Stochastic Endogenous Time Preference ∗

Youichiro Higashi, Kazuya Hyogo, and Norio Takeoka†

May 12, 2010

Abstract

In the literature of dynamic models under uncertainty, it is commonly assumedthat uncertainty is resolved gradually over time and its resolution is independent ofthe actions taken by a decision maker up to that time period. However, contingenciesin the decision maker’s mind may change and even increase over time according tohabits that have evolved by experiences, actions, or consumptions in the past. Weprovide an axiomatic model where the decision maker faces uncertainty about herown time preference or discount factors in future and her belief over these uncertaintymay evolve according to histories of the past consumption. Moreover, we provide abehavioral comparison about attitude toward flexibility across different histories, andcharacterize a property of the representation where the decision maker’s belief overfuture time preference at some history is more patient than her belief at anotherhistory in some stochastic sense.

Keywords: endogenous time preference, subjective states, preference for flexibility,comparative impatience, increasing convex [concave] stochastic order.JEL classification: D11, D81, D91,

∗An earlier version of this paper is circulated under the title “Random Discounting with Habits.” Wewould like to thank Chiaki Hara, Katsutoshi Wakai, and the audiences of the 2009 JEA Spring Meeting,FESAMES 2009, SWET 2009, ICNAAM 2009, Osaka GCOE Workshop, and the seminar participants atOsaka Keizai, Osaka Prefecture, and Toyama Universities, and Otaru University of Commerce for usefulcomments. Hyogo gratefully acknowledges financial support from a Grant-in-Aid for Young Scientists (B).All remaining errors are the author’s responsibility.

†Higashi is at the Faculty of Economics, Okayama University, Okayama, Japan, higashi@e.okayama-u.ac.jp; Hyogo is at the Faculty of Economics, Ryukoku University, 67 Fukakusa Tsukamoto-cho, Fushimi-ku, Kyoto 612-8577, Japan, hyogo@econ.ryukoku.ac.jp; Takeoka is at the Faculty of Economics, YokohamaNational University, 79-3 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan, takeoka@ynu.ac.jp.

1

1 Introduction

1.1 Objective

In a dynamic decision making, a state space with a filtration on it is a standard tool formodeling uncertainty. It is commonly assumed that uncertainty is resolved gradually overtime and its resolution is independent of the actions taken by a decision maker (DM) up tothat time period. However, contingencies in the DM’s mind may change and even increaseover time according to habits that have evolved by experiences, actions, or consumptions inthe past. For example, a DM who has the habits of thrift may surely anticipate that facinga trade-off between a later but larger reward and an immediate but smaller reward, shewill be patient enough to choose the former. But, according to having a higher standardof life, she may become less certain about her own time preference in future and anticipatethat the immediate but smaller reward may become more attractive in some contingencies.

Our purpose is to provide an axiomatic model where the DM faces uncertainty abouther own time preference or discount factors in future and these uncertainty may evolveaccording to histories of the past consumption. In decision theoretic models, expectationsof taste shocks, whether they are about discount factors or other aspects of tastes, havebeen associated with preference for flexibility, which can be modeled by preference overopportunity sets (for example, Kreps [16, 17] and Dekel, Lipman, and Rustichini [5]).We incorporate histories of past consumption into this framework and model a DM asa collection of preferences depending on various histories. Then, a change of the DM’sbelief over future time preference according to histories should be revealed as a change inattitudes toward flexibility regarding consumption plans.

In the framework without uncertainty, endogenous determination of time preferencehas been investigated in economics. Since Fisher [9], it is admitted that time preferencemay be affected by habits and change over time. Becker and Mulligan [2] consider amodel in which a time preference depends on the quantity of resources the DM investsinto making future pleasures less remote, for example, investment for health or education.However, past consumption may affect current time preference in a stochastic manner asmentioned in the first paragraph, and investment decisions may involve uncertainty in theirown nature. Thus, our model provides a general framework for endogenous time preferenceaccommodating uncertainty.

In case of endogenous time preference without uncertainty, if one’s discount factor atsome history is greater than that at another history, she is said to be more patient at theformer history than the latter. We would like to have a comparative notion of patiencealso in case of stochastic endogenous time preference. In our decision theoretic framework,it is possible to provide a behavioral comparison of patience across various histories. Weshow that this comparison characterizes the case where the DM’s belief over future timepreference at some history is dominated by her belief at another history in the sense of theincreasing convex and concave order, which is a generalization of the first-order stochasticdominance. Hence, if beliefs are deterministic and degenerate, this condition is simply

2

saying that the discount factor at some history is greater than that at another.For the purpose, we adopt the same domain of choice used by Gul and Pesendorfer [11].

Let C be the consumption set, which is a compact metric space. They show that thereexists a compact metric space Z such that Z is homeomorphic to K(∆(C × Z)), where∆(X) is the set of all Borel probability measures over X and K(X) is the set of all non-empty compact subsets of X. An element of Z, called a menu, is an opportunity set oflotteries over pairs of current consumption and future menus.

We have in mind the following timing of decisions: Imagine a DM who has experienced

0 1− 1 1+

x α l ∈ x (c, x′)

2− 2

α′ l′ ∈ x′c1c2

−1−2

· · ·

Figure 1: Timing of Decisions

a history of consumption h = (· · · , c2, c1). In period 0, the DM chooses a menu x accordingto preference %h. At the moment of this choice, the DM is uncertain about future discountfactors and has a belief about them. This belief is reflected somehow on %h and will beelicited as a component of a representation (see below). In period 1− (which is not a partof the primitives), current discount factor α becomes known to the DM, and she choosesa lottery l out of the menu x in period 1. In period 1+, the DM receives a pair (c, x′)according to realization of the lottery l, and consumption c takes place. At this point intime, the DM evaluates x′ according to preference %hc, where hc = (· · · , c2, c1, c) is anupdated history. The DM expects another discount factor α′ to be realized in the followingperiod. Subsequently, she chooses another lottery l′ out of the menu x′, and so on.

We consider a collection of preferences {%h}h∈H over Z, where h = (· · · , c2, c1) is ahistory of past consumption and H is the set of all histories. We axiomatize {%h}h∈H havingthe following representation: there exist a non-constant continuous mixture linear functionu : ∆(C) → R and a history-dependent belief µ(·|h) about discount factors α ∈ [0, 1] suchthat

U(x|h) =

∫[0,1]

maxl∈x

∫C×Z

((1 − α)u(c) + αU(z|hc)

)dl(c, z) dµ(α|h),

where hc = (· · · , c2, c1, c) if h = (· · · , c2, c1). Moreover, we show that u is unique up to apositive affine transformation and µ(·|h) is uniquely identified for all h ∈ H.

In this model, the DM surely anticipates that her instantaneous utility function u willnot change over time, while she is uncertain about her future discount factor α and has abelief µ(α|h) on it. After observing α, the DM will choose a lottery l ∈ x so as to maximizethe expected utility of the weighted average between an instantaneous utility u(c) and acontinuation value U(z|hc). Notice that an immediate consumption, c, has two effects: Oneis a direct effect on an instantaneous utility u(c), and the other is an indirect effect on a

3

continuation utility U(·|hc) (or future belief µ(·|hc)) through history dependence. A menux is evaluated according to its expected maximum value with respect to her belief µ(·|h).

1.2 Changing in Preference for Flexibility

We illustrate how histories of past consumption affect beliefs about time preference infuture and how these beliefs are in turn reflected on behavior of the DM. A key behavioris an attitude toward flexibility and its change according to habits.

Let C stand for a set of monetary payoffs. For simplicity, we use a usual notation(c1, c2, c3, · · · ) instead of (c1, {(c2, {(c3, · · · )})}). Let 0 = (0, 0, · · · ).

Imagine a DM who has experienced a low level of consumption in her life and haspreference %L over menus. Now consider two options, (0, 110,0) and (100, 0,0). Supposethat the DM is patient and prefers a later but lager reward to an immediate but smallerreward, that is, {(0, 110,0)} %L {(100, 0,0)}. Moreover, since the DM has experienced alow level of consumption in her life, she may have no contingency in her mind where shewill become impatient and choose (100, 0,0) over (0, 110,0). Hence it is not beneficial tokeep (100, 0,0) with (0, 110,0) as an option. That is,

{(100, 0,0), (0, 110,0)} ∼L {(0, 110,0)} %L {(100, 0,0)}.

However, once the DM experiences some higher level of consumption for certain timeperiods, she may become less certain about her time preference and expect that (100, 0,0)may be chosen over (0, 110,0) in some contingencies. That is, in terms of preference %LH ,the DM may have preference for flexibility:

{(100, 0,0), (0, 110,0)} ÂLH {(0, 110,0)} %LH {(100, 0,0)}.

1.3 Related Literature

1.3.1 Endogenous Time Preference

Fisher [9] discussed how time preference or discount factor depends on individual character-istics and argued that one of the determinants of time preference is “habits”. He providedintuitive arguments for the endogenous time preference depending on habits and its changeover time:1

It has been noted that a person’s rate of preference for present over future in-come, given a certain income stream, will be high or low according to the pasthabits of the individual. If he has been accustomed to simple and inexpensiveways, he finds it fairly easy to save and ultimately to accumulate a little prop-erty. The habits of thrift being transmitted to the next generation, by imitationor by heredity or both, result in still further accumulation. The foundations ofsome of the world’s greatest fortunes have been based upon thrift.

1See Fisher [9, pp. 337-338]. This quote is the same as in Shi and Epstein [23].

4

Reversely, if a man has been brought up in the lap of luxury, he will havea keener desire for present enjoyment than if he had been accustomed to thesimple living of the poor. The children of the rich, who have been accustomed toluxurious living and who have inherited only a fraction of their parent’s means,may spend beyond their mean and thus start the process of the dissipationof their family fortune. In the next generation this retrograde movement islikely to gather headway and to continue until, with the gradual subdivisionof the fortune and the reluctance of the successive generations to curtail theirexpenses, the third or fourth generation may come to actual poverty.

The accumulation and dissipation of wealth do sometimes occur in cycles.Thrift, ability, industry and good fortune enable a few individuals to rise towealth from the ranks of the poor. A few thousand dollars accumulation underfavorable circumstances may grow to several millions in the next generation ortwo. Then the unfavorable effects of luxury begin, and the cycle of poverty andwealth begins anew. The old adage, “From shirt sleeves to shirt sleeves in fourgenerations,” has some basis in fact.

Uzawa [26] formulates a model of endogenous rate of time preference in which thediscount function depends on current consumption and future consumption.2 The Uzawautility function has been widely used in application where the implications of constant timepreference are unappealing. Shi and Epstein [23] assume that a DM’s discount functiondepends not only on current consumption and future consumption, but also on an index ofpast consumption or habit. In a closed economy, their model generates the cyclical patternof wealth accumulation. In our model, the discount factor is random and independent offuture consumption, but depends on habits. We analyze the changes of the DM’s attitudetoward flexibility over time that arises from random discounting with habits.

1.3.2 Axiomatic Models

We study preference on opportunity sets (or menus) in order to provide a foundation forrandom discounting with habit. Opportunity sets are first considered as choice objectsby Koopmans [15] to model sequential decision making. He points out that if a DManticipates preference change in future, she may strictly prefer to keep some flexibility inher options rather than to choose a completely specified future plan. Kreps [16, 17] derivethe set of anticipated future preferences from preference over opportunity sets and call itthe subjective state space. By assuming richer choice objects than in Kreps [16, 17], Dekel,Lipman, and Rustichini [5] (henceforth DLR) uniquely identify the subjective state space.wTheir argument on the additive representation is modified by Dekel, Lipman, Rustichini,and Sarver [6].

2Its axiomatic foundation is provided by Epstein [8], in which the DM’s preference is defined on lotteriesover consumption stream, and her von Neumann-Morgenstern utility index follows Uzawa functional form.

5

Higashi, Hyogo, and Takeoka [14] provide an axiomatic model of random discounting,where DLR’s model is extended to an infinite-horizon setting and the subjective state spaceis specified to the set of sequences of discount factors. More formally, preference over Zhaving the following representation is axiomatized:

U(x) =

∫[0,1]

maxl∈x

∫C×Z

((1 − α)u(c) + αU(z)

)dl(c, z) dµ(α).

In their model, the belief about future discount factors, that is, µ, is uniquely identified,but is assumed to be i.i.d. and independent of past consumption. Hence the DM’s attitudetoward flexibility is the same over time. In contrast, in the present paper, the belief aboutfuture discount factors depends on past consumption and hence is not restricted to be i.i.d.

Gul and Pesendorfer [11] originally consider preference over Z and model a DM whosuffers from self-control problem. Their DM anticipates that she will experience temptationor craving at the moment of choice, and hence, may prefer to limit her options in advancerather than to keep flexibility. Gul and Pesendorfer [12] consider a collection of preferences{%h} on Z which depend on histories of the last t-period consumption, and extend theirself-control model to accommodate history dependence of degree of self-control. Theyaxiomatize a recursive self-control representation where the strength of temptation dependson past consumption.

Rozen [21] provides a foundation for intrinsic habit model, which has been widely ap-plied in asset pricing models, by considering a DM who has preference over deterministicconsumption streams. If the DM’s preference is affected by habit, she may not prefer aconsumption stream yielding high consumption at each date to one yielding low consump-tion at each date. The key axiom in her paper is that there is a collection of consumptionstreams under which the DM behaves as if she is habit-free.

2 Model

2.1 Domain

Let C be the outcome space (consumption set), which is assumed to be compact and metric.Let ∆(C) be the set of lotteries, that is, all Borel probability measures over C. Under theweak convergence topology, ∆(C) is also compact and metric. For any compact metricspace X, let K(X) be denoted by the set of all non-empty compact subsets of X. The setK(X) is endowed with the Hausdorff metric. 3

Gul and Pesendorfer [11] show that there exists a compact metric space Z such thatZ is homeomorphic to K(∆(C × Z)). Generic elements of Z are denoted by x, y, z, · · · .Each such object is called a menu (or an opportunity set) of lotteries over pairs of currentconsumption and menu for the rest of the horizon.

3Details are relegated to Appendix A.

6

An important subdomain of Z is the set L of perfect commitment menus where DM iscommitted in every period. We identify a singleton menu with its only element. Then aperfect commitment menu can be viewed as a multistage lottery, that is, L is a subdomainof Z satisfying L ' ∆(C × L). A formal treatment is found in the Appendix of Higashi,Hyogo, and Takeoka [14].

Another special subclass is a multistage lottery where a level of consumption is i.i.d. overtime. Formally, for all ` ∈ ∆(C), ⊗` denotes the perfect commitment lottery such that

` ⊗ {` ⊗ {· · · }}.

Such lotteries are called i.i.d. lotteries. Let ⊗∆c ⊂ L denote the set of all i.i.d lotteries. Aconsumption stream with a constant level of consumption, that is, (c, c, c, · · · ), is includedin this subdomain under a suitable identification.

