Interim efficiency with MEU-preferences

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Interim efficiency with MEU-preferences

V. Filipe Martins-da-Rocha∗

Graduate School of Economics, Getulio Vargas Foundation, Praia de Botafogo 190, Rio de Janeiro – RJ, 22250-900 Brazil

Abstract

Recently Kajii and Ui (2009) proposed to characterize interim efficient allocations in an ex-change economy under asymmetric information when uncertainty is represented by multiple posteriors. When agents have Bewley’s incomplete preferences, Kajii and Ui (2009) proposed a necessary and sufficient condition on the set of posteriors. However, when agents have Gilboa–Schmeidler’s MaxMin expected utility preferences, Kajii and Ui (2009) only propose a sufficient condition.

The objective of this paper is to complete Kajii and Ui’s work by proposing a necessary and sufficient condition for interim efficiency for various models of ambiguity aversion and in particular MaxMin expected utility. Our proof is based on a direct application of some results proposed by Rigotti, Shannon, and Strzalecki (2008).

JEL Classification:D81; D82; D84

Key words: Interim efficiency, Multiple priors and posteriors, MaxMin expected utility, No-trade

1. Introduction

We borrow from Kajii and Ui (2009) the following model of an exchange economy with a single good and a finite set of possible states of nature. Finitely many agents exchange contingent contracts. There are two stages: ex-ante each agent’s perception of uncertainty is represented by a family of priors; at the interim stage each agent receives a private signal about which states will not occur and his interim perception of uncertainty is then repre-sented by a family of posteriors. Each agent is endowed with a concave utility index func-tion from which he derives either Bewley’s incomplete preferences or Gilboa–Schmeidler’s MaxMin expected utility preferences. The set of priors induces preferences at the ex-ante stage (before agents receive their private signal) and the set of posteriors induces prefer-ences in the interim stage depending on the private signal agents receive.

✩_{The paper has been improved by the detailed suggestions of a referee of this journal. I would like to thank}

Rose-Anne Dana, Jürgen Eichberger, Jayant Ganguli, Jean Philippe Lefort, Atsushi Kajii, Frank Riedel, Chris Shannon, Takashi Ui and Jan Werner for valuable discussions and comments.

∗_{Corresponding author}

Email address:[email protected](V. Filipe Martins-da-Rocha)

It is well-known that ex-ante efficiency can be characterized through a necessary and sufficient condition imposed on agents’ priors. Bewley (2001) and Rigotti and Shannon (2005) characterized ex-ante efficiency for agents with Bewley’s incomplete preferences.1 Billot, Chateauneuf, Gilboa, and Tallon (2000) and Rigotti, Shannon, and Strzalecki (2008) characterized ex-ante efficiency for agents with Gilboa–Schmeidler’s MaxMin expected utility preferences.2

In the standard Bayesian models, Morris (1994) and Feinberg (2000) provided a charac-terization of interim efficiency in terms of agents’ posteriors. Kajii and Ui (2009) proposed to address the same question when agents have multiple posteriors. They identified a key concept, called thecompatible prior set, that plays a crucial role in their analysis. The com-patible prior set of an agent is the collection of all probability measures which, conditional to each private signal, coincides with a posterior.3 _{When agents have Bewley’s incomplete}

preferences, Kajii and Ui (2009) succeeded to characterize interim efficiency by providing a necessary and sufficient condition in terms of compatible prior sets. For the particular case of linear utility index functions, they proved that an allocation is interim efficient if and only if the compatible prior sets of all agents have a non-empty intersection.

When agents have Gilboa–Schmeidler’s MaxMin expected utility preferences, Kajii and Ui (2009) proposed a condition that is only sufficient. The objective of this paper is to show that it is possible to find a necessary and sufficient condition for interim efficiency when agents have Gilboa–Schmeidler’s MaxMin expected utility preferences. The condition that we propose is closely related to the one introduced by Kajii and Ui (2009). Actually we show that the concept of compatible prior set is central not only for models where agents have Bewley’s incomplete preferences or Gilboa–Schmeidler’s MaxMin expected utility pref-erences, but also for any model with general convex preferences. More precisely, we provide a general necessary and sufficient condition for interim efficiency in terms of compatible pri-ors associated to interim subjective beliefs as introduced by Rigotti, Shannon, and Strzalecki (2008). All the characterization results in Kajii and Ui (2009) follow as corollaries of our general characterization. In particular, having a complete characterization of ex-ante and interim efficiency, we can provide conditions under which there is no speculative trade as first studied by Milgrom and Stokey (1982) for standard Bayesian models.

The paper is organized as follows. Section 2 sets up the formal framework, notation and some preliminary definitions. The characterization results proposed by Kajii and Ui (2009) are presented in Section 3. Our necessary and sufficient condition for interim efficiency is stated and proved in Section 4. We illustrate in Section 5 how the results in Kajii and Ui (2009) can be deduced from our general characterization. Section 6 is devoted to no speculative trade and Section 7 shows how our results can be extended to encompass general convex preferences. We explore a slightly different concept of interim efficiency in Section 8 and Section 9 presents the corresponding no speculative trade results.

1_{See also Dana (2000).}

2_{See also Dana (1998), Samet (1998), Tallon (1998), Chateauneuf, Dana, and Tallon (2000), Dana (2002) and}

Dana (2004).

