Noname manuscript No.(will be inserted by the editor)

A Multiattribute Decision Time Theory

Nobuo Koida

This version: July 9, 2016

Abstract In this study, we analyze choice in the presence of some conflictthat affects the decision time (response time), a subject that has been docu-mented in the literature. We axiomatize a multiattribute decision time (MDT)representation, which is a dynamic extension of the classic multiattribute ex-pected utility theory that allows potentially incomplete preferences. Underthis framework, one alternative is preferred to another in a certain period ifand only if the weighted sum of the attribute-dependent expected utility in-duced by the former alternative is larger than that induced by the latter forall attribute weights in a closed and convex set. MDT uniquely determines thedecision time as the earliest period at which the ranking between alternativesbecomes decisive. The comparative statics result indicates that the decisiontime provides useful information to locate indifference curves in a specific set-ting. MDT also explains various empirical findings in economics and otherrelevant fields.

Keywords Multiattribute utility, Incomplete preference, Decision time,Choice under conflict

JEL Classification Numbers: D03, D81

I would like to thank Atsushi Kajii, Takashi Ui, Norio Takeoka, Kazuya Hyogo, YouichiroHigashi, Hitoshi Matsushima, Koki Oikawa, and numerous seminar and conference partici-pants for helpful discussions and comments. I also thank two anonymous referees for valuablecomments that significantly improve the paper. Financial support from the Japan Societyfor the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (Grant Numbers24730168, 26380241) is gratefully appreciated.

N. KoidaFaculty of Policy Studies, Iwate Prefectural University, 152-52 Sugo, Takizawa, Iwate 020-0693, JapanTel.: +81-19-694-2814, Fax: +81-19-694-2701E-mail: nobuo@iwate-pu.ac.jp

1 Introduction

A decision maker (DM) often experiences a tradeoff when choosing betweenalternatives. One example is the choice between the attribute values of goods,such as cars. For instance, car x is a luxury model but expensive, whereas cary provides average performance but is reasonably priced. Another example isthe tradeoff between current and future profits: investment project x yields asubstantial current profit but a trivial future profit, whereas investment projecty yields an enormous future profit but a negligible current profit. As a thirdexample, a tradeoff may exist between self-interest and emotion (Loewensteinand O’Donoghue 2004): option x is more self-serving, whereas option y is moreemotionally appealing. Finally, this type of tradeoff is also likely to occur incollective decision making, wherein DMs are individuals who have differentobjectives.

If alternatives x and y in these examples appear almost equally attractiveto the DM but some difference is still discernible, the tradeoff is intensifiedand referred to as a conflict. Because the DM finds it difficult to resolve such aconflict, he or she may carefully examine each alternative before reaching a de-cision, which presumably involves a longer decision time.1 Such a relationshipbetween conflict and decision time is well known in psychology (Berlyne 1960;Dhar 1997; Diederich 2003; Festinger 1964; Tversky and Shafir 1992; Tyebjee1979; Weber et al. 2000), and has been confirmed by recent empirical studieson economic topics, such as time preference (Chabris et al. 2009), the ultima-tum game (Knoch et al. 2006), search (Gabaix and Laibson 2005; Gabaix etal. 2006), and lottery choice (Wilcox 1993).2

Accordingly, the objectives of this paper are threefold. First, to formalizeconflict, decision time, and their relationships within a binary choice frame-work, we axiomatize a multiattribute decision time (MDT) representation,which is a dynamic extension of the classic multiattribute expected utilitytheory (e.g., Debreu 1959; Fishburn 1965; Pollak 1967) that allows for poten-tially incomplete preferences. Consider the preferences %τ for periods τ = 0,1, · · · , and regard the ranking between alternatives (i.e., multiattribute lotter-ies) at τ as decisive if one is preferred to the other, and as indecisive if neitheris preferred. Our main theorem, Theorem 1, indicates that MDT renders theranking between alternatives decisive, that is, alternative p is preferred to q atperiod τ , if and only if the weighted sum of the attribute-dependent expectedutility induced by p is greater than that induced by q for all attribute weightsin some closed and convex set Λτ . Conversely, the ranking between p and q isindecisive if there is a disagreement in the evaluation, that is, the weighted suminduced by p is larger for some attribute weights, whereas that induced by qis larger for other weights. This latter scenario captures the conflict describedabove.

1 The decision time is also referred to as the response time, reaction time, and contem-plation time.

2 Spiliopoulos and Ortmann (2015) provided a comprehensive survey on the economicanalyses of decision time in other contexts.

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An advantage of the MDT approach over the aforementioned existing stud-ies is that our axiomatization can characterize conflict not only descriptivelybut also normatively. Our first key axiom, single-attribute regularity, requiresthat a ranking between two alternatives that differ with respect to only oneattribute is decisive, and independent of values in other attributes because noconflict exists between such alternatives. This implies that conflict only occursin our model if the alternatives’ values differ in two or more attributes. Second,our consistency axiom requires that the ranking between alternatives is neverindecisive once it has been decisive in an earlier period. Corollary 1 indicatesthat this axiom uniquely determines the decision time τ∗(p, q) between alter-natives p and q as the earliest period at which the ranking between p and qbecame decisive.

Second, we conduct comparative statics with respect to the decision time,that is, a strictly positive decision time in MDT not only indicates whetherconflict exists between alternatives, but also derives information on how diffi-cult it is to reach a decision, which also sets it apart from atemporal incompletepreference approaches, such as those discussed by Bewley (1986) and Dubraet al. (2004). First, we conduct an interpersonal comparison and find that theset Λτ of attribute weights serves as an index of the DM’s susceptibility toconflict, that is, the preference becomes less decisive (i.e., the decision timebetween any given alternative pair becomes longer, while preserving the rank-ing over the alternative pair) if and only if Λτ expands for all τ (Theorem2). Second, we provide two conflict intensity indices relevant to the decisiontime. One may assume that a longer decision time between two alternatives isassociated with the alternatives being closer to indifference. Thus, we defineour first conflict intensity index by the angle formed between the long-run in-difference curve (i.e., an indifference curve induced by an attribute weight λ∞

included in all Λτ ) and the difference in the expected utility vectors inducedby the alternatives. A similar reasoning is used to derive our second conflictintensity index, the difference of weighted expected utility induced by alterna-tives, given the long-run attribute weight λ∞. Theorems 3 and 4 indicate thatthese indices are indeed the predictors of decision time, and the decision timeis also the predictor of these indices. This result agrees with those in the em-pirical literature and shows that the decision time provides useful informationfor specifying indifference curves for potentially incomplete preferences.

Finally, we demonstrate that MDT can explain many empirical findingsrelevant to the decision time in the literature, such as choice under conflict(Tversky and Shafir 1992), time preference (Chabris et al. 2009), and theultimatum game (Knoch et al. 2006). Our model typically predicts a zerodecision time when one alternative dominates the other. This is derived fromour monotonicity axiom, and carries intuitive and appealing implications.

The remainder of this paper is organized as follows. In Section 2, we presentour basic framework, state the main representation theorem, and formalize thedecision time. In Section 3, we provide an interpersonal comparison result anddefine two conflict intensity indices relevant to the decision time. In Section 4,we illustrate some applications of MDT that accommodate empirical results.

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In Section 5, we review the related literature, and provide concluding remarksin Section 6.

2 A multiattribute decision time model

For a finite number n and I = {1, · · · , n}, let Xi be the set of values inattribute i ∈ I, each of which is compact and metric. We assume that each Xi

has at least two elements. Let X be the Cartesian product of Xi for all i ∈I, that is, X ≡ ×n

i=1Xi, which is similar to the counterpart of the standardmultiattribute expected utility theory (e.g., Fishburn 1965). We specify thedomain of choice as P(X), which is the set of Borel probability measures(or lotteries) over X that are endowed with the weak convergence topology.3

Thus, we also refer to P(X) as the set of alternatives. Generic elements inX are denoted, for example, by x, y, z, and those in P(X), for example, byp, q, r. For all p ∈ P(X), we denote p’s marginal probability distribution ofXi as p|i, and that of X−i ≡ ×j 6=iXj as p|−i. We naturally identify x ∈ Xwith a degenerated lottery δx ∈ P(X) that induces x with probability one.A mixture λp + (1 − λ)q of alternatives p and q ∈ P(X) with λ ∈ [0, 1] isdefined by the mixture of two probability distributions over X induced by pand q. For all periods τ = 0, 1, 2, · · · , we assume there are preferences %τ onP(X). We define the strict preferences �τ and indifferences ∼τ in the usualmanner. For all p, q ∈ P(X), we say that the preference between p and q isindecisive at period τ if neither p %τ q nor q %τ p (which we denote by p./τ q), whereas we say that the preference is decisive at τ if either p %τ qor q %τ p (or both). We denote the (n − 1)-dimensional simplex by ∆n−1 ≡{λ = (λ1, · · · , λn) ∈ <n :

∑ni=1 λi = 1, λi ≥ 0 for all i ∈ I}.

