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Recursive Vector Expected Utility PRELIMINARY DRAFT: comments welcome!!! Latest version: http://faculty.wcas.northwestern.edu/~msi661/recveu.pdf

Marciano Siniscalchi∗ March 27, 2011

Abstract This paper proposes and axiomatizes a recursive version of the vector expected utility (VEU) decision model (Siniscalchi, 2009). Recursive VEU preferences are dynamically consistent and “consequentialist.” Dynamic consistency implies standard Bayesian updating of the baseline (reference) prior in the VEU representation, but imposes no constraint on the adjustment functions and one-step-ahead adjustment factors. This delivers both tractability and flexibility. Recursive VEU preferences are also consistent with a dynamic, i.e. intertemporal extension of atemporal VEU preferences. Dynamic consistency is characterized by a timeseparability property of adjustments—the VEU counterpart of Epstein and Schneider (2003)’s rectangularity for multiple priors. A simple exchangeability axiom ensures that the baseline prior admits a representation à la de Finetti, as an integral of i.i.d. product measures with respect to a unique probability µ. Jointly with dynamic consistency, the same axiom also implies that µ is updated via Bayes’ Rule to provide an analogous representation of baseline posteriors. Finally, an application to a dynamic economy à la Lucas (1978) is sketched.

Keywords: Ambiguity, reference prior, dynamic consistency, exchangeability. ∗ Economics

Department, Northwestern University, Evanston, IL 60208; [email protected].

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1

Introduction

Siniscalchi (2009) (henceforth S09) proposes and axiomatizes the “vector expected utility” decision model—VEU for short. According to this model, the individual evaluates uncertain prospects, or acts, by a process suggestive of anchoring and adjustment (Tversky and Kahneman, 1974). The “anchor” is the expected utility of the prospect under consideration, computed with respect to a baseline probability; the “adjustment” depends upon its exposure to distinct sources of ambiguity, as well as its variation away from the anchor at states that the individual deems ambiguous. Formally, an act f , mapping each state ω ∈ Ω to a consequence x ∈ X , is evaluated via the functional V (f ) = Ep [u ◦ f ] + A

 € Š Ep [ζi · u ◦ f ] 0≤i t , say that A ∈ Ft is ¼t ,ω –null if (f t 0 (ω0 ))t 0 ≥0 = ( f t00 (ω0 ))t 0 ≥0 for all ω0 6∈ A implies that f ∼t ,ω f 0 . The key axiom is the standard consistency requirement, applied to decision trees based upon the filtration F . Axiom 14 (Dynamic Consistency—DC) For every node (t , ω) and f , f 0 ∈ F p : if f τ = f τ0 for all τ ≤ t , and f ¼t +1,ω0 f 0 for all ω0 , then f ¼t ,ω ; and the latter ranking is strict if the former is strict at every ω0 in a ¼t ,ω –non-null event. Finally, ensure that all conditioning events “matter”: Axiom 15 (Full Support–FS) Every A ∈

S t ≥0

Ft is ¼0 –non-null.

Note that, since there is a single time-0 preference, we can write ¼0 in lieu of ¼0,ω . The main characterization result can now be stated. Theorem 1 Fix a preference system (¼t ,ω ) on F p . The following statements are equivalent: (1) Axioms 1–15 hold
(2) (¼t ,ω ) admits a (sharp) recursive VEU representation u , β , p, (n t ,ω , ζt ,ω , A t ,ω )t ≥0,ω∈Ω . € Š (3) (¼t ,ω ) admits a dynamic VEU representation u , β , p, (n˜ t ,ω , ζ˜ t ,ω , A˜t ,ω )t ≥0,ω∈Ω with timeseparable adjustments. € Š In (2), p and β are unique, and if u¯ , β , p, (n¯ t ,ω , ζ¯ t ,ω , A¯t ,ω )t ≥0,ω∈Ω is another recursive VEU representation of (¼t ,ω ), then p 0 = p , u¯ = αu +γ for some α, γ ∈ R with α > 0, and for every (t , ω) 17

β

β

there is a linear surjection Tt ,ω : E ( 1−β u¯ ◦ Ft +1 ; p (·|Ft (ω)), ζ¯ t ,ω ) → E ( 1−β u ◦ Ft +1 ; p (·|Ft (ω)), ζ) such that ∀a¯ ∈

β u ◦Ft +1 , 1−β

€ Š 1 Tt ,ω Ep [ζ¯ t ,ω · a¯ ] = Ep [ζt ,ω ·a¯ ] and α

€ Š € € ŠŠ A¯t ,ω Ep [ζ¯ t ,ω · a¯ ] = αA t ,ω Tt ,ω Ep [ζ¯ t ,ω · a¯ ] . (14)

Finally, if the representation in (2) is sharp, then n t ,ω ≤ |Ft +1 | − 1. For elaboration on the uniqueness statement, please refer to S09.

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Exchangeability

In order to analyze exchangeability for recursive VEU preferences, assume that the underlying uncertainty concerns the realization of a stochastic process (X t )t ≥0 , where each random variable X t takes values in the finite set X . This is a special case of the environment considered in Sec. 2, with Ft = σ(X 0 , . . . , X t −1 ) for all t ≥ 0; it is also, essentially, the environment considered by Epstein and Seo (2010). Recall that, for every E t ∈ Ft , E t = {(x 0 , . . . , x t −1 )} × X ∞ .

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S We also let Σ = σ( t ≥0 Ft ), the product sigma-algebra on X ∞ . Now fix t > 0 and (x 0 , . . . , x t −1 ) ∈ X t . For y , z ∈ X , the notation (y , (x 0 , . . . , x t −1 ); z ) indicates the plan f ∈ F p such that f τ = z for τ 6= t , f t (ω) = y for ω ∈ {(x 0 , . . . , x t −1 )} × X ∞ , and f t (ω) = y for ω 6∈ {(x 0 , . . . , x t −1 )}×X ∞ . For y > z , every such plan is a “bet” on the event “X 0 = x 0 , . . . , X t −1 = x t ” that “pays” y at time t if it is successful, and z otherwise; the bet also pays z at all other times. Finally, for t > 0, let P t denote the class of permutations of {0, . . . , t − 1}. The main axiom can now be stated. First, although this paper focuses on recursive VEU preferences, the characterization of exchangeability applies as long as ¼0 is a dynamic VEU preference, i.e. the time-0 component of a dynamic VEU representation as per Def. 6. Thus, the focus will be on a dynamic VEU preference ¼0 , and the axiom will concern ¼0 alone, rather than an entire preference system. For a (time-0) EU preference ¼∗0 , the intuition behind exchangeability is simple: a bet on some time-t realization (x 0 , . . . , x t −1 ) should be just as good as a bet on any permutation (x π(0) , . . . , x π(t −1) ), 18