3_{Technically, the compatible prior set of an agent is the convex hull of all sets of the agent’s posteriors.}

2. Set up

We consider a model of an exchange economy E under uncertainty with asymmetric information as presented in Kajii and Ui (2009). There is a finite setΩof states. The set of all probability measures overΩis denoted by Prob(Ω)and we letP be the collection of all non-empty, convex and closed subsets of Prob(Ω). The expectation of a vector x∈RΩunder a probability measurep∈Prob(Ω)is denoted byEp[x], i.e.,

Ep[x]≡X ω∈Ω

p(ω)x(ω).

IfPis a set inP then we let

E_P[x]≡min

p∈P

Ep[x].

There is a finite setIof agents. Each agenti’s information is characterized by a partition

Πi_{ofΩ. Any eventπ∈Π}i _{can be interpreted as a signal received by agentiat the interim}

stage. The unique eventπ ∈ Πi _{containing ω is denoted by Π}i_{(ω). The information is}

assumed to be correct in the sense that if the state of nature isω, agentiknows that thetrue

state does not belong toΩ\Πi(ω)but cannot discern among the states inΠi(ω)which one is the true state. Each agentihas a set of priorsPi∈ P which represents his prior beliefs, and a set of posteriorsΦi_{(π)∈ P for each signal π∈Π}i_{, which represents his posterior}

beliefs after observingπ. The collection of posteriors{Φi_(π)}

π∈Πi is denoted byΦi.

Assumption 1. For every agentiand every signalπ∈Πi_, (a) there exists at least one priorp∈Pisuch thatp(π)>0;

(b) every posteriorp∈Φi_{(π)satisfiesp(π) =1.}

For notational convenience, given a subsetπ⊂Ω, we denote byP(π)the subset ofP

defined as follows: a setP∈ P belongs toP(π)if and only if the support of any probability inPis a subset ofπ, i.e.,

∀p∈P, p(π) =1.

Observe that for every agentiand every interim signalπ∈Πi, the set of posteriorsΦi(π)

belongs toP(π).

There is a single good in the economy, and agent i has a concave, strictly increasing and continuous differentiable utility index functionui:[0,∞)→Rwhich induces MaxMin expected utility preferences as defined by Gilboa and Schmeidler (1989).

Definition 1. Agenti(strictly) prefers the contingent consumption bundley∈RΩ₊tox∈RΩ₊ at theex-antestage if

E_Pi[ui(y)]>E_Pi[ui(x)].

The set of all contingent consumption bundles that are (strictly) preferred toxat the ex-ante stage is denoted by Prefi_Ω(x).

Similarly, we can define agenti’s preference relation at the interim stage.

Definition 2. Agenti(strictly) prefers the contingent consumption bundley∈RΩ

+tox∈R Ω + at theinterimstage with private informationπ∈Πi_if

E_Φi_(π)[ui(y)]>E_Φi_(π)[ui(x)].

The set of all contingent consumption bundles that are (strictly) preferred toxat the interim stage with private informationπ∈Πi_{is denoted by Pref}i

π(x).

An allocationx_{is a family}x= (xi)_i∈I where xi is a vector inRΩ

+ representing a con-tingent consumption bundle. We fix from now on an allocatione_{= (e}i₎

i∈I whereeican be

interpreted as the current endowment of agenti.

Assumption 2. For each agent i, the contingent consumption bundleei is interior in the sense that

∀ω∈Ω, ei(ω)>0.

A familyt_{= (t}i₎

i∈Iwheretiis a vector inRΩis called a feasible trade (from the allocation

e_{) if}

∀i∈I, ei+ti∈RΩ

+ and

X

i∈I

ti=0. Each vectorti_{correspond to the net trade of agenti.}

We follow Kajii and Ui (2009) and introduce the concepts ofex-ante andinterim effi-ciency.

Definition 3. The allocatione_is

• ex-ante efficientif there does not exist a feasible tradet_{such that each agentiprefers}

at the ex-ante stage the contingent consumptionei_+ti_toei_;

• interim efficientif there does not exist a feasible tradet_{such that each agentiprefers}

at every interim stageπ∈Πi_{the contingent consumptione}i_+ti_toei_.

In other words, the allocatione_{is not ex-ante efficient if and only if there exists a feasible}

tradet_{such that}

∀i∈I, ei+ti∈Pref_Ωi(ei).

The allocatione_{is not interim efficient if and only if there exists a feasible trade}t_{such that}

∀i∈I, ∀π∈Πi, ei+ti∈Pref_πi(ei).

In order to provide a characterization of efficiency in terms of primitives it is important to characterize the set of net tradestisuch that the associated contingent consumptionei+ti

is strictly preferred to the initial endowmenteiwhen agents have MEU-preferences. For that purpose, we need to introduce the concept of active belief.

Definition 4. Fix an agent iand a setQ∈ P of beliefs. We denote by Acti(Q)the set of beliefsp∈Qthat minimizeEp[ui(ei)]overQ, i.e.,

Acti(Q)≡argmin{Ep[ui(ei)] : p∈Q}. Any belief in Acti(Q)is called anactive belief inQatei.

Since we allow for averse agents, we also need to introduce the concept of risk-adjusted belief.

Definition 5. Fix an agentiand a beliefp∈Prob(Ω). Therisk-adjustedbelief RAi(p)is the probability measure in Prob(Ω)defined by4

∀ω∈Ω, RAi_(p)(ω)≡ p(ω)∇u

i_(ei_(ω))

Ep_[∇ui_(ei_)] .