In the following analysis, we impose six axioms on %τ for all periods τ . Forsimplicity, the quantifier “for all τ” is omitted from the statement of the firstfive axioms. Our first axiom is standard, except that it permits incompletepreferences.

Axiom 1 (Partial order) %τ are reflexive and transitive.

The continuity axiom is also standard.

Axiom 2 (Continuity) {q : q ∈ P(X), p %τ q} and {q : q ∈ P(X), q %τ p}are closed for all p ∈ P(X).

The third is the independence axiom, whose implication is straightforward.

3 Because we consider the set of all Borel probability measures over X, our framework mayappear to permit a correlation among attribute values. However, the following analysis ofthis study discards this possibility (i.e., it identifies two alternatives that have the identicalmarginal probability distributions of all Xi) because we impose the axioms called indepen-dence and single-attribute regularity. Alternatively, we may consider P(X1)× · · · × P(Xn),that is, the product set of Borel probability measures over each attribute set Xi, as the do-main of choice, which would simplify some of our analyses. However, for reasons of generality,we assume P(X) as our domain of choice.

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Axiom 3 (Independence) For all p, q, r ∈ P(X) and µ ∈ (0, 1), p %τ q ifand only if µp+ (1− µ)r %τ µq + (1− µ)r.

The next axiom ensures that the preferences have a more concrete structurefor a specific class of alternative pairs.

Axiom 4 (Single-attribute regularity)For all i ∈ I and p, q, p′, and q′ ∈ P(X) such that p|i = p′|i, q|i = q′|i, p|−i

= q|−i, and p′|−i = q′|−i, the following two statements hold:

(a) The preference between p and q is decisive.(b) p %τ q if and only if p′ %τ q′.

Statement (a) implies that, for a pair of alternatives p and q that onlydiffer with respect to the marginal probability distributions of Xi, (i.e., sharethe same marginal probability distributions of X−i), either p %τ q or q %τ p(or both).4 Statement (b) implies that the ranking between such alternativesis independent of their marginal probability distributions of X−i.

The intuition behind this axiom is as follows. First, as mentioned in theintroduction, our model assumes that conflict between alternatives p and qoccurs if p is better than q in some attribute i and q is better than p inanother attribute j. Accordingly, it is not too difficult for the DM to reacha decision if the alternatives only differ with respect to a single attribute i,for which the ranking is decisive, as in statement (a). Second, for such a pairof alternatives, it is also natural to assume that the ranking between themdepends only on their marginal probability distributions in attribute i, whichjustifies statement (b). Note that Axiom 4(b) is similar to those imposed byDebreu (1959), Fishburn (1965), and Pollak (1967), which derive the additiveseparability of each attribute’s utility.

The following axiom states the monotonicity of preference.

Axiom 5 (Monotonicity) For all p, q ∈ P(X), let pi, qi ∈ P(X) for all i ∈I be such that pi|i = p|i, qi|i = q|i, pi|−i = qi|−i. If p

i %τ qi for all i ∈ I, p%τ q.

Although its representation may seem unfamiliar, this axiom’s implicationis similar to that of the standard monotonicity axiom. Intuitively, alternativespi and qi above are the proxies of alternatives p and q for attribute i in thesense that the former alternative pair shares identical marginal probabilitydistributions of Xi with the latter, and pi and qi share a common marginalprobability distribution of X−i. Axiom 5 implies that, if such proxies pi arepreferred to qi for all attributes i, alternative p should be preferred to q.Note that we need to define the proxy alternatives pi and qi to state thisaxiom because Axiom 4(a) implies that marginal probability distributions p|i

4 We require Axiom 4(a) because the preferences are possibly incomplete. This statementtrivially holds with complete preferences.

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and q|i are only comparable if p|−i = q|−i. Moreover, Axiom 4(b) rendersthe rankings between the proxies pi and qi independent of their marginalprobability distributions pi|−i and q

i|−i.The final axiom, which is crucial to derive a well-defined decision time in

our model, relates the preferences %τ at different periods.

Axiom 6 (Consistency) For all τ and τ ′ such that τ < τ ′ and all p, q ∈P(X), if p %τ q, then p %τ ′

q.

Note that the ranking between p and q is not necessarily decisive, as indi-cated by Axiom 1. Accordingly, Axiom 6 implies that, once the DM reachesa decision between alternatives, not only does the same preference hold overtime, but the ranking between them will also never be indecisive again. Asimilar axiom was employed by Kopylov (2009) and Gilboa et al. (2010) in thecontext of ambiguity.

We now consider the following preference representation. We denote theinner product of vectors s = (s1, · · · , sn) ∈ <n and t = (t1, · · · , tn) ∈ <n ass · t ≡

∑ni=1 siti.

Definition 1 We say that preferences {%τ}∞τ=0 admit anMDT representation(u,Λ) if the following conditions hold:

(a) u ≡ (u1, · · · , un), where ui : Xi → [0, 1] is a continuous function such thatmaxxi∈Xi ui(xi) = 1 and minxi∈Xi ui(xi) = 0 for all i ∈ I;(b) Λ ≡ {Λτ}∞τ=0, where Λ

τ ⊆ ∆n−1 is closed and convex, and such that Λτ

⊇ Λτ ′for all τ < τ ′;

(c) for all p, q ∈ P(X), p %τ q whenever

λτ · Ep[u(x)] ≥ λτ · Eq[u(x)] for all λτ ∈ Λτ ,

where E(·)[u(x)] = (E(·)|1 [u1(x1)], · · · , E(·)|n [un(xn)]) ∈ [0, 1]n denotes a vec-tor that comprises the expected utility E(·)|i [ui(xi)] in each attribute i givenalternative (·)’s marginal probability distributions of Xi for i = 1, · · · , n.

This representation ensures that the ranking between two alternatives isdecisive at period τ if and only if the weighted sum of expected utility inducedby one alternative is higher than that induced by the other alternative for allattribute weights λτ in Λτ . By contrast, the ranking is indecisive if there is aconflict between alternatives, that is, one alternative is ranked higher than theother for some attribute weights in Λτ and ranked lower for other attributeweights. Moreover, the set Λτ of attribute weights shrinks (in the sense ofset inclusion) over time, which can be interpreted as a result of external orinternal information acquisition, or information processing. We discuss theseinterpretations in Section 5. Finally, we normalize ui for all i and Λ

τ for all τto obtain uniqueness results.

The following theorem indicates that our axioms derive a unique MDTrepresentation.

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Theorem 1 Preferences {%τ}∞τ=0 satisfy Axioms 1–6 if and only if {%τ}∞τ=0

admit a unique MDT representation (u,Λ).

The proof is provided in the appendix; however, we provide a sketch here.First, we fix τ . Then, partial order, continuity, independence, and single-attribute regularity can be used to derive a continuous von Neumann–Morgensternutility function ui for each attribute i. Alternatives that only differ with respectto the marginal probability distributions of Xi are then evaluated in terms oftheir expected utility E(·)|i [ui(xi)], which is independent of the marginal prob-ability distribution of X−i.

Next, for all attributes i, we denote the set of expected utility generated byui and P(Xi) (i.e., the set of marginal probability distributions in attribute i)as Ji, and let J = ×n

i=1Ji ⊆ <n. For all p ∈ P(X), we denote the correspondingexpected utility vector as Ep[u(x)] ≡ (Ep|1 [u1(x1)], · · · , Ep|n [un(xn)]) ∈ J .From the monotonicity axiom, it follows that we can define a binary relation%∗ on J such that Ep[u(x)] %∗ Eq[u(x)] if and only if p %τ q, and define K ={µ(s− t) : µ ≥ 0, s, t ∈ J, s %∗ t} ⊆ <n, which is proved to be a closed convexcone in <n.

Finally, we derive a closed and convex set Λτ ≡ {λ ∈ ∆n−1 : λ · (s− t) ≥0 for all s, t ∈ J such that s− t ∈ K} from K in a similar manner to Bewley(1986). The uniqueness results are obtained by construction.