for π ∈ P t . This basic intuition underlies the exchangeabiity axiom(s) considered, for instance, in Epstein and Seo (2010). The proof of Theorem 2 shows that, indeed, such condition is equivalent to the assumption that the probability measure p ∗ representing the EU preference ¼∗0 is indeed exchangeable, and hence admits a de Finetti-style representation as an average of product probability measures. While ¼0 is not an EU preference in general, the objective of this section is to indentify a condition that ensures that the baseline prior p in the VEU representation of ¼0 is exchangeable. The restriction just described would be much too strong; however, a suitable assumption can be formulated leveraging the notion of complementarity, which is central to VEU preferences. The basic intuition is as follows. Recall that complementary plans (or, in S09, acts) are ranked solely via their baseline EU evaluation, as adjustment terms cancel out. Now suppose that a bet f “on” a sequence (x 0 , . . . , x t −1 ) is (weakly) preferred to a complementary plan f¯, which must then be, essentially, a bet “against” the same sequence, possibly involving different prizes. Then, if g , g¯ are complementary, g represents a bet “on” a permutation (x π(0) , . . . , x π(t −1) ) of the original sequence, and g¯ represents a bet “against” that permutation, the exchangeability intuition suggests that g ¼0 g¯ —provided, of course, the prizes involved in the bets f , g and, respectively, f¯, g¯ are the same. This is precisely what the following axiom requires. Axiom 16 (Complementary Exchangeability) For all t > 0, (x 0 , . . . , x t −1 ) ∈ X t , π ∈ P t , and y , z ∈ X with y  z : if (f , f¯) and (g , g¯ ) are complementary pairs with f = (y , (x 0 , . . . , x t −1 ); z ), g = (y , (x π(0) , . . . , x π(t −1) ); z ), and 21 f τ + 12 f¯τ = 12 g τ + 12 g¯ τ for all τ ≥ 0, then f ¼0 f¯ implies g ¼0 g¯ . The main result of this section can now be stated. Denote ba1 (2X ) simply by ∆(X ); for ` ∈ ∆(X ), let `∞ be the corresponding i.i.d. product measure on X ∞ , endowed with the product sigma-algebra. Theorem 2 Let ¼0 be a dynamic VEU preference, with baseline prior p . Then Axiom 16 holds if R and only if there exists a (unique) µ ∈ ∆(X ) such that p = ∆(X ) `∞ d µ(`). The result holds a fortiori for recursive VEU preferences. Thus, Complementary Exchangeability is compatible with full dynamic consistency; by comparison, as noted in the Introduction, Epstein and Seo (2010) are careful to point out that their notion of exchangeability can

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only be accommodated in a setting where a weak form of dynamic consistency holds. Recall that Bayesian updating of the baseline prior is a key feature of recursive VEU preferences, and one that is ensured by Axiom DC. Theorem 2 then has an immediate,3 but important Corollary: Corollary 1 Let (¼t ,ω ) be a recursive VEU preference system, with p ∈ ca1 (Σ) as baseline prior; also let µ be as in Theorem 2. Then, for every t > 0, (x 0 , . . . , x t −1 ) ∈ X t , and rectangle A = A 0 × A 2 × . . . ∈ Σ, Z

Y

p (A|X 0 = x 0 , . . . , X t −1 = x t −1 ) =

`(A τ )d µ(x 0 ,...,x t −1 ) (`),

∆(X ) τ≥t

where

Qt −1

d µ(x 0 ,...,x t −1 ) (`) = R

`(x τ )d µ(`) τ=0 . Qt −1 ¯ τ )d µ(`) ¯ `(x τ=0 ∆(X )

In other words, once Dynamic Consistency and Complementary Exchangeability are combined, one concludes that (1) all conditional preferences are exchangeable, and (2) the beliefs µ(x 0 ,...,x t −1 ) are obtained from µ via Bayes’ Rule. This provides a straightforward behavioral foundations for specifications of recursive VEU preferences that feature “learning about parameters.”

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Applications: Lucas (1978)

I now sketch an application of recursive VEU preferences to the celebrated dynamic stochastic general-equilibrium model developed by Lucas (1978). The objective of the discussion is to highlight the similarities with the usual EU analysis of this well-known model, and the tractability and flexibility of recursive VEU preferences. The economy features a single, stand-in consumer, characterized by recursive VEU preferences. The stochastic environment is Markovian, with infinite horizon; states of nature can be represented as ω = (x 0 , x 1 , . . .), where x t ∈ X ≡ R+J , for some J ≥ 1 (the interpretation will be provided momentarily). The filtration F0 , F1 , . . . reveals the information that, at time t and R 3 Just

note that p (A|X 0 = x 0 , . . . , X t −1 = x t −1 ) =

∆(X )

Qt −1 R

τ≥t `(A τ )d µ(`) τ=0 `(x τ ) Qt −1 ¯ τ )d µ(`) ¯ `(x τ=0 ∆(X )

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Q

.

state ω = (x τ )τ≥0 , only the initial realizations (x 0 , x 1 , . . . , x t −1 ) is known; this filtration is defined in the usual way. To interpret, imagine that there are J firms, denoted j = 1, . . . , J , producing a single conj

j

sumption good. Firm j ’s ouput at time t is a random variable x˜t , with realization x t . The vector of random output and realized output for all firms are denoted x˜t and x t respectively. Thus, each state encodes the sequence of output realizations for each firm in the economy. R To simplify notation, it is convenient to write expectations in the form Ex [a ] = a (x˜ ) d π(x˜ |x ), where I denote the representative agent’s prior by π and reserve the lowercase letter p for prices. Similarly, let I x (a (x˜ )) = Ex [a ] + A(Ex [a · ζx ]): this indicates that we assume stationary adjustment functions; on the other hand, it is natural to let the one-period-ahead adjustment factors ζx = (ζxi )0≤i λt +1 > 0 for all t ≥ 0, and for any f ∈ F p , V0 (f ) =

X

λt Epˆ [u ◦ f t ◦ πΩ |t ] + Aˆ

t ≥0

X

! λt Epˆ [ζt · u ◦ f t ◦ πΩ |t ] .

t ≥0

ˆ to p on Ω. The next step is to show that the conditional expectations with reFrom pˆ on Ω spect to pˆ can be replaced with unconditional ones with respect to a measure p on Ω. To simplify the notation, let pˆ (E |t ) ≡ pˆ (E × T |Ω × {t }). A preliminary result is required; just like atemporal VEU preferences represent the DM’s ranking of complementary acts via their baseline EU evaluation, recursive VEU preferences represent the ranking of complementary pairs via their baseline discounted EU evaluation. This requires a proof, because the notion of complementary plans is weaker than is assumed in S09. The proof relies upon Eq. (19), and thus on Axiom 8 and the strengthened Axiom 9. Claim: if (f , f¯) are complementary plans, then f ¼0 f¯

⇔

X

λt Epˆ [u ◦ f t ◦ πΩ |t ] ≥

t ≥0

X

λt Epˆ [u ◦ f¯t ◦ πΩ |t ].