Given a setQ∈ P of beliefs, we denote by RAi_{(Q)the set of risk-adjusted beliefs defined by}

RAi(Q)≡[

q∈Q

{RAi(q)}.

Adapting the arguments in Rigotti, Shannon, and Strzalecki (2008) we can prove the following lemma. This is the crucial technical result of this paper. We provide a detailed proof for more general preferences in Section 7.

Lemma 1. Fix an agent i, a set of beliefs Q∈ P and a net trade ti∈RΩsuch that ei+ti¾0.

• If ei+ti satisfiesE_Q[ui(ei+ti)]>E_Q[ui(ei)]then

∀p∈RAi◦Acti(Q), Ep[ti]>0. (1)

• Reciprocally, if ti is such that (1) is satisfied then there existsǫ >0small enough such thatE_Q[ui(ei+ηti)]>E_Q[ui(ei)]for everyη∈(0,ǫ).

A direct consequence of this lemma and the fundamental theorem of welfare economics is the following characterization of ex-ante efficiency.5

Proposition 1. The allocatione_{is ex-ante efficient if and only if}

\

i∈I

RAi◦Acti(Pi)6=;.

It is natural to investigate whether a similar characterization is possible for interim effi-ciency.

3. The characterization proposed by Kajii and Ui

Kajii and Ui (2009) introduced the key concept of compatible priors which is a natural way to construct a family of priors when starting from a family of posterior beliefs.

4_{Iff :[0,∞)→Ris differentiable on(0,∞), we denote by∇f(α)the differential off atα >0.}

5_{See Proposition 2 and Proposition 7 in Rigotti, Shannon, and Strzalecki (2008) or Proposition 5 Kajii and Ui}

(2009). We also refer to Dana (1998), Samet (1998), Tallon (1998), Chateauneuf, Dana, and Tallon (2000), Billot, Chateauneuf, Gilboa, and Tallon (2000), Dana (2002) and Dana (2004).

Definition 6. Fix an agentiand a familyQ= (Q(π))_π∈Πi of posterior beliefs where for each

interim signalπ∈Πi_{, the setQ(π)belongs toP(π). A probability p∈Prob(Ω)is said to}

be aQ-compatible priorif for every interim signalπ∈Πi_{, the conditional probabilityp(·|π),}

when it exists, belongs to the setQ(π). The set of all Q-compatible priors is denoted by CPi_(Q).

Kajii and Ui (2009) showed that the set ofQ-compatible priors is actually the convex hull of the union of all setsQ(π), i.e.,

CPi(Q) =co [

π∈Πi Q(π).

In particular the set CPi_{(Q)is non-empty, convex and closed, i.e., it belongs toP.}

Unfortunately, Kajii and Ui (2009) did not propose a “general” characterization of in-terim efficiency similar to the characterization given in Proposition 1 for ex-ante efficiency. They propose a sufficient condition and prove that this condition is necessary provided that the interim utility of the initial endowment is independent of the signal received, i.e., the following mapping

π7→E_Φi_(π)[ui(ei)]

is constant overΠifor every agenti.

Definition 7. The allocation e _{is said to haveconstant interim utilityif for every agent i,}

the utility at an interim stage of the contingent consumptionei_{is independent of the signal}

received, i.e.,

∀i∈I, ∀π,σ∈Πi, E_Φi_(π)[ui(ei)] =E_Φi_(σ)[ui(ei)].

Kajii and Ui (2009) proved the following (partial) characterization.

Proposition 2.

(a) The allocatione_{is interim efficient if}6

\

i∈I

RAi◦Acti◦CPi(Φi_{)6=;. (2)}

(b) If the allocatione_{has constant interim utility then condition (2) is also necessary.}

In order to provide a necessary and sufficient condition for interim efficiency, Kajii and Ui (2009) introduced the concept of full insurance in the interim stage.

Definition 8. A contingent consumption bundlex has thefull-insurance property at the in-terim stagefor agentiif it isprivately measurablein the sense that the restriction ofxto each signalπ∈Πi_{is constant.}

Kajii and Ui (2009) proposed the following necessary and sufficient condition when the allocatione_{is privately measurable.}

Proposition 3. Assume that the allocatione_{is privately measurable in the sense that for}

each agenti, the contingent consumption bundleei _{has the full-insurance property at the}

interim stage. Then the allocatione_{is interim efficient if and only if}

\

i∈I

RAi◦CPi(Φi_{)6=;. (3)}

6_{We abuse notations by writing RA}i_◦Acti_◦CPi_(Φi_{)instead of RA}i”_Acti”_CPi_(Φi₎——_.

4. A necessary and sufficient condition for interim efficiency

We propose to improve the latter results by exhibiting a necessary and sufficient con-dition. For expositional reasons we abuse notations by writing CPi_◦RAi_◦Acti_(Φi_)instead

of

CPi”€RAi”Acti(Φi(π))—Š

π∈Πi

— .

Theorem 1. The allocatione_{is interim efficient if and only if}

\

i∈I

CPi◦RAi◦Acti(Φi)6=;. (4)

In other words, the allocation e _{is interim efficient if and only if there exists a probability}

measure q∈Prob(Ω)and for each i, a family ri_{= (r}i

π)π∈Πi with ri

πan active belief ofΦi(π)at

ei_{such that}

∀i∈I, q= X

π∈Πi

q(π)RAi(r_πi).