As the proof indicates, our representation may be reminiscent of Bewley’s(1986) Knightian uncertainty model. However, our model differs from his forthe following three reasons. First, Bewley attributed incomplete preferences touncertainty, whereas we primarily focus on the indecisiveness caused by con-flict rather than uncertainty. Second, one of his axioms required decisivenessbetween alternatives involving no uncertainty, that is, alternatives generatingconstant utility across states. By contrast, our single-attribute regularity as-sumes decisiveness between alternatives generating utility that differs in onlya single attribute. Finally, the aforementioned difference in axioms derives astate-independent utility function and a constant risk attitude across states inBewley’s model, whereas our model obtains attribute-dependent utility func-tions ui and allows for different risk attitudes across attributes.

To provide a graphical interpretation, it is useful to define the expectedutility vector difference induced by alternative pair (p, q) ∈ P(X)×P(X) andu by Ep−q[u(x)] ≡ Ep[u(x)]−Eq[u(x)] ∈ <n, which implies that λ ·Ep−q[u(x)]≥ 0 if and only if λ · Ep[u(x)] ≥ λ · Eq[u(x)]. We also define the hyperplanewith normal vector λ ∈ ∆n−1 and a ∈ < by h = {s ∈ <n : λ · s = a}, whichcorresponds to an indifference curve for the given attribute weight λ.

Figure 1 illustrates how MDT determines the ranking between alterna-tives. In the n-dimensional Euclidean space <n induced by the expected util-ity vectors, fix τ and let Λτ ⊆ ∆n−1 be a closed and convex set of attributeweights, wherein λ and λ are its extreme points. For some p, p′, p′′, q ∈ P(X),we denote Ep[u(x)], Ep′ [u(x)], Ep′′ [u(x)], and Eq[u(x)] as p, p′, p′′, and qin the figure for simplicity. Hyperplanes with normal vectors λ and λ thatpass through Eq[u(x)] are denoted by h and h, respectively. The MDT rep-

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p’’

hh

hλ

λ

p

p’τΛ

q

Fig. 1 Indecisive region

resentation implies that, if the expected utility vector difference Ep−q[u(x)]= Ep[u(x)] − Eq[u(x)] ∈ <n lies in the shaded region (which we refer to asthe indecisive region), then conflict occurs and the ranking between p and qis indecisive: Ep−q[u(x)] is in the positive half-space defined by hyperplane h,which implies that λ · Ep[u(x)] > λ · Eq[u(x)], whereas it is in the negativehalf-space defined by h, which implies that λ · Ep[u(x)] < λ · Eq[u(x)]. Thus,p ./τ q.

Next, consider the vector difference Ep′−q[u(x)] = Ep′ [u(x)] − Eq[u(x)]induced by p′ and q. For this case, the figure indicates that p′ %τ q becauseEp′−q[u(x)] is in the positive half-space defined by both h and h, as wellas all other hyperplanes with normal vectors in Λτ . Similarly, for the vectordifference Ep′′−q[u(x)] = Ep′′ [u(x)] − Eq[u(x)] induced by p′′ and q, q %τ p′′

because Ep′′−q[u(x)] lies in the intersection of the negative half-spaces definedby all the hyperplanes with normal vectors in Λτ .

It is natural to define the decision time in our framework as the earliestperiod at which the ranking between alternatives changes from indecisive to

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decisive. In the following definition, we also require that the ranking does notchange once it has become decisive in some period.

Definition 2 For all p, q ∈ P(X), the decision time τ∗(p, q) between p and qis a non-negative integer such that

(a) p ./τ q for all τ < τ∗(p, q), and(b) p %τ q for all τ ≥ τ∗(p, q), or q %τ p for all τ ≥ τ∗(p, q).

The consistency axiom implies that the decision time τ∗(p, q) is well-defined.5 We set τ∗(p, q) = 0 if p %0 q or q %0 p holds, and τ∗(p, q) = ∞if p ./τ q for all τ . Accordingly, we can consider atemporally complete (ex-pected utility) preferences as a special case of our model in which the decisiontime equals zero for all alternative pairs, whereas atemporally incomplete pref-erences can be interpreted as a special case in which the decision time is eitherzero or infinity for all alternative pairs.

The previous discussion implies that, as the set Λτ of attribute weightsshrinks over time, so does the indecisive region. Accordingly, the vector dif-ference Ep−q[u(x)] in Fig. 1 may be outside this region at some period τ , andnever be inside again. The following corollary summarizes this result.

Corollary 1 Suppose that preferences {%τ}∞τ=0 admit an MDT representation(u,Λ). Then, for all p, q ∈ P(X), τ∗(p, q) is the smallest non-negative integerτ for which one of the following conditions holds:

(a) λτ · Ep[u(x)] ≥ λτ · Eq[u(x)] for all λτ ∈ Λτ , or(b) λτ · Ep[u(x)] ≤ λτ · Eq[u(x)] for all λτ ∈ Λτ .

To summarize, the decision time can be defined as the earliest period τat which conflict is resolved, that is, one of two alternatives is unanimouslysupported by all of the attribute weights in Λτ .

To conclude this section, we provide some possible examples of the sequenceΛ.

Example 1 (ε-indecisiveness) For some λ ∈ ∆n−1, we define Λτ = {(1−ετ )λ+ετλ : λ ∈ ∆n−1}, where ετ ≥ ετ

′for all τ ≤ τ ′.

Intuitively, each Λτ consists of convex combinations of the “true” attributeweight λ and all weights in the (n−1)-dimensional simplex, which approaches{λ} in the long run. This describes a situation in which the DM initiallyperceives that any attribute weight is possible to degree ε0, but he or she gainsa more precise idea over time. This may be reminiscent of the ε-contaminationmodel, which is a special case of the maxmin expected utility (Gilboa andSchmeidler 1989), but we should note that, unlike ε-contamination, ετ herecaptures the degree of indecisiveness due to conflict rather than ambiguity.

5 Under the consistency axiom, Definition 2(b) is equivalent to the statement that“τ∗(p, q) equals the smallest non-negative integer τ such that p %τ q or q %τ p.”

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Example 2 (Hierarchical aggregation) For all τ and j = 1, · · · , mτ , let Iτ ={Iτj }m

τ

j=1 be a partition of I (i.e., ∪mτ

j=1Iτj = I and Iτj ∩ Iτk = φ for all j 6= k).

We define

Λτ = {(λ1, · · · , λn) ∈ ∆n−1 :λi∑

k∈Iτjλk

= constant for all i, j such that i ∈ Iτj }

for all τ . Moreover, Iτ ′is coarser than Iτ (i.e., J ′ ∈ Iτ ′

implies that there issome J ∈ Iτ such that J ⊆ J ′) for all τ ≤ τ ′.

The formulation implies that, whereas the attribute weights within a groupIτj of attributes are uniquely determined, the weights across groups Iτj and Iτkwith j 6= k may not be. This is the case, for example, when some attributesare similar and easily comparable, whereas others are dissimilar and incompa-rable, which is a scenario that drives the DM to bundle similar attributes intogroups.6

Note that the sequence {Iτ}∞τ=0 of partitions coarsening over time is com-patible with the consistency axiom. This can be interpreted as the DM usinghis or her better understanding of the nature of the attributes to graduallyintegrate groups of similar attributes into a smaller number of groups withinwhich the attribute weights are uniquely determined.

3 Decision time and conflict

Thus far, we have qualitatively characterized the behavioral implications ofconflict between two alternatives. In this section, however, we quantify conflictfor the following two aspects: first, in Section 3.1, we conduct an interpersonalcomparison of the decision time with respect to the sets Λτ of attribute weights.Second, in Sections 3.2 and 3.3, we define two conflict intensity indices for givenalternative pairs, both of which are closely related to the decision time.

3.1 Interpersonal comparison of decision time

Recall that, in MDT, the ranking between alternatives p and q is indecisive atperiod τ if some attribute weights in Λτ generate conflicting rankings betweenthe alternatives, that is, there exist λ, λ′ ∈ Λτ such that λ · Ep[u(x)] > λ ·Eq[u(x)] and λ

′ · Ep[u(x)] < λ′ · Eq[u(x)]. Accordingly, one may assume thatan expansion of the attribute weight sets Λτ will prolong the decision timebecause it creates greater conflict between alternatives. To formalize this idea,we first define the comparative decisiveness of a preference as follows.

Definition 3 For preferences {%τ1}∞τ=0 and {%τ

2}∞τ=0, we say that {%τ1}∞τ=0

are more decisive than {%τ2}∞τ=0 if p %τ

2 q implies p %τ1 q for all p, q ∈ P(X)

and τ ∈ N ∪ {0}.6 Chapter 2 of Keeney and Raiffa (1993) discusses multiattribute utility models in a

similar form.