(20)

t ≥0

To see this, suppose that (γt )t ≥0 ⊂ R is such that, for every t ≥ 0, u ◦ f t +u ◦ f¯t = γt . Then, for every t ≥ 0, λt Epˆ [ζt · u ◦ f¯t ◦ πΩ |t ] = Eπ [ζt · u ◦ f¯t ; t ] = Eπ [ζt · γt ; t ] − Eπ [ζt · u ◦ f t ; t ] = −Eπ [ζt · u ◦ f t ; t ] = −λt Eπ [ζt · u ◦ f t ◦ πΩ |t ], where the third equality follows from Eq. (19). Since Aˆ is symmetric €P Š €P Š ¯t ◦ πΩ |t ] , and ˆ around 0, it follows that A λ E [ζ · u ◦ f ◦ π |t ] = A λ E [ζ · u ◦ f ˆ ˆ t p t t Ω t p t t ≥0 t ≥0 so V0 (f ) ≥ V0 ( f¯) reduces to Eq. (20). Return to the main step. I claim that, for all t ≥ 0 and E ∈ Ft , if τ ≥ t then pˆ (E |τ) = pˆ (E |t ). To see this, note first that it is wlog to assume that 0 ∈ int u (X ). Now fix t , E ∈ Ft and find x , y ∈ X such that u (x ), u (y ) > 0 and u (x )λt = u (y )λt +1 . To see that such prizes exist, recall that λt > λt +1 > 0, and pick y such that u (y ) > 0. Then u (y )λt > u (y )λt +1 > 0, so there is x with u (x ) ∈ (0, u (y )) such that u (x )λt = u (y )λt +1 . Finally, define z = λ

λt

t +λt +1

t −1 X τ=0

λτ u (z )+λt u (x )+λt +1 u (y )+

X τ>t +1

λτ u (z ) =

t −1 X τ=0

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t +1 x + λ λ+λ y . Note that t

t +1

λτ u (z )+(λt +λt +1 )u (z )+

X τ>t +1

λτ u (z ) = u (z ).

Since 0 ∈ int u (X ), it is then possible to construct plans f , f¯ such that, for some α > 0, 

∀τ, ω,

α[u (x ) − u (z )] τ = t , ω ∈ E     u ◦ f τ (ω) = α[u (y ) − u (z )] τ = t + 1, ω ∈ E     0 otherwise,

and u ◦ f¯τ (ω) = −u ◦ f τ (ω).


Clearly, f , f¯ are complementary. Furthermore, the above calculation implies that f τ (ω) τ≥0 ∼ P P P
f¯τ (ω) τ≥0 for all ω, because τ≥0 λτ u ◦ f τ (ω) = 0 = − τ≥0 λτ u ◦ f τ (ω) = τ≥0 λτ u ◦ f¯τ (ω). Axiom 7 then implies that f ∼0 f¯. Therefore, λt α[u (x ) − u (z )]pˆ (E |t ) + λt +1 α[u (y ) − u (z )]pˆ (E |t + 1) = =λt α(−1)[u (x ) − u (z )]pˆ (E |t ) − λt +1 α(−1)[u (y ) − u (z )]pˆ (E |t + 1) which simplifies to λt [u (x ) − u (z )]pˆ (E |t ) + λt +1 [u (y ) − u (z )]pˆ (E |t + 1) = 0. Move u (z ) to the rhs and divide both sides by λt pˆ (E |t ) + λt +1 pˆ (E |t + 1): λt pˆ (E |t ) λt +1 pˆ (E |t + 1) λt λt +1 u (x )+ u (y ) = u (z ) = u (x )+ u (y ). λt pˆ (E |t ) + λt +1 pˆ (E |t + 1) λt pˆ (E |t ) + λt +1 pˆ (E |t + 1) λt + λt +1 λt + λt +1 Since x 6∼ y because λt > λt +1 , this requires λt pˆ (E |t ) λt = λt pˆ (E |t ) + λt +1 pˆ (E |t + 1) λt + λt +1

⇔

(λt + λt +1 )pˆ (E |t ) = λt pˆ (E |t ) + λt +1 pˆ (E |t + 1)

which implies that pˆ (E |t ) = pˆ (E |t + 1), as required. Now consider S = {;} ∪

S t ≥0

Ft . Then S is a semiring;6 Define p : S → [0, 1] by p (E ) =

pˆ (E |t ) for E ∈ Ft , and p (;) = 0. It remains to be shown that p thus defined is countably additive on S , as then the Caratheodory extension theorem will imply that p is countably additive on σ(S ). Thus, suppose (E k )k ≥0 is a collection of pairwise disjoint elements of S such that S Sk E = E ∈ S (assume E 6= ;, or there is nothing to show). Let E¯k = E \ `=0 E k ; if all such sets k k T are non-empty, (E¯k ) satisfies the conditions of Eq. (4), and therefore k E¯k 6= ;, contradiction. 6 By definition ; ∈ S

. If A, B ∈ S , then A ∩B is either empty or it coincides with one of A or B . Finally, if A, B ∈ S ,

then A \ B is either empty or, if A ∈ Ft and B ∈ Ft 0 with t 0 > t , it is a union of elements of Ft 0 ⊂ S .

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Thus, E¯k = ; for some k , and hence E =

Sk

`=0

E ` , E k = ; for ` > k . Let t ` be such that E ` ∈ Ft ` for

each ` = 0, . . . , k , and define t = max`=0,...,k t ` . Then pˆ (E ` |t ) = pˆ (E ` |t ` ) = p (E ` ), and furthermore, Pk if E ∈ Ft¯ , then also t¯ ≤ t and so pˆ (E |t ) = pˆ (E |t¯) = p (E ). Hence p (E ) = pˆ (E |t ) = `=0 p (E ` |t ) = Pk P p (E ) = p (E k ), as required. This completes the proof of the claim. ` k ≥0 `=0 P P Finally, for every plan f ∈ F p , Epˆ [u ◦ f t ◦ πΩ |t ] = E ∈Ft pˆ (E |t )u ◦ f t (ωE ) = E ∈Ft p (E )u ◦ f t (ωE ) = Ep [u ◦ f t ], where ωE is a generic element of E . Similarly, Epˆ [ζi t · u ◦ f t ◦ πΩ |t ] = Ep [ζt · u ◦ f t ]. Thus, we can write V0 (f ) =

X

λt Ep [u ◦ f t ]+Aˆ

t ≥0

X

!



