called ‘‘binary’’. To establish a direct correspondencebetween properties of extended relations and those of their induced choice-functions, b and g need to be strengthened to conditions b` and g` as follows.

(b`) C (Mx, yN)"Mx, yNN [(C (AXMxNUx8C (AXMyN)Uy )]

for all x, y, A .

(g`) x3ACC (A)& y3BCC (B) & y3A]NxNC ((AXB )CMyN)

for all x, y, A, B .

b is b` is restricted to x, y, A such that x, y3A, and g is g` restricted toA, B such that A"B. While b` seems intuitively natural a strengthening of b,g` seems to lack intuitive meaning as a choice-consistency condition proper,that is, unless it is interpreted as transitivity of revealed preference P

C.

We now present and discuss the corresponding EPR properties. They owetheir names to their precise analogy to the corresponding properties of binaryrelations, as will be shown later.

Acyclicity (ACY):

&x3A : not APx, for all A .

Transitivity (TRA):

AXMyNPx &BPyNAXBPx, for all x, y, AXMyN , B .

Symmetric Negative Transitivity (SNT):

Not xPy & not yPxN (APx8APy ) , for all x, y, A .

Connectedness (CON):

not APx & y3ANxPy, for all A, x, y such that xOy .

Binariness (BIN):

APxN&y3A : yPx , for all x, A .

ACY is straightforward: a set cannot be superior (in terms of indirectutility) to all of its elements.

TRA says that if an alternative in a superior set is replaced by a superiorset of alternatives, the resulting set must also be superior.

SNT asserts that if ‘‘indifference’’ is negatively defined from P (xIy8 (notxPy & not yPx)), indifferent acts are treated equivalently by P.

If an EPR P is binary, it can be reduced to its binary base relationPb"PWX]X (where X and the set of its singletons are identified); corres-

pondingly, rationalization by the EPR P reduces to rationalization by Pb

410 K. Nehring

ordinarily defined. For irreflexive Q-X]X, define the choice-function DQby D

Q(A)"Mx3A D for no y3A : yQxN, for A3F (X ).

The term ‘‘binary’’ and ‘‘binary relation’’ are used here in the traditionalsense to denote choice-functions and EPRs that can be fully described bya relation between elements. Of course, EPRs can be viewed as ‘‘binary’’relations between sets and elements; this is indeed required by the proposedindirect-utility interpretation of EPRs. Thus, ‘‘binariness’ as such cannot be atissue, and whether or not ‘‘binariness’’ in the narrow, traditional sense issatisfied may not seem a big deal. Indeed: one thing this paper intends to showis that nothing of importance depends on it.

Fact 3. An EPR P is binary if and only if DPb"C

P.

The following standard properties can alternatively be viewed as proper-ties of binary relations or of extended relations.

(IRR@) For no x3X, xPx .

(TRA@) xPy& yPzNxPz, for all x, y, z .

(ACY@ ) For no finite sequence MxiNi/0,1 ,2 ,n

in X such that x0"x

n:x

iPx

i~1for all i, 14i4n .

(SNT@ ) not xPy & not yPxN (zPx8 zPy), for all x, y, z .

(CON@) not yPxNxPy, for all x, y such that xOy .

It is straightforward to verify:

Fact 4. For all monotone and binary extended relations:(i) IRR8 IRR@ ,(ii) ¹RA8¹RA@,(iii) AC½8AC½ @,(iv) SN¹8SN¹ @,(v) CON8CON @ .

Properties of preference and choice translate as follows.

Fact 5. ¹he choice-function C satisfies ‘‘?’’ if and only if PC

satisfies ‘‘???’’ :(i) u 8AC½,(ii) g 8¹RA ,(iii) s 8CON ,(iv) b 8SN¹ ,(v) c 8BIN .

Fact 5 helps clarify the intuitive meaning of different choice-consistencyconditions. In particular, it becomes transparent that a, g (or g`) and c arequalitatively entirely heterogeneous; this is less evident without the help of fact5, as Moulin’s [11, p. 154] assessment of these conditions as ‘‘in the same veinyet logically unrelated’’ indicates. Note also that only part (iii) of fact 5 makesuse of the maintained assumption that C satisfies a, respectively that P

Csatisfies MON.Fact 5 enables one to read implications among consistency conditions

on choice in terms of preference; this is done in the subsequent Fact 6 which


translates Lemma 1 below. The implications among choice-consistency condi-tions seem intuitively much less natural than the corresponding implicationsamong properties of extended relations.

The following implications among the basic EPR properties hold.

Fact 6. For any EPR P:(i) ¹RANAC½ ,(ii) CONNSN¹ ,(iii) AC½ & SN¹N¹RA & BIN .

Facts 4 and 6 justify the following definitions; note in particular thatFact 6(iii) allows one to define weak and linear orders without reference tothe notion of binariness.