The proof will be a very simple consequence of Proposition 1. To see this, we need the following intermediate result.

Lemma 2. The allocatione_{is interim efficient if and only if there does not exist a feasible trade} t_{such that}

∀i∈I, ∀p∈ [

π∈Πi

RAi_◦Acti_(Φi_{(π)), E}p_[ti_{]>0. (5)}

Proof of Lemma 2. We first prove the “if” part. Assume that there does not exist a feasible trade satisfying (5) but the allocatione_{is not interim efficient. Then there exists a feasible}

tradet_{such that}

∀i∈I, ∀π∈Πi, E_Φi_(π)[ui(ei+ti)]>E_Φi_(π)[ui(ei)].

Following Lemma 1 we must have

∀i∈I, ∀π∈Πi_{, ∀p∈RA}i_◦Acti_(Φi_{(π)), E}p_[ti_]>0.

This contradicts the assumption that there does not exist a feasible trade satisfying (5). We now prove the “only if” part. Assume that the allocatione is interim efficient but there exists a feasible tradeτ _{such that}

∀i∈I, ∀p∈ [

π∈Πi

RAi◦Acti(Φi(π)), Ep[τi]>0.

Fix an agentiand a signalπ∈Πi. We have

∀p∈RAi◦Acti(Φi(π)), Ep[τi]>0.

It follows from Lemma 1 that there existsǫi

π>0 such that

∀η∈(0,ǫ_πi), E_Φi_(π)[ui(ei+ητi)]>E_Φi_(π)[ui(ei)].

We letǫ >0 be defined by

ǫ≡min{ǫi

π : i∈I and π∈Πi}

and for eachi, we poseti_=ǫτi_{. The allocationt= (t}i₎

i∈I is a feasible trade such that

∀i∈I, ∀π∈Πi, ei+ti∈Pref_πi(ei)

which leads to a contradiction.

The proof of Theorem 1 follows from Lemma 2 and Proposition 1. We provide the straightforward details hereafter.

Proof of Theorem 1. We consider themodifiedeconomyEbwhere

• each agentiis risk-neutral in the sense that his utility indexu_bi_{is linear, i.e.,bu}i_{(c) =c;}

• each agenti’s set of priorsbPi _{is defined by}

b

Pi≡co [

π∈Πi

RAi◦Acti(Φi(π)).

According to Lemma 2, the allocatione_{is interim efficient for the economyE if and only if}

it is ex-ante efficient for the economyEb. Observe thatbPi_{coincides with CP}i_◦RAi_◦Acti_(Φi_).

Applying Proposition 1, we obtain the desired result.

We provide hereafter an example where Theorem 1 can be applied but not the results in Kajii and Ui (2009).

Example 1. Consider an exchange economy with two risk-neutral agents I = {i1,i2} and

four states of natureΩ ={ω₁, . . . ,ω₄}.7 Agenti1may receive two signalsΠi1={a,b}with

a={ω₁,ω₂}andb={ω₃,ω₄}. His posterior beliefs are given by

Φi1_{(a) ={p∈Prob(Ω): p(ω}

1) +p(ω2) =1 and 1/4¶p(ω1)¶1/2}

and

Φi1_{(b) ={p∈Prob(Ω): p(ω}

3) +p(ω4) =1 and 1/4¶p(ω3)¶1/2}.

Agent i1’s contingent plan is ei1 = (1, 3, 3, 1). Agent i2 has no private information, i.e., Πi2_{={Ω}and his posterior beliefs are represented by a single probability}

Φi2_{(Ω) ={(1/4, 1/4, 1/8, 3/8)}.}

Agenti₂’s contingent plan is any interior vectorei2_∈RΩ

++. We can compute the set of active priors for agenti₁:

Acti1(Φi1(a)) ={(1/2, 1/2, 0, 0)} and Acti1(Φi1(b)) ={(0, 0, 1/4, 3/4)}. It follows that

CPi1◦Acti1(Φi1_{) ={p∈Prob(Ω): p(ω}

1) =p(ω2) and p(ω4) =3p(ω3)}.

7_{The utility index functionu}i_{of each agentiis defined byu}i_{(c) =cfor eachc¾0.}

Since the unique posterior belief of agenti2belongs to CPi1◦Acti1(Φi1), we can apply

Theo-rem 1 to conclude that the allocatione_{is interim efficient. Since the interim expected utility}

of agenti1is not constant, we cannot apply Proposition 6 in Kajii and Ui (2009).8 Neither

can we apply Proposition 7 in Kajii and Ui (2009) sinceei1 does not have the full-insurance property at the interim stage.

5. Relation with the literature

In order to simplify the comparison between our characterization result and those pre-sented in Kajii and Ui (2009), we propose to state some properties satisfied by the operators Acti_{, CP}i_{and RA}i_{. The details of the proofs are postponed to Appendices.}

Lemma 3.For every agent i and every family Q= (Q(π))π∈Πiof posterior beliefs Q(π)∈ P(π), we have

CPi◦RAi(Q) =RAi◦CPi(Q).

Remark1. As a consequence of Theorem 1 and Lemma 4 we obtain the following equivalent characterization: The allocationeis interim efficient if and only if

\

i∈I

RAi◦CPi◦Acti(Φi)6=;. (6)

Lemma 4.For every agent i and every family Q= (Q(π))_π∈Πiof posterior beliefs Q(π)∈ P(π), we always have

Acti◦CPi(Q)⊂CPi◦Acti(Q).