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Note that preferences {%τ1}∞τ=0 being more decisive than preferences {%τ

2

}∞τ=0 imply not only that the decision time over any alternative pair is shorterfor the former preferences than for the latter, but also that they share a com-mon baseline preference (i.e., the preferences rank alternatives in the samedirection if they are both decisive).

Now, the following theorem indicates that the aforementioned assumptionis correct.

Theorem 2 Assume that preferences {%τ1}∞τ=0 and {%τ

2}∞τ=0 admit MDT rep-resentations (u, {Λτ

1}∞τ=0) and (u, {Λτ2}∞τ=0). Then, the following statements

are equivalent:

(a) {%τ1}∞τ=0 are more decisive than {%τ

2}∞τ=0.(b) Λτ

1 ⊆ Λτ2 for all τ ∈ N ∪ {0}.

The proof is provided in the appendix. This theorem implies that, if thedecision time is solely determined by the conflict between the alternatives (i.e.,the utility functions are identical among DMs and no other factors or noiseare involved), an interpersonal difference in decision time can be attributed tothe difference in size of the sets Λτ for all τ , under a common baseline prefer-ence over alternatives. Conversely, the larger Λτ becomes for all τ , the morelikely that attribute weights in Λτ will generate opposite rankings betweenalternatives, in which case our model predicts a longer decision time.

Note that the set Λ = {Λτ}∞τ=0 can also be used to explain a variance in thedecision time under different baseline preferences over alternatives. Experimen-tal studies indicate that the decision time may vary significantly depending onthe type of DM; that is, subjects who make a particular choice and those whomake another choice may exhibit distinctively different decision times (Knochet al. 2006; Rubinstein 2007, 2013). In Section 4, we demonstrate that theseresults can be explained by assuming a different Λ for each type of DM.

3.2 Direction of conflict

In this and the next subsections, we define conflict intensity indices that arerelevant to the decision time for given alternative pairs while fixing the set Λτ

of attribute weights.

To define our first conflict intensity index, we fix λ∞ ∈ ∩∞τ=0Λ

τ for a givenMDT representation (u,Λ) and consider a hyperplane h∞ with normal vectorλ∞.7 Because λ∞ is included in all Λτ , we refer to λ∞ as the long-run attributeweight and h∞ as the long-run indifference curve. It follows from constructionthat, if λ∞ ·Ep−q[u(x)] = 0 for some p, q ∈ P(X) (that is, the expected utilityvector difference Ep−q[u(x)] = Ep[u(x)]−Eq[u(x)] induced by p and q parallelsthe long-run indifference curve h∞), the ranking between alternatives p andq remains indecisive until the rankings between all alternative pairs become

7 Consistency implies that ∩∞τ=0Λ

τ is nonempty.

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decisive.8 Accordingly, if the expected utility vector difference Ep−q[u(x)] forsome p, q ∈ P(X) is close to the long-run indifference curve h∞ (or, moreprecisely, to the projection of Ep−q[u(x)] onto h

∞, as we will formalize below),conflict tends to be persistent and difficult to resolve. Conversely, if Ep−q[u(x)]deviates significantly from (its projection onto) the long-run indifference curveh∞, the ranking between p and q promptly becomes decisive, and the conflictcan be easily resolved. Hence, the relationship between the expected utilityvector difference induced by alternative pairs and the long-run indifferencecurve—more specifically, the angle formed between them—can be consideredas an index of conflict intensity.

To this end, we focus on the n-dimensional Euclidean space <n induced bythe expected utility vectors, which we considered in the previous section. Forall s ∈ <n, ‖s‖ denotes the Euclidean norm of vector s. For all vectors s ∈ <n

and hyperplanes h with normal vectors λ ∈ ∆n−1, the projection t ∈ <n of sonto h is such that λ · t = 0 and s − t = ψλ for some ψ ∈ < \ {0}; that is,vector t is parallel to hyperplane h, and the vector difference between s andt is orthogonal to h. For all s, t ∈ <n, let the function θ : <n × <n → [0, π]denote the angle formed between s and t, that is, θ is such that cos(θ(s, t)) =

s·t‖s‖‖t‖ .

We focus on the following classes of alternative pairs.

Definition 4 Let (u,Λ) be an MDT representation of preferences {%τ}∞τ=0

and h be a hyperplane with normal vector λ ∈ Λτ for some τ . Then, alternativepairs (p, q), (p′, q′) ∈ P(X)× P(X) are referred to as follows:

(a) congruent with respect to h if there exist some s ∈ <n and ψ ∈ {−1, 1}such that λ · s = 0, and Ep−q[u(x)] and Ep′−q′ [u(x)] are conical combinationsof s and ψλ; that is, Ep−q[u(x)] = µ1s+µ2ψλ and Ep′−q′ [u(x)] = µ′

1s+µ′2ψλ

for some µ1, µ2, µ′1, µ

′2 ≥ 0;

(b) strongly congruent with respect to h if they are congruent with respect toh, and for µ1, µ2, µ

′1, µ

′2 defined above, (µ2 − µ′

2)(µ2/µ1 − µ′2/µ

′1) > 0.

Statement (a) requires that congruent alternative pairs induce expectedutility vector differences Ep−q[u(x)] and Ep′−q′ [u(x)] that lie in the convexcone spanned between two vectors: vector s, which is parallel to the hyper-plane (i.e., indifference curve) h, and vector ψλ, which is the normal vectorto h or its opposite. For strongly congruent alternative pairs, statement (b)additionally requires that the coefficients µ2 and µ′

2 of vector ψλ be sorted bythe corresponding slopes µ2/µ1 and µ′

2/µ′1 of the expected utility vector dif-

ferences. By construction, µ1, µ2, µ′1, µ

′2 in these statements can be uniquely

determined for a given s ∈ <n and ψ ∈ {−1, 1}.Note that if n = 2, vectors Ep−q[u(x)], Ep′−q′ [u(x)], s, and λ specified

above are trivially on the identical hyperplane for all alternative pairs (p, q)and (p′, q′). Accordingly, the alternative pairs are congruent in this case, pro-vided they are “not too different” (formally, p′ = (1 − µ)p + µp for some p

8 Such alternatives p and q are indifferent once the ranking between them has becomedecisive, which explains the terminology of the long-run indifference curve.

12

∈ P(X) and sufficiently small µ ∈ (0, 1), and q′ = q).9 This property ren-ders congruent alternative pairs easily applicable to empirical settings becausemany experimental studies on the decision time focuses on the two attributecase (see Section 4).

The following theorem indicates that, in our model, the conflict intensityindex defined by the angle between the long-run indifference curve and theexpected utility vector difference induced by congruent alternatives serves asa predictor of the decision time, and vice versa.

Theorem 3 Assume that preferences {%τ}∞τ=0 admit an MDT representation(u,Λ). For all hyperplanes h∞ with the normal vector λ∞ ∈ ∩∞

τ=0Λτ , congruent

alternative pairs (p, q), (p′, q′) ∈ P(X) × P(X) with respect to h∞, and theprojection t of Ep−q[u(x)] onto h

∞,

(a) τ∗(p, q) < τ∗(p′, q′) implies θ(Ep−q[u(x)], t) > θ(Ep′−q′ [u(x)], t).(b) θ(Ep−q[u(x)], t) ≥ θ(Ep′−q′ [u(x)], t) implies τ∗(p, q) ≤ τ∗(p′, q′).

The proof is provided in the appendix. Theorem 3 implies that the deci-sion time between alternatives provides significant information for specifyinglong-run indifference curves h∞ (or equivalently, the long-run attribute weightλ∞); that is, the longer the decision time between alternatives, the closer theexpected utility vector difference induced by the alternatives will be to (its pro-jection onto) the long-run indifference curve h∞ (statement (a)). Conversely,if the expected utility vector difference forms a smaller angle with the long-runindifference curve h∞, the decision time between alternatives becomes longer(statement (b)). Note that statement (b) implies that the decision time is pos-sibly insensitive to the change in expected utility vector difference induced byalternatives (i.e., we may have θ(Ep−q[u(x)], t) > θ(Ep′−q′ [u(x)], t) and τ

∗(p, q)= τ∗(p′, q′)) because the set Λτ of attribute weights shrinks so rapidly thatthe difference in choice difficulty between (p, q) and (p′, q′) produces no differ-ence in decision time. Moreover, the congruence of alternative pairs (p, q) and(p′, q′) with respect to h∞ ensures that the projections t and t′ have the samedirection, and thus both statements (a) and (b) refer to vector t in place of t′.Finally, if we relax the congruence condition of the alternative pairs, that is,the expected utility vectors induced by the alternative pairs are not includedin a single convex cone, we do not generally obtain a comparative statics resultbecause the set Λτ may not contract uniformly.10

Figure 2 illustrates Theorem 3 for the case n = 2. For an MDT represen-tation (u,Λ), take the horizontal axis u1(x1) and vertical axis u2(x2). For p,p′, q, q′ ∈ P(X) in the statement of Theorem 3, let p and p′ be indifferent

9 A similar argument can hold for the case n > 2, under the assumption that the consideredalternatives have the identical marginal probability distributions in all but two attributes.10 If Λτ comprises circular cones, that is, Λτ = {λ ∈ ∆n−1 : dτ ≤ λ·λ∞

‖λ‖‖λ∞‖ ≤ 1} for all τ

and some dτ ∈ [0, 1] such that dτ ≤ dτ′for all τ ≤ τ ′, the contraction of set Λτ is uniform in

the sense that each Λτ consists of all the vectors in ∆n−1 that form an angle with λ∞ thatis less than a certain value. By assuming such a case, statements (a) and (b) in Theorem 3hold for non-congruent alternative pairs.