λt Ep [ζt · u ◦ f t ] = Ep 

t ≥1

 X



λt u ◦ f t +Aˆ Ep 

t ≥0

X

! ζt · λt u ◦ f t 

t ≥1

where Ep [ζi t ] = 0 for all i , t due to Eq. (19). Since ζi 0 = 0, the t = 0 term in the adjustment can be omitted. The argument can now be replicated for each (t , ω), leading to representations !    X X λtτ,ω ζtτ,ω · u t ,ω ◦ f t  . Vt (f , ω) = Ep t ,ω  λτt ,ω u t ,ω ◦ f t  + Aˆt ,ω Ep t ,ω  τ≥t +1

τ≥0

The final part of the argument used in the proof of ES’s Lemma A.1 applies here, too; see also Appendix B therein and note that, while I do not assume ES’s Axiom BW, I do restrict attention to plans (and constant streams) that are bounded above and below. Conclude that, under Axioms RP, CP and IMP, for some Bernoulli utility u and discount factor β ∈ (0, 1),  !   X X Ut (f , ω) = Ep t ,ω  β τ−t u ◦ f τ  + A t ,ω Ep t ,ω  β τ−t ζtτ,ω · u ◦ f τ  , τ≥t

(21)

τ≥t +1

where the adjustment function has been adjusted to account for the normalization of the discount factors; this is a dynamic VEU representation (the properties of A t ,ω and ζt ,ω in Def. (6) correspond directly to properties of adjustment functions/factors in the atemporal VEU representation). Furthermore, Ep t ,ω [ζti τ,ω ] = 0 for all i , τ, and for all complementary pairs f , f¯,  f ¼t ,ω f¯

⇔

Ep t ,ω 

 X



β τ−t u ◦ f τ  ≥ Ep t ,ω 

τ≥t

 X

β τ−t u ◦ f¯τ  .

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τ≥t

Axiom DC and first recursive formulation. Now consider the implications of Axiom DC. Fix a node (t , ω) and a plan f ; then, define a plan g as follows. Let g τ = g t for τ ≤ t ; then, for each ω0 , 29

choose g t +1 (ω0 ) ∈ X such that g t +1 (ω0 ) ∼t +1,ω0 f [note that the lhs represents a constant stream equal to g t +1 (ω0 ) in every state and period]; finally, for τ > t + 1, let g τ = g t +1 . Hence, by construction (and using the representation derived above), g ∼t +1,ω0 f for all ω0 , and Axiom DC then implies that f ∼t ,ω g . Furthermore, by Eq. (21) and the fact that g τ (ω0 ) = P g t +1 (ω) for τ > t + 1 and ω0 ∈ Ft (ω), τ≥t +1 β τ−t −1 u ◦ g τ (ω0 ) = Ut +1 (f , ω0 ) for all τ ≥ t + 1 and all ω0 ∈ Ft (ω). Hence, —Š € ”   · βU (f , ·) = Ut (f , ω) =Ut (g , ω) = Ep t ,ω u ◦ f t + βUt +1 (f , ·) + A t ,ω Ep t ,ω ζtt ,ω t +1 +1 € ” —Š   =u ◦ f t (ω) + β Ep t ,ω Ut +1 (f , ·) + A t ,ω β Ep t ,ω ζtt ,ω · Ut +1 (f , ·) . +1 As in Def. 4, henceforth denote the size of the vector ζt ,ω by n t ,ω . This is almost the recursive representation in Eq. (7), except for the appearence of the so far arbitrary measures p t ,ω . Bayesian updating and full support. I now show that p t ,ω = p 0 (·|Ft (ω)). As above, it is wlog to assume that 0 ∈ int u (X ); let z ∈ X be such that u (z ) = 0. Also choose ε > 0 so that [−ε, 2ε] ⊂ u (X ), and let x ∈ X be such that u (x ) = ε. Now fix a node (t , ε) and E ∈ Ft +1 (ω). Define a plan f by letting f τ = z for τ ≤ t + 1 or τ > t + 2, f t +2 (ω0 ) = x for ω0 ∈ E , and f t +2 (ω0 ) = z otherwise. Finally, let f¯ be a plan complementary with f and such that f ∼t +1,ω f¯. To see that such a plan exists, let y ∈ X be such that u (y ) = 2ε, and w ∈ X be such that u (w ) = −ε; then the plan g 0 that yields x at time t + 2 for ω0 ∈ E , and y elsewhere, is complementary with f and at least as good as f by Monotonicity; similarly, the plan g 00 that yields w at time t +2 for ω0 ∈ E and z elsewhere is also complementary with f and at most as good as f . Mixtures of g 0 and g 00 are complementary to f , and one such mixture f¯ will be conditionally indifferent to it given (t + 1, ω). By Eq. (22) we then have € Š β p t +1,ω (E )ε = β γ − p t +1,ω (E )ε

⇔

1 p t +1,ω (E ) = γ, 2

where u (f τ (ω0 )) + u ( f¯τ (ω0 )) = γ for all (τ, ω0 ). Let z¯ ∈ X be such that u (z¯ ) = 12 γ. Now consider complementary plans (g , g¯ ) such that g τ = g¯ τ = z¯ for τ ≤ t + 1 or τ > t + 2, g t +2 (ω0 ) = f t +2 (ω0 ) and g¯ t +2 (ω0 ) = f¯t +2 (ω0 ) for ω0 ∈ Ft +1 (ω), and g t +2 (ω0 ) = g¯ t +2 (ω0 ) = z¯ elsewhere. Then by Axiom CP we again have g ∼t +1,ω g¯ . Furthermore, by Axiom DC, since g

30

and g¯ agree outside Ft +1 (ω), g ∼t ,ω g¯ . Again by Eq. (22),   1 1 β2 1 t ,ω t ,ω t ,ω γ(1 + β ) + β p (E )ε + p (Ft +1 (ω) \ E ) · 0 + [1 − p (Ft +1 (ω))] γ + γ= 2 2 1−β 2   1 β2 1 1 t ,ω t ,ω t ,ω γ. = γ(1 + β ) + β p (E )(γ − ε) + p (Ft +1 (ω) \ E ) · γ + [1 − p (Ft +1 (ω))] γ + 2 2 1−β 2 Canceling common terms yields p t ,ω (E )ε = p t ,ω (E )(γ − ε) + p t ,ω (Ft +1 (ω) \ E ) · γ

⇔

1 p t ,ω (E ) = p t ,ω (Ft +1 (ω)) γ, 2

i.e., substituting for 12 γ, p t ,w (E ) = p t ,ω (Ft +1 (ω)) · p t +1,ω (E ). The result will follow once it is shown that p 0 (Ft (ω)) > 0 for all (t , ω). But this follows from Axiom FS. Suppose that p 0 (Ft (ω)) = 0. If the plans f , f 0 satisfy f τ (ω0 ) = f τ0 (ω0 ) for all τ and ω0 6∈ Ft (ω), it follows that Ep 0 [u ◦ f t ] = Ep 0 [u ◦ f t 1Ω\Ft (ω) ] = Ep 0 [u ◦ f t0 1Ω\Ft (ω) ] = Ep 0 [u ◦ f t0 ], and similarly Ep 0 [ζt +1 · u ◦ f t ] = Ep 0 [ζt +1 · u ◦ f t0 ], so f ∼0 f 0 : that is, Ft (ω) is ¼0 -null, contradiction. Summing up, henceforth we let p ≡ p 0 and obtain the recursive representation ” — € ” —Š Ut (f , ω) = u ◦ f t (ω) + β Ep Ut +1 (f , ·) Ft (ω) + A t ,ω β Ep ζtt ,ω · Ut +1 (f , ·) Ft (ω) ; +1