The converse inclusion is true ife_{has constant interim utility.}

The partial characterization proved by Kajii and Ui (2009) (and presented in Proposi-tion 2) follows from our main characterizaProposi-tion result (Theorem 1) and the two preceding lemmas. One may want to compare Proposition 3 with our general necessary and sufficient condition. The following characterization is a straightforward consequence of Theorem 1.

Proposition 4. Assume that the allocatione_{is privately measurable. Then the allocation}e

is interim efficient if and only if _\

i∈I

CPi(Φi)6=;. (7)

Proof of Proposition 4. Assume that the allocatione_{is privately measurable. It is sufficient}

to prove that for each agentiand each private signalπ∈Πi_{, we have RA}i_◦Acti_(Φi_{(π)) =} Φi_{(π). Since e}i _{is constant on π, we denote by e}i_{(π) its value. Observe that for each}

rπ∈Φi(π)we haveErπ[ui(ei)] =ui(ei(π)). In particular, any posterior belief is active, i.e.,

Acti[Φi(π)] = Φi(π).

8_{We can check that}

E_Φi_1(a)[ui1(ei1)] =2 and E_Φi_1(b)[ui1(ei1)] =3/2.

Observe moreover that

Acti1_◦CPi1_(Φi1_{) ={(0, 0, 1/4, 3/4)}.}

Since the unique posterior belief of agenti2does not belong to Acti1◦CPi1(Φi1), condition (7) in Kajii and Ui (2009)

cannot be necessary.

We propose now to prove that there is no need to adjust for risk. This is very intuitive since there is no risk. More precisely, letκ_π be a probability measure in RAi(Φi_{(π)). There}

exists a posterior beliefrπ∈Φi(π)such that

∀ω∈Ω, κ_π(ω) = 1

Erπ[∇ui(ei)]∇u

i_(ei_(ω))r

π(ω).

Sinceeiis constant onπ, the functionω7→ ∇ui(ei(ω))is also constant onπ. We denote by

∇ui(ei(π))its value. It follows thatErπ_[∇ui_(ei_{)] =∇u}i_(ei_{(π))implying thatκ} π=rπ.

Our necessary and sufficient condition (7) seems to be different from (3) the condition proposed by Kajii and Ui (2009). Actually, when the allocatione _{has the full-insurance}

property at the interim stage, there is no need to adjustΦ-compatible priors to risk.

Lemma 5. If the allocatione_{is privately measurable then}

∀i∈I, RAi◦CPi(Φi_{) =CP}i_(Φi_).

The proofs of Lemma 3, Lemma 4 and Lemma 5 can be found in Appendix A, Appendix B and Appendix C respectively.

6. No speculative trade

Following Kajii and Ui (2009) we propose to investigate under which conditions specu-lative trade is impossible.

Definition 9. We say that there is nospeculative tradeif ex-ante efficiency of the allocation

e_{implies that it is also interim efficient.}

As a direct consequence of Proposition 1 and Theorem 1 we get the following general characterization.

Proposition 5. There is no speculative trade if and only if \

i∈I

RAi◦Acti(Pi)6=;=⇒\

i∈I

CPi◦RAi◦Acti(Φi_{)6=;. (8)}

The two general characterization results (Corollary 8 and Corollary 9) proposed by Ka-jii and Ui (2009) follow from the previous result together with Lemma 3, Lemma 4 and Lemma 5.

Proposition 6. Assume that the allocatione_{has constant interim utility. Then there is no}

speculative trade if and only if \

i∈I

RAi◦Acti(Pi)6=;=⇒\

i∈I

RAi◦Acti◦CPi(Φi_{)6=;. (9)}

Assume that the allocatione_{is privately measurable. Then there is no speculative trade if}

and only if _\

i∈I

RAi◦Acti(Pi)6=;=⇒\

i∈I

CPi(Φi_{)6=;. (10)}

Remark2. In (Kajii and Ui, 2009, Corollary 9), the condition (10) is replaced by \

i∈I

RAi◦Acti(Pi)6=;=⇒\

i∈I

RAi◦CPi(Φi)6=;. (11)

We proved (see Lemma 5) that when the allocatione_{is privately measurable then for}

ev-eryi, we have RAi_◦CPi_(Φi_{) =CP}i_(Φi_{). This implies that both conditions (11) and (10) are}

equivalent.

The necessary and sufficient condition (8) imposes an abstract relation between the set of priorsPiand the familyΦi= (Φi(π))π∈Πi of posteriors. We propose to investigate sufficient

conditions relating priors and posteriors to preclude speculative trade. A straightforward sufficient condition is proposed hereafter.

Proposition 7. Assume that for each agenti, for every active prior pi _∈Acti_(Pi_{)and for}

every signalπ∈Πi _{the condition probabilityp}i_{(·|π), when it exists, is an active posterior,}

i.e.,

∀pi∈Acti(Pi), ∀π∈Πi, pi(π)>0=⇒pi(·|π)∈Acti(Φi(π)). (12)

Then there is no speculative trade.

Proof of Proposition 7. We only have to check that (8) is satisfied. Assume that there exists a probabilityq∈Prob(Ω)and for each agentian active priorpi∈Acti(Pi)such that

∀i∈I, q=RAi(pi).