At each time period, preference is defined on Z, and depends on past consumption. Leth = (· · · , c2, c1) be a history of past consumption and

H ≡ {h = (· · · , ct, · · · , c2, c1) | ct ∈ C for all t = 1, 2, · · · }

be the set of all histories, which is endowed with the metric

ρ(h, h′) =∞∑

t=1

1

2t· d(ct, c

′t)

1 + d(ct, c′t),

where h = (c1, c2, · · · ), h′ = (c′1, c′2, · · · ), and d is a metric on C.4 For all c ∈ C and

h = (· · · , c2, c1) ∈ H, the updated history (· · · , c2, c1, c) will be denoted by hc. For eachhistory h ∈ H, let %h be preference over Z. We consider a collection of preferences{%h}h∈H .

2.2 Functional Form

Let u : ∆(C) → R be a non-constant continuous mixture linear function. The set ofdiscount factors is denoted by [0, 1] and its generic element is denoted by α. We modela DM who is uncertain about future discount factors and whose belief depends on pastconsumption. Formally, for all h ∈ H, let µ(·|h) be a probability measure on [0, 1]. Letαh be the average of µ(·|h), that is, αh ≡

∫[0,1]

α dµ(α|h). A set of probability measures

{µ(·|h)}h∈H is called a history-dependent belief and assumed to satisfy sup{αh |h ∈ H} < 1and continuity of µ(·|h) in h.

We formulate the following functional form: For each pair of an instantaneous utilityfunction u and a history-dependent belief {µ(·|h)}h∈H , a history-dependent random dis-counting utility U(·|h) : Z → R, h ∈ H, is defined as

U(x|h) =

∫[0,1]

maxl∈x

∫C×Z

((1 − α)u(c) + αU(z|hc)

)dl(c, z)dµ(α|h). (1)

4Under the metric ρ, H is complete (see Aliprantis and Border [1, p.90, Theorem 3.37]).

7

In this model, the DM surely anticipates that her instantaneous utility function u willnot change over time, while she is uncertain about her future discount factor α and has abelief µ(α|h) on it. After observing α, the DM will choose a lottery l ∈ x so as to maximizethe expected utility of the weighted average between an instantaneous utility u(c) and acontinuation value U(z|hc). Notice that an immediate consumption, c, has two effects: Oneis a direct effect on an instantaneous utility u(c), and the other is an indirect effect on acontinuation utility U(·|hc) (or future belief µ(·|hc)) through history dependence. A menux is evaluated according to its expected maximum value with respect to her belief µ(·|h).

Our functional form can accommodate several specifications used in macroeconomicmodels. Becker and Mulligan [2] suggest a model in which a discount factor (determinis-tically) depends on the quantity of resources the DM invests into making future pleasuresless remote. For example, investment in education or health-related decisions may focusone’s attention on the future. Notice that these activities can be included into the set Cof consumptions. If the outcome of investment is uncertain, consequently, discount factorswill change stochastically.

An alternative interpretation of infinite-lived agents is to think of a dynasty, in whichcase a discount factor is regarded as a degree of altruism. The bequest motives of thecurrent generations may depend on whether they are married or not, or whether they havedescendants or not.

If a belief µ(·|h) is independent of h, (1) is reduced to the history-independent randomdiscounting model of Higashi, Hyogo, and Takeoka [14]:

U(x) =

∫[0,1]

maxl∈x

∫C×Z

((1 − α)u(c) + αU(z)

)dl(c, z) dµ(α),

or equivalently,

U(x) =

∫[0,1]

maxl∈x

((1 − α)

∫C

u(c) dlc + α

∫Z

U(z) dlz

)dµ(α),

where lc and lz denote the marginal distributions of l on C and Z respectively.

3 Foundations

3.1 Axioms

The axioms which we consider on {%h}h∈H are the following. The first four axioms havecounterparts in DLR. The only difference is that we impose the same axioms on preferencesover infinite horizon decision problems Z rather than a two-period domain K(∆(C)).

Axiom 1 (Order). For all h ∈ H, %h is complete and transitive.

Axiom 2 (Continuity).

8

(i) For all h ∈ H and x ∈ Z, {z ∈ Z|x %h z} and {z ∈ Z|z %h x} are closed.

(ii) For all x, y ∈ Z, {h ∈ H | x %h y} is closed.

Part (i) is a standard continuity condition. Part (ii) requires continuity with respect tohistories, and has no counterpart in DLR.

For any x, x′ ∈ Z and λ ∈ [0, 1], define the mixture of two menus by considering themixtures element by element between x and x′, that is,

λx + (1 − λ)x′ ≡ {λl + (1 − λ)l′|l ∈ x, l′ ∈ x′} ∈ Z.

Axiom 3 (Independence). For all h ∈ H, x, y, z ∈ Z and λ ∈ (0, 1],

x Âh y ⇒ λx + (1 − λ)z Âh λy + (1 − λ)z.

Independence can be justified by two steps as in DLR. The first step is a standardargument of the von-Neumann and Morgenstern Independence: If x is preferred to y, thenthe lottery yielding x and z with probabilities λ and 1 − λ, denoted by λ ◦ x + (1 − λ) ◦ z,is preferred to λ ◦ y + (1 − λ) ◦ z.

In the next step, we argue that λ ◦ x + (1− λ) ◦ z is indifferent to λx + (1− λ)z. Noticethat these two alternatives differ in timing of resolution of randomization λ. For the former,the DM makes choice out of a menu (either x or z) after the resolution of λ, while for thelatter, this order is reversed, that is, the choice out of the menu λx + (1 − λ)z is madebefore the resolution of λ. Suppose that a DM is uncertain about future preference over∆(C ×Z), yet she surely anticipates that it will satisfy the expected utility axioms. Let land l′ be a rational choice from x and z, respectively, with respect to a future preference.Therefore, λl + (1 − λ)l′ is the expected choice from λ ◦ x + (1 − λ) ◦ z. On the otherhand, if the future preference satisfies the expected utility axioms, λl + (1 − λ)l′ is also arational choice from λx+(1−λ)z. Therefore, irrespective of the future preference, the twoalternatives will ensure indifferent consequences.

Axiom 4 (Monotonicity). For all h ∈ H, x, y ∈ Z, if y ⊂ x, then x %h y.

This axiom says that a bigger menu is always weakly preferred, and is consistent withpreference for flexibility.

Say that preferences %1 and %2 on Z are equivalent on ∆(C) if, for all `, `′ ∈ ∆(C),

{⊗`} Â1 {⊗`′} ⇔ {⊗`} Â2 {⊗`′}.

The next three axioms concern risk preference over ∆(C).

Axiom 5 (History-Independent Risk-Preference). For all h, h′ ∈ H, %h and %h′ areequivalent on ∆(C).

9

Suppose that the DM surely expects her attitude toward static risk, that is, preferenceover ∆(C), to be independent of histories of past consumption. Since a compound lotterysuch as ⊗` provides the same risky consumption in every period, the ranking between{⊗`} and {⊗`′} should reflect solely static risk preference and have nothing to do withexpectations for future time preference.

Axiom 6 (Nondegeneracy). For all h ∈ H, there exist `, `′ ∈ ∆(C) such that {⊗`} Âh

{⊗`′}.

The ranking in the axiom states that receiving a lottery ` ∈ ∆(C) in every period ispreferred to receiving `′ in every period, which means that an atemporal lottery ` is betterthan `′. The existence of such `, `′ implies that the DM’s risk preference over ∆(C) issensitive.

Axiom 7 (IID-Separability). For all h ∈ H, `, `′ ∈ ∆(C), and λ ∈ [0, 1],

{λ ⊗ ` + (1 − λ) ⊗ `′} ∼h {⊗(λ` + (1 − λ)`′)}.

As explained above, if the DM surely anticipates that her risk preference is history-independent, a compound lottery such as ⊗` is viewed as a static lottery `. Thus, the DMshould be indifferent between λ ⊗ ` + (1 − λ) ⊗ `′ and ⊗(λ` + (1 − λ)`′).5

Axiom 8 (Best-Worst). For all h ∈ H, x ∈ Z, there exist `, `′ ∈ ∆(C) such that{⊗`} %h x %h {⊗`′}.

For example, let ⊗`+ be a maximal lottery among ⊗∆c. If the DM surely anticipatesher future risk preference to be time-invariant and is only concerned about uncertaintyabout intertemporal trade-offs, she is willing to make a commitment to ⊗`+. On the otherhand, she will never benefit from the commitment to a minimal lottery among ⊗∆c.

Axiom 9 (Strong Archimedean). For all `, `′ ∈ ∆(C) such that {⊗`} Âh {⊗`′}, thereexists `′′ ∈ ∆(C) such that {⊗`} Âh {⊗`′′} %h {`′ ⊗ {⊗`}} for all h ∈ H.

By assumption of the axiom, an atemporal lottery ` is better than `′. Thus, in `′⊗{⊗`},the better option is delayed until second period, and it should be strictly worse than ⊗`.The axiom requires that irrespective of histories, it is even worse than some lottery ⊗`′′

which is strictly worse than ⊗`. Intuitively, this condition states that a loss from receivinga better lottery with one-period delay is uniformly bounded across histories.

The last two axioms concern intertemporal decision making and impose some restric-tions on preferences having different histories.

Axiom 10 (Dynamic Consistency). For all h ∈ H, x, y ∈ Z, and c ∈ C,

x %hc y ⇔ {(c, x)} %h {(c, y)}.5This axiom excludes a DM who may care about timing of resolution of multistage lottery. A distinction

between early and late resolution of risk is examined in Kreps and Porteus [18]

10

Dynamic Consistency requires that the ranking between {(c, x)} and {(c, y)} under ahistory h reflects how the DM evaluates x and y under the updated history hc.

Finally, we introduce a key axiom called “Period-Wise Dominance”. Consider twooptions (c, x) and (c′, x′). At an intuitive level, if c is preferred to c′ and x is preferred tox′, then (c, x) is definitely better than (c′, x′) no matter what time preference the DM has.If this is the case, (c, x) is said to dominate (c′, x′), and the DM will never choose suchdominated options. It is reasonable to assume that keeping dominated options within amenu adds no value of flexibility, that is,

{(c, x), (c′, x′)} ∼h {(c, x)}.

We use this idea in Higashi, Hyogo and Takeoka [14] to characterize the history-independentrandom discounting model. However, in the current setup, the immediate consumptionlevel, c, has two effects; one is a direct effect from its consumption and the other is anindirect effect on evaluation of future opportunity sets through habit formation. In orderto define an appropriate dominance notion, we need to separate these two effects.

To illustrate the dominance notion, take any two options (c, x) and (c′, x′). The ranking

{⊗c} %h {⊗c′} (2)

should solely reflect the direct effect of immediate consumption. On the other hand, tocapture the effect of habit formation, let `, `′ ∈ ∆(C) satisfy6

x ∼hc {⊗`} and x′ ∼hc′ {⊗`′}.

The first indifference relation says that at history hc, the DM is indifferent between havingx and committing ` in every time period. We can interpret ⊗` as an smoothing equivalentof x at history hc. Similarly, ⊗`′ is an smoothing equivalent of x′ at history hc′. Since therankings over iid lotteries are assumed to be identical across histories, if

{⊗`} %h {⊗`′}, (3)

then we know that the menu x after consumption c takes place is preferred to the menux′ after consumption c′ takes place. That is, (3) reflects the ranking of the two menus, xand x′, with taking into account habit formation from difference in immediate consumptionlevel, c and c′. We say that (c, x) dominates (c′, x′) if conditions (2) and (3) hold.

Now we extend the above dominance notion to lotteries with finite supports. Fix ahistory h. Take two lotteries l and l′ in ∆(C × Z) with finite supports, which can bedenoted by l = ((ci, xi), λi)

ni=1 and l′ = ((c′i, x

′i), λ

′i)

mi=1. Let lc and l′c denote the marginal

distributions of l and l′ on C, respectively. Since a comparison between ⊗lc and ⊗l′csolely reflects static risk attitude, the ranking {⊗lc} %h {⊗l′c} presumably means that l ispreferred to l′ in terms of current consumption.

6Under the other axioms, each menu x ∈ Z admits a smoothing equivalent. See Appendix B for details.

11

On the other hand, let ⊗`i (resp. ⊗`′i) be a smoothing equivalent of xi (resp. x′i) at

history hci (resp. hc′i). Since the DM receives xi at history hci with probability λi, by defi-nition of smoothing equivalent, she should be indifferent between this situation and insteadreceiving ⊗`i with probability λi. Moreover, under Independence and IID-Separability, thelatter would be indifferent to receive its reduced lottery

∑ni=1 λi`i every time period. Hence,

the ranking {⊗(∑n

i=1 λi`i)} %h {⊗(∑m

i=1 λ′i`

′i)} reveals that l is preferred to l′ in terms of

future opportunity sets with taking into account the effect of habit formation from currentconsumption.

Definition 3.1. For all h and l, l′ ∈ ∆(C × Z) with finite supports, say that l dominatesl′ at history h if {⊗lc} %h {⊗l′c} and {⊗(

∑ni=1 λi`i)} %h {⊗(

∑mi=1 λ′

i`′i)}.

Let ∆s(C ×Z) be the set of lotteries with finite supports.

Axiom 11 (Period-Wise Dominance). For all h ∈ H, x ∈ Z, and l, l′ ∈ ∆s(C ×Z), ifl ∈ x and l dominates l′ at history h, then

x ∪ {l′} ∼h x.

Period-Wise Dominance states that the DM should not care about dominated lotteries.Since dominated lotteries give less utilities in the future, both immediate and remote, theyare useless for the DM who has in mind subjective contingencies concerning intertemporaltrade-offs, and hence never chosen over a dominant lottery. As stated in the axiom, addinga dominated lottery to a menu does not change its ranking.

3.2 Representation Results

Now we state our main representation theorem. A proof is relegated to Appendix D.

Theorem 3.1. If a set of preferences {%h}h∈H satisfies Order, Continuity, Independence,Monotonicity, History-Independent Risk-Preference, Nondegeneracy, IID-Separability, Best-Worst, Strong Archimedean, Dynamic Consistency, and Period-Wise Dominance, then itadmits a history-dependent random discounting representation (u, {µ(·|h)}h∈H).

Conversely, for all history-dependent random discounting utilities (u, {µ(·|h)}h∈H), thereexists a unique functional form U(·|·) that satisfies functional equation (1) and the prefer-ence it represents satisfies all the axioms.