(23)

since Ut (f , ω) is bounded by monotonicity and the assumption that f ∈ F p , the above equation shows that (2) holds, taking ζt ,ω = ζtt ,ω +1 . Furthermore, we obtain the non-recursive representation    ! X X Ut (f , ω) = Ep  β τ−t u ◦ f τ Ft (ω) + A t ,ω Ep  β τ−t ζtτ,ω · u ◦ f τ Ft (ω) . τ≥t

τ≥t +1

Time-separability. Now use Eq. (24) to substitute for Ut +1 (f , ω0 ) in Eq. (23): 

 X

Ut (f , ω) = u ◦ f t (ω) + β Ep Ep 

τ≥t +1





+ β Ep A t +1,· Ep 

X τ≥t +2

  β τ−t −1 u ◦ f τ Ft +1 (·) Ft (ω) +

!  β τ−t −1 ζtτ+1,· · u ◦ f τ Ft +1 (·) Ft (ω) +

€ ” —Š Ft (ω) . + A t ,ω β Ep ζtt ,ω · U (f , ·) t +1 +1

31

(24)

Comparing the above expression with Eq. (24) yields !  X — Š € ” τ−t t ,ω Ft (ω) , ω + · U (f , ·) A t ,ω Ep  β ζτ · u ◦ f τ Ft (ω) = A t ,ω β Ep ζtt ,ω t +1 +1

(25)

τ≥t +1





+ β Ep A t +1,· Ep 

X τ≥t +2

!  β τ−t −1 ζtτ+1,· · u ◦ f τ Ft +1 (·) Ft (ω) .

Therefore, (3) holds as well. In particular, note that ζt ,ω in Def. 4 is ζ˜ tt ,ω +1 in Def. 6, and A t ,ω in β

Def. 4 coincides with the restriction of A˜t ,ω in Def. 6 to E ( 1−β u ◦ Ft +1 , p, ζt ,ω ). Finally, iterating Eq. (25) yields !  • ˜ X X —Š € ” τ,· τ−t τ−t t ,ω   = β Ep A τ,· β Ep ζτ+1 · Uτ+1 (f , ·) Fτ (·) Ft (ω) . β ζτ · u ◦ f τ Ft (ω) A t ,ω Ep τ≥t +1

τ≥t

(26)

Necessity of the axioms. It is clear that (3) ⇒ (2), defining A t ,ω , ζt ,ω as indicated after Eq. (25) above, so assume (2) and show that it implies (1). Suppose that the preference system (¼t ,ω ) has a recursive VEU representation as per Def. 4. A VEU-like representation over F p will be constructed first; then, the latter will be employed to verify that the axioms hold. [Note that, alternatively, one could construct a time-separable, dynamic VEU representation from the intermediate one obtained here, and hence show that (2) implies (3); then, showing that (3) implies (1) would be straightforward.] Preliminaries. Since the baseline p is fixed in this argument, write Et [a ] in lieu of Ep [a |Ft ] and Et ,ω [a ] in lieu of Ep [a |Ft (ω)]. Similarly, write I t ,ω (a ) = Et ,ω [a ] + A t ,ω (Et ,ω ζtt +1 · a ]). Since I t ,ω is a niveloid, it has a (minimal) niveloidal extension from (1 − β )−1 β u ◦ Ft +1 = B 0 (σ(Ft +1 ), (1 − β )−1 β u (X )) to all of B 0 (σ(Ft +1 )) (Maccheroni et al., 2006a, p. 1476); denote this extension by Iˆt ,ω . Also recall that a niveloid satisfies a Lipschitz condition with Lipschitz constant 1. Finally, for every a ∈ B 0 (σ(Ft +1 )), denote by Iˆt (a ) the Ft -measurable map ω 7→ I t ,ω (a ). Existence of utility over F p . Define the set U of sequences υ = (u t )t ≥0 of real-valued functions on Ω such that each u t is Ft –measurable, and kυk ≡ supt ,ω |u t (ω)| < ∞. There is a natural ˆ ˆ = T × Ω, where Ω ˆ mapping between U and the set of bounded, Σ-measurable functions on Ω ˆ are as in the proof of sufficiency. In particular, U is a Banach, hence complete metric and Σ space. 32

The following result is the key step in the proof of sufficiency; as noted in the main text, it implies that, once one-step-ahead VEU preferences have been specified, a bounded utility process Ut (·, ω) that satisfies the recursive relation in Eq. (23) is uniquely pinned down. The proof is standard. Proposition 2 Fix β ∈ (0, 1), a Bernoulli utility function u : X → R, and an adapted collection of niveloids (J t ,ω )t ≥0,ω∈Ω , where J t ,ω is defined on B (σ(Ft +1 )) for every node (t , ω). Then, for every f ∈ F p , the map T f : U → U defined by letting ∀υ = (u t )t ≥0 ∈ U ,

Ttt (υ)(ω) = u ◦ f t (ω) + J t ,ω (β u t +1 )

has a unique fixed point υ∗ = (u t∗ )t ≥0 in U ; if ξ, ξ0 ∈ R are such that u ◦ f t (ω) ∈ [ξ, ξ0 ] for all ξ ξ , 1−β ] for all (t , ω). Finally, for every υ0 ∈ U , u ∗ = limN →∞ [Tf ]N (υ0 ), (t , ω), then u t∗ (ω) ∈ [ 1−β 0

where [Tf ]N denotes the N -fold application of T f . Proof: Fix f ∈ F p . Note first that T f indeed maps into U , because u ◦ f t (ω) ∈ [ξ, ξ0 ] for some ξ, ξ0 ∈ R and min a ≤ J t ,ω (a ) ≤ max a for all a ∈ B (σ(Ft +1 )). Next, for all υ = (u t )t ≥0 , υ0 = (u t0 )t ≥0 ∈ U , and all nodes (t , ω), f

f

|Tt (υ)(ω) − Tt (υ0 )(ω)| = | J t ,ω (β u t +1 ) − J t ,ω (β u t0 +1 )| ≤ β ku t +1 − u t0 +1 k because the niveloid J t ,ω has Lipschitz constant 1. Since this is true for all t ≥ 0 and ω ∈ Ω, kTf (υ) − Tf (υ0 )t k ≤ β supt ≥0 ku t +1 − u t0 +1 k = β kυ − υ0 k, i.e. T f is a contraction. Therefore, by the Contraction Mapping Theorem (e.g Aliprantis and Border, 1994, Theorem 3.48) it has a unique fixed point u ∗ ∈ U (ξ, ξ0 ), that can be approached by iterating T f . Now, adapting an argument due to Marinacci and Montrucchio (2010, p. 24), let U (ξ, ξ0 ) = {υ = (u t )t ≥0 ∈ U : ∀(t , ω), u t (ω) ∈ [(1 − β )−1 ξ, (1 − β )−1 ξ0 ]}, where ξ, ξ0 are as in the statement. Then U (ξ, ξ0 ) is a closed subset of U , and hence it is complete; furthermore, if υ = (u t )t ≥0 ∈ ξ ξ ξ = 1−β ; similarly, Tt (υ)(ω) ≥ 1−β . In U (ξ, ξ0 ), then Tt (υ)(ω) = u ◦ f t (ω) + J t ,ω (β u t +1 ) ≤ ξ0 + β 1−β 0

f

0

f

other words, T f maps U (ξ, ξ0 ) into itself. Hence, T f has a unique fixed point υ∗∗ ∈ U (ξ, ξ0 ); but since U (ξ, ξ) ⊂ U , υ∗∗ = υ∗ .