Definition 1. An extended sub-order is an acyclic, an extended partial order(EPO) is a transitive EPR. A partial order/weak order/linear order is an EPOthat is binary/symmetrically negatively transitive/connected.

Fact 5 yields immediate characterizations of the class of choice-functionsthat are rationalizable by various types of orders. The resulting characteriza-tion of extended partial and of weak orders can be significantly improved. Inparticular, g` can be weakened to g in the former, and even to u in the latter.Furthermore, b` can be weakened to b in the characterization of weakorders.

These assertions follow from the following Lemma 1 on the cross-implica-tions among choice-consistency conditions which contains the entire math-ematical content of the abstract theory of choice of this section (to the degreeone wishes to speak of such content at all).

Lemma 1. (i) u1

& gNu .(ii) a & gNg` .(iii) a & u & bN g .(iv) a & g & bNb` .(v) a & u & bN c .

Remark 1. Parts (i), (ii) and (iv) are new, since the involved conditions u1,

g` and b` are; (v) is standard but (iii) seems new as a ‘‘direct’’ implication,i.e. as derived without the help of c obtained from (v).

Remark 2. Note that the binariness condition c does not occur in the presup-position of any of the five implications; it thus turns out to be redundant fromthe deductive (‘‘syntactic’’) point of view. c occurs only in (v), which is used toensure agreement of the proposed definition of a weak order with the standardone.

Fact 5 and Lemma 1 imply immediately:

Theorem 1. (i) Rationalizability by an extended sub-order is equivalent to a & u .(ii) Rationalizability by an extended partial order is equivalent to a & g .(iii) Rationalizability by a partial order is equivalent to a & g & c .(iv) Rationalizability by a weak order is equivalent to a & u & b .(v) Rationalizability by a linear order is equivalent to a & u & s .

412 K. Nehring

Theorem 1(iii)—(v) are new only in terms of exposition. In particular, theprogression from acyclic binary relations (a & u & c, Sen [17]) to partialorders (a & g & c, Schwartz [16]) by means of g is shown to be the binaryversion of the more general progression from acyclic EPRs to EPOs. Part (iv)is a version of Arrow’s result [4] as reformulated by Sen [17].

Theorem 1(ii) is a key result of this paper, as it provides a choice-functional(‘‘revealed preference’’) characterization of extended partial orders. It isalso the basis for two further choice-functional characterizations of EPOs(Theorems 2 and 7). The class of choice-functions satisfying a & g has beencharacterized before in a variety of ways by A-M; see Sects. 5 and 8 forextensive discussions of the relation of their characterizations to ours.

Theorem 1(i) appears to be the first (mathematically trivial but concep-tually meaningful) rationalization of the class of ‘‘a & u’’ choice-functions.Its interest is normative, in that the interpretation of u as ACY yields a newjustification for its acceptance. The normative appeal of ACY seems compell-ing under the assumed IU interpretation of EPRs. By comparison, acyclicitydefined in the usual way for binary relations as ACY@ is much less compelling,since it imposes restrictions on a possibly heterogeneous set of preference-comparisons. Note, though, that ACY@ follows from ACY with the help ofMON, a condition that genuinely pertains to EPRs and has no equivalent interms of binary relations.

The result also explains why acyclic binary relations cannot be character-ized in an intuitively more appealing manner than by the conjunction of a,u and condition c. Kreps [9, p. 15] declares himself dissatisfied with the latteras ‘‘not particularly intuitive’’; such dissatisfaction should vanish once onerealizes that c accounts for binariness of the revealed preference relation buthas nothing at all to do with acyclicity, the latter being due to u, and hence‘‘invisible’’ in the standard approach which incorporates u into the definitionof a choice-function.

Theorem 1 shows that the concept of an extended preference relationmakes possible a ‘‘semantic’’ unification of the abstract theory of choice byproviding a uniform notion of rationalization of choice in terms of extendedorders of various kinds. By comparison, Moulin [11] distinguishes at leastthree modalities of rationalization, ‘‘sub-’’ and ‘‘pseudo-’’rationalizationbesides the ordinary kind; the place of these non-standard models within thepresent theory is shown in Sect. 5.

The derivation of Theorem 1 from Fact 5 and Lemma 1 amounts also toa syntactic unification of abstract choice theory. Binariness turns out to be ofmarginal importance from both points of view, unnecessary as it is for eitherthe definition or the characterization of various types of orders.

4. Axioms of consistent preference revelation

Maximization of a weak order according to o1

implies the following ‘‘maximi-zation property’’ M

5that any P-maximal alternative is superior in direct

comparison to any P-inferior alternative.

(M5) [x3C

P(A)& y3ACC

P(A)]N xPy .


M5

can be viewed as a completeness property; this is confirmed by theobservation that M

5is only satisfied if P is a weak order (Theorem 3 below).