The sufficiency is closely related to DLR’s study, and the role of the axioms may bewell understood when compared with their axioms. DLR show that preference over menusof lotteries admits the additive representation of the form that

U(x) =

∫S

maxl∈x

V (l, s) dµ(s),

12

where S is a state space, µ is a non-negative measure on S, and V (·, s) is a state-dependentexpected utility function if and only if preference satisfies Order, Continuity-(i), Indepen-dence, and Monotonicity.7 Thus, the other six axioms mainly play a role to specify a generalstate space S to the set [0, 1] of discount factors, and to convert V (l, s) to the desired form.By History-Independent Risk-Preference, u is independent of h, and by IID-Separability, itsatisfies mixture linearity. Best-Worst, Dynamic Consistency, and Period-Wise Dominancejointly deliver the desired recursive structure of V across different histories. By Nodegen-eracy, u is non-constant, and Strong Archimedean implies that αh is uniformly boundedfrom one. Continuity-(ii) ensures that the representation is continuous in histories.

The next result concerns uniqueness of the representation. A proof can be found inAppendix E.

Theorem 3.2. If history-dependent random discounting representations, U and U ′, withcomponents (u, {µ(·|h)}h∈H) and (u′, {µ′(·|h)}h∈H) respectively, represent the same set ofpreferences {%h}h∈H , then:

(i) u and u′ are cardinally equivalent; and

(ii) µ(·|h) = µ′(·|h) for all h ∈ H.

Theorem 3.2 pins down a subjective probability measure µ(·|h) over the set of futurediscount factors, which is interpreted as a belief over subjective states. Thus, our result isin contrast to Kreps [16, 17] and DLR, where probability measures over subjective statesare not identified because the ex post utility functions are state-dependent, and hence,probabilities assigned to those states can be manipulated arbitrarily. Here, it is a combi-nation of the additive recursive structure and the normalization of discount factors thatensures uniqueness of subjective beliefs. The argument is the same as in Higashi, Hyogo,and Takeoka [14].

3.3 Special Case: Dependence on Average Consumption

In the history-dependent random discounting model, beliefs over future time preferencedepend on histories of past consumption in an arbitrary way. In the existing habit models,this history dependence has been often specified to dependence on the average consumptionin a history. In this subsection, we provide such a specification.

From now on, let C = [0,M ] for some M > 0. The metric on C is understood to bed(c, c′) = |c − c′|. For some discount factor δ ∈ (0, 1), define the average consumption athistory h = (· · · , c2, c1) by

ch = (1 − δ)∞∑

t=1

δt−1ct.

If there exists δ ∈ (0, 1) such that for all h, h′ ∈ H,

ch = c′h =⇒ %h=%h′ ,7Dekel, Lipman, Rustichini, and Sarver [6] fill a gap in DLR surrounding this representation result.

13

then {%h}h∈H is said to be an average-dependent preference.For all c ∈ C, let c denote the history where c has been consumed in every time period,

that is, c = (· · · , c, c, c). For all h = (ct)∞t=1, h

′ = (c′t)∞t=1 ∈ H, the λ-mixture between h and

h′ is defined asλh + (1 − λ)h′ ≡ (λct + (1 − λ)c′t)

∞t=1.

For all h ∈ H, define its equivalence class by

[h] ≡ {h′ ∈ H | %h′=%h}. (4)

These equivalence classes define an equivalence relation on H. We impose the followingconditions on this equivalence relation.

Axiom 12 (Sensitivity). For all c, c′ ∈ C with c 6= c′, c′ /∈ [c].

This axiom requires that preferences differ between at histories c = (· · · , c, c) andc′ = (· · · , c′c′).

Axiom 13 (Smoothing). For all h ∈ H, there exists c ∈ C such that c ∈ [h].

If %h is the same as %c for some c ∈ C, c is interpreted as a “smoothed” history ofh, which induces the same habit as at the original history h. Thus, the Smoothing axiomrequires that each history admits its smoothed history.

Axiom 14 (Continuity). If hn → h and cn → c with cn ∈ [hn], then c ∈ [h].

This is a technical requirement, and says that the equivalence relation satisfies upperhemi-continuity.

Axiom 15 (Linearity). For all h, h′ ∈ H, c, c′ ∈ C, and λ ∈ [0, 1],

c ∈ [h] and c′ ∈ [h′] =⇒ λc + (1 − λ)c′ ∈ [λh + (1 − λ)h′].

This axiom states that a mixed history between two histories h and h′ admits a smoothedhistory which is the mixture between their own smoothed histories.

Axiom 16. There exists c ∈ C such that if c ∈ [cc] and c′ ∈ [c′c], then

c > c′ ⇐⇒ c > c′.

Axiom 17. For all h ∈ H and c, c1 ∈ C,

c ∈ [h] ⇐⇒ cc1 ∈ [hc1].

The above two axioms capture “stationarity”. Since cc and c′c share the same consump-tion level in the immediate past, the difference between their smoothed histories shouldreflect the difference between c and c′. A similar explanation can be applied to Axiom 17.

14

Axiom 18 (Separability). Suppose that c ∈ [hc1], c′ ∈ [hc′1], c ∈ [h′c1], and c′ ∈ [h′c′1].Then, (i) c > c′ =⇒ c > c′, and (ii) c > c =⇒ c′ > c′.

To interpret part (i), let c > c′. By assumption, c (resp. c′) is a smoothed history of hc1

(resp.hc′1). Since hc1 and hc′1 differ only in the immediate past, c > c′ suggests c1 > c′1,which in turn implies that h′c1 corresponds to a higher level of past consumption than thatof h′c′1. Thus, their smoothed histories, c and c′ should satisfy c > c′. A similar argumentcan be applied to part (ii).

Theorem 3.3. If the equivalence relation given as in (4) satisfy Axioms 12-18 if and onlyif {%h} is an average-dependent preference.

This result is obtained by translating the axioms for stationary cardinal utilities ofEpstein [8] into the equivalence relation on histories. See Appendix F for a formal proof.

4 Comparative Impatience across Histories

We would like to analyze the situation where a DM is more patient at some history thanat another. In case of deterministic discounting, the more patient the DM is, the closerto one of her discount factor. We provide a behavioral comparison about attitudes towardflexibility and characterize intuitive properties of subjective beliefs.

Consider a DM having history-dependent preference {%h}h∈H . Suppose that {%h}h∈H

admits a history-dependent random discounting representation. Let

LP ≡ {` ⊗ {⊗`′} ∈ L | {⊗`′} %h {⊗`} for some `, `′ ∈ ∆(C)},LI ≡ {` ⊗ {⊗`′} ∈ L | {⊗`} %h {⊗`′} for some `, `′ ∈ ∆(C)}.

A multi-stage lottery l ∈ LP (LI) gives the DM a “worse (better)” lottery in the immediatefuture and a “better (worse)” lottery for the rest of the horizon. Since {%h}h∈H satisfiesHistory-Independent Risk Preference, LP and LI are independent of h. Define

ZP ≡ K(LP ) ⊂ Z and ZI ≡ K(LI) ⊂ Z.

Definition 4.1. A DM is more patient at history h1 than at history h2 if the following twoconditions hold:

(P1) For all x ∈ ZP and l ∈ LI ,

x Âh2 {l} =⇒ x Âh1 {l}.

(P2) For all l ∈ LP and x ∈ ZI ,

{l} Âh2 x =⇒ {l} Âh1 x.

15

(P1) states that if the DM at history h2 strictly prefers a menu x consisting of lotterieswith later but larger rewards to a commitment lottery {l} yielding the immediate butsmaller reward, then so does she at history h1. (P2) states that if the DM at history h2

strictly prefers a commitment lottery {l} yielding a later but larger reward to a menu xconsisting of lotteries with the immediate but smaller rewards, then so does she at historyh1.

We consider the implication of the behavioral comparison expressed by Definition 4.1.We show that Definition 4.1 characterizes the increasing convex and concave order onsubjective beliefs.

Definition 4.2. Consider probability measures µ1 and µ2 over [0, 1]. Say that µ2 is smallerthan µ1 in the increasing convex [concave] order if, for all continuous, increasing and convex[concave] functions v : [0, 1] → R,8∫

[0,1]

v(α) dµ2(α) ≤∫

[0,1]

v(α) dµ1(α). (5)

One immediate observation is that Eµ2 [α] ≤ Eµ1 [α] if µ2 is smaller than µ1 in theincreasing convex [concave] order because v(α) = α is increasing and convex [concave].Thus, these orders imply that the belief µ1 is more patient on average than the belief µ2.

The increasing convex and concave orders have a close relation to other stochastic ordersused in economic analysis. A probability measure µ1 on [0, 1] is said to dominate µ2 in thefirst-order stochastic dominance if the inequality (5) holds for all increasing functions. Sim-ilarly, µ1 is said to dominate µ2 in the second-order stochastic dominance if the inequality(5) holds for all continuous concave functions. By definition, the domination according tothe increasing convex [concave] order is a generalization of the first-order stochastic domi-nance. Moreover, if µ1 dominates µ2 in the second-order stochastic dominance, then µ2 issmaller (resp. larger) than µ1 in the increasing concave (resp. convex) order.

In the literature of stochastic orders, some equivalent conditions of Definition 4.2 areinvestigated. Let Fi be the cumulative distribution function of a probability measure µi

on [0, 1] and F i(α) = 1 − Fi(α) be its survival distribution function. From Shaked andShanthikumar [22, p.182], the condition that µ2 is smaller than µ1 in the increasing convexorder is equivalent to the following condition: for all t ∈ [0, 1],∫ 1

t

F 2(α) dα ≤∫ 1

t

F 1(α) dα.

On the other hand, the condition that µ2 is smaller than µ1 in the increasing concave orderis equivalent to the following: for all t ∈ [0, 1],∫ t

0

F2(α) dα ≥∫ t

0

F1(α) dα.

8Notice that continuity is not redundant because a concave or convex function is continuous in theinterior of the domain. In Hadar and Russell [13] and Rothschild and Stiglitz [20], continuity is notimposed.

16

Furthermore, from Shaked and Shanthikumar [22, p.197], if Eµ1 [α] = Eµ2 [α], then theincreasing convex [concave] order reduces to the convex [concave] order.9

The increasing concave order has been applied in economics. In choice under risk, thepreferences of risk averse DMs can be expressed in terms of the increasing concave order. SeeHadar and Russell [13], and Rothschild and Stiglitz [20]. Moreover, in Shorrocks [24], theincreasing concave order is shown to be equivalent to the generalized Lorenz order whichhas been used in ranking income distributions; µ2 is smaller than µ1 in the generalizedLorenz order: for all k ∈ [0, 1],∫ k

0

F−12 (t)dt ≤

∫ k

0

F−11 (t)dt,

where Fi is the distribution function of µi and F−1i = sup{x|Fi(x) ≤ k}.

As an illustration of the increasing concave order, following Ramos, Ollero, and Sordo[19], consider the Gamma distribution with density

f(x) =λα

Γ(α)xα−1e−λx, x > 0,

where λ > 0, α > 0, and Γ(·) denotes the complete Gamma function. If λ1 > λ2 and α1

λ1≥

α2

λ2, F2 is smaller than F1 in the generalized Lorenz order, or equivalently in the increasing

concave order. This example shows that unlike the first-order stochastic dominance, twodistribution functions with single crossing property may be ranked by the increasing concaveorder.

We may now provide a characterization result.

Theorem 4.1. Assume that {%h}h∈H satisfies all the axioms of Theorem 3.1 and admits ahistory-dependent random discounting representation (u, {µ(·|h)}h∈H). Then the followingconditions are equivalent:

(a) The DM is more patient at history h1 than at history h2.

(b) µ(·|h2) is smaller than µ(·|h1) in the increasing convex and concave orders.

A proof is relegated to Appendix G. The intuition behind (a) ⇒ (b) in case of theincreasing convex order is as follows: (P1) implies that U(x|h1) ≥ U(x|h2) for all x ∈ ZP .Notice that for all x ∈ ZP , the function

maxl∈x

∫C×Z

((1 − α)u(c) + αU(z|hc)) dl(c, z) = maxl∈x

((1 − α)u(`′) + αu(`′′)) , (6)

where l = `′ ⊗ {⊗`′′}, is increasing and convex in α because of {⊗`′′} %h {⊗`′}. Hencethe ranking U(x|h1) ≥ U(x|h2) means that the integral of an increasing convex function

9We say that µ2 is smaller than µ1 in the convex [concave] order if the inequality (5) holds for allcontinuous convex [concave] functions.

17

of the form (6) with respect to µ1 is always bigger than that corresponding to µ2. As thelast step, we show that any continuous, increasing, and convex function v on [0, 1] can bearbitrarily approximated by a function of the form (6) if an affine transformation of u ischosen appropriately.

The following are some examples of Theorem 4.1.

Example 1 (Fisher (1930)). Let C be a compact interval of R+. Suppose that {%h}h∈H

is an average-dependent preference with a discount factor δ ∈ (0, 1). Define the averageconsumption of history h = (· · · , c2, c1) by

ch = (1 − δ)∞∑

t=1

δtct.

Fisher’s discussion corresponds to the case that the DM is more patient at h than at historyh′ if ch < ch′ . That is, the DM tends to be more patient if the average consumption up tothat time period is lower maybe because of the habit of thrift.

Example 2 (Becker and Mulligan (1997)). Let C = A × B, where A ⊂ R+ is a setof regular consumptions and B ⊂ R+ is a set of investment actions. Their assumptioncorresponds to the case that the DM is more patient at h = (· · · , (a1, b1)) than at historyh′ = (· · · , (a′

1, b′1)) if b1 > b′1.

In the history-independent random discounting model, Higashi, Hyogo, and Takeoka [14]provide an inter-personal comparison toward flexibility which is associated with the second-order stochastic dominance on subjective beliefs over discount factors. Their condition isstronger than Definition 4.1 (P1): agent 1 is said to be more averse to commitment thanagent 2 if for all x ∈ K(L) and l ∈ L,

x Â2 {l} =⇒ x Â1 {l}. (7)

Unlike Definition 4.1, this comparison is not restricted to the trade-offs between soonerbut smaller rewards and later but larger rewards. Condition (7) is shown to be equivalentto the condition that µ2 dominates µ1 in the second-order stochastic dominance, which isin turn equivalent to saying that µ2 is smaller than µ1 in the convex order. Theorem 4.1shows that (P1) is equivalent to the condition that µ(·|h2) is smaller than µ(·|h1) in theincreasing convex order.

The notion of comparative impatience has been already examined in a different frame-work from the current paper. Benoit and Ok [3] provide a comparison of the attitudetowards time delay under the assumptions that agents have additively separable utilityfunctions over streams of deterministic outcomes and that their discount functions are notnecessarily exponential. Consider two agents, agent 1 and agent 2, facing the same endow-ment stream. In Benoit and Ok [3], agent 1 is more delay averse than agent 2 if agent2 prefers receiving small additional consumption at an earlier period to large additionalconsumption at a later period, so does agent 1.