33

VEU-like representation of utility. Return to the proof of necessity. Since by assumption
Ut (f , ·) t ≥0 is bounded and recursive for every f ∈ F p , by Prop. 2 it is the unique such representation of (¼t ,ω ). Now let x , x 0 ∈ X be such that x 0 ¼ f t 0 (ω0 ) ¼ x for all (t 0 , ω0 ), and let φ = 21 u (x ) + 12 u (x 0 ). Define υ0 = (φ) (the constant function equal to φ). Since φ ∈ u (X ), υ0 ∈ U (u (x ), u (x 0 )), so that ([T f (0)]N ) ∈ U (u (x ), u (x 0 )) for all N , as shown in the proof of Prop. 2. But this implies that Iˆt ,ω (β ([T f (0)]N −1 )t +1 ) = I t ,ω (β ([T f (0)]N −1 )t +1 ), which is a VEU functional (Iˆt ,ω need not be). For every n ≥ 0, define the adapted processes (a tn ), (u tn ) as follows: for every (t , ω), a t0 (f , ω) = 0 u t0 (f , ω) = u ◦ f t (ω) + β φ −1 −1 (f , ·)]) (f , ·)] + A t ,ω (β E t ,ω [ζt ,ω · u tn+1 a tn (f , ω) = E t ,ω [β a tn+1

u tn (f

, ω) =

t +n X

Et ,ω [β τ−t u ◦ f τ ] + β n +1 φ + a tn (f , ω).

τ=t

I now inductively rewrite u tn and at the same time verify that the definition of a tn is well−1 posed (in particular, β Et ,ω [ζtt +1 · u tn+1 (f , ·)] is in the domain of A t ,ω ). For n = 0, a t0 (f , ω) = 0,

which is obviously well-defined; moreover, note that, for all τ, ω0 , u τ0 (f , ω0 ) = u ◦ f τ (ω0 ) + β φ ≤ (x ) u (x 0 ) + β u1−β = 0

u (x 0 ) 1−β

and similarly u τ0 (f , ω0 ) ≥

functions with values in

β (u (X )), 1−β

u (x 0 ) . 1−β

By assumption, I t ,ω is well-defined for

hence in particular for β u t0+1 (f , ·); thus, A t ,ω (β Et ,ω [ζt ,ω · β

β

u t0+1 (f , ·)]) is indeed well-defined. Inductively, assume that u τn −1 (f , ω0 ) ∈ [ 1−β (x ), 1−β (u (x 0 ))] for all τ, ω0 ; then a tn (f , ·) is well-defined; furthermore, rewrite u tn (f , ω) as follows, breaking up the summation and substituting for a tn (f , ·): u tn (f

, ω) =u ◦ f t (ω) + β

(t +1)+(n X −1) τ=t +1

−1 Et ,ω [β τ−t −1 u ◦ f τ ] + β n φ + E t ,ω [β a tn+1 (f , ·)] + A t ,ω (β E t ,ω [ζt ,ω · u tn−1 (f , ·)]) = +1

=u ◦ f t (ω) + Et ,ω [β u tn−1 (f , ·)] + A t ,ω (Et ,ω [ζt ,ω · β u tn−1 (f , ·)]) = +1 +1

(27)

−1 =u ◦ f t (ω) + I t ,ω [β u tn+1 (f )]; (x ) (x ) (x ) It then follows that u tn (f , ω) ≥ u (x ) + β u1−β = u1−β and similarly u tn (f , ω) ≤ u1−β , by monotonicity 0

of I t ,ω and the induction hypothesis. This completes the induction step.
Eq. (27) also shows that, letting υn = u tn (f , · t ≥0 , u tn (f , ·) = T f (υn −1 ) = . . . = [T f (u 0 )]n , 34

and therefore u tn (f , ·) → Ut (f , ·). Since

Pt +n τ=t

Et ,ω [β τ−t u ◦ f τ ] + β n +1 φ →

P

τ≥t

Et ,ω [β τ−t u ◦ f τ ],

a tn (f , ·) also has a well-defined limit, denoted a t (f , ·), and one can write Ut (f , ·) =

X

Et ,ω [β τ−t u ◦ f τ ] + a t (f , ·).

(28)

τ≥t

Finally, observe that, since Et ,ω [ζt ,ω · β φ] = 0,  (f , ·)] + A t ,ω β E t ,ω ζt ,ω · a tn (f , ω) =E t ,ω [β a tn−1 +1  (f , ·)] + A t ,ω E t ,ω  =E t ,ω [β a tn−1 +1

t +n X

(t +1)+(n−1) X τ=t +1

!! −1 (f , ·)  = Et +1 [β τ−(t +1) u ◦ f τ ] + a tn+1

 ζt ,ω · β τ−t u ◦ f τ  + Et ,ω

τ=t +1

”

! — −1 (f , ·) . ζt ,ω · β a tn+1 (29)

Verification of the axioms. I now employ Eqs. (28) and (29) to show that the axioms hold. For the basic preference axioms, focus on ¼0 (the argument for other conditional preferences is analogous). Note first that, if f , g ∈ F p are such that u ◦ f − u ◦ g is a deterministic process (γt )t ≥0 , then inductively, using Eq. (29), a tn (f , ω) = a tn (g , ω); hence, the same is true in the limit. Similarly, if f t (ω) ≥ g t (ω) for all (t , ω), then inductively, using Eq. (27), u 0n (f , ω) ≥ u 0n (g , ω), and ˆ = T × Ω and Fˆb is the set of so again U0 (f , ω) ≥ U0 (g , ω). These observations imply that, if Ω ˆ corresponding to adapted sequences in F p , the functional Iˆ : u ◦ Fˆb → R bounded acts on Ω defined by Iˆ(u ◦ fˆ) = (1 − β )U0 (f , ω) (any ω), is a normalized niveloid [if x ∈ X , then U0 (x , ω) = u (x ) , 1−β

ˆ 0 on so the adjustment factor (1 − β ) is required]. Hence, the corresponding preference ¼