13

00

λh∞

∞

00

λh∞

∞

qp

p’ α’

α

2 2

u (x )1 1

u (x )

t

Fig. 2 Direction of conflict

in the long run, that is, the vector difference Ep[u(x)] − Ep′ [u(x)] inducedby p and p′ is parallel to the long-run indifference curve h∞, and assume q= q′ without loss of generality. For simplicity, we denote Ep[u(x)], Eq[u(x)],and Ep′ [u(x)] as p, q, and p′ in the figure. Now, let t ∈ <n be the projectionof Ep−q[u(x)] onto h

∞. Figure 2 indicates that θ(Ep−q[u(x)], t) = α > α′ =θ(Ep′−q[u(x)], t), and so Theorem 3 implies that τ∗(p, q) ≤ τ∗(p′, q). Note thatalternative pairs (p, q) and (p′, q) are congruent if 0 ≤ α′ < α ≤ π/2 becauseEp−q[u(x)] and Ep′−q[u(x)] are included in the convex hull spanned by t and−λ∞. This result indicates that, even if alternatives p and p′ are indifferent inthe long run, the decision times for alternative pairs (p, q) and (p′, q) may varysignificantly depending on the vector differences Ep−q[u(x)] and Ep′−q[u(x)].As we will see in Section 4, this discussion is a generalization of the choicedeferral under conflict considered by Tversky and Shafir (1992).

3.3 Proximity of attractiveness

In this subsection, we present our second conflict intensity index, the proximityof attractiveness. Intuitively, the DM may find it difficult to choose betweenconflicting alternatives p and q if they possess similar attractiveness, evenafter careful consideration. This implies that a smaller difference in the long-run weighted sum of expected utility induced by alternatives will result in a

14

longer decision time between the alternatives. The following theorem confirmsthis implication.

Theorem 4 Assume that preferences {%τ}∞τ=0 admit an MDT representation(u,Λ). For all hyperplanes h∞ with the normal vector λ∞ ∈ ∩∞

τ=0Λτ and

strongly congruent alternative pairs (p, q), (p′, q′) ∈ P(X)×P(X) with respectto h∞,

(a) τ∗(p, q) < τ∗(p′, q′) implies |λ∞ · Ep−q[u(x)]| > |λ∞ · Ep′−q′ [u(x)]|.(b) |λ∞ · Ep−q[u(x)]| ≥ |λ∞ · Ep′−q′ [u(x)]| implies τ∗(p, q) ≤ τ∗(p′, q′).

The proof is provided in the appendix. Intuitively, |λ∞ · Ep−q[u(x)]| and|λ∞ · Ep′−q′ [u(x)]| denote the differences in the long-run weighted sum of ex-pected utility given alternative pairs (p, q) and (p′, q′), respectively. Thus, The-orem 4 implies that the decision time is negatively correlated with the differ-ence in the long-run weighted sum of expected utility induced by stronglycongruent alternative pairs. This formalizes the argument in economics andother fields (typically psychology) that the decision time is a decreasing func-tion of the difference between expected values/utility induced by alternatives(Berlyne 1960; Chabris et al. 2009; Dhar 1997; Festinger 1964; Gabaix andLaibson 2005; Gabaix et al. 2006; Tyebjee 1979; Wilcox 1993). As in Theorem3, note that statement (b) implies that the decision time may be unresponsiveto a change in the weighted sums of expected utility, that is, we may have|λ∞ · Ep−q[u(x)]| > |λ∞ · Ep′−q′ [u(x)]| but τ∗(p, q) = τ∗(p′, q′).

Figure 3 illustrates Theorem 4 for n = 2. For p, p′, q ∈ P(X), let the long-run indifference curves (i.e., hyperplanes) h, h′, and h∞ with normal vectorλ∞ pass through Ep[u(x)], Ep′ [u(x)], and Eq[u(x)], respectively. Assume that|λ∞ ·Ep−q[u(x)]| > |λ∞ ·Ep′−q[u(x)]|, that is, indifference curve h′ is closer toh∞ than h, and π/2 ≥ θ(Ep−q[u(x)], t) = α > α′ = θ(Ep′−q[u(x)], t) ≥ 0 for theprojection t of Ep−q[u(x)] onto h

∞. Note that alternative pairs (p, q) and (p′, q)are strongly congruent: (p, q) and (p′, q) are congruent because Ep−q[u(x)] andEp′−q[u(x)] are in the convex hull spanned by t and −λ∞. Thus, let µ1, µ2,µ′1, µ

′2 ≥ 0 be such that Ep−q[u(x)] = µ1t + µ2(−λ∞) and Ep′−q[u(x)] =

µ′1t+ µ′

2(−λ∞). It follows from |λ∞ ·Ep−q[u(x)]| > |λ∞ ·Ep′−q[u(x)]| that µ2

> µ′2, whereas α > α′ implies that µ2/µ1 > µ′

2/µ′1, which satisfies the strong

congruence condition. Accordingly, Theorem 4 implies that τ∗(p, q) ≤ τ∗(p′, q)for such p, p′, and q.

We should note, however, that the proximity of the weighted sums of ex-pected utility alone cannot predict the decision time in our model unless al-ternative pairs are strongly congruent. For example, suppose that alternativepairs (p, q) and (p′, q) are congruent with respect to h∞, but |λ∞ ·Ep−q[u(x)]|< |λ∞ ·Ep′−q[u(x)]| and θ(Ep−q[u(x)], t) = α > α′ = θ(Ep′−q[u(x)], t) for theprojection t of Ep−q[u(x)] onto h

∞ (Fig. 4). These conditions imply that µ2 <µ′2 and µ2/µ1 > µ′

2/µ′1, respectively, for µ1, µ2, µ

′1, and µ

′2 defined in the previ-

ous paragraph, which violates the strong congruence condition. Now, Theorem3 predicts that τ∗(p, q) ≤ τ∗(p′, q) because α > α′, even if |λ∞ ·Ep−q[u(x)]| <|λ∞ ·Ep′−q[u(x)]|. Moreover, MDT generally predicts a zero decision time when

15

00

λh∞

∞

00

λh∞

∞

q

p

h’h

p’

α

2 2

u (x )1 1

u (x )

α’t

Fig. 3 Proximity of attractiveness

one alternative dominates the other (e.g., τ∗(p, q) = 0 for alternatives p andq in Fig. 4, as the latter dominates the former) because of the monotonicityaxiom; that is, moving one alternative toward the other while maintaining thedominance relation never increases the decision time between them, regardlessof the difference in the weighted sum of expected utility.

To summarize, the proximity of the weighted sum of expected utility isa predictor of decision time in MDT only if the choice between alternativesinvolves conflict (i.e., there is no dominance relation) and the prediction con-forms with that discussed in the previous subsection. In this sense, our modelcharacterizes the relationship between decision time and the proximity of at-tractiveness more precisely than existing studies, wherein conflict between al-ternatives is generally presupposed and the direction of conflict is disregarded.

4 Applications

In the previous sections, we discussed the implications of MDT preference ina general multiattribute setting. In this section, we focus on some possibleapplications of our model.

First, consider the scenario of choice under conflict discussed by Tverskyand Shafir (1992). Assume that multiattribute alternatives x = (x1, x2), y =(y1, y2), and x

′ = (x′1, x′2) ∈ < × < are such that x1 > y1, x2 < y2, x1 > x′1,

16

00

λh∞

∞

00

λh∞

∞

qp

α’

α

p’

2 2

u (x )1 1

u (x )

t

hh’

Fig. 4 Proximity of attractiveness (cont’d)

x2 > x′2. Accordingly, x dominates x′, whereas neither x nor y dominates theother. Moreover, alternatives x′ and y are assumed to be indifferent. Tverskyand Shafir reported that the DM is likely to defer his or her choice (which pre-sumably involves a strictly positive decision time) between alternatives underconflict, such as x and y, whereas the deferred choice seldom occurs betweenalternatives under dominance, such as x and x′.