By comparison, maximization of an EPO yields only the weaker propertythat the set of P-maximal alternatives is superior to any P-inferior alternative:

(M4) [C

P(A)O0 & y3ACC

P(A)]NC

P(A )Py .

The maximization properties of EPRs M4

and M5

can be translated into‘‘revealed preference properties’’ o

4and o

5of choice-functions.

(o4) [C (A)O0 & y3ACC (A)]NC (A) P

Cy .

(o5) [x3C (A) & y3ACC (A)]NxP

Cy .

If C satisfies o5, for instance, inferiority of some alternative y in a given

choice-situation A reveals (PC-) inferiority to any alternative that is acceptable

in the same choice-situation. While o5

is the natural result of maximizinga weak order, it is never satisfied otherwise.

The revealed preference conditions o4

and o5

lead naturally to revealed-preference-based choice-consistency conditions. For example, o

5and a imply

the classical axiom ‘‘WARP’’.

(WARP) [x3C (A) & y3ACC (A ) &x3B]N yNC (B )

for all x, y, A, B .

This implication is easily seem as follows: By o5, x3C (A) & y3ACC (A)

imply xPCy. By a, x3B yields BP

Cy , and thus yNC (B ) by the definition

of PC.

Analogously, o4, a and u imply the following ‘‘Generalized Axiom of

Revealed Preference’’ (GARP).

(GARP) [y3ACC (A) &C (A )-B]N yNC (B ) for all y, A, B .

GARP says that if y is inferior in A, and ‘‘thereby’’ revealed to be inferiorto C (A), it must also be inferior in any situation in which the entire set C (A) isavailable. GARP characterizes rationalizability by an EPO, just as WARP(plus u) characterizes rationalizability by a weak order.

Theorem 2. ¹he following conditions on a choice-function are equivalent:(i) P

Cis a extended partial order.

(ii) a & o4

& u .(iii) GARP.

Theorem 3. ¹he following conditions on a choice-function are equivalent:(i) P

Cis a weak order.

(ii) a & o5

& u .(iii) ¼ARP & u .

For the record, we note the following dual statement for EPRs as acorollary,

Corollary 1. An extended sub-order is an extended partial order (respectivelya weak order) if and only if it satisfies M

4(respectively M

5).

414 K. Nehring

Remark. Properties corresponding to M5/o

5and their equivalence to WARP

have been thematized in Clark [7], whereas M4/o

4, GARP, and Theorem 2

are new.

5. Characterization of extended sub- and partial orders

Two questions are natural to ask of an incomplete preference relation: First,can it be ‘‘completed’’ to an order? And second, can absence of preference beaccounted for as ‘‘possible’’ converse preference within a complete order?

In the present context, these questions translate to the following twoextension properties, with PL denoting the set of linear orders.

(EXT1) &Q3PL : Q.P .

(EXT2) P"YMQ3PL DQ.PN .

It is well-known that, due to Szpilrajn’s theorem, for binary relations,EXT1 characterizes acyclic and EXT2 characterizes transitive asymmetricrelations, i.e. partial orders. At least for the case of a finite universe X (to whichwe shall confine ourselves in the next two sections), both characterizationscarry over to EPRs. In fact, it turns out that both results have already beenproven in the literature in dual form as results on refinements of choice-functions.

Let CL denote the class of choice-functions rationalizable by a linear order(i.e. characterized by a &u & s ) . The dual of the above extension propertiesare the following refinement properties of choice-functions.

(REF1) &G3CL : G-C .

(REF2) C"ZMG3CL DG-CN .

It is straightforward to verify

Fact 7. (i) P satisfies EXT18CP

satisfies REF1.(ii) P satisfies EXT28C

Psatisfies REF2 .

In view of a result of Deb [8], Fact 7(i) and Theorem 1 yield immediately

Theorem 4. An EPR is an extended sub-order if and only if it satisfies EX¹1 .

We also want to establish

(H) IRR&MON&TRA8EXT2.

From Fact 7(ii) and Theorem 1 one obtains as a dual equivalence

(A/M) a & g8REF2.

That is to say, Fact 7(ii) and Theorem 1 imply the following translationresult.

Proposition 1. H8A/M .

Since Aizerman and Malishevski [2] already established A/M, H can beasserted as a theorem.

Theorem 5. An extended relation is an EPO if and only if it satisfies EX¹2 .


Conceptually, Proposition 1 plus Theorem 5 allow to understand‘‘pseudo-rationalizability’’ (condition REF2) of a choice-function as the con-junction of rationalizability by an EPO plus the representability of an EPO asintersection of its extensions; in particular, this clarifies the status of thepseudo-rationalizing set of linear orders MP

G3PL DG3CL&G-CN as rep-

resentation of the incompleteness of PC.