18

Finally, one might wonder what is the gap between Theorem 4.1 and characterizingthe first-order stochastic dominance on subjective beliefs. A difficulty of behavioral char-acterization of the first order stochastic dominance arises from the fact that an increasingcontinuous function may not be approximated by the sum of an increasing convex and anincreasing concave function. See Thon and Thorlund-Petersen [25, Lemma 1] for details.

Appendices

A Hausdorff Metric

Let X be a compact metric space with a metric d. Let K(X) be the set of all non-emptycompact subsets of X. For x ∈ X and A, B ∈ K(X), let

d(x, B) ≡ minx′∈B

d(x, x′), d(A,B) ≡ maxx∈A

d(x,B).

For all A,B ∈ K(X), define the Hausdorff metric dH by

dH(A,B) ≡ max[d(A,B), d(B,A)].

Then, dH satisfies (i) dH(A,B) ≥ 0, (ii) A = B ⇔ dH(A, B) = 0, (iii) dH(A,B) = dH(B,A),and (iv) dH(A,B) ≤ dH(A,C)+dH(C,B). Moreover, K(X) is compact under the Hausdorffmetric.

B Some Implications of Period-Wise Dominance

B.1 Dominance Operator for Finite Menus

Fix any history h. If %h is restricted on iid lotteries, it induces a preference over ∆(C),denoted by %∗. By History-Independent Risk Preference, %∗ is independent of h. ByOrder, Continuity, Independence, and IID-separability, %∗ satisfies all the VNM axioms.Thus, there exists a continuous mixture linear representation v : ∆(C) → R. Let `+ and`− be a maximal and a minimal lottery with respect to v. We can assume that v(`−) = 0.

Take any c ∈ C and x ∈ Z. By Best-Worst, {⊗`+} %hc x %hc {⊗`−}. By Continuity,Independence and IID-Separability, there exists a unique λ(x, hc) ∈ [0, 1] such that

x ∼hc {⊗(λ(x, hc)`+ + (1 − λ(x, hc))`−)}. (8)

Hence, all x ∈ Z admits a smoothing equivalent at history hc.Now we can define a dominance operation for all l ∈ ∆s(C × Z) and all finite subsets

x ⊂ ∆s(C ×Z) as follows:

Dsh(l) ≡ {l′ ∈ ∆s(C ×Z) | l dominates l′ at history h},

Dsh(x) ≡ ∪l∈xcl(D

sh(l)),

19

where cl(·) is the closure operation in ∆(C × Z). That is, Dsh(x) is the set of all lotteries

dominated by some lottery in x at history h (and their accumulation points). Since Dsh(x)

is a nonempty compact set in ∆(C ×Z), it is a well-defined choice object.The following lemma shows that the DM does not care about dominated lotteries. Since

x ⊂ Dsh(x) when %h satisfies Order, the DM having preference for flexibility weakly prefers

Dsh(x) to x. Thus, the lemma is a counterpoint to Monotonicity, and shows that it is not

useful to keep dominated lotteries (and their accumulation points) within the menu, thatis, x %h Ds

h(x).

Lemma B.1. For all h ∈ H and all finite menus x ⊂ ∆s(C ×Z), x ∼h Dsh(x).

Proof. First, take any menu x ∈ Z such that there exists l ∈ x with finite support. Wewill claim that x ∪ cl(Ds

h(l)) ∼h x. As a first step, take any finite menu y ⊂ Dsh(l). By

applying Period-Wise Dominance finite times, we have x ∪ y ∼h x. Next, since cl(Dsh(l))

is compact, by Lemma 0 of Gul and Pesendorfer [10, p.1421], there exists a sequenceyn ⊂ ∆(C × Z) such that each yn is finite with yn ⊂ cl(Ds

h(l)) and yn → cl(Dsh(l)). Since

yn is finite, it can be denoted by yn = {ln1 , · · · , lnKn}. By definition of closure, we can find

yn ≡ {ln1 , · · · , lnKn} ⊂ Ds

h(l) such that d(lni , lni ) < 1n

for all i = 1, · · · , Kn. By definition ofHausdorff metric dH , dH(yn, yn) → 0 as n → 0. By the triangle inequality,

dH(yn, cl(Dsh(l))) ≤ dH(yn, yn) + dH(yn, cl(Ds

h(l))).

Thus, dH(yn, cl(Dsh(l))) → 0 as n → 0. Since the first step implies x ∪ yn ∼h x, by

Continuity, x ∪ cl(Dsh(l)) ∼h x as desired.

Now take any finite menu x ⊂ ∆s(C ×Z). By applying the above claim finite times,

Dsh(x) = ∪l∈xcl(D

sh(l)) = x ∪ (∪l∈xcl(D

sh(l))) ∼h x.

B.2 Extension of Dominance Operator to General Menus

We consider a dominance operator for a general menu x ∈ Z as we now describe below.Fix any history h ∈ H. Recall a smoothing equivalent of x as given by (8). Definefh : C ×Z → ∆(C) by

fh(c, z) ≡ λ(x, hc)`+ + (1 − λ(x, hc))`−

and Fh : ∆(C ×Z) → ∆(∆(C)) by

Fh(l) ≡ l ◦ f−1h ,

that is, Fh(l)(G) = l(f−1h (G)) for all Borel subsets G of ∆(C). Lemma B.2 (i) and (ii)

below show that fh is continuous, and Fh is continuous and mixture linear.

20

Finally, let E : ∆(∆(C)) → ∆(C) be the reduction operator, defined as the functionassigning η ∈ ∆(∆(C)) with its reduced lottery in ∆(C), that is, E(η) ∈ ∆(C) satisfies∫

C

g(c) dE(η)(c) =

∫∆(C)

∫C

g(c) d`(c) dη(`)

for all continuous functions g : C → R. For example, if η =∑

i λi ◦ `i ∈ ∆(∆(C)) denotesthe lottery yielding `i with probability λi, then E(η) =

∑i λi`i ∈ ∆(C). See Lemma B.2

(iii) and (iv) for properties of E.For all h and l ∈ ∆(C ×Z), define lh ≡ E(Fh(l)) ∈ ∆(C).

Definition B.1. For all h and l, l′ ∈ ∆(C × Z), say that l dominates l′ at history h if{⊗lc} %h {⊗l′c} and {⊗lh} %h {⊗l′h}.

For all h ∈ H, l ∈ ∆(C ×Z), and x ∈ Z, let

Dh(l) ≡ {l′ ∈ ∆(C ×Z) | l dominates l′ at history h},Dh(x) ≡ ∪l∈xDh(l).

That is, Dh(x) is the set of all lotteries dominated by some lottery in x at history h. Asshown in Lemma B.2 (viii), Dh(x) coincides with Ds

h(x) if x ⊂ ∆s(C × Z) is finite. Thus,Dh is regarded as an extension of the dominance operator to general menus.

B.3 Properties of Dominance Operator

Lemma B.2. Fix an arbitrary h ∈ H.

(i) fh : C ×Z → ∆(C) is continuous.

(ii) Fh : ∆(C ×Z) → ∆(∆(C)) is continuous and mixture linear.

(iii) E : ∆(∆(C)) → ∆(C) is well-defined, continuous, and mixture linear.

(iv) For all `i ∈ ∆(C) and λi ∈ [0, 1], i = 1, · · · , n, with∑

i λi = 1, E(∑

i λ◦`i) =∑

i λi`i.

(v) For all x ∈ Z, Dh(x) ∈ Z.

(vi) For all x ∈ Z, if x is convex, so is Dh(x).

(vii) Dh : Z → Z is Hausdorff continuous.

(viii) For all finite menus x ⊂ ∆s(C ×Z), Dh(x) = Dsh(x).

(ix) For all x ∈ Z, x ∼h Dh(x).

21

Proof. (i) Let (cn, zn) → (c, z). We want to show that fh(cn, zn) → fh(c, z). Suppose

otherwise. Then, there exists an open neighborhood O of fh(c, z) such that fh(cn, zn) /∈ O

for infinitely many n. Let {fh(cm, zm)} be a subsequence satisfying fh(c

m, zm) /∈ O. Since∆(C) is compact, we can assume that fh(c

m, zm) → ` for some ` ∈ ∆(C). Since zm ∼hcm

{⊗fh(cm, zm)}, Dynamic Consistency implies that {(cm, zm)} ∼h {(cm, {⊗fh(c

m, zm)})}.By Continuity, {(c, z)} ∼h {(c, {⊗`})}. Again, by Dynamic Consistency, z ∼hc {⊗`}, Sinceλ(z, hc) is unique, we have fh(c, z) = `, which contradicts the fact that ` /∈ O.

(ii) From (i), fh is continuous. Continuity of Fh follows from Billingsley [4, p.20].To show that Fh is mixture linear, take any lotteries l, l′ ∈ ∆(C ×Z) and λ ∈ [0, 1]. By

definition, for all Borel subsets G of ∆(C),

Fh(λl + (1 − λ)l′)(G) = (λl + (1 − λ)l′)(f−1h (G)) = λl(f−1

h (G)) + (1 − λ)l′(f−1h (G))

= λFh(l)(G) + (1 − λ)Fh(l′)(G).

(iii) For all η ∈ ∆(∆(C)) and real-valued continuous functions g on C, define

T (g) ≡∫

∆(C)

∫C

g d` dη.

Since T is a bounded linear functional on the set of all real-valued continuous functions onC, the Riesz representation theorem ensures that there exists a unique E(η) ∈ L such that

T (g) =

∫C

g dE(η).

Hence E is well-defined.To show continuity, let ηn → η. We want to show that E(ηn) → E(η). Suppose

otherwise. Then, there exists an open neighborhood O of E(η) such that E(ηn) /∈ Ofor infinitely many n. Let {E(ηm)} be a subsequence satisfying E(ηm) /∈ O for all m.Since ∆(C) is compact, we can assume that {E(ηm)} converges to a limit `∗ ∈ ∆(C). Bydefinition, for all real-valued continuous functions g on C,∫

C

g dE(ηm) =

∫∆(C)

∫C

g d` dηm. (9)

As m → ∞, we have ∫C

g d`∗ =

∫∆(C)

∫C

g d` dη.

Since E is well-defined, E(η) = `∗, which contradicts the fact that `∗ /∈ O.Finally, we show mixture linearity of E. Take any η, η′ ∈ ∆(∆(C)) and λ ∈ [0, 1]. By

definition, ∫C

g dE(λη + (1 − λ)η′) =

∫∆(C)

∫C

g d` d(λη + (1 − λ)η′)(`). (10)

22

The RHS of (10) is equal to

λ

∫∆(C)

∫C

g d` dη + (1 − λ)

∫∆(C)

∫C

g d` dη′ = λ

∫C

g dE(η) + (1 − λ)

∫C

g dE(η′)

=

∫C

g d(λE(η) + (1 − λ)E(η′)).

Thus, for all continuous functions g on C,∫C

g dE(λη + (1 − λ)η′) =

∫C

g d(λE(η) + (1 − λ)E(η′)).

Since E is well-defined, E(λη + (1 − λ)η′) = λE(η) + (1 − λ)E(η′).

(iv) For all continuous functions g on C,∫C

g dE(∑

i

λi ◦ `i) =

∫∆(C)

∫C

g d` d(∑

i

λi ◦ `i) =∑

i

λi

∫C

g d`i =

∫C

g d(∑

i

λi`i).

Thus, E(∑

i λi ◦ `i) =∑

i λi`i.

(v) Since ∆(C × Z) is compact, it is enough to show that Dh(x) is closed. Take anysequence ln ∈ Dh(x) converging to a limit l ∈ ∆(C ×Z). There exists l

n ∈ x such that

{⊗ln

c } %h {⊗lnc } and {⊗E(Fh(ln))} %h {⊗E(Fh(l

n))}.

Since x is compact, without loss of generality we assume that ln → l for some limit l ∈ x.

Since ⊗lnc → ⊗lc, ⊗ln

c → ⊗lc and since Fh and E are continuous, the Continuity axiomimplies that

{⊗lc} %ht {⊗lc} and {⊗E(Fh(l))} %h {⊗E(Fh(l))}.Thus l ∈ x dominates l at history h and, hence, l ∈ Dh(x).

(vi) Take any two points l, l′ ∈ Dh(x) and λ ∈ [0, 1]. There exist l, l′ ∈ x such that l

dominates l, and l′dominates l′. Let lλ ≡ λl + (1 − λ)l′ and l

λ ≡ λl + (1 − λ)l′. Since

x is convex, lλ ∈ x. We show that l

λdominates lλ, which results in lλ ∈ Dh(x). Since

{⊗lc} %h {⊗lc} and {⊗l′c} %h {⊗l′c}, Independence and IID-Separability imply that

{⊗lλ

c} ∼h {λ ⊗ lc + (1 − λ) ⊗ l′c} %h {λ ⊗ lc + (1 − λ) ⊗ l′c} ∼h {⊗lλ}.

Moreover, since Fh and E are mixture linear, Independence and IID-Separability implythat

{⊗E(Fh(lλ))} = {⊗(λE(Fh(l)) + (1 − λ)E(Fh(l

′)))}

∼h {λ ⊗ E(Fh(l)) + (1 − λ) ⊗ E(Fh(l′))}

%h {λ ⊗ E(Fh(l)) + (1 − λ) ⊗ E(Fh(l′))}

∼h {⊗(λE(Fh(l)) + (1 − λ)E(Fh(l′)))} = {⊗E(Fh(l

λ))}.

23

Thus lλ

dominates lλ.

(vii) Let xn → x. We want to show Dh(xn) → Dh(x). By contradiction, suppose

otherwise. Then, there exists a neighborhood U of Dh(x) such that Dh(x`) /∈ U for infinitely

many `. Let {x`}∞`=1 be the corresponding subsequence of {xn}∞n=1. Since xn → x, {x`}∞`=1

also converges to x. Since {Dh(x`)}∞`=1 is a sequence in a compact metric space Z, there

exists a convergent subsequence {Dh(xm)}∞m=1 with a limit y 6= Dh(x). As a result, now we

have xm → x and Dh(xm) → y. In the following argument, we will show that y = Dh(x),

which is a contradiction.

Step 1: Dh(x) ⊂ y.

Take any l ∈ Dh(x). Then, there exists l ∈ x such that {⊗lc} %h {⊗lc} and {⊗lh} %h

{⊗lh}. Since xm → x, we can find a sequence {lm}∞m=1 such that lm ∈ xm and lm → l.Now we will construct a sequence {lm}∞m=1 with lm ∈ Dh(x

m) satisfying lm → l. Forall sufficiently large k, let B1/k(l) be the 1/k-neighborhood of l with respect to the weakconvergence topology. There exists 0 < λk < 1 such that lk ≡ λkl+(1−λk)⊗ `− ∈ B1/k(l).By construction, lk → l.