To explain this result in MDT, assume that n = 2, X = [0, 1] × [0, 1],and ui(xi) = xi for i = 1 and 2. Then, Fig. 2 precisely depicts the situationunder the assumptions that p = x′, p′ = y, and q = x, each of which can beconsidered as a degenerated lottery. By assuming that alternative pairs (y, x)and (x′, x) are congruent, without loss of generality, Theorem 3 predicts thatchoice with respect to the former alternative pair will involve a longer decisiontime than over the latter, which conforms to the observations of Tversky andShafir. In particular, because we impose the monotonicity axiom, the decisiontime over (x′, x) equals zero, whereas choice over (y, x) generally involves astrictly positive decision time, which is also consistent with their findings.

Second, consider time preference. Let (x1, x2) ∈ <×< denote a project thatyields $x1 current and $x2 future payoffs, the latter of which is discounted byδ ∈ (0, 1).11 In their experimental study, Chabris et al. (2009) demonstrated

11 For simplicity, we assume two calendar dates and exponential discounting, although theconclusion of this example can be extended to models with three or more calendar dates

17

that the decision time between such projects becomes longer if the differencebetween their present values is smaller. To explain this result in MDT, wenormalize the current and future payoffs so that X = [0, 1] × [0, 1], u1(x1) =x1, and u2(x2) = x2. For all x = (x1, x2), y = (y1, y2) ∈ X, let x %τ y if andonly if λ1x1+λ2x2 ≥ λ1y1+λ2y2 for all λ = (λ1, λ2) ∈ Λτ . Now, for some x =(x1, x2), y = (y1, y2), x

′ = (x′1, x′2), y

′ = (y′1, y′2) ∈ X, we assume that conflict

exists between x and y and between x′ and y′ (i.e., x1 > y1, x2 < y2, x′1 > y′1,

and x′2 < y′2), the long-run utility of the project is determined by its presentvalue (i.e., λ∞ = (1/(1+δ), δ/(1+δ))), and alternative pairs (x, y) and (x′, y′)are strongly congruent. Then, Theorem 4 implies that |(x1+ δx2)− (y1+ δy2)|≥ |(x′1 + δx′2)− (y′1 + δy′2)| only if τ∗(x, y) ≤ τ∗(x′, y′), which agrees with theresults of Chabris et al.

Finally, in the ultimatum game experiment conducted by Knoch et al.(2006), a substantial variance in the responders’ decision time was attributedto the difference in their decisions, that is, subjects who reject an unfair offerexhibit a shorter decision time, whereas those who accept it exhibit a longer de-cision time. By employing the dual-self model (Loewenstein and O’Donoghue2004), Knoch et al. interpreted this result as the former group activating onlythe affective process (i.e., fairness judgment), whereas the latter requires thedeliberative process (i.e., utility maximization) to suppress the affective pro-cess.

To explain this result, consider the following adaptation of Fehr and Schmidt’s(1999) preference: for all x = (x1, x2), y = (y1, y2) ∈ X = [0, 1]× [0, 1], whereinx1, y1 denote self-interest utility and x2, y2 denote fairness utility, x %τ y ifand only if λ1x1+λ2x2 ≥ λ1y1+λ2y2 for all λ = (λ1, λ2) ∈ Λτ . Assume that,for a total prize of one, the proposer offers fractions 1− ε to himself and ε tothe responder, with ε ≤ 1/2. If the responder accepts the offer, her self-interestutility equals the responder’s payoff ε times a normalizing factor two, that is,x1 = 2ε, whereas her fairness utility equals one minus the difference betweenthe proposer’s and responder’s dividends, that is, x2 = 1− ((1− ε)− ε) = 2ε,which is also normalized to [0, 1] for all ε ∈ [0, 1/2]. If the responder rejects theoffer, a similar argument concludes that her self-interest utility equals y1 = 0,whereas her fairness utility equals y2 = 1− (0− 0) = 1 because neither partygets anything. Moreover, we assume that DMs are classified into the followingtwo groups: the first group is unambiguously fairness-driven, that is, Λτ = {λ}with λ = (0, 1) for all τ . Then, MDT predicts the outright rejection of an un-fair offer, such as ε = 0.1; that is, the choice of y = (0, 1) over x = (2ε, 2ε) witha zero decision time. The second group of DMs, in comparison, is such thatΛτ = {(λ1, λ2) ∈ ∆n−1 : λ2 ≤ 1/(τ + 1)} for τ = 0, 1, · · · ; that is, they placemore weight on self-interest in the long run (i.e., λ∞ = (1, 0)), while sufferingfrom a long-lasting conflict between self-interest and fairness. Presumably, thisgroup then exhibits a significantly longer decision time between acceptance x

and hyperbolic discounting, by increasing the number of attributes (i.e., calendar dates) andsetting an appropriate discounting rate for each of them.

18

and rejection y than the first group, which explains Knoch et al.’s result.12

Note that the difference in decision time between the two groups derives fromthe difference in Λ = {Λτ}∞τ=0, which is reminiscent of Theorem 2.

5 Related literature

In this section, we discuss the literature relevant to the present study. First, weelaborate on the relationship between MDT and the decision time literature ineconomics and other fields by employing Spiliopoulos and Ortmann’s (2015)framework, which classified decision-time-related cognitive processes into sub-processes, such as external/internal information acquisition and informationintegration.13

External information acquisition processes examine alternatives in the choiceset one-by-one and/or collect external signals. For example, search models(Caplin and Dean 2011; Gabaix and Laibson 2005; Gabaix, et al. 2006) allowsome alternatives in the choice set to be ignored (especially if the number ofalternatives is large), and identify the decision time with the number of al-ternatives considered until the search is terminated. Another example is thedelayed decision caused by resolving uncertainty in the external state thataffects the evaluation of alternatives (Bewley 1986; Kopylov 2009; Levi 1986).

Next, internal information acquisition processes either retrieve informa-tion relevant to the evaluation of alternatives in the DM’s memory, or resolveuncertainty about the subjective state. Diederich (2003) formalized these pro-cesses using a variant of decision field theory (Busemeyer and Townsend 1993),wherein information favorable to one alternative over the other is collectedfrom memory over time, and a decision is made (i.e., the decision time isdetermined) once substantial evidence has been accumulated.

Finally, information integration processes consolidate all the informationobtained in the information acquisition processes. An important implicationis that internal conflict may still exist in these processes, even after informa-tion acquisition has been completed. Such a conflict may arise, for example,among moral values (Levi 1986), dual selves (Knoch et al. 2006; Loewensteinand O’Donoghue 2004), and competing algorithms that vary in quality andduration (Rubinstein 2007, 2013; Wilcox 1993).

The gradual contraction of the set Λτ in MDT can be interpreted as eachof the aforementioned processes. That is, the contraction may be attributedto uncertainty resolution regarding the external or internal state, the sup-pression of the affective self by activating the deliberative self, or settlingconflict among moral values. Specifying the sources of contraction is beyond

12 A simple calculation yields that τ∗(x, y) equals the smallest integer larger than or equalto (1− 2ε)/2ε for the second group, which is strictly greater than zero if ε < 0.5.13 They also considered strategic processes, which analyze the structure of a game, and themotor function response, which actually implements the act of choice. However, a discussionof these processes is omitted here because the former is irrelevant to our individual choiceframework and the latter is a non-decision process.

19

the scope of this study; however, in future work, we could conduct a moreprecise characterization, for example, by introducing additional components,such as externally provided information or emotional states, and the relevantaxioms to our model. Moreover, it should be noted that our model may makea different prediction from those of search models, because the former focuseson a binary choice framework, whereas the latter generally assumes more thanthree alternatives in the choice set. If there are only two alternatives in thechoice set, search models generally predict a zero (or nearly zero) decisiontime, whereas MDT may predict a significantly larger decision time.

Second, incomplete preferences have been widely considered in economics(e.g., Bewley 1986; Dubra et al. 2004). In addition to the difference in inter-pretation (i.e., uncertainty or conflict), our approach can specify how difficult(or how easy) it is to reach a decision, as well as the eventual preference.Suppose that the rankings over alternative pairs (p, q) and (p′, q′) are bothinitially indecisive, but a decision is reached earlier over the first alternativepair than over the second. Whereas the aforementioned literature cannot dis-tinguish between the two pairs of alternatives, our model predicts that thesecond pair involves a greater internal conflict than the first. As Theorems 3and 4 suggest, we can also infer long-run indifference curves from alternativepairs that involve the longest decision time.