We conclude this section by commenting on the interpretation of the A/Mtheorem. According to the representation of choice-functions obtained fromA/M, i.e. REF2, an alternative is acceptable in X if and only if it is the bestalternative for some linear order P

iin a representing set MP

iNi3I

.This representation is of significant interest under a behavioral interpreta-

tion of choice-functions. For instance, if one agent’s choices are observed overan extended period of time, and if his preferences are non-stationary, a plusg exactly characterize the restrictions on choices made at some point in timei3I by an agent whose choices are always based on linear orders. In view ofour characterization of this class, the moral is that the observability of anagent’s conforming to transitivity is not affected by non-stationariness ofpreference, while, on the other hand, non-stationariness makes preferencecompleteness unobservable. Alternatively, I may denote a heterogeneous setof agents, and C (A ) the set of alternatives chosen by some agents. Again,aggregation preserves transitivity but not completeness; the aggregate choice-function reveals the consensus preference EPO P

C"YMP

iNi3I

, but not theunderlying set of preferences MP

iNi3I

.We will argue in the next section that the A/M representation is less

compelling as a rationalization proper.

6. Multi-preference rationalization of extended partial orders

Incompleteness can be understood in two kinds of ways: on the one hand,it may be due to an only incomplete, partial (self-)elicitation of the deci-sion-maker’s preferences that are complete in principle. Alternatively, onan exhaustive interpretation, incompleteness reflects a considered suspensionof judgment. Thus, on a partial interpretation, absence of preference (neitherxPy nor yPx) corresponds to ‘‘non-comparedness’’ of x and y, while itcorresponds to judged non-comparability on an exhaustive interpretation.Considered suspension of judgment assumes the decision-maker to be as‘‘rational’’ as ever, whereas incompleteness of elicitation arises naturally fromits costliness, and thus, in this sense, from the decision-maker’s ‘‘boundedrationality’’.

On a partial interpretation, Theorems 4 and 5 have straightforwardsignificance. Theorem 4 characterizes the EPRs that result from partialelicitation of an underlying complete (or even linear) order — formally: that aresub-relations of complete/linear orders — as those that are acyclic.

Each acyclic EPR P determines a smallest strict EPO that incorporatesall implications of the preference axioms transitivity, monotonicity and irreflex-ivity, its ‘‘strict transitive closure’’ ¹s P"YMQ DQ.P, Q is an EPON. Notethat for non-binary extended relations, the strict transitive closure may strictlyinclude the transitive closure ¹r P"YMQ D Q.P, Q is a transitive extendedrelationN, as the following example shows.

416 K. Nehring

Example 2. Define an extended sub-order P on X by APx8AUy,APy8A.Mx, zN, and APz for no A3F (X). From TRA, Mz, yN ¹s P MyN.From IRR, MzN ¹s P MyN . However, it is easily checked that not MzN ¹r P MyN.

It is natural to ask whether the knowledge that the underlying ordering iscomplete entitles to any further inference: Theorem 5 shows this never to bethe case.

On an exhaustive interpretation, the following question arises naturally: isthe ‘‘effective’’ (choice-determining) EPO P (whose incompleteness reflectsconsidered suspension of judgment) rationalizable in terms of a set of ‘‘pos-sible’’ ‘‘relevant’’ linear or weak orders Q between which the decision-makersuspends judgment? Such ‘‘multi-preference rationalization’’ of P by Q in-volves at least two things: first, each Q3Q must be compatible with P(i.e. Q.P ). Secondly, each absence of preference is to be accounted for aspossible converse preference. In formal terms, the condition

‘‘not APxN &Q3Q :xQy ∀y3A’’

must hold for all A, x ; the right-hand expression stands in for the undefinedexpression ‘‘&Q3Q : xQA’’. These two conditions are clearly equivalent to re-quiring that P be the unanimity relation of Q, i.e. P"YQ . Thus, a seeminglyaffirmative answer to the question of multiple-preference rationalizability ofextended partial orders is given by Theorem 5.

However, there is a glitch. It arises from the question whether an arbitrarygiven set of orderings Q canonically determines a unique effective EPOPQ according to PQ"YQ . For Q to properly rationalize PQ , this would need tobe the case; otherwise, if Q were compatible with more than one EPO,something beyond Q must be involved in the decision-maker’s determinationof the effective preference ordering P. The skepticism is the following: isunanimous preference with respect to potential preferences A (YQ )x sufficientfor effective preference APx? In terms of choice, it translates into skepticism asto whether acceptability of x in AXMxN with respect to C necessarily pre-supposes acceptability of x in AXMxN with respect to some C

Q(Q3Q ) .