Fix a stateω∈Ωand an agenti. We denote byπthe associated signalΠi_{(ω). Ifp}i_(π)>0

then

q(ω) = 1

Epi[∇ui_(ei_)]∇u

i_(ei_(ω))pi_(ω)

= p

i_(π)

Epi[∇ui_(ei_)]∇u

i_(ei_(ω))pi_(ω|π)

= λi_πRAi(pi(·|π))(ω)

where

λi

π≡

pi_(π)Epi_(·|π)

[∇ui_(ei_)]

Epi[∇ui_(ei_)] .

Since the family(λi

π)π∈Πi belongs to Prob(Πi), it follows that for each agenti, the probability qbelongs to the set CPi_◦RAi_◦Acti_(Φi_).

Condition (12) is still an abstract relation between the set of priors and the family of posteriors. However, it is now simple to provide explicit conditions on the way agents “up-date” their beliefs to guarantee that no speculative trade is possible. Kajii and Ui (2009) used the concepts offull Bayesian updating. We consider a weaker concept.

Definition 10. We say that the set of posteriorsΦi_{isBayesian consistentwithP}i_{if for every}

posterior belief p ∈ Pi and for every private signal π∈ Πi _{plausible according to p, i.e.,}

p(π)>0, the conditional probabilityp(·|π)is a possible posterior, i.e.,

∀p∈Pi, ∀π∈Πi_{, p(π)>0=⇒p(·|π)∈Φ}i_{(π). (13)}

As a simple corollary of Proposition 7 we obtain the following no-trade result.

Corollary 1. Assume that for each agent i the set of posteriors is Bayesian consistent with the set of priors. If the allocatione_{is privately measurable (or equivalently satisfies the full}

insurance property at the interim stage) then there is no speculative trade.

This is a slight generalization of Proposition 10 in Kajii and Ui (2009) since we only assume that the set of posteriors is Bayesian consistent with priors, while Kajii and Ui (2009) assumed that posteriors are the full Bayesian updating of priors in the sense that for every agenti,

∀π∈Πi, Φi(π) =cl{p(·|π) : p∈Pi and p(π)>0}.

Proof of Corollary 1. We only have to check that (12) is satisfied. Fix an agent i, an active priorpi∈Acti(Pi)and a private signalπ∈Πi_withpi_{(π)>0. SinceP}i_{is Bayesian consistent}

withΦi_{, the conditional beliefp}i_{(·|π)belongs toΦ}i_{(π). We should now prove that p}i_(·|π)

is active. Actually, a direct consequence of the measurability ofeiis that any prior is active, i.e., Acti_(Φi_{(π)) = Φ}i_{(π). Indeed, for every posteriorr}

π∈Φi(π), we have

Erπ_[ui_(ei_{)] =u}i_(ei_(π)).

We can also obtain the no-trade result of Epstein and Schneider (2003) and Wakai (2008) as a simple corollary of Proposition 7. These authors proved that if priors are rectangular sets with respect to posteriors, then MEU-preferences are dynamically consistent which in turn implies that there is no speculative trade.9 In order to recall the concept of a rectan-gular prior set, we introduce some notations. Fix a probabilityλ = (λ_π)_π∈Πi in Prob(Πi)

representing beliefs about the private signals of agenti. Let r = (rπ)π∈Πi be a family of

posteriors. The probability in Prob(Ω)defined by X

π∈Πi

λπrπ

is denoted byλ⊗r. IfΛi is a set of probabilities in Prob(Πi), we denote byΛi⊗Φithe set of all probabilitiesλ⊗rwhereλ∈Λi _andr

π∈Φi(π)for eachπ∈Πi. Observe that the set

CPi(Φi_)ofΦi_{-compatible priors coincides with the set Prob(Π}i_)⊗Φi_.

Definition 11. The set of priors isrectangular with respect to posteriors(or equivalentlyPi_is Φi_{-rectangular) if there exists a setΛ}i _{of beliefs in Prob(Π}i_{)about the realization of private}

signals such that

Pi= Λi⊗Φi

and such that for every private signalπ∈Πi_{, there existsλ∈Λ}i_{such thatλ}

π>0.

IfPi_{= Λ}i_⊗Φi _thenΛi_{must coincide withP}i_(Πi_{)the set of all probabilities(p(π))}

π∈Πi

defined by all priorsp∈Pi_{. Observe that ifΦ}i_{is Bayesian consistent withP}i_{then we have}

Pi⊂Pi(Πi)⊗Φi.

9_{For the readers interested in dynamically consistent update rules for decision making under ambiguity, we refer}

to Hanany and Klibanoff (2007) and the literature therein.

When the inclusion is replaced by an equality, we obtain that Pi is Φi_{-rectangular. It is}

straightforward that check if the set of priors is rectangular with respect to posteriors then posteriors are the full Bayesian updating of priors.10

As a simple corollary of Proposition 7 we obtain the following no-trade result which corresponds to Proposition 11 in Kajii and Ui (2009).

Corollary 2. If for each agent the set of priors is rectangular with respect to posteriors, then there is no speculative trade.

Proof of Corollary 2. We only have to check that (12) is satisfied. Fix an agent i, an active priorpi ∈Acti_(Pi_{)and a private signalπ∈Π}i _{with p}i_{(π)> 0. Since the set of priorsP}i

isΦi_{-rectangular, it is Bayesian consistent withΦ}i_{. Therefore, the conditional beliefp}i_(·|π)

belongs toΦi_{(π). We should now prove that the posterior p}i_{(·|π)is active, i.e., belongs to}

Acti_(Φi_{(π)). We know thatp}i_{is an active prior, i.e.,}

∀q∈Pi, Epi[ui(ei)]¶ Eq[ui(ei)]. (14)

Fix an arbitrary posteriorrπ∈Φi(π)and letqbe the probability in Prob(Ω)defined by

q=pi(π)rπ+

X

σ6=π

pi(σ)p(·|σ).