To the best of our knowledge, Kopylov (2009) was the first to study choicedeferral using incomplete binary relations. However, his main focus was onproviding a foundation for maxmin expected utility (Gilboa and Schmeidler1989) using Bewley’s (1986) Knightian uncertainty model as the ex ante pref-erence, rather than the decision time. Kraus and Sagi (2006) also characterizedchoice deferral in the context of temporal menu preference. However, they at-tributed choice deferral to a preference for flexibility, whereas we focus onchoice deferral caused by conflict.

Finally, multiattribute expected utility theory has long been developed byauthors such as Debreu (1959), Fishburn (1965), and Pollak (1967). Keeneyand Raiffa (1993) provided a comprehensive survey on the literature in thisfield. The present study can be considered as a temporal extension of previousmodels that allows for incomplete preferences.

6 Concluding remarks

In this study, we have formally characterized the decision time in a binarychoice framework. Caplin (2008) argued that non-choice data, such as thedecision time, can enrich economic predictions, whereas Gul and Pesendorfer(2008) criticized the fact that such data often have no direct translationsinto economic models, which renders their use less tractable. In this study,we have addressed this issue by providing a decision theoretic foundation forthe decision time, conflict, and their relationships. Thus, we believe that thisapproach elucidates the use of such non-choice data.

20

Appendix

Proof of Theorem 1

The sufficiency part is straightforward. To show the necessity, we first fix τand note that Xi is separable for all i ∈ I because it is compact. Becauseeach Xi is separable and metric, the partial order, continuity, independence,and single-attribute regularity axioms imply that there is a continuous vonNeumann–Morgenstern utility function ui for each attribute i that representsthe preference over the alternatives that differ with respect to only one at-tribute (using an argument similar to that of Fishburn (1965)). Consistencyimplies that ui is independent of τ ; that is, for all p, q ∈ P(X) such that p|−i

= q|−i, p %τ q if and only if Ep|i [ui(xi)] ≥ Eq|i [ui(xi)]. (E(·)|i [ui(xi)] denotesthe expected utility in attribute i given alternative (·)’s marginal probabilitydistribution of Xi.) Because the Xi are compact, we can normalize each util-ity function so that maxxi∈Xi ui(xi) = 1 and minxi∈Xi ui(xi) = 0, which alsoderives the uniqueness of ui. Define u ≡ (u1, · · · , un).

Next, let Ji = {η ∈ < : ∃p ∈ P(X), η = Ep|i [ui(xi)]} = [0, 1], and J =×n

i=1Ji = [0, 1]n ⊆<n. For all p ∈ P(X), we define Ep[u(x)]≡ (Ep|1 [u1(x1)], · · · ,Ep|n [un(xn)]) ∈ J . We also define a binary relation %∗ on J such that p %τ

q if and only if Ep[u(x)] %∗ Eq[u(x)]. From the monotonicity axiom, and byconstruction, it follows that %∗ is well-defined (i.e., p ∼τ q if Ep|i [ui(xi)] =Eq|i [ui(xi)] for all i ∈ I) and satisfies the independence and continuity axioms.

Now, define a cone K = {µ(s − t) : µ ≥ 0, s, t ∈ J, s %∗ t} ⊆ <n. Thefollowing lemmas specify the characteristics of K.

Lemma 1 For all s, t ∈ J , s %∗ t if and only if s− t ∈ K.

Proof We prove this by showing that for all s, t ∈ J , s %∗ t if and only ifthere are s′, t′ ∈ J and µ > 0 such that s′ %∗ t′ and s− t = µ(s′− t′). Assumethe latter statement. Then, it follows from s− t = µ(s′ − t′) that

1

1 + µs+

µ

1 + µt′ =

1

1 + µt+

µ

1 + µs′

%∗ 1

1 + µt+

µ

1 + µt′,

where the last preference relation holds because of the independence axiom.Now, 1

1+µs+µ

1+µ t′ %∗ 1

1+µ t+µ

1+µ t′, which implies that s %∗ t by independence.

The converse is straightforward. utLemma 2 K is convex.

Proof Suppose that s, t, s′, t′ ∈ J are such that s − t, s′ − t′ ∈ K. Lemma1 implies that s %∗ t and s′ %∗ t′. For some µ ∈ (0, 1), independence impliesthat µs+(1−µ)s′ %∗ µt+(1−µ)s′ and µt+(1−µ)s′ %∗ µt+(1−µ)t′, fromwhich it follows that µs+(1−µ)s′ %∗ µt+(1−µ)t′. Again, Lemma 1 impliesthat µ(s− t) + (1− µ)(s′ − t′) ∈ K. utLemma 3 K is closed.

21

Proof The proof is an adaptation of that given by Ok (2007, Proposition H1,Claim 2). Let {µm}∞m=1, {sm}∞m=1, and {tm}∞m=1 be such that µm ≥ 0, sm, tm

∈ J , sm %∗ tm (i.e., µm(sm − tm) ∈ K) for all m, and limm→∞ µm(sm − tm)= v ∈ <n. We show that v ∈ K. If sm = tm for infinitely many m, it is trivialthat v = 0 ∈ K. Accordingly, we can assume that sm 6= tm for all m withoutloss of generality.

Because J ⊆ <n, we apply the Euclidean distance d(s, t) = (∑n

i=1 |si −ti|2)1/2 to all s = (s1, · · · , sn), t = (t1, · · · , tn) ∈ J . Fix an interior point s ofJ (note that the interior of J is nonempty because J = [0, 1]n). Then, thereis some ε > 0 such that Nε,<n(s) ≡ {s ∈ <n : d(s, s) < ε} ⊆ J . For some δ< ε, define U = {r ∈ J : r %∗ s, d(r, s) ≥ δ}, which is closed because %∗ iscontinuous, and is compact because J is also compact.

Let dm = d(sm, tm). Then, because

µm(sm − tm) =dmµm

δ· δ

dm(sm − tm)

for all m, we obtain µm(sm − tm) = µ′m(rm − s) by defining µ′

m = dmµm/δand rm = s+ δ(sm − tm)/dm. Because d(rm, s) = δ < ε, rm ∈ Nε,<n(s) ⊆ Jfor all m. Moreover, because K is a cone, sm − tm ∈ K implies rm − s ∈ K,thus, it follows from Lemma 1 that rm %∗ s for all m. Accordingly, rm ∈ U forall m, which implies that, from the compactness of U , there is a subsequenceof {rm}∞m=1 that converges to some r ∈ J such that r %∗ s. Additionally,because limm→∞ µm(sm − tm) = limm→∞ µ′

m(rm − s) = v and d(rm, s) = δ,{µ′

m}∞m=1 is a bounded sequence. Thus, there is a subsequence of {µ′m}∞m=1

that converges to some µ′ ≥ 0, which implies that v = µ′(r − s). The lastcondition implies that v ∈ K. ut

Because K is convex and closed, it can be denoted by the intersection ofclosed half-spaces induced by the hyperplanes supporting it. Moreover, becauseK is a cone, each of the closed half-spaces is denoted by the set {γ ∈ <n :λ · γ ≥ 0}, where λ ∈ <n satisfies λ · (s − t) = 0 for some s, t ∈ J suchthat s − t ∈ K. Eventually, by defining a closed and convex set C ≡ {λ ∈<n : λ · (s − t) ≥ 0 for all s, t ∈ J such that s − t ∈ K}, K = {s − t : s, t ∈J, λ · (s − t) ≥ 0 for all λ ∈ C}. The following lemma indicates that C onlyconsists of non-negative vectors.

Lemma 4 C ⊆ <n+ ≡ {λ = (λ1, · · · , λn) ∈ <n : λi ≥ 0 for all i ∈ I}.