For arbitrary sets Q, such skepticism seems unanswerable. Indeed, whilex may fail to be Q-optimal with respect to any Q3Q , it may nonetheless bea legitimate compromise among the different Q3Q . More specifically, x maybe optimal with respect to some compromise preference Q@. In example 1, forinstance, if 5(s(10, y seems attractive as a compromise choice if criteriaare completely non-comparable; selection of y results from a compromisepreference Q@ characterized by the weight vector (0.5, 0.5). Hence, unless Q@is guaranteed to be contained in Q, that is to say: unless Q is ‘‘closedunder compromise’’, Q-unanimous preference need not imply effectivepreference (in which case choice fails to be determined canonically byQ through PQ) .

To convincingly answer the skepticism raised requires giving precisemeaning to the phrase ‘‘closed under compromise’’. As a sufficient condi-tion for closedness under compromise, we propose the following convexitycondition on Q:

Definition 2. Q is convex if it is representable by a convex set U of utilityfunctions, i.e. if there exists a convex subset U of RX such that


Q"MQU

Dº3UN ,

with AQU

x8maxy3A

º (y )'º (x).

Remark 1. Within the present ordinal framework, ‘‘convexity’’ as definedlooks rather ad hoc. This ceases to be the case when convexity is understoodas embeddability in a set QI of von Neumann—Morgenstern preferences on DX,the set of probability-mixtures on X, such that Q"MRest

XQ D Q3QI N, and

such that QI is representable (in the natural sense) by a convex set of vonNeumann—Morgenstern utility functions U : XPR (equivalently: U3RX).This (and more, see Remark 3) is left to future work.

Remark 2. In a cardinal setting, the convexity-condition on QI is easilyformulated directly, without reference to representing utility functions. Wesuspect that a direct characterization of convexity should be possible in anordinal setting as well, but again leave this to future work.

Remark 3. We believe convexity to be the right, i.e. necessary as well assufficient, formalization of the notion of ‘‘closedness under compromise’’.Of course, justification of this claim requires a theory on DX. In the presentcontext, however, in which the issue is answering the skeptic, such justificationis not needed, since it is sufficiency rather than necessity that matters.

Remark 4. In a multi-criterion context with additively separable criteria suchas Example 1, Q is convex if and only if Q is obtained from a convex set ofweight-vectors. In particular, this means that each ordering Q3Q representsan aggregate preference that takes all criteria into account, rather thana criterion-specific ranking as in A-M’s interpretation.

Our discussion suggests the following definition.

Definition 3. An extended relation P is multi-preference rationalizable if thereexists a convex set of weak orders Q-F (X)]X such that P"YQ .

Note that, unless Q is a singleton, any convex set of weak orders Q will infact contain at least one weak order that is not a linear one.

Theorem 6. An extended relation P is multi-preference rationalizable if and onlyif P is an extended partial order.

Multi-preference rationalizability gives rigorous expression to the un-resolved-preference interpretation of EPOs suggested in Sect. 2; this followsfrom noticing the equivalence of ‘‘AQx , for all Q3Q’’ and ‘‘for all Q3Q , thereexists y3A : yQx’’. Thus for any Q rationalizing P :

APx8∀Q3Q&y3A : yQx . (1)

(1) explains the non-binariness of P through the failure of the quantifiers‘‘∀’’ and ‘‘&’’ to interchange.

7. A choice-functional axiomatization of indirect utility

The crucial role of the IU interpretation of EPOs suggests that additionalinsight can be obtained by directly looking at indirect-utility-based choices

418 K. Nehring

over sets. This turns out be true; in particular, it will be shown that the keyaxioms a and g can be derived as natural consequences of the requirementthat the value of a set must be monotonic with respect to set-inclusion.

An augmented choice function G is a choice function in the sense of Sect.2 for the universe of alternativesF (X) rather than X. Thus, G maps arbitraryfinite sets S3F (F (X)) of finite subsets of X to subsets G (S )-S such thatG (MAN)"MAN for all A3F (X) . G is to be understood as describing first-stagechoices in a two-stage decision problem; for our purposes, it turns out not tobe necessary to describe choices at the second stage.

Augmented extended relations P are defined accordingly as subsetsof F (F (X))]F(X) . For A-X, let S (A) be the set of singletons in X,S (A)"MMxN D x3AN. G restricted to F (S (X)) defines a standard one-stagechoice-function C

Gunder the obvious identification of S (X) and X; analog-

ously, P restricted to F(S (X))]S (X) defines an extended relation P0, its

base relation.In the following, choice and preference properties applied to augmented

choice-functions and relations will be denoted by adding a hat (‘‘\’’) ontop; the properties restricted to one-stage choice-functions respectively baserelations will be referred to by their original symbol. Thus, a, for instance,denotes

A3SCG (S) &¹.SNANG (¹ ) for all A, S, ¹ ,

while a denotes the restriction of a to F (S (X)) .

Let ZS"ZMA D A3SN ; in particular, ZMAN"A. The augmented choice-function G is indirect-utility based if it satisfies the ‘‘reduction condition’’

(IUC) A3G (S )8AWCG(ZS )O0 for all A, S .