Since the set of priors isΦi_{-rectangular, the probabilityqalso belongs to P}i_{. Applying (14)}

we get

Epi(·|π)[ui(ei)]¶ Erπ_[ui_(ei_)].

We have proved thatpi(·|π)is an active posterior.

We propose hereafter an example where Proposition 7 can be applied but the results in Kajii and Ui (2009) cannot.

Example2. Consider the economy described in Example 1. We now fix prior beliefs for both agents. Since agent i2 has no private information, we assume that he has a unique prior

belief which coincides with his posterior belief, i.e.,Pi2={(₁/_{4, 1}/_{4, 1}/_{8, 3}/₈)}_{. To describe}

the priors of agenti1, we propose the following parametrization of his posterior beliefs:

∀π∈Πi1_{={a,b}, Φ}i1_{(π) ={r}θ

π : θ∈Θ}

whereΘ = [0, 1/4]and

r_aθ= (1/2−θ, 1/2+θ, 0, 0) and rθ_b = (0, 0, 1/4+θ, 3/4−θ).

The parameterθ can be interpreted as a scenario that agent i1 considers plausible. This

parameter describes agenti1’s ambiguity in the sense thatθ cannot be observed and agenti1

has no beliefs about its realization. We assume that contingent to a scenario θ, agent i1

10_{Actually we also have thatΦ}i_{is the maximum likelyhood updating ofP}i_{in the sense that}

∀π∈Πi, Φi(π) =

n

p(·|π): p∈argmax_q∈Piq(π)

o .

believes he will receive the signalawith probabilityλθ

a ∈(0, 1). We poseλ

θ

b=1−λ

θ

a such

thatλθ _{belongs to Prob(Π}i1_{). The prior beliefs of agenti}

1are constructed in the following

way:

Pi1_={λθ_⊗rθ _{: θ∈Θ}.}

For each plausible scenarioθ, agenti1’s prior belief is represented by the probability measure

pθ=λθ⊗rθ= (λθ_a((1/2)−θ),λθ_a(1/2+θ),λθ_b(1/4+θ),λθ_b(3/4−θ)).

Observe thatΦi1 _{is the full Bayesian updating ofP}i1_{in the sense thatΦ}i1_{(π) ={p(·|π):p∈}

Pi1_{}. However,P}i1 _{is notΦ}i1_{-rectangular.}

We assume thatθ7→λθ

a is strictly increasing withλ 0 a=1/2.

11 _Since

Epθ[ui1_(ei1_{)] =2θ+}3

2+

λθ_a

2

it follows that the only active prior belief isp0, i.e., Acti1(_Pi1) ={λ0⊗_r0}_{. Sincep}0_coincides

with the unique prior of agenti2, the allocationeis ex-ante efficient. Observe that

p0(·|a) =r0_a∈Acti1_(Φi1_{(a)) and p}0_{(·|b) =r}0 b∈Act

i1_(Φi1_(b))

implying that we can apply Proposition 7 to conclude that there is no speculative trade. Since the interim expected utility of agenti1is not constant, we cannot apply Corollary 8

in Kajii and Ui (2009). Neither can we apply Corollary 9 or Corollary 10 sinceei1 _{does not}

have the full-insurance property at the interim stage.

7. General convex preference and subjective beliefs

Until now we assumed that agents have MaxMin expected utility preferences. When uncertainty is represented by multiple priors and posteriors, there are other modelings of preferences: the incomplete preferences model of Bewley (2002), the convex Choquet model of Schmeidler (1989), the smooth second-order prior models of Klibanoff, Marinacci, and Mukerji (2005) and Nau (2006), the second-order expected utility model of Ergin and Gul (2009), the confidence preferences model of Chateauneuf and Faro (2006), the multiplier model of Hansen and Sargent (2001), and the variational preferences model of Maccheroni, Marinacci, and Rustichini (2006). We propose to follow the approach initiated by Rigotti, Shannon, and Strzalecki (2008) by considering a broad class of convex preferences which encompasses as special cases all the aforementioned models.

Each agentihas an ex-ante preference relation≻_Ωi on contingent consumption bundles inRΩ

+ defining the correspondence Pref

i

Ω :R Ω + → R

Ω

+ of strictly preferred contingent con-sumption bundles:

∀x∈RΩ

+, Pref

i

Ω(x)≡ {y∈R Ω + : y≻

i

Ω x}.

Similarly, for each possible interim signalπ∈Πi_{, agent iis endowed with an interim}

pref-erence relation≻i

π on consumption bundles contingent to the informationπ, defining the

correspondence Pref_πi :Rπ

+→Rπ+of strictly preferred contingent consumption bundles at the 11_{Take for exampleλ}θ_{= (1/2+θ, 1/2−θ)for everyθ∈[0, 1/4].}

interim stage. Ifxis a vector inRΩandσis a subset ofΩ, we denote byx|σthe restriction ofxtoσ, i.e.,x|σis the vector inRσdefined by(x|σ)(ω) =x(ω)for eachω∈σ.