By Independence and IID-Separability, {⊗lc} Âh {⊗lkc} if {⊗lc} Âh {⊗`−}, and{⊗lc} ∼h {⊗lkc} if {⊗lc} ∼h {⊗`−}. In the case of former, since lm → l, by Continu-ity, there exists mk

1 such that for all m ≥ mk1, {⊗lmc } Âh {⊗lkc}. In the case of latter, for all

m, {⊗lmc } %h {⊗`−} ∼h {⊗lkc}. In both cases, we have {⊗lmc } %h {⊗lkc} for all m ≥ mk1.

On the other hand, by Lemma B.2 (ii) and (iii), {⊗lh} Âh {⊗lkh} if {⊗lh} Âh {⊗`−},and {⊗lh} ∼h {⊗lkh} if {⊗lh} ∼h {⊗`−}. Similarly, since lm → l, by Lemma B.2 (ii)and (iii), there exists mk

2 such that for all m ≥ mk2, {⊗lmh } %h {⊗lkh}. Therefore, for all

m ≥ mk ≡ max[mk1,m

k2],

{⊗lmc } %h {⊗lkc}, {⊗lmh } %h {⊗lkh},

that is, lk ∈ Dh(lm) ⊂ Dh(x

m) for all m ≥ mk. Since we have mk+1 ≥ mk for all k, definelm ≡ lk for all m satisfying mk ≤ m < mk+1. Then, {lm}∞m=1 is a required sequence.

Since lm → l and Dh(xm) → y with lm ∈ Dh(x

m), we have l ∈ y. Thus, Dh(x) ⊂ y.

Step 2: y ⊂ Dh(x).

Take any l ∈ y. Since Dh(xm) → y, we can find a sequence lm ∈ Dh(x

m) with lm → l.By definition, there exists lm ∈ xm such that {⊗lmc } %h {⊗lmc } and {⊗lmh } %h {⊗lmh }.Since ∆(C × Z) is compact, we can assume that {lm} converges to a limit l ∈ ∆(C × Z).Since lm → l and xm → x with lm ∈ xm, we have l ∈ x. From Continuity and Lemma B.2(ii) and (iii), {⊗lc} %h {⊗lc} and {⊗lh} %h {⊗lh}. Thus, l ∈ Dh(x).

(viii) It suffices to show that for all l ∈ ∆s(C ×Z), Dh(l) = cl(Dsh(l)) because then

Dsh(x) = ∪l∈xcl(D

sh(l)) = ∪l∈xDh(l) = Dh(x).

Step 1: Let l = ((ci, xi), λi)ni=1 ∈ ∆s(C × Z) and ⊗`i is a smoothing equivalent of xi at

history hci. Then, {⊗(∑

i λi`i)} ∼h {⊗E(Fh(l))}.

24

Since ⊗fh(ci, xi) is also a smoothing equivalent of xi at history hci, {⊗`i} ∼h {⊗fh(ci, xi)}.By Independence and IID-Separability, {⊗(

∑i λi`i)} ∼h {⊗(

∑i λifh(ci, xi))}. Thus, by

part (iv),

{⊗(∑

i

λi`i)} ∼h {⊗(∑

i

λifh(ci, xi))} = {⊗E(∑

i

λi ◦ fh(ci, xi))} = {⊗E(Fh(l))}.

Step 2: cl(Dsh(l)) ⊂ Dh(l).

Take any l′ ∈ Dsh(l). Since l has a finite support, it can be rewritten as l = ((ci, xi); λi)

ni=1.

By definition, there exists a smoothing equivalent ⊗`i of xi at history hci. Similarly, l′ iswritten as l′ = ((c′i, x

′i); λ

′i)

mi=1, and let ⊗`′i be a smoothing equivalent of x′

i at historyhc′i. Then, by Step 1, {⊗(

∑i λi`i)} %h {⊗(

∑i λ

′i`

′i)} is equivalent to {⊗E(Fh(l))} %h

{⊗E(Fh(l′))}, and hence l′ ∈ Dh(l). Since Dh(l) is closed as shown in part (v), cl(Ds

h(l)) ⊂Dh(l) as desired.

Step 3: Dh(l) ⊂ cl(Dsh(l)).

As a preliminary, we first show the following:Step 3-1: For all l ∈ ∆(C × Z), there exists a sequence ln ∈ ∆s(C × Z) converging to lsuch that the support of ln is included in that of l.

Let E denote the support of l. Since C × Z is compact, E is also compact. For all n,there exists a finite partition {En

i }Mni=1 of E such that the diameter of En

i is smaller than1/n. Take (c, x)n

i ∈ Eni arbitrarily. Let ln be the probability measure with probability

l(Eni ) at the point (c, x)n

i . Take any uniformly continuous function g : C × Z → R. Letαn

i = infEni

g and βni = supEn

ig. Since g is uniformly continuous and since the diameter of

Eni converges to zero as n → ∞ uniformly in i, supi(β

ni − αn

i ) → 0 as n → ∞. Thus, asn → ∞, ∣∣∣∣∫ g dln −

∫g dl

∣∣∣∣ =

∣∣∣∣∣∑i

∫En

i

(g − g((c, x)ni )) dl

∣∣∣∣∣ ≤ supi

(βni − αn

i ) → 0.

Hence, ln → l as desired.

To show that Dh(l) ⊂ cl(Dsh(l)), take any l′ ∈ Dh(l). By definition, {⊗lc} %h {⊗l′c}

and {⊗lh} %h {⊗l′h}. Define ln = n−1n

l′ + 1n(⊗`−) for each n. By Step 3-1, we can find

ln ∈ ∆s(C ×Z) such that d(ln, ln) < 1n. Then, ln → l′ because d(ln, l′) ≤ d(ln, ln) + d(ln, l′)

and d(ln, ln) → 0 and d(ln, l′) → 0.To show that {⊗lc} %h {⊗lnc }, we consider several cases as below.Case 1: {⊗lc} Âh {⊗l′c}. Since ln → l′, by Continuity, we have {⊗lc} Âh {⊗lnc }.Case 2: {⊗l′c} Âh {⊗`−}. We have lnc = n−1

nl′c + 1

n`−. By Independence and IID-

Separability, {⊗l′c} Âh {⊗lnc }. Thus, by Continuity, {⊗lc} Âh {⊗lnc }.

25

Case 3: {⊗lc} ∼h {⊗l′c} ∼h {⊗`−}. By assumption, v(l′c) = v(`−) = 0. Let fc :C ×Z → C denote the projection mapping on C. Since

0 = v(l′c) =

∫v(c) dl′c =

∫C

v(c) d(l′ ◦ f−1c )(c)

=

∫C×Z

v(fc(c, x)) dl′(c, x) =

∫C×Z

v(c) dl′(c, x),

by Theorem 11.16 (3) of Aliprantis and Border [1, p.412], v(c) = 0 almost surely withrespect to l′. That is, for all (c, x) within the support of l′, {⊗c} ∼h {⊗`−}. Since ln has afinite support, ln is written as ln = ((cn

i , xni ), λn

i )mi=1. By Independence and IID-Separability,

{⊗lnc } = {⊗(∑

i

λifc(cni , xn

i ))} ∼h {⊗`−}.

Hence, for all n, {⊗lc} ∼h {⊗lnc }.Next, we show {⊗lh} %h {⊗lnh} by a case-by-case argument.Case 1: {⊗lh} Âh {⊗l′h}. Since ln → l′, by the Continuity axiom and continuity of E

and Fh, we have {⊗lh} Âh {⊗lnh}.Case 2: {⊗l′h} Âh {⊗`−}. Since E and Fh are mixture linear, lnh = E(Fh(l

n)) =n−1

nl′h + 1

n`−. By Independence and IID-Separability, {⊗l′h} Âh {⊗lnh}. Thus, by the

Continuity axiom and continuity of E and Fh, {⊗lh} Âh {⊗lnh}.Case 3: {⊗lh} ∼h {⊗l′h} ∼h {⊗`−}. By assumption, v(l′h) = v(`−) = 0. Since

0 = v(l′h) =

∫v(c) dE(Fh(l

′)) =

∫∆(C)

∫C

v(c) d` d(l′ ◦ f−1h )(`)

=

∫C×Z

∫C

v(c) dfh(c, x) dl′(c, x) =

∫C×Z

v(fh(c, x)) dl′(c, x),

by Theorem 11.16 (3) of Aliprantis and Border [1, p.412], v(fh(c, x)) = 0 almost surelywith respect to l′. That is, for all (c, x) within the support of l′, {⊗fh(c, x)} ∼h {⊗`−}.Since ln has a finite support, ln is written as ln = ((cn

i , xni ), λn

i )mi=1. By Independence and

IID-Separability,

{⊗lnh} = {⊗(∑

i

λifh(cni , x

ni ))} ∼h {⊗`−}.

Hence, for all n, {⊗lh} ∼h {⊗lnh}.Therefore, ln ∈ Ds

h(l), and hence l′ ∈ cl(Dsh(l)).

(ix) First we will claim that for all x ∈ Z, there exists a sequence xn ⊂ ∆s(C × Z)such that xn is finite and xn → x. By Lemma 0 of Gul and Pesendorfer [10, p.1421], thereexists a sequence xn ⊂ ∆(C ×Z) such that each xn is finite and xn → x. Since xn is finite,it can be denoted by xn = {ln1 , · · · , lnKn

}. Since the set of lotteries with finite supportsis dense in ∆(C × Z) (see Aliprantis and Border [1, p.513, Theorem 15.10]), we can find

26

xn ≡ {ln1 , · · · , lnKn} such that d(lni , lni ) < 1

nfor all i = 1, · · · , Kn. By definition of Hausdorff

metric dH , dH(xn, xn) → 0 as n → 0. By the triangle inequality,

dH(xn, x) ≤ dH(xn, xn) + dH(xn, x).

Thus, dH(xn, x) → 0 as n → 0.Now we show that for all x ∈ Z, x ∼h Dh(x). Take any x ∈ Z. By the above claim,

there exists a sequence xn ⊂ ∆s(C ×Z) such that xn is finite and xn → x. By Lemma B.1and property (viii), Dh(x

n) = Dsh(x

n) ∼h xn. By the Continuity axiom and continuity ofDh, Dh(x) ∼h x as desired.

C Preliminaries

Lemma C.1. Let X be a compact metric space. Let fn, f : X → R be continuous functionsand pn, p be Borel probability measures on X. If fn → f with respect to the sup-norm andpn → p with respect to the weak convergence topology, then

∫fn dpn →

∫f dp.

Proof. For all n, we have∣∣∣∣∫ fn dpn −∫

f dp

∣∣∣∣ ≤∣∣∣∣∫ fn dpn −

∫f dpn

∣∣∣∣ +

∣∣∣∣∫ f dpn −∫

f dp

∣∣∣∣≤

∫|fn − f | dpn +

∣∣∣∣∫ f dpn −∫

f dp

∣∣∣∣≤ sup

x∈X|fn(x) − f(x)| +

∣∣∣∣∫ f dpn −∫

f dp

∣∣∣∣ . (11)

By assumption, (11) converges to zero as n → ∞.

Lemma C.2. Let L be a compact metric space and f : L → RN be a continuous function.For all A ∈ K(L), define f(A) ≡ {f(l) ∈ RN | l ∈ A}. If An → A with An, A ∈ K(L), thenf(An) → f(A) with respect to the Hausdorff topology.

Proof. First of all, since L is compact and f is continuous, f(L) is a compact subset of RN .Since f(A) is a compact subset of f(L), we have f(A) ∈ K(f(L)) for all A ∈ K(L).

To show that f(An) → f(A) with respect to the Hausdorff metric, by contradiction,suppose otherwise. Then, there exists a neighborhood O of f(A) such that f(Am) /∈ Ofor infinitely many m. Let {Am}∞m=1 be the corresponding subsequence of {An}∞n=1. SinceAn → A, {Am}∞m=1 also converges to A. Since {f(Am)}∞m=1 is a sequence in a compactmetric space K(f(L)), there exists a convergent subsequence {f(A`)}∞`=1 with a limit V 6=f(A). As a result, now we have A` → A and f(A`) → V .

In the following argument, we will show that f(A) = V , which is a contradiction. Toshow f(A) ⊂ V , take any a ∈ f(A). There exists l ∈ A such that a = f(l). Since A` → A,we can find {l`}∞`=1 such that l` → l with l` ∈ A`. Let a` ≡ f(l`). Since a` → a andf(A`) → V with a` ∈ f(A`), we have a ∈ V . Thus, f(A) ⊂ V .

27

To show V ⊂ f(A), take any a ∈ V . Since f(A`) → V , we can find {a`}∞`=1 such thata` → a with a` ∈ f(A`). There exists l` ∈ A` satisfying a` = f(l`). Since L is compact,there exists a convergent subsequence {lk}∞k=1 with a limit l. By continuity of f , f(l) = a.Moreover, since lk → l, Ak → A with lk ∈ Ak, we have l ∈ A. Thus a ∈ f(A), whichimplies V ⊂ f(A).

D Proof of Theorem 3.1

D.1 Necessity

History-Independent Risk Preference: First notice that for all h,

U({⊗`}|h) =

∫[0,1]

∫C

((1 − α)u(c) + αU({⊗`}|hc)) d`(c)dµ(α|h)

= (1 − αh)u(`) + αh

∫U({⊗`}|hc) d`(c),

where u(`) =∫

Cu(c) d`(c). Hence, for all h,

U({⊗`}|h) = u(`)

{1 − αh + αh

∫1d` + αh

∫ (−αhc + αhc

∫1d`

)d`(c)

+αh

∫αhc

(∫ (−αhcc′ + αhcc′

∫1d`

)d`(c′)

)d`(c) + · · ·

}= u(`).

Thus, for all `, `′ and all histories h, h′,

U({⊗`}|h) > U({⊗`′}|h) ⇔ u(`) > u(`′) ⇔ U({⊗`}|h′) > U({⊗`′}|h′).

We show that for all (u, {µ(·|h)}), there exists U(·|h) satisfying the functional equation.Let W be the Banach space of all real-valued continuous functions on Z × H with thesup-norm metric. For all (x, h) ∈ Z × H and W ∈ W , define

T (W )(x, h) ≡∫

[0,1]

maxl∈x

∫C×Z

((1 − α)u(c) + αW (z, hc)

)dl(c, z)dµ(α|h). (12)

Lemma D.1. T (W ) ∈ W for all W ∈ W.

Proof. We want to show that if (xn, hn) → (x0, h0), then T (W )(xn, hn) → T (W )(x0, h0).For all (l, h) ∈ ∆(C × Z) × H, define f(l, h) ∈ R2 by f1(l, h) ≡ u(lc) and f2(l, h) ≡∫

C×Z W (z, hc) dl. It is easy to see that f1 : ∆(C ×Z) × H → R is continuous.

Step 1: f2 : ∆(C ×Z) × H → R is continuous.