Analogously, P is indirect-utility based if it satisfies the reduction condition

SPA8∀x3A : (ZS)P0MxN for all A, S .

Defining PG

from G via o1, it is easily verified that

Fact 8. G satisfies IºC if and only if PG

satisfies IºP .

A subset A of X has two ‘‘copies’’ in F (F (X)), the set of singletonsS (A)3F (S (X)), and the singleton set MAN. Accordingly, S (A)P

0MxN and

MANP0MxN denote two different preference judgments. The former is naturally

interpreted as ‘‘unresolved preference’’ of the alternatives in S (A) over x, thelatter as IU-preference of A as a set over MxN. It is easily seen that IUP impliesthe equivalence of both, S (A)P

0MxN8 MANP

0MxN, thereby underwriting the

dual interpretation of EPOs in terms of indirect utility and in terms ofunresolved preference.

Note also that IUC implies u due to the assumed non-emptiness of G onS (F (X)); similarly, for augmented relations P, IUP combined withIRRY implies AC½Y . It is also easily verified that any non-empty-valued choice-function on F (S (X)) can be uniquely extended to an IU-based choice-function on F (F (X )). These implications underline the force of theindirect-utility based justification of acyclicity in Sect. 3.


Of special interest are the following two monotonicity condition a* and g*.

(a*) [D3SXMAN)CG (SXMAN)&B.A]NDNG (SXMBN) for allA, B, D, S .

(g*) A3G (SXMAN)&B.ANB3G (SXMBN) for all A, B, S .

g* asserts that if a choice-situation A is acceptable in the context of the set S,then any situation B that includes A is also acceptable: a*, in complementaryfashion, asserts that the substitution of A by a larger set B cannot makea previously inferior situation D acceptable.

The key observation of this section is the following.

Lemma 2. (i) For G satisfying IºC, a*, a and a are equivalent.(ii) For G satisfying IºC, g*, g and gL are equivalent.

A partial indirect utility (PIº-) ordering is an augmented extended relationsatisfying IUP, IRR, MON, and TRA; Lemma 2 implies that a PIU orderingsatisfies IRRY , MONY and ¹RAY as well.

One immediately obtains from Theorem 1, Lemma 2 and Fact 8 thefollowing characterization result:

Theorem 7. An augmented choice-function G is rationalizable by a PIº order-ing if and only if G satisfies IºC, a* and g*.

A PIU ordering is ‘‘linear’’ if its base relation is. In view of the reducibilityof a PIU ordering to its base relation due to IUP, the following result is astraightforward corollary to Theorem 5.

Theorem 8. An augmented relation is a PIº ordering if and only if it is theintersection of a set of linear PIº orderings.

The dynamic treatment of this section attains direct relevance in a one-shot setting via an embedding argument which requires that the one-stagechoice-function C be extendible to a two-stage choice-function G satisfyingIUC as well as a* and g*. Theorem 7 shows that the class of such choice-functions coincides with those rationalizable by an EPO.

The dynamic structure involved in the hypothesized extension is naturallyinterpreted as that of a sequential algorithm of selecting alternatives(in ‘‘logical time’’), as compared to a truly dynamic setting in which differentstages correspond to distinct ‘‘real time’’ dates. In a genuinely dynamicinterpretation, IUC is restrictive in that it rules out ‘‘intrinsic preferenceof freedom of choice’’. Note, however, that a* and g* appeal even in that case,as long as there are no problems of changing tastes, weakness of will, etc.

On a logical time interpretation, opportunities cannot be kept open in realtime by definition. IUC can then be viewed as a ‘‘path independence’’ propertyof sorts. On such an interpretation, the force of Theorem 7 resides primarilyin the monotonicity conditions a* and g* which represent straightforwardcontext-independence requirements on first-stage choices.

It is of some interest to compare Theorem 7 to the equivalence of Plott’s[15] classic axiom of path-independence IP and the conjunction of a and

420 K. Nehring

g (shown by A-M as well as Blair et al. [5]).

(IP) C (AXB)"C (C (A)XB) ) for all A, B .

Whereas IUC, a* and g* all refer to first-stage choices, IP effectively refersto the outcomes of a sequential choice in two stages. While its compactappearance and logical form are appealing, on closer inspection IP turns outto be rather complex and not very transparent in its normative presupposi-tions as well as its logical implications.

8. Relation to the work of Aizerman and Malishevski

The idea of explaining choice-functions in terms of extended relations (underthe name of hyperrelation) as well as the recognition of the importance ofaxiom g appeared first in the highly seminal work of A-M and their colla-borators. There are significant and interrelated differences both in terms ofinterpretation as well as technical execution between their work and ours. Interms of interpretation, A-M present choice based on extended relations as aninstance of context-dependent choice, while context-independence is a crucialfeature of the indirect-utility interpretation proposed in this paper. We willturn to the differences of interpretation later on in more detail, and begin bydescribing the major technical differences.1. A first important difference arises from their definition of acyclicity(‘‘ahypercyclicity’’) in A-M’s terminology, see Aizerman [1, p. 246] whichimplies an irreflexivity condition IRRH (not explicitly defined by A-M) that isstronger that IRR.