Proof Suppose that there exists λ = (λ1, · · · , λn) ∈ C such that λj < 0 forsome j ∈ I. Then, by defining s = (s1, · · · , sn), t = (t1, · · · , tn) ∈ J such thatsj < tj and si = ti for all i 6= j, s − t ∈ <n

− ≡ {r = (r1, · · · , rn) ∈ <n : ri ≤0 for all i ∈ I} and λ · (s − t) > 0. This also implies that t − s ∈ <n

+ andλ · (t− s) < 0. It follows from the monotonicity axiom and t− s ∈ <n

+ that t�∗ s. However, λ · (t− s) < 0 implies t− s 6∈ K by the construction of C andK, which contradicts Lemma 1. ut

22

Now, we define Λτ ≡ C∩∆n−1, which is closed and convex because both Cand∆n−1 are closed and convex. It follows from Lemma 4 and the constructionof K and C that for all p, q ∈ P(X), p %τ q if and only if λ · Ep[u(x)] ≥λ · Eq[u(x)] for all λ ∈ Λτ . To prove the uniqueness of Λτ , suppose that bothΛτ and Λ′ (along with u) represent %τ . Without loss of generality, we assumeΛ′ \ Λτ 6= φ. Then, because Λτ is closed and convex, it follows from oneversion of the separating hyperplane theorem (Dunford and Schwartz 1957,V2.12) that there exist p, q ∈ P(X) such that λ · (Ep[u(x)] − Eq[u(x)]) > 0for all λ ∈ Λτ , but λ′ · (Ep[u(x)]−Eq[u(x)]) < 0 for some λ′ ∈ Λ′ \Λτ . Thus,the preference represented by Λτ implies p %τ q, whereas that represented byΛ′ implies p ./τ q, which is a contradiction.

Finally, the following lemma characterizes the relationship among the Λτ

for different τ .

Lemma 5 For all τ , τ ′ ∈ N ∪ {0}, if p %τ q implies p %τ ′q for all p, q ∈

P(X), then Λτ ⊇ Λτ ′.

Proof Suppose, on the contrary, that Λτ ′ \Λτ 6= 0. Then, because Λτ is closedand convex, one version of the separating hyperplane theorem (Dunford andSchwartz 1957, V2.12) implies that there exist p, q ∈ P(X) such that λ ·(Ep[u(x)]−Eq[u(x)]) > 0 for all λ ∈ Λτ and λ′ · (Ep[u(x)]−Eq[u(x)]) < 0 for

some λ′ ∈ Λτ ′ \ Λτ . From this, it follows that p %τ q and p ./τ′q, which is a

contradiction. ut

By the consistency axiom, Lemma 5 implies that Λτ ⊇ Λτ ′if τ < τ ′, which

concludes the proof. ut

Proof of Theorem 2

First, we show that statement (a) implies (b). Assume that for all τ ∈ N∪{0}and p, q ∈ P(X), p %τ

2 q implies p %τ1 q. Thus, an argument similar to that

applied for Lemma 5 implies that Λτ1 ⊆ Λτ

2 for all τ ∈ N ∪ {0}. Conversely,assume that statement (b) holds, that is, Λτ

1 ⊆ Λτ2 for all τ ∈ N∪{0}. It follows

from the definitions of Λτ1 and Λτ

2 that, for all τ ∈ N ∪ {0} and p, q ∈ P(X),p %τ

2 q implies p %τ1 q, which implies statement (a). ut

Proof of Theorem 3

First, because alternative pairs (p, q) and (p′, q′) are congruent with respectto the hyperplane h∞, which has normal vector λ∞ ∈ ∩∞

τ=0Λτ , there exist s

∈ <n and ψ ∈ {−1, 1} such that λ∞ · s = 0, Ep−q[u(x)] = µ1s+ µ2ψλ∞, and

Ep′−q′ [u(x)] = µ′1s + µ′

2ψλ∞ for some µ1, µ2, µ

′1, µ

′2 ≥ 0. Without loss of

generality, we assume that ψ = 1 in the following. (The case ψ = −1 can beproved similarly.)

23

Now, we prove statement (a). Assume that τ∗(p, q) < τ∗(p′, q′), whichimplies τ∗(p, q) < ∞. Because ψ = 1 implies that λ∞ · Ep−q[u(x)] ≥ 0 and

λ∞ · Ep′−q′ [u(x)] ≥ 0, there exists λ ∈ Λτ∗(p,q) \ Λτ∗(p,q)+1 such that

λ · Ep−q[u(x)] ≥ 0 > λ · Ep′−q′ [u(x)]; (A.1)

that is, a hyperplane h with normal vector λ separates vectors Ep−q[u(x)] andEp′−q′ [u(x)]. We also have µ′

1 > 0 to satisfy (A.1); otherwise, µ′1 = 0 and

λ ·Ep′−q′ [u(x)] = λ · (µ′2λ

∞) ≥ 0, which is a contradiction. (Note that λ · λ∞≥ 0 because λ, λ∞ ∈ Λτ∗(p,q) ⊆ ∆n−1.) It follows from ψ = 1 that there exists

r ∈ < such that r is parallel to h (i.e., λ · r = 0) and r = µ1s+ µ2λ∞ for some

µ1, µ2 ≥ 0. Now, we consider the following cases.

Case 1: µ1 = 0

It follows from r = µ2λ∞ and λ · r = 0 that λ · (µ2λ

∞) = 0. Because µ2

> 0 (otherwise, we must have r = 0, which is a contradiction), λ · λ∞ = 0.

Thus, (A.1) implies that λ · (µ1s) ≥ 0 > λ · (µ′1s). Because we already showed

that µ′1 > 0, we must have λ · s < 0 and thus µ1 = 0. Accordingly, π/2 =

θ(Ep−q[u(x)], s) > θ(Ep′−q′ [u(x)], s).

Case 2: µ1 > 0

First, we consider µ1 > 0. By construction, λ · r = µ1λ · s+ µ2λ · λ∞ = 0,or

λ · s = − µ2

µ1λ · λ∞. (A.2)

Now, λ·λ∞ ≥ 0 because λ, λ∞ ∈ Λτ∗(p,q) ⊆∆n−1. Suppose λ·λ∞ = 0, which fol-lows from (A.2) that λ·s = 0. This implies that λ·Ep−q[u(x)] = λ·Ep′−q′ [u(x)]

= 0, contradicting (A.1). Thus, we conclude that λ·λ∞ > 0. Substituting (A.2)

into λ · Ep−q[u(x)] = µ1λ · s + µ2λ · λ∞ ≥ 0 gives µ2/µ1 ≥ µ2/µ1, whereas

substituting (A.2) into λ ·Ep′−q′ [u(x)] = µ′1λ · s+µ′

2λ ·λ∞ < 0 obtains µ′2/µ

′1

< µ2/µ1. (Note that we already have µ′1 > 0.) Accordingly, µ2/µ1 ≥ µ2/µ1 >

µ′2/µ

′1. Thus, the slope of the vector difference Ep−q[u(x)] is greater than that

of Ep′−q′ [u(x)], that is, θ(Ep−q[u(x)], s) > θ(Ep′−q′ [u(x)], s). Second, considerµ1 = 0. Then, we obtain π/2 = θ(Ep−q[u(x)], s) > θ(Ep′−q′ [u(x)], s), as inCase 1.

In both cases, we conclude that θ(Ep−q[u(x)], t) > θ(Ep′−q′ [u(x)], t) forthe projection t of Ep−q[u(x)] onto h∞ because the vector t has the samedirection as s by construction. Statement (b) is straightforward because it isthe contraposition of (a). ut

Proof of Theorem 4

We first note that, if the alternative pairs (p, q) and (p′, q′) are strongly congru-ent with respect to some h∞ that has normal vector λ∞ ∈ ∩∞

τ=0Λτ , then (p, q)

24

and (p′, q′) are congruent with respect to h∞; that is, there exist s ∈ <n, ψ ∈{−1, 1}, µ1, µ2, µ

′1, µ

′2 ≥ 0 such that λ∞ · s = 0, Ep−q[u(x)] = µ1s+ µ2ψλ

∞,and Ep′−q′ [u(x)] = µ′

1s+ µ′2ψλ

∞.Now, we prove statement (a). Assume that τ∗(p, q) < τ∗(p′, q′). Because

(p, q) and (p′, q′) are congruent with respect to h∞, Theorem 3 implies thatθ(Ep−q[u(x)], t) > θ(Ep′−q′ [u(x)], t) for the projection t of Ep−q[u(x)] onto h

∞,which in turn implies that θ(Ep−q[u(x)], s) > θ(Ep′−q′ [u(x)], s) for the vectors defined in the previous paragraph because s has the same direction as t.From the last condition, it follows that µ2/µ1 > µ′

2/µ′1. The strong congruence

of (p, q) and (p′, q′) then implies that µ2 > µ′2, and thus |λ∞ · Ep−q[u(x)]| =

|µ2ψ|‖λ∞‖2 > |µ′2ψ|‖λ∞‖2 = |λ∞ ·Ep′−q′ [u(x)]| because λ∞ ·s = 0. Statement

(b) is straightforward because it is the contraposition of (a). ut

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