28

First ,we will claim that if hn → h, then sup(c,z)∈C×Z |W (z, hnc)−W (z, hc)| → 0. Sup-pose otherwise. Then, there exists ε > 0 such that sup(c,z)∈C×Z |W (z, hmc) − W (z, hc)| ≥ε for infinitely many m. There exists a corresponding sequence {(cm, zm)} satisfying|W (zm, hmcm) − W (zm, hcm)| ≥ ε

2for all m. Since C ×Z is compact, we can assume that

{(cm, zm)} converges to a limit (c, z). Since W is continuous, |W (zm, hmcm)−W (zm, hcm)| →|W (z, hc) − W (z, hc)| = 0, which is a contradiction. By Lemma C.1, f2 is continuous.

Step 2: For all (x, h) ∈ Z × H, let f(x, h) ≡ {f(l, h) ∈ R2 | l ∈ x}. Then, f(xn, hn) →f(x0, h0) with respect to the Hausdorff metric.

Let H∗ ≡ {hn}∞n=1 ∪ {h0}. Since hn → h0, H∗ is a compact subset of H. Sincef = (f1, f2) : ∆(C ×Z)×H → R2 is a continuous function, the desired result follows fromLemma C.2.

Notice that

σ(α; xn, hn) ≡ maxl∈xn

∫C×Z

((1 − α)u(c) + αW (z, hnc)

)dl(c, z)

= maxl∈xn

(1 − α)u(lc) + α

∫C×Z

W (z, hnc) dl = max(a1,a2)∈f(xn,hn)

(1 − α)a1 + αa2.

By Lemma C.2 and the same argument of HHT[Lemma C.4 (i), Step 2], we can show that

supα∈[0,1]

|σ(α; xn, hn) − σ(α; x0, h0)| → 0

as n → ∞. Thus, by Lemma C.1, T (W )(xn, hn) → T (W )(x0, h0).

By Lemma D.1, the operator T : W → W is well-defined. To show that T is acontraction mapping, it suffices to verify that (i) T is monotonic, that is, T (W ) ≥ T (V )whenever W ≥ V , and (ii) T satisfies the discounting property, that is, there exists δ ∈ [0, 1)such that for any W and β ≥ 0, T (W + β) ≤ T (W ) + δβ (see Aliprantis and Border [1,p.97, Theorem 3.53]).

Step 1: T is monotonic.

Take any W,V ∈ W with W ≥ V . Since (1 − α)u(c) + αW (z, hc) ≥ (1 − α)u(c) +αV (z, hc) for all (c, z, h) ∈ C ×Z × H and and α ∈ [0, 1], we have∫

C×Z((1 − α)u(c) + αW (z, hc))dl(c, z) ≥

∫C×Z

((1 − α)u(c) + αV (z, hc))dl(c, z)

for all l, and hence, T (W )(x, h) ≥ T (V )(x, h).

Step 2: T satisfies the discounting property.

Let δ ≡ sup{αh |h ∈ H} < 1. For all W ∈ W and β ≥ 0,

T (W + β)(x, h) =

∫[0,1]

maxl∈x

∫C×Z

((1 − α)u(c) + α(W (z, htc) + β))dl(c, z)dµ(α|h)

= T (W )(x, h) + αhβ ≤ T (W )(x, h) + δβ.

29

By Steps 1 and 2, T is a contraction mapping. Thus, the fixed point theorem (See Gul andPesendorfer [12, Lemma 6]) ensures that there exists a unique U ∈ W satisfying U = T (U).This U satisfies the equation (1).

D.2 Sufficiency

Let co(x) denote the closed convex hull of x. As in DLR, Order, Continuity, and Indepen-dence imply x ∼h co(x). Hence we can pay attention to the sub-domain

Zc ≡ {x ∈ Z|x = co(x)}.

Since Zc is a mixture space, Order, Continuity, and Independence ensure that %h can berepresented by a continuous mixture linear function U(·|h) : Zc → R.

For all h and ` ∈ ∆(C), let u(`|h) ≡ U({⊗`}|h). By definition, u(·|h) is continuous andrepresents %h on ⊗∆c.

Lemma D.2. u(·|h) is mixture linear.

Proof. Take all `, `′ ∈ ∆(C) and λ ∈ [0, 1]. Since U(·|h) is mixture linear and %h satisfiesIID-Separability,

u(λ` + (1 − λ)`′|h) = U({⊗(λ` + (1 − λ)`′)}|h) = U({λ ⊗ ` + (1 − λ) ⊗ `′}|h)

= λU({⊗`}|h) + (1 − λ)U({⊗`′}|h) = λu(`|h) + (1 − λ)u(`′|h).

Lemma D.3. History-dependent continuous mixture linear representations {U(·|h)}h∈H

can be taken to satisfyU({⊗`}|h) = U({⊗`}|h′)

for all h, h′ ∈ H.

Proof. Fix a history h arbitrarily. By History Independent Risk Preference, for all h, %h and%h are equivalent on ⊗∆c. By Lemma D.2, u(·|h) and u(·|h) are continuous mixture linearfunctions that represent the same preference over ∆(C). Thus there exist β(h) > 0 andγ(h) ∈ R satisfying u(`|h) = β(h)u(`|h)+γ(h). Let U ′(x|h) ≡ (U(x|h)−γ(h))/β(h). Then,U ′(·|h) is a continuous mixture linear function representing %h. Moreover, by definition,for all ` ∈ ∆(C), U ′({⊗`}|h) = u(`|h). Redefine U ′(·|h) as U(·|h).

Fix a history h arbitrarily. Let u(`) ≡ u(`|h). By Lemma D.3, u(`) = u(`|h) for all h.By Nondegeneracy, u is not constant. Since u is continuous and ∆(C) is compact, thereexist a maximal and a minimal lottery `+, `− ∈ ∆(C) with respect to u. We can normalizeu(`−) = 0.

Lemma D.4. For all η ∈ ∆(∆(C)),

U({⊗E(η)}|h) =

∫∆(C)

U({⊗`}|h) dη(`).

30

Proof. For all η ∈ ∆(∆(C)), there exists a sequence {ηn} converging to η such that eachη has a finite support. Thus let ηn be denoted by (λn

1 ; `n1 , · · · , λn

mn; `n

mn) with

∑i λ

ni = 1.

Since U({⊗·}|h) is mixture linear,

U({⊗E(ηn)}|h) = U({⊗(∑

i

λni lni )}|h) =

∑i

λni U({⊗`n

i }|h) =

∫L

U({⊗`}|h) dηn.

Since U({⊗·}|h) and E are continuous, we have the desired result as ηn → η.

Lemma D.5. (i) For all l ∈ ∆(C ×Z) and h,

U({⊗lh}|h) =

∫C×Z

U(z|hc) dl(c, z).

(ii) For all l ∈ ∆(C ×Z) and h, h′ ∈ H,

U({⊗lh}|h) = U({⊗lh}|h′).

Proof. (i) By definition, z ∼hc {⊗fh(c, z)}. Since U({⊗`}|h) = U({⊗`}|h′) = u(`) for allhistories h, h′ ∈ H and ` ∈ ∆(C), we have

U(z|hc) = U({⊗fh(c, z)}|h′)

for all histories h′. By Lemma D.4,

U({⊗lh}|h) = U({⊗E(Fh(l))}|h) =

∫∆(C)

U({⊗`′}|h) dFh(l)(`′)

=

∫∆(C)

U({⊗`′}|h) dl ◦ f−1h (`′) =

∫C×Z

U({⊗fh(c, z)}|h) dl(c, z)

=

∫C×Z

U(z|hc) dl(c, z).

(ii) Since fh(c, z) ∈ ∆(C), U({⊗fh(c, z)}|h) = U({⊗fh(c, z)}|h′) for all h, h′ ∈ H.Hence

U({⊗E(Fh(l))}|h) =

∫C×Z

U({⊗fh(c, z)}|h) dl(c, z)

=

∫C×Z

U({⊗fh(c, z)}|h′) dl(c, z) = U({⊗E(Fh(l))}|h′).

LetZh ≡ {x ∈ Zc |x = Dh(x)}.

31

From Lemma B.2 (vi), Zh is compact. Moreover, Lemma B.2 (iv) and (v) imply that allx ∈ Zh are compact and convex.

Let C([0, 1]) be the set of all continuous functions on [0, 1] with the sup-norm. For allx ∈ Zh, define the function σ : Zh → C([0, 1]) by

σx(α) ≡ maxl∈x

(1 − α)u(lc) + αU({⊗lh}|h) (13)

for all α ∈ [0, 1]. By Lemma D.5 (i), (13) can be rewritten as

maxl∈x

∫C×Z

((1 − α)u(c) + αU(z|hc)

)dl(c, z).

LetVh(x) ≡ {(u,w) ∈ R2

+ |u = u(lc), w = U({⊗lh}|h) for some l ∈ x}.Lemma D.6. Vh(x) is compact and convex.

Proof. Notice that u is continuous and mixture linear. Moreover, by Lemma B.2 (ii) and(iii), U({⊗E(Fh(·))}|h) is continuous and mixture linear. Since Vh(x) is the image of acompact convex set under a continuous and mixture linear function, Vh(x) is compact andconvex.

Let

uM ≡ max{u(lc) ∈ R+ | l ∈ x}, wM ≡ max{U({⊗lh}|h) ∈ R+ | l ∈ x}.

Lemma D.7. (i) (0, 0) ∈ Vh(x), (ii) (uM , 0) ∈ Vh(x), and (iii) (0, wM) ∈ Vh(x).

Proof. (i) Since any lottery dominates {⊗`−}, {⊗`−} ∈ Dh(x) = x. Notice that u(`−) = 0and U({⊗E(Fh(⊗`−))}|h) = U({⊗`−}|h) = u(`−) = 0. That is, (0, 0) ∈ Vh(x).

(ii) There exists l ∈ x with u(lc) = uM . Take the lottery l∗ ≡ lc ⊗ {⊗`−} ∈ ∆(C × L).Then, u(l∗c) = u(lc) = uM and

U({⊗E(Fh(l∗))}|h) =

∫C×Z

U({⊗fh(c, z)}|h) d(lc ⊗ {⊗`−})

=

∫C

U({⊗fh(c, {⊗`−})}|h) dlc =

∫C

U({⊗`−}|hc) dlc = 0.

Since u(lc) = u(l∗c) and U({⊗E(Fh(l))}|h) ≥ U({⊗E(Fh(l∗))}|h), l dominates l∗, and hence

l∗ ∈ Dh(x) = x. Therefore, (uM , 0) ∈ Vh(x).(iii) There exists l ∈ x satisfying U({⊗lh}|h) = wM . Take the lottery l∗ ≡ `−⊗{⊗lh} ∈

∆(C × L). Then, u(l∗c) = u(`−) = 0. Moreover, by Lemma D.5 (ii),

U({⊗E(Fh(l∗))}|h) =

∫C×Z

U({⊗fh(c, z)}|h) d(`− ⊗ {⊗lh})

=

∫C

U({⊗fh(c, {⊗lh})}|h) d`− =

∫C

U({⊗lh}|hc) d`−

=

∫C

U({⊗lh}|h) d`− = U({⊗lh}|h) = U({⊗E(Fh(l))}|h).

32

Since u(lc) ≥ u(l∗c) = 0 and U({⊗E(Fh(l))}|h) = U({⊗E(Fh(l∗))}|h), l dominates l∗, and

hence l∗ ∈ Dh(x) = x. Therefore, (0, wM) ∈ Vh(x).

Lemma D.8. Take any point (u,w) ∈ Vh(x). If u ≥ u′ ≥ 0 and w ≥ w′ ≥ 0, then(u′, w′) ∈ Vh(x).

Proof. Since uM ≥ u and wM ≥ w, Lemmas D.6 and D.7 imply that (u, 0), (0, w) ∈ Vh(x).Again, by Lemmas D.6 and D.7,

w′

w(u,w) +

(1 − w′

w

)(u, 0) = (u,w′) ∈ Vh(x),

w′

w(0, w) +

(1 − w′

w

)(0, 0) = (0, w′) ∈ Vh(x).

Henceu′

u(u,w′) +

(1 − u′

u

)(0, w′) = (u′, w′) ∈ Vh(x).

Lemma D.9. σ : Zh → C([0, 1]) is injective.

Proof. Take x, x′ ∈ Zh with x 6= x′. Without loss of generality, assume x 6⊂ x′. Takel ∈ x \ x′. Let u = u(lc) and w = U({⊗lh}|h).

We will claim that ({(u, w)}+ R2+)∩Vh(x

′) = ∅. Suppose otherwise. Then, there existsl′ ∈ x′ such that u(l′c) ≥ u and U({⊗l′h}|h) ≥ w. By Lemma D.8, (u, w) ∈ Vh(x

′), that is,l ∈ Dh(x

′) = x′. This is a contradiction.By Lemma D.6 and the above claim, the separating hyperplane theorem ensures that

there exists α ∈ [0, 1] and γ ∈ R such that (1 − α)u + αw > γ > (1 − α)u′ + αw′ for all(u′, w′) ∈ Vh(x

′). Equivalently,

(1 − α)u(lc) + αU({⊗lh}|h) > γ > (1 − α)u(l′c) + αU({⊗l′h}|h),

for all l′ ∈ x′. Hence,

σx(α) = maxl∈x

(1 − α)u(lc) + αU({⊗lh}|h) ≥ (1 − α)u(lc) + αU({⊗lh}|h)

> maxl′∈x′

(1 − α)u(l′c) + αU({⊗l′h}|h) = σx′(α).

Therefore, σx 6= σx′ .

Lemma D.10. (i) σ is continuous.

(ii) For all x, y ∈ Zh and λ ∈ [0, 1], λσx + (1 − λ)σy = σDh(λx+(1−λ)y).

33

Proof. Since u and U({⊗E(Fh(·))}|h) are continuous and mixture linear by Lemma B.2(ii) and (iii), the desired result follows from the same argument of Lemma C.4 (i) and(ii) of HHT. Replace V (x) and W in Lemma C.4 (i) and (ii) of HHT by Vh(x) andU({⊗E(Fh(·))}|h) respectively.

Let Ch ⊂ C([0, 1]) be the range of σ.

Lemma D.11. (i) Ch is convex.

(ii) The zero function belongs to Ch.

(iii) The constant function equal to a positive number c > 0 belongs to Ch.

(iv) The supremum of any two points f, f ′ ∈ Ch belongs to Ch. That is, max[f(α), f ′(α)]belongs to Ch.

(v) For all f ∈ Ch, f ≥ 0.

Proof. (i) Take any f, f ′ ∈ Ch and λ ∈ [0, 1]. There are x, x′ ∈ Z2 satisfying f = σx andf ′ = σx′ . From Lemma D.10 (ii),

λf + (1 − λ)f ′ = λσx + (1 − λ)σx′ = σDh(λx+(1−λ)x′) ∈ Zh.