(IRRH) AUxNnot APx for all x, A .

The difference between IRR and IRRH is non-trivial, since only the formeris compatible with MON. Thus IRRH seems unacceptable for an IU-inter-pretation, while it might well be deemed sensible under a context-dependentinterpretation. However, even then IRRH seems unnecessarily strong; inparticular, it fails to follow from IRR@ when P is binary, in contrast to IRR.2. Coupled with different definitions of acyclicity and irreflexivity, perhapsthe root difference is in the relation between extended relations and choice, forwhich A-M assume o

2instead of o

1. By consequence, there is no analogue to

Fact 5 in A-M. The resulting non-uniqueness problem is addressed (in ‘‘anti-monotone’’ spirit) as a matter of convenience with an eye toward economizingon the cardinality of the extended relation (A-M, p. 1037/8). Indeed, quiteremarkably, A-M do not consider any kind of monotonicity condition!3. While A-M’s definition of acyclicity is stronger than ours, their definition of‘‘hyper-transitivity’’ is weaker than the conjunction of TRA and IRR.

(TRAH) [APx& BPy&AUy& yOx]N &D-(AXBXMxN)Cy :DPxfor all x, y, A, B .

Malishevski, using o2

rather than o1

as link between preference andchoice, has obtained results analogues to parts(i) and (ii) of Theorem 1, withACYH replacing ACY and TRAH replacing TRA (quoted in Aizerman[1, pp. 246—247] without reference to a publication).


As a fundamental definition of transitivity, TRAH has some drawbackswhich TRA does not share. First of all, there does not seem to exist an obviousway to define the ‘‘hyper-transitive closure’’ of a given extended relation P; inparticular, a smallest hyper-transitive superrelation of P may easily fail toexist (take, for example, the extended relation P defined by Mx, yNPw andMwNPx). Moreover, TRAH is generally inapplicable to weak EPOs (strictEPO without irreflexivity), used in the analysis of preferences for opportuni-ties and diversity in Nehring and Puppe [14].

Furthermore, due to the absence of monotonicity, there is no intersection-representation of hypertransitive extended relations in the manner ofTheorem 5. Indeed, A-M as well as Moulin [11] in their wake emphasize thedifference between ‘‘Pareto-’’ and ‘‘pseudo-’’rationalization, while the twocoincide if Pareto rationalization is defined in terms of EPOs rather than theirbinary base relation, as shown in Sect. 5.4. A-M do not define a negative transitivity condition for extended relationssuch as SNT.

Throughout, A-M [2] emphasize a context-dependent interpretation ofextended relations; moreover, these do not seem to be privileged as struc-tures/mechanisms of choice, especially compared to a pseudo-rationalizationof choice by multiple preferences.

In our account, multi-preference representations may be viewed as ‘‘deepstructures’’ of EPOs, thus further substantiating the interpretation of EPOs asgeneralizations of binary partial orders. The availability of this interpretationas well that of the intersection representation of EPOs hinges crucially on themonotonicity property of EPRs, a property eschewed by A-M. The morespecific role played by multiple preferences in our approach is reflected ina more specific interpretation of sets of preferences as sets of ‘‘aggregate’’orderings that are closed under compromise; there is no mention of a compro-mise problem in A-M.

Finally, there is some difference in the stance towards the classical binaryconception of rationalization. While we concur with A-M in their rejection ofbinariness as postulate of rationality, the classical intuition of the canonical statusof choice-rationalization by means of a single preference-relation emerges essen-tially unscathed. Our analysis merely adds that incompleteness of preference hasa previously unrecognized dimension, that of unresolvedness of preference. In-deed, one might even argue that the classical intuition is strengthened sincerationalization by EPOs respectively weak orders arises from the notion ofcontext-independent choice alone, without reference to the condition of binari-ness which cannot be thus motivated, and indeed seems hard to justify withoutassuming rationalizability by a (binary) preference relation in the first place.

Appendix: proofs

Proof of Fact 2. The fact is contained in the following four implications‘‘p implies q due to xyz’’; pN

irefers to the N-part of o

i, o2

1to the =-part.

1. o21

implies o13

due to IRR.2. o1

1implies o2

3due to MON.

3. o23

implies o11

due to IRR.4. o1

3implies o2

1due to MON. h

422 K. Nehring

Proof of Lemma 1.(i) ua & gNu: straightforward.(ii) a& gN g` : Take x, y, A and B such that x3ACC (A) and

y3BCC (B ). By twofold application of a, x3(AXB )CC (AXB) andy3(AXB)CC (AXB). Hence by g, xNC ((AXB)CMyN).