Fˆb satisfies the atemporal Weak Order, Monotonicity, Continuity, Non-Degeneracy and weak constant Independence axioms in Maccheroni et al. (2006a), which implies that ¼0 satisfies Axioms 1–5. By induction using Eq. (29), a 0n ((x t )t ≥0 , ω) = 0 for all deterministic consumption streams (x t )t ≥0 , so the same is true in the limit. This implies Axioms 8. Again, by induction, if (f , f¯) are complementary then inductively, using Eq. (29), a 0n (f , ω) = a 0n ( f¯, ω), and so a 0 (f , ω) = a 0 ( f¯, ω); this implies that Axioms 9 and 10 hold. Axiom 7 also holds, because EU preferences satisfy this axiom and complementary plans are evaluated using their baseline EU evaluation. ˆ 0 satisfies CompleBecause ¼0 satisfies Axioms 9 and 10, the corresponding preference ¼ memtary Independence and Complementary Translation Invariance, so Lemma 5 and the ar35

ˆ 0 satisfies Monotone Continuity, so that ¼0 satisfies Axiom gument on p. 842 of S09 imply that ¼ 6 (see the proof of sufficiency for details on how to map between the two domains). Axioms 11, 12, 15 and 14 are immediate; for the first two, again recall that a 0 ((x t )t ≥0 , ω) = 0 for all consumption streams (x t )t ≥0 . For 13 (IMP), observe that, if f ∗ , f , f n , f ∗∗ , x are as in the Axiom and φ is as in the above construction of the process u t (f , ·), we have a tm (f n , ω) = 0 = a t0 (f , ω) for all ω, t ≥ n and m ≥ 0 by induction on m and t ; therefore, a 0m −n (f n , ω) = a 0n (f , ω) for m ≥ n , so a 0 (f n , ω) = a 0n (f , ω). Therefore, n X

u (x ) + a 0 (f n , ω) = 1 − β t =0   n X u (x ) t n +1 n +1 = E0 [β u ◦ f t ] + β φ + β − φ + a 0n (f , ω) = 1 − β t =0   n +1 β u (x ) n = u 0 (f , ω) + β − φ → u 0 (f , ω), 1−β

u 0 (f , ω) = n

E0 [β t u ◦ f t ] + β n +1

which implies that IMP holds. Sharp representation and uniqueness. Any recursive VEU representation u , β , p, (n t ,ω , ζt ,ω , A t ,ω )t ,ω
β yields a VEU representation 1−β u , p (·|Ft (ω)), n t ,ω , ζt ,ω , A t ,ω on Ft +1 at each (t , ω); if this representation is not sharp, Theorem 1 in S09 shows that a sharp representation


β u¯ , p (·|Ft (ω)), n¯ t ,ω , ζ¯ t ,ω , A¯t ,ω 1−β

can be constructed. In particular, since preferences over X must be preserved, one may as well β assume that u¯ = u , so that A¯t ,ω (Et ,ω [ζ¯ t ,ω ·a ]) = A t ,ω (Et ,ω [ζt ,ω ·a ]) for all a ∈ 1−β u ◦Ft +1 . But then,

one can indifferently construct the sequences u tn , a tn as in the proof of necessity in terms of A¯t ,ω and ζ¯ t ,ω , or in terms of the original A t ,ω and ζt ,ω : hence, the utility indices U¯ t generated using the sharp VEU representations at each node (t , ω) coincide with the original ones, Ut . Hence, € Š u , β , p, (n¯ t ,ω , ζ¯ t ,ω , A¯t ,ω )t ,ω is a sharp recursive VEU representation of the same preferences. Also, since acts f t +1 ∈ Ft +1 can be seen as maps fˆt +1 : Ft +1 → X on a finite “state space” Ft +1 , Theorem 1 in S09 implies that n¯ t ,ω ≤ |Ft +1 | − 1. The uniqueness claim follows directly from S09, again after noticing that every recursive VEU representation induces a VEU representation on Ft +1 at each (t , ω).

36




B

Proof of Theorem 2

Sufficiency. Note first that, if f , f¯ and g , g¯ are as in the statement of the axiom, then one can exchange the role of (f , f¯) and (g , g¯ ), using π−1 as the permutation of {0, . . . , t − 1}, to conclude that f ¼0 f¯ if and only if g ¼0 g¯ . P ¯ h ¼0 h¯ iff Ep [ βt u◦ As shown in the proof of Theorem 1, for any complementary pair (h, h), t ≥0 P h t ] ≥ Ep [ t ≥0 β t u ◦ h¯ t ]. Assume wlog that 0 ∈ int u (X ), and let z ∈ u −1 (0); also fix ε > 0 such that [−ε, 2ε] ⊂ u (X ) and let y ∈ u −1 (ε). Now construct (f , f¯) and (g , g¯ ) as follows: f , g are as in Axiom 16; f¯τ = g¯ τ = z for τ 6= t ; and 1 u 2

◦ f t + 12 u ◦ f¯t = 12 u ◦ g t + 21 u ◦ g¯ t ≡ γt , a parameter that shall be determined momentarily. Then

f ¼0 f¯ iff Ep [u ◦ f t ] ≥ 2γt − Ep [u ◦ f t ], i.e. iff Ep [u ◦ f t ] ≥ γt ; similarly, g ¼0 g¯ iff Ep [u ◦ g t ] ≥ γt . Now let γt = Ep [u ◦ f t ]; I claim that this is possible, i.e. that 2γt − u ◦ f t (ω) ∈ u (X ) for all ω. To see this, note that Ep [u ◦ f t ] ∈ [0, ε], so γt ∈ [0, 2ε]; hence, 2γt − u ◦ f t ∈ [−ε, 2ε] ⊂ u (X ) by the choice of ε > 0. This completes the definition of f , f¯, g , g¯ ; by the above arguments, f ∼0 f¯, and so Axiom 16 implies that g ∼0 g¯ as well. But then 1 1 p (X 0 = x 0 , . . . , X t −1 = x t −1 ) = Ep [u ◦ f t ] = Ep [u ◦ g t ] = p (X 0 = x π(0) , . . . , X t −1 = x π(t −1) ). ε ε

(30)

Since this holds for all t > 0 and π ∈ P t , this implies that p is exchangeable, as e.g. per the definition of Hewitt and Savage (1955) (who actually use the term “symmetric”); the result then follows. For completeness, I provide the details. Now fix a finite permutation of T = {t : t ≥ 0}, i.e. a map π∞ : T → T that is one-to-one and such that π∞ (t ) 6= t for finitely many t ∈ T . Then there is T ≥ 0 such that t > T implies π∞ (t ) = t ; furthermore, for all t ≤ T , π∞ (t ) ≤ T : otherwise, if t 0 = π∞ (t ) > T for some t ≤ T , we would have π∞ (t 0 ) = t 0 = π∞ (t ) and t 0 > t , a contradiction. Hence, for every such π∞ there is T ≥ 0 and π ∈ P T such that π∞ (t ) = π(t ) for t ≤ T , and π∞ (t ) = t for t > T . Qt Let C = {( τ=0 C τ ) × X ∞ : t ≥ 0,C τ ⊂ X ∀τ}, the class of “cylinders.” Note that C generates Σ (it contains all sets E t as in Eq. (15), and every A ∈ C is a finite union of such sets E t ), and ! ! t Y X p C τ × X ∞ = p (X 0 ∈ C 0 , . . . , X t ∈ C t ) = p (X 0 = x 0 , . . . , X t = x t ) τ=0