Hence, Ch is convex.(ii) Let x ≡ Dh({⊗`−}) ∈ Zh. Then, for all α,

σx(α) = maxl∈Dh({⊗`−})

(1 − α)u(lc) + αU({⊗E(Fh(l))}|h)

= (1 − α)u(`−) + αU({⊗E(Fh(⊗`−))}|h) = (1 − α)u(`−) + αU({⊗`−}|h) = 0.

(iii) Take a lottery ¯∈ ∆(C) such that u(¯) > 0. Let c ≡ u(¯). Since U({⊗E(Fh(⊗`+))}|h) =u(`+), (u(`+), u(`+)) ∈ Vh(Dh({⊗`+})). Since u(`+) ≥ u(¯), Lemma D.8 ensures that(c, c) ∈ Vh(Dh({⊗`+})). Hence, there exists l∗ ∈ Dh({⊗`+}) such that u(l∗c) = c andU({⊗E(Fh(l

∗))}|h) = c. Let x ≡ Dh({l∗}) ∈ Zh. Then, for all α,

σx(α) = maxl∈x

(1 − α)u(lc) + αU({⊗E(Fh(l))}|h)

= (1 − α)u(l∗c) + αU({⊗E(Fh(l∗))}|h) = c.

(iv) There exist x′, x ∈ Zh such that f = σx and f ′ = σx′ . Let f ′′ ≡ σDh(co(x∪x′)) ∈ Ch.Then, f ′′(α) = max[σx(α), σx′(α)].

(v) There exists x ∈ Zh such that f = σx. Since Dh({⊗`−}) ⊂ x, Lemma D.11 (ii)implies f(α) = σx(α) ≥ σDh({⊗`−})(α) = 0, for any α.

Define Th : Ch → R by Th(f) ≡ U(σ−1(f)|h). By Lemma D.9, Th is well-defined.Notice that Th(0) = 0 and Th(c) = c, where 0 and c are identified with the zero functionand the constant function equal to c > 0, respectively. Since U(·|h) and σ is continuousand mixture linear, so is Th.

34

By the same argument as in DLR and Dekel, Lipman, Rustichini, and Sarver [6], we willextend T to C([0, 1]) step by step. For any r ≥ 0, let rCh ≡ {rf |f ∈ Ch} and Hh ≡ ∪r≥0rCh.For any f ∈ Hh \ 0, there is r > 0 satisfying (1/r)f ∈ Ch. Define Th(f) ≡ rTh((1/r)f).By the same argument of C.6 of HHT, we can show that Th is linear on Ch, which impliesthat Th(f) is well-defined. It is easy to see that Th on Hh is mixture linear. By the sameargument of C.6 of HHT, Th is also linear.

LetH∗

h ≡ Hh − Hh = {f1 − f2 ∈ C([0, 1])|f1, f2 ∈ Hh}.For any f ∈ H∗

h, there are f1, f2 ∈ Hh satisfying f = f1−f2. Define Th(f) ≡ Th(f1)−Th(f2).We can verify that Th : H∗

h → R is well-defined because of linearity of Th on Hh.

Lemma D.12. H∗h is dense in C([0, 1]).

Proof. From the Stone-Weierstrass theorem, it is enough to show that (i) H∗h is a vector

sublattice, (ii) H∗h separates the points of [0, 1]; that is, for any two distinct points α, α′ ∈

[0, 1], there exists f ∈ H∗h with f(α) 6= f(α′), and (iii) H∗

h contains the constant functionsequal to one. By the exactly same argument as Lemma 11 (p.928) in DLR, (i) holds.To verify condition (ii), take α, α′ ∈ [0, 1] with α 6= α′. Without loss of generality, α′ >α. Take a lottery ` ∈ ∆(C) such that u(`) > 0. Since U({⊗E(Fh(⊗`+))}|h) = u(`+),(u(`+), u(`+)) ∈ Vh(Dh({⊗`+})). Since u(`+) ≥ u(`), Lemma D.8 ensures that (u(`), 0) ∈Vh(Dh({⊗`+})). Hence, there exists l∗ ∈ Dh({⊗`+}) such that U({⊗E(Fh(l

∗))}|h) = 0and u(l∗c) = u(`). Let x ≡ Dh({l∗}) ∈ Zh. Then, σx ∈ Ch ⊂ H∗

h and

σx(α) = (1 − α)u(l∗c) + αU({⊗E(Fh(l∗))}|h) = (1 − α)u(l∗c)

> (1 − α′)u(l∗c) = (1 − α′)u(l∗c) + α′U({⊗E(Fh(l∗))}|h) = σx(α

′).

Finally, condition (iii) directly follows from Lemma D.11 (iii) and the definition of Hh.

By adopting the same argument as in Theorem 2 of Dekel, Lipman, Rustichini, andSarver [6], we can show that there exists a constant K > 0 such that Th(f) ≤ K‖f‖ forany f ∈ H∗

h. See Lemma C.8 of HHT for more details.By the Hahn-Banach theorem, we can extend Th : H∗

h → R to T h : C([0, 1]) → R in alinear, continuous and increasing way. Since H∗

h is dense in C([0, 1]) by Lemma D.12, thisextension is unique.

Since T h is a positive linear functional on C([0, 1]), the Riesz representation theoremensures that there exists a unique countably additive probability measure µ(·|h) on [0, 1]satisfying

T h(f) =

∫[0,1]

f(α) dµ(α|h),

for all f ∈ C([0, 1]). Thus, by Lemma D.5 (i), we have

U(x|h) = T h(σ(x)) =

∫[0,1]

maxl∈x

((1 − α)u(lc) + αU({lh}|h)) dµ(α|h)

=

∫[0,1]

maxl∈x

∫C×Z

((1 − α)u(c) + αU(z|hc)) dl(c, z)dµ(α|h).

35

Lemma D.13. sup{αh |h ∈ H} < 1.

Proof. Take any `, `′ with U({⊗`}|h) > U({⊗`′}|h). By Strong Archimedean, there exists`′′ ∈ ∆(C) such that U({⊗`}|h) > U({⊗`′′}|h) ≥ U({`′⊗{⊗`}|h). Since U({`′⊗{⊗`}|h) =(1 − αh)u(`′) + αhu(`) > u(`′) = U({⊗`′}|h), we have u(`) > u(`′′) > u(`′). There existsλ ∈ (0, 1) such that λu(`) + (1 − λ)u(`′) > u(`′′), and hence, λu(`) + (1 − λ)u(`′) >(1 − αh)u(`′) + αhu(`). The latter condition implies (u(`) − u(`′))(λ − αh) > 0. Sinceu(`) − u(`′) > 0, we have αh < λ. This property holds for all h, and hence, we havesup{αh |h ∈ H} ≤ λ < 1.

Lemma D.14. For all x ∈ Z, U(x|·) : H → [0, 1] is continuous.

Proof. Since U({⊗`}|h) = u(`) for all ` ∈ ∆(C), by Continuity-(ii), {h ∈ H |U(x|h) ≥u(`)} and {h ∈ H |u(`) ≥ U(x|h)} are closed. Since %h satisfies Order, this is equivalentto say that {h ∈ H |U(x|h) < u(`)} and {h ∈ H |u(`) < U(x|h)} are open for all ` ∈ ∆(C).Since u(`) varies over [0, 1], this property says that the inverse images of all subbasic opensets of [0, 1] are open in H. By Dugundji [7, p.83, Theorem 10.1], U(x|·) is continuous.

Let L1 = {` ⊗ {⊗`′} ∈ L | `, `′ ∈ ∆(C)}. For x ∈ K(L1) ⊂ Z, define the functionσ : K(L1) → C([0, 1]) by

σx(α) = max`⊗{⊗`′}∈x

(1 − α)u(`) + αu(`′).

Denote C1 = {σx |x ∈ K(L1)}, H1 = ∪r≥0rC1, and H∗1 = H1 − H1.

Lemma D.15. For all f ∈ H∗1 ,

∫f(α) dµ(α | ·) : H → R is continuous.

Proof. Since u(`) = U({⊗`} |h) for all h ∈ H, we have for all x ∈ K(L1),∫σx(α) dµ(α |h) =

∫max

`⊗{⊗`′}∈x(1 − α)u(`) + αU({⊗`′} |h) dµ(α |h) = U(x |h).

Let f ∈ H∗1 . Then there exists r, r′ ≥ 0 and x, x′ ∈ K(L1) such that f = rσx − r′σx′ .

Thus,∫

f(α) dµ(α | ·) = rU(x | ·) − r′U(x′ | ·), and the result directly follows from LemmaD.14.

Lemma D.16. H∗1 is dense in C([0, 1]).

Proof. The same argument as the proof of Lemma D.12 applies. We show only that (ii)H∗

1 separates the points of [0, 1], and (iii) H∗1 contains the constant functions equal to one.

For (ii), let x = {`+ ⊗ {⊗`−}}, where `+ and `− are a maximal and a minimal lotterywith respect to u. Notice that u(`+) = 1 and u(`−) = 0 by normalization. Then σx ∈ H∗

1

and σx(α) = (1−α)u(`+) separates the points. For (iii), σ{⊗`+} is the required function.

Lemma D.17. µ(· |h) is continuous with respect to h.

36

Proof. Suppose hn, h ∈ H and hn → h. Let f ∈ C([0, 1]). For abbreviation, denoteµn = µ(· |hn) and µ = µ(· |h). We show that

∫f dµn →

∫f dµ.

Let ε > 0. From Lemma D.16, we can take fε ∈ H∗1 satisfying sup |fε−f | < ε

3, and thus∣∣∣∣∫ fε dµn −

∫f dµn

∣∣∣∣ ≤ ∫|fε − f | dµn <

ε

3, and∣∣∣∣∫ fε dµ −

∫f dµ

∣∣∣∣ ≤ ∫|fε − f | dµ <

ε

3.

Moreover, from Lemma D.15, there exists n such that for all n ≥ n,∣∣∣∣∫ fε dµn −∫

fε dµ

∣∣∣∣ <ε

3.

Therefore, we have for all n ≥ n,∣∣∣∣∫ f dµn −∫

f dµ

∣∣∣∣<

∣∣∣∣∫ f dµn −∫

fε dµn

∣∣∣∣ +

∣∣∣∣∫ fε dµn −∫

fε dµ

∣∣∣∣ +

∣∣∣∣∫ fε dµ −∫

f dµ

∣∣∣∣<

ε

3+

ε

3+

ε

3= ε.

E Proof of Theorem 3.2

(i) Since mixture linear functions u and u′ represent the same preference over ∆(C), by thestandard argument, u′ is rewritten as an affine transformation of u. That is, u and u′ arecardinally equivalent.

(ii) From (i), there exist γ > 0 and ζ ∈ R such that u′ = γu + ζ. Since U(·|h) andU ′(·|h) are mixture linear functions representing the same preference, there exist γh > 0and ζh ∈ R such that U ′(·|h) = γhU(·|h) + ζh. Let xc be the perfect commitment menuto c, that is, xc ≡ {(c, {(c, {· · · })})}. Since U(xc|h) = u(c) and U ′(xc|h) = u′(c), we haveU ′(xc|h) = γU(xc|h) + ζ, which implies γh = γ and ζh = ζ. Now we have

U ′(x|h) =

∫[0,1]

maxl∈x

∫C×Z

((1 − α)u′(c) + αU ′(z|hc)) dl(c, z)dµ′(α|h)

=

∫[0,1]

maxl∈x

∫C×Z

((1 − α)(γu(c) + ζ) + α(γU(z|hc) + ζ)) dl(c, z)dµ′(α|h)

= γ

∫[0,1]

maxl∈x

∫C×Z

((1 − α)u(c) + αU(z|hc)) dl(c, z)dµ′(α|h) + ζ.

37

Hence,

U ′′(x|h) ≡∫

[0,1]

maxl∈x

∫C×Z

((1 − α)u(c) + αU(z|hc)) dl(c, z)dµ′(α|h)

also represents the same preference. Since U ′(·|h) = γU(·|h)+ζ and U ′(·|h) = γU ′′(·|h)+ζ,we must have U(x|h) = U ′′(x|h) for all x. For all x ∈ Z and α ∈ [0, 1], let

σx(α) ≡ maxl∈x

∫C×Z

((1 − α)u(c) + αU(z|hc)) dl(c, z).

Then,

U(x|h) =

∫σx(α) dµ(α|h) =

∫σx(α) dµ′(α|h) = U ′′(x|h). (14)

If x is convex, σx is its support function. Equation (14) holds also when σx is replaced withaσx − bσy for any convex menus x, y and a, b ≥ 0. From Lemma D.12, the set of all suchfunctions is a dense subset of the set of real-valued continuous functions over [0, 1]. Hence,equation (14) holds when σx is replaced with any real-valued continuous function. Hence,the Riesz representation theorem implies µ(·|h) = µ′(·|h).

F Proof of Theorem 3.3

Define the binary relation º∗ on H by

h º∗ h′ ⇐⇒ c ≥ c′ for some c ∈ [h] and c′ ∈ [h′].

For all h ∈ H, Sensitivity and Smoothing ensure that there exists a unique c ∈ C suchthat c ∈ [h]. Define V : H → R+ by V (h) = c according to this property. Then, V is arepresentation of º∗. It is enough to show that V (h) is rewritten as

V (h) = (1 − δ)∞∑

t=1

δt−1ct

for some δ ∈ (0, 1). Indeed, under this claim, for all h, h′ ∈ H,

ch = ch′ = c =⇒ V (h) = V (h′) = c =⇒ c ∈ [h], [h′] =⇒ %h=%h′ .

First of all, V is a functional defined on a mixture space H. We will claim that Vsatisfies all the assumptions of Epstein [8, Theorem 2]. By the Linearity axiom, for all h, h′

and λ ∈ [0, 1],

V (λh + (1 − λ)h′) = λc + (1 − λ)c′ = λV (h) + (1 − λ)V (h′),

where c ∈ [h] and c′ ∈ [h′]. Thus, V is mixture linear.Let hn → h. there exist c, cn ∈ C such that c ∈ [h] and cn ∈ [hn]. We want to show that

V (hn) → V (h). Suppose otherwise. Then, there exists ε > 0 such that |V (hm)−V (h)| ≥ ε

38

for infinitely many m, or equivalently, |cm − c| ≥ ε. Let {hm} denote a correspondingsubsequence of {hn}. Since {cm} is a sequence in C = [0,M ], there exists a subsequence {c`}converging to a point c∗ ∈ C with c∗ 6= c. Let {h`} denote the corresponding subsequence of{hm}. Notice that {h`} also converges to h. Since h` → h and c` → c∗, Continuity impliesthat c∗ ∈ [h], or V (h) = c∗. This contradicts V (h) = c 6= c∗. Thus, V is continuous.

Take any c, c′ ∈ C with c 6= c′. By Sensitivity, V (c) = c 6= c′ = V (c′). Thus,Assumption 1 holds.