(iii) a& b&uN g : Assume, by way of contradiction, that there existA-X, x, y3XCA such that x3C (AXMxN) and C (AXMx, yN)WMx, yN"0 .By u, there exists z3C (AXMx, yN)WA. By a, C (Mx, zN)"Mx, zN. By b,x3C (AXMx, yN), the desired contradiction.

(iv) a& g &bNb`: Take any x, y and A such that Mx, yN"C (Mx, yN)and x3C (MxNXA). By g, Mx, yNWC (Mx, yNXA)O0 . Hence by b,Mx, yN-C (Mx, yNXA ). It follows from a that y3C (MyNXA ).

(v) a& g &bN c: Take any x and A such that x3C (Mx, yN) for all y3A.Assume that not x3C (MxNXA). By u, for some y3A, y3C (MxNXA). By a,y3C (Mx, yN), in contradiction to b. h

Proof of Fact 6. (ii) Is trivial. (i) Follows from Lemma 1(i), thus exploiting theirreflexivity of EPRs, and (iii) follows from parts (ii), (iii), and (v) of the samelemma. h

Proof of Theorem 1. (i) Follows immediately from Fact 5. (ii) Follows fromFact 5 and Lemma 1(ii). (iii) Is a direct consequence of (ii). (iv) Follows fromFact 5 and the equivalence a &u&b8a& g`&b`, which is implied byLemma 1(i)—(iv). (v) Is a direct consequence of (iv). h

Proof of Theorem 2. In view of Theorem 1, the assertion follows from thestraightforward implications

(i) a& o4&uNGARP.

(ii) GARPN a& g.(iii) gNo

4.

As to the second, note that a corresponds to GARP restricted to A, B suchthat A-B, and that g corresponds to GARP restricted to A, B such thatB-A and d (ACB)"1. h

Proof of Theorem 3. In view of Theorem 1, the assertion follows from thestraightforward implications

(i) a& o5NWARP.

(ii) WARPNa&b.(iii) a& bNo

5.

To verify the third, take x, y, A such that x3C (A) and y3ACC (A). Bya, x3C (Mx, yN), and thus by b, MxN"C (Mx, yN), i.e. xP

Cy. h

Proof of Fact 7. (i) Follows directly from the fact that Q.P8CQ-C

P.

(ii) REF2NEXT2: Take A, x such that not APx. By o1, x3C

P(AXMxN). By

REF2, x3G (AXMxN) for some G-CP, G3CL . By o

1, not AP

Gx, with

PG.P, P

G3PL , thus verifying EXT2.

EXT2NREF2: is shown analogously. h

Proof of Theorem 4. ACY=EXT1 is obvious, since any P3PL satisfiesACY.

To show the converse implication, note that if P satisfies MON and ACY,CP

satisfies a &u. Deb’s [8] result (Theorem 2.10) implies a&uNREF1. Itfollows that C

Psatisfies REF1. Hence, by Fact 7(i), P satisfies EXT1. h


Proof of Theorem 6. Necessity is immediate from the definition of multi-preference rationalizability. For sufficiency, letQ be any set of linear extensionsof P such that YQ"P; such a set exists by Theorem 5. Fix e: 0(e(1/dQ, anddefine, for Q3Q and x3X :º

Q(x)"edMy3X DyQxN. Then U defined as

the convex hull of MºQDQ3QN does the desired job, that is, it satisfies:

APx8maxy3A

º (y)'º (x), ∀º3U, for all A3F (X) and x3X.The validity of the ‘‘=’’ implication is straightforward from the definition

of U.To show the converse, assume by way of contradiction that, for some

A3F (X), x3X, and j3*Q : APx and

+Q3Q

jQº

Q(x)5 +

Q3Q

jQº

Q(y) ∀y3A . (2)

Define j*3*Q by

j*Q"

jQº

Q(x)

+Q{3Q

jQ{

ºQ{

(x).

By (2),

15 +Q3Q

j*Q

ºQ

(y)

ºQ(x)

∀y3A . (3)

Now for every Q3Q , there exists y3A such that

ºQ(y)

ºQ(x)

5

1

e'dQ . (4)

Since ºQ{

(z) is non-negative for all Q@ and z, (3) and (4) imply thatj*Q(1/dQ , for all Q3Q, the desired contradiction since the j*

Qadd up to

one. h

Proof of Lemma 2. We show (ii); (i) is analogous.1. gN g*: Iterated application of g yields

[AWCG((XS)XA)O0&B.A]NBWC

G((XS)XB)O0 .

Given IUC, this amounts to g*.2. g*N g : g may be written as

A3G (SXMAN)N MA, BNWG (SXMA, BN)O0 .

By IUC, this is equivalent to

A3G (SXMAN)NMAXBN3G (SXMAXBN), an instance of g*.

3. gN g is immediate. h

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