(x 0 ,...,x t )∈X t +1 : x τ ∈C τ ∀τ

37

For A ∈ Σ, let π∞ [A] = {(x t )t ≥0 : (x π∞ (t ) )t ≥0 ∈ A}. In particular, for t ≥ T ,  (  ) ! t t Y Y Cτ π∞  C τ × X ∞  = (x τ )τ≥0 : (x π(0) , . . . , x π(T ) , x T +1 , . . . , x t ) ∈ τ=0

τ=0

and therefore !  ! t Y p π∞  C τ × X ∞ = τ=0

X

p (X 0 = x 0 , . . . , X t = x t ) =

(x 0 ,...,x T ,...,x t )∈X t +1 : x π(τ) ∈C τ ∀0≤τ≤T, x τ ∈C τ ∀T +1≤τ≤t

X

=

p (X 0 = x π(0) , . . . , X T = x π(T ) , . . . , X t = x t ) =

(x 0 ,...,x T ,...,x t )∈X t +1 : x π(τ) ∈C τ ∀0≤τ≤T, x τ ∈C τ ∀T +1≤τ≤t

X

= (x 0 ,...,x t

=p

p (X 0 = x 0 , . . . , X t = x t ) =

)∈X t +1 : x t Y

τ ∈C τ ∀τ

!

!

Cτ × X ∞ .

τ=0

The second equality follows by considering π0 ∈ P t with π0 (τ) = π(τ) = π∞ (τ) for τ ≤ T , and π0 (τ) = τ = π∞ (τ) for τ > T , and applying Eq. (30). Finally, the third equality follows by simply relabeling x π(τ) as x τ for 0 ≤ τ ≤ T , as we have both X τ = x π(τ) and x π(τ) ∈ C τ . Hence, p (π∞ A) = p (A) for all A ∈ C with T ≥ t non-trivial components; since some C τ ’s can be equal to X , this is true for all A ∈ C . Then, by standard arguments (e.g. Aliprantis and Border, 2007, Theorem 10.10), this is true for all A ∈ σ(C ) = Σ. Necessity. Consider t > 0, (x 0 , . . . , x t −1 ), y , z , f , f¯, g , g¯ as in the Axiom. Letting 21 u ◦ f τ + 12 u ◦ f¯τ = γτ for all τ, we have f ¼0 f¯ iff X

β τ u (z ) + β t {u (z ) + [u (y ) − u (z )]p (X 0 = x 0 , . . . , X t −1 = x t −1 )} ≥

τ6=t

≥

X

β τ [2γτ − u (z )] + β t {2γt − u (z ) − [u (y ) − u (z )]p (X 0 = x 0 , . . . , X t −1 = x t −1 )}

τ6=t

and therefore iff β t [u (y ) − u (z )]p (X 0 = x 0 , . . . , X t −1 = x t −1 ) ≥

X

β τ [γτ − u (z )].

τ≥0

Similarly, g ¼0 g¯ iff β t [u (y ) − u (z )]p (X 0 = x π(0) , . . . , X t −1 = x π(t −1) ) ≥

X τ≥0

38

β τ [γτ − u (z )].

If p (X 0 = x 0 , . . . , X t −1 = x t −1 ) =

Qt −1

R ∆(X )

τ=0

`(x τ ) d µ(`), then clearly Eq. (30) holds, so that f ¼0 f¯

iff g ¼0 g¯ , as required.

References C.D. Aliprantis and K.C. Border. Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer Verlag, 2007. Charalambos Aliprantis and Kim Border. Infinite Dimensional Analysis. Springer Verlag, Berlin, 1994. Frank J. Anscombe and Robert J. Aumann. A definition of subjective probability. Annals of Mathematical Statistics, 34:199–205, 1963. Aurélien Baillon, Olivier L’Haridon, and Laetitia Placido. Ambiguity models and the machina paradoxes. American Economic Review, forthcoming. Ramon Casadesus-Masanell, Peter Klibanoff, and Emre Ozdenoren. Maxmin expected utility over savage acts with a set of priors. Journal of Economic Theory, 92:33–65, 2000. Larry G. Epstein and Martin Schneider. Recursive multiple-priors. Journal of Economic Theory, 113:1–31, 2003. L.G. Epstein and K. Seo. Symmetry of Evidence without Evidence of Symmetry. Theoretical Economics, forthcoming, 2010. Paolo Ghirardato and Marciano Siniscalchi. A more robust definition of multiple priors. November 2009. URL http://faculty.wcas.northwestern.edu/~msi661. Paolo Ghirardato, Fabio Maccheroni, Massimo Marinacci, and Marciano Siniscalchi. A subjective spin on roulette wheels. Econometrica, 71:1897–1908, 2003. Paolo Ghirardato, Fabio Maccheroni, and Massimo Marinacci. Differentiating ambiguity and ambiguity attitude. Journal of Economic Theory, 118:133–173, October 2004. 39

Itzhak Gilboa and David Schmeidler. Maxmin expected utility with a non-unique prior. Journal of Mathematical Economics, 18:141–153, 1989. E. Hewitt and L.J. Savage. Symmetric measures on Cartesian products. Transactions of the American Mathematical Society, 80(2):470–501, 1955. N. Ju and J. Miao. Ambiguity, learning, and asset returns. Boston University and Hong Kong University of Science and Technology, 2009. Alexander Kechris. Classical Descriptive Set Theory. New York, Springer-Verlag, 1995. Peter Klibanoff, Massimo Marinacci, and Sujoy Mukerji. Recursive smooth ambiguity preferences. Journal of Economic Theory, 144(3):930 – 976, 2009. ISSN 0022-0531. doi: DOI: 10.1016/j.jet.2008.10.007. Robert Lucas. Asset prices in an exchange economy. Econometrica, 46:1429–1446, 1978. F. Maccheroni, M. Marinacci, and A. Rustichini. Ambiguity Aversion, Robustness, and the Variational Representation of Preferences. Econometrica, 74(6):1447–1498, 2006a. F. Maccheroni, M. Marinacci, and A. Rustichini. Dynamic variational preferences. Journal of Economic Theory, 128(1):4–44, 2006b. Mark J. Machina. Risk, ambiguity, and the rank-dependence axioms. American Economic Review, 99:385–392, 2009. M. Marinacci and L. Montrucchio. Unique solutions for stochastic recursive utilities. Journal of Economic Theory, 2010. David Schmeidler. Subjective probability and expected utility without additivity. Econometrica, 57:571–587, 1989. Marciano Siniscalchi. Vector Expected Utility and Attitudes Toward Variation. Econometrica, 77, May 2009. Amos Tversky and Daniel Kahneman. Judgement under uncertainty: Heuristics and biases. Science, 185:1124–1131, 1974. 40