Take any h, h′ ∈ H. Let c ∈ [h] and c′ ∈ [h′]. Suppose V (h) = c ≥ c′ = V (h′). ByAxiom, cc ∈ [hc] and c′c ∈ [h′c]. Let c ∈ [cc] and c′ ∈ [c′c]. Since c ≥ c′, Axiom impliesthat c ≥ c′. Since c ∈ [hc] and c′ ∈ [h′c], V (hc) = c ≥ c′ = V (h′c). Thus, Assumption 2holds.

Take any c1, c′1 ∈ C and h, h′ ∈ H. Let V (hc1) ≥ V (h′c1). There exist c, c′ ∈ C such

that c ∈ [hc1] and c′ ∈ [h′c1] with c ≥ c′. Let c, c′ satisfy c ∈ [hc′1] and c′ ∈ [h′c′1]. BySeparability (ii), c ≥ c′, that is, V (hc′1) ≥ V (h′c′1). Thus, Assumption 3 holds.

Take any c1, c′1 ∈ C and h, h′ ∈ H. Let V (hc1) ≥ V (hc′1). There exist c, c′ ∈ C such

that c ∈ [hc1] and c′ ∈ [hc′1] with c ≥ c′. Let c, c′ satisfy c ∈ [h′c1] and c′ ∈ [h′c′1]. BySeparability (i), c ≥ c′, that is, V (h′c1) ≥ V (h′c′1). Thus, Assumption 5 holds.

By Theorem 2 of Epstein [8], there exists δ ∈ (0, 1) and v : C → R such that V (h) =∑∞t=1 δt−1v(ct). For all c ∈ C,

c = V (c) =v(c)

1 − δ.

Thus, v(c) = (1 − δ)c as desired.

G Proof of Theorem 4.1

Lemma G.1. (i) (P1) implies that, for all x ∈ ZP and l ∈ LI , x %h2 {l} ⇒ x %h1 {l}.

(ii) (P2) implies that, for all l ∈ LP and x ∈ ZI , {l} %h2 x ⇒ {l} %h1 x.

Proof. (i) It suffices to show that x ∼h2 {l} ⇒ x %h1 {l}. By assumption, l is denoted by` ⊗ {⊗`′} for some `, `′ with {⊗`} %h2 {⊗`′}. Let `+ and `− be a maximal and minimallottery in ∆(C) with respect to u. If either {⊗`} Âh2 {⊗`−} or {⊗`′} Âh2 {⊗`−}, wehave (1 − αh2)u(`) + αh2u(`′) > (1 − αh2)u(λ` + (1 − λ)`−) + αh2u(λ`′ + (1 − λ)`−) for allλ ∈ (0, 1), that is, x ∼h2 {`⊗ {⊗`′}} Âh2 {(λ` + (1− λ)`−)⊗ {⊗(λ`′ + (1− λ)`−)}} for allλ ∈ (0, 1). By assumption, x Âh1 {(λ`+(1−λ)`−)⊗{⊗(λ`′+(1−λ)`−)}} for all λ ∈ (0, 1).Thus Continuity implies x %h1 {l} as λ → 1. If both {⊗`} and {⊗`′} are indifferent to{⊗`−}, define xλ ≡ {(λ` + (1 − λ)`+) ⊗ {⊗(λ`′ + (1 − λ)`+)} | ` ⊗ {⊗`′} ∈ x} ∈ ZP . Fromthe representation, U(xλ|h2) > U(x|h2) = U({l}|h2) for all λ ∈ (0, 1). By assumption,xλ Âh1 {l}. Thus by Continuity, x %h1 {l}.

(ii) The result follows from the same argument as above.

Lemma G.2. (i) (P1) implies that, for all x ∈ ZP , U(x|h2) ≤ U(x|h1).

39

(ii) (P2) implies that, for all x ∈ ZI , U(x|h1) ≤ U(x|h2).

Proof. (i) Notice that for all ` ∈ ∆(C), U({⊗`}|h1) = u(`) = U({⊗`}|h2). Since {%h}h∈H

satisfies Best-Worst, there exist l, l′ ∈ L such that {l} %h2 x %h2 {l′}. Let `+ and `−

be a maximal and minimal lottery in ∆(C) with respect to u. From the representation,{⊗`} %h2 {l} and {l′} %h2 {⊗`−}. Thus, {⊗`+} %h2 x %h2 {⊗`−}, or equivalently,u(`+) ≥ U(x|h2) ≥ u(`−). By continuity of u, there exists λ ∈ [0, 1] such that U(x|h2) =u(λ`+ + (1 − λ)`−), or x ∼h2 {⊗(λ`+ + (1 − λ)`−)}. Since l ≡ ⊗(λ`+ + (1 − λ)`−) ∈ LI ,x ∼h2 {l} implies that x %h1 {l}. Thus, U(x|h1) ≥ U({l}|h1) = U({l}|h2) = U(x|h2).

(ii) By the same argument as above, we have l ∈ LP such that U({l}|h1) = U({l}|h2)and x ∼h2 {l}. Then, (P2) implies that {l} %h1 x. Thus, U(x|h1) ≤ U({l}|h1) =U({l}|h2) = U(x|h2).

Lemma G.3. Assume that {%h}h∈H satisfies all the axioms of Theorem 3.1 and admits ahistory-dependent random discounting representation (u, {µ(·|h)}h∈H). Then (P1) holds ifand only if (ICX) µ(·|h2) is smaller than µ(·|h1) in the increasing convex order.

Proof. (P1) =⇒ (ICX). We show that, for all continuous, increasing and convex functionsv of α, there is a sequence {vn} of functions such that v ≥ vn,

supα

|v(α) − vn(α)| <1

n, and

∫vn(α) dµ(α|h2) ≤

∫vn(α) dµ(α|h1)

for all n = 1, 2, · · · . Then the result follows from the dominated convergence theorem.Let v : [0, 1] → R be a continuous increasing convex function. Then, for every α ∈ [0, 1],

there exists a vector (pα,1, pα,2) ∈ R2 such that pα,2 ≥ pα,1 and for all α ∈ [0, 1],

v(α) ≥ (1 − α)pα,1 + αpα,2

with equality for α.Fix n. Since v(α) − {(1 − α)pα,1 + αpα,2} is continuous with respect to α, there exists

an open neighborhood B(α) of α such that for every α ∈ B(α)

0 ≤ v(α) − {(1 − α)pα,1 + αpα,2} <1

n.

It follows from the compactness of [0, 1] that there exists a finite set {αi}Mi=1 ⊂ [0, 1] such

that {B(αi)}Mi=1 is a covering of [0, 1].

We define vn : [0, 1] → R by

vn(α) = max1≤i≤M

[(1 − α)pαi,1 + αpαi,2].

Then it is straightforward that v(α) ≥ vn(α) for every α ∈ [0, 1]. Moreover, we see that

supα

|v(α) − vn(α)| <1

n.

40

In fact, pick an arbitrary α ∈ [0, 1]. Then there is j ∈ M such that α ∈ B(αj). This implies

0 ≤ v(α) − vn(α) ≤ v(α) − {(1 − α)pαj ,1 + αpαj ,2} <1

n.

Finally we will see that∫vn(α) dµ(α|h2) ≤

∫vn(α) dµ(α|h1).

Since u(∆(C)) is a closed nondegenerate interval, we can assume, without loss of generality,that there exist `i, `

′i ∈ ∆(C), i = 1, · · · , M , satisfying u(`i) = pαi,1 and u(`′i) = pαi,2. Thus

vn is rewritten as

vn(α) = maxi

(1 − α)u(`i) + αu(`′i).

Now consider the menu xn = {`i⊗{⊗`′i}|i = 1, · · · ,M}. Since u(`′i) = pαi,2 ≥ pαi,1 = u(`i),{⊗`′i} %h {⊗`i} for all h. Thus, by definition, xn ∈ ZP . Then it follows from Lemma G.2that ∫

vn(α) dµ(α|h2) = U(xn|h2) ≤ U(xn|h1) =

∫vn(α) dµ(α|h1),

which completes the proof.(ICX) =⇒ (P1). First, we will claim that for all x ∈ ZP , U(x|h2) ≤ U(x|h1). Let

vx(α) ≡ max`⊗{⊗`′}∈x

(1 − α)u(`) + αu(`′).

Then, vx is convex. Moreover, since u(`′) ≥ u(`), vx is increasing. Since µ(·|h2) is smallerthan µ(·|h1) in the increasing convex order, we have

U(x|h2) =

∫vx(α) dµ(α|h2) ≤

∫vx(α) dµ(α|h1) = U(x|h1).

Now assume U(x|h2) > U({l}|h2) for some x ∈ ZP and l ∈ LI . By definition, l iswritten as ` ⊗ {⊗`′} for some `, `′ with {⊗`} % {⊗`′}. Notice that αh2 ≤ αh1 because ofthe increasing convex order. From the above claim and this observation,

U(x|h1) ≥ U(x|h2) > U({l}|h2) = (1 − αh2)u(`) + αh2u(`′)

≥ (1 − αh1)u(`) + αh1u(`′) = U({l}|h1),

as desired.

Lemma G.4. Assume that {%h}h∈H satisfies all the axioms of Theorem 3.1 and admits ahistory-dependent random discounting representation (u, {µ(·|h)}h∈H). Then (P2) holds ifand only if (ICV) µ(·|h2) is smaller than µ(·|h1) in the increasing concave order.

41

Proof. (P2) =⇒ (ICV ). We show that, for all continuous, decreasing and convex functionsw of α, there is a sequence {wn} of functions such that w ≥ wn,

supα

|w(α) − wn(α)| <1

n, and

∫wn(α) dµ(α|h1) ≤

∫wn(α) dµ(α|h2)

for all n = 1, 2, · · · . Then the result follows from the dominated convergence theorem.Let w : [0, 1] → R be a continuous decreasing convex function. Then, for every α ∈ [0, 1],

there exists a vector (pα,1, pα,2) ∈ R2 such that pα,1 ≥ pα,2 and for all α ∈ [0, 1],

w(α) ≥ (1 − α)pα,1 + αpα,2

with equality for α.By the same argument as Lemma G.3, there exists a finite set {αi}M

i=1 ⊂ [0, 1] such that

wn(α) = max1≤i≤M

[(1 − α)pαi,1 + αpαi,2], and supα

|w(α) − wn(α)| <1

n.

Finally we will see that∫wn(α) dµ(α|h1) ≤

∫wn(α) dµ(α|h2).

By the same argument as Lemma G.3, there exists a menu xn = {`i ⊗{⊗`′i}|i = 1, · · · , M}such that u(`i) = pαi,1 ≥ pαi,2 = u(`′i), or {⊗`i} %h {⊗`′i} for all h. Thus, by definition,xn ∈ ZI . Then it follows from Lemma G.2 that∫

wn(α) dµ(α|h1) = U(xn|h1) ≤ U(xn|h2) =

∫wn(α) dµ(α|h2),

which completes the proof.(ICV ) =⇒ (P2). The result follows from the same argument as in (ICX)=⇒(P1).

References

[1] Aliprantis, C. D., and K. C. Border. Infinite Dimensional Analysis, third edition,Springer, 2006.

[2] Becker, G. S., and C. B. Mulligan. The endogenous determination of time preference.Quarterly Journal of Economics, Vol. 112, 729-758, 1997.

[3] Benoit, J. P., and E. A. Ok. Delay aversion. Theoretical Economics, Vol. 2, 71-113,2007.

[4] Billingsley, P. Convergence of Probability Measures, second edition, Wiley, 1999.

42

[5] Dekel, E., B. Lipman, and A. Rustichini. Representing preferences with a uniquesubjective state space. Econometrica, Vol. 69, 891-934, 2001.

[6] Dekel, E., B. Lipman, A. Rustichini, and T. Sarver. Representing preferences with aunique subjective state space: a corrigendum. Econometrica, Vol. 75, 591-600, 2007.

[7] Dugundji, J. Topology, Allyn and Bacon,INC., Boston, 1966.

[8] Epstein, L, G. Stationary cardinal utility and optimal growth under uncertainty. Jour-nal of Economic Theory, Vol. 31, 133-152, 1983.

[9] Fisher, I. The Theory of Interest, Macmillan, New York, 1930.

[10] Gul, F., and W. Pesendorfer. Temptation and self-control. Econometrica, Vol. 69, 1403-1435, 2001.

[11] Gul, F., and W. Pesendorfer. Self-control and the theory of consumption. Economet-rica, Vol. 72, 119-158, 2004.

[12] Gul, F., and W. Pesendorfer. Harmful addiction. Review of Economic Studies, Vol. 74,147-172, 2007.

[13] Hadar, J., and W. Russell. Rules for ordering uncertain prospects. American EconomicReview, Vol. 59, 25-34, 1969.

[14] Higashi, Y., K. Hyogo, and N. Takeoka. Subjective random discounting and intertem-poral choice. Journal of Economic Theory, Vol. 144, 1015-1053, 2009.

[15] Koopmans, T. C. On flexibility of future preference. In M. W. Shelley and G. L. Bryaneditors, Human Judgments and Optimality, chapter 13. Academic Press, New York,1964.

[16] Kreps, D. M. A representation theorem for preference for flexibility. Econometrica,Vol. 47, 565-578, 1979.

[17] Kreps, D. M. Static choice and unforeseen contingencies. In Dasgupta, P. and Gale,D. and Hart, O. and Maskin, E., editors, Economic Analysis of Markets and Games:Essays in Honor of Frank Hahn, 259-281. MIT Press, Cambridge, MA, 1992.

[18] Kreps, D. M. and Porteus, E. L. Temporal resolution of uncertainty and dynamicchoice theory. Econometrica, Vol. 46, 185-200, 1978.

[19] Ramos, H. M., Ollero, J., and M. A. Sordo. A sufficient condition for generalizedLorenz order. Journal of Economic Theory, Vol. 90, 286-292, 2000.

[20] Rothschild, M., and J. E. Stiglitz. Increasing risk: I. a definition. Journal of EconomicTheory, Vol. 2, 225-243, 1970.

43

[21] Rozen, K. Foundations of intrinsic habit formation. Working paper, 2009.

[22] Shaked, M., and J. G. Shanthikumar. Stochastic Orders, Springer, 2007.

[23] Shi, S., and L. G. Epstein. Habits and time preference. International Economic Review,Vol. 34 61-84, 1993.

[24] Shorrocks, A. F. Ranking income distributions. Economica, Vol. 50, 3-17, 1983

[25] Thon, D., and L. Thorlund-Petersen. Sums of increasing convex and increasing concavefunctions. Operations Research Letters, Vol. 5, 313-316, 1986.

[26] Uzawa, H. Time preference, the consumption function, and optimum asset holdings.In J. N. Wolfe, ed., Capital and Growth: Papers in Honour of Sir John Hicks, Aldine,Chicago, 1968.

44