Preference relations are assumed to satisfy the following properties

Assumption 3. For each agenti, for eachσ∈ {Ω} ∪Πi_{, the binary relation≻}i

σis

(a) irreflexive, i.e., x6∈Pref_σi(x)for all x∈Rσ

+;

(b) convex, i.e., the set Prefi_σ(x)is convex for allx∈Rσ

+; (c) monotone, i.e.,x+h∈Prefi_σ(x)for all x,h∈Rσ

+andhinterior;

12

(d) continuous, i.e., the set Prefi_σ(x)is open inRσ₊.

Following Rigotti, Shannon, and Strzalecki (2008) we introduce the concepts of ex-ante and interim subjective beliefs.

Definition 12. The set Subi_Ωofex-ante subjective beliefs(orsubjective priors) of agentiatei

is

Subi_Ω≡ {p∈Prob(Ω) : Ep[x]¾ Ep[ei], ∀x∈Pref_Ωi(ei)}.

The set Sub_πi ofinterim subjective beliefs(orsubjective posteriors) of agentiateiwith private informationπ∈Πiis

Subi_π≡ {r∈Prob(π): Er[y]¾ Er[ei_{|π], ∀y∈Pref}i

Ω(ei|π)}. For anyσ∈ {Ω} ∪Πi_{, the set Sub}i

σis non-empty. Indeed, the vectore

i_{|σdoes not belong}

to the convex set Pref_Ωi(ei|σ). Applying the Separating Hyperplane Theorem there exists a non-zero vectorξ∈Rσ₊supporting the set Prefi_Ω(ei|σ)atei|σ, i.e.,

∀y∈Prefi_σ(ei|σ), ξ·y≡X

ω∈σ

ξ(ω)y(ω)¾X

ω∈σ

ξ(ω)ei(ω) =ξ·(ei|σ).

Fix a stateω∈σ,ǫ >0 and lethǫ be the vector inRσ++ defined byhǫ(ω′) =ǫ for every ω′6=ωandhǫ(ω) =1. Since the binary relation≻σi is monotone we getξ·hǫ¾0. Passing

to the limit whenǫtends to 0 we can conclude that the vectorξis a non-zero vector inRσ

+. Sinceξis not zero we can normalizeξsuch thatrdefined by

∀ω∈σ, r(ω) = P ξ(ω)

ω′_∈σξ(ω′)

can be assimilated to a probability measure in Prob(σ). For any vector y ∈Rσ, the inner productr·yis then denoted byEr[y]. Abusing notations we will assimilate the probability measurer∈Prob(σ)to its natural extension in Prob(Ω)by posing r(ω) =0 ifωdoes not belong toσ.

The set Subi_σis also compact and convex, therefore it belongs toP(σ). We can adapt the arguments in Rigotti, Shannon, and Strzalecki (2008) to prove the following intermediary result which is the counterpart of Lemma 6 for general convex preferences.

12_{The vectorh∈R}σ

+is interior if it is strictly positive, i.e.,h(ω)>0 for everyω∈σ.

Lemma 6. Fix an agent i, a signalσ∈ {Ω} ∪Πi_{and a vector t}i _∈Rσ _{such that(e}i_{|σ) +t}i

belongs toRσ

+.

• If(ei|σ) +tibelongs toPrefi_σ(ei|σ)then

∀p∈Subi_σ, Ep[ti]>0. (15)

• Reciprocally, if ti_{is such that (15) is satisfied then there existsǫ >0small enough such}

that

∀η∈(0,ǫ), (ei|σ) +ηti∈Prefi_σ(ei|σ).

For the sake of completeness, we provide a detailed proof.

Proof of Lemma 6. Assume that (ei|σ) +ti belongs to Pref_σi(ei|σ). We propose to prove that (15) is satisfied. Since(ei|σ)is strictly positive and Pref_σi(ei|σ)is open inRσ

+, there existsα∈(0, 1]close enough to 1 such that(ei_{|σ) +αt}i_{is strictly positive and still belongs}

to Prefi_σ(ei_{|σ). Since the set Pref}i

σ(ei|σ)is open inRσ+, there existsǫ >0 small enough such that13

(ei|σ) +αti−ǫ1σ∈Prefiσ(ei|σ).

It follows from the definition of Subi_σ that

∀p∈Subi_σ, Ep[ti]¾ ǫ

α >0.

Assume now thatti_∈Rσ_{is such that(e}i_|σ)+ti_{belongs toR}σ

+and (15) is satisfied. Since

ei|σis strictly positive, there existsǫsuch that for everyǫ ∈(0,ǫ] the vector(ei|σ) +ǫti

belongs toRσ₊. We claim that there exists at least oneǫ∈(0,ǫ]such that(ei|σ)+ǫtibelongs to Prefi_σ(ei_{|σ). Assume by way of contradiction that}

{(ei|σ) +ǫti : ǫ∈(0,ǫ]} ∩Pref_σi(ei|σ) =;.

Applying the Separating Hyperplane Theorem there exists a non-zero vectorξ∈Rσ such that

∀ǫ∈(0,ǫ], ∀x∈Prefi_σ(ei|σ), ξ·x¾ξ·”(ei|σ) +ǫti—.

Lettingǫ tend to 0, we obtain thatξsupports Prefi_σ(ei|σ)at(ei|σ). Following a previous discussion we can prove thatξbelongsRσ₊. Normalizing if necessary, we can assume thatξ

is a subjective belief, i.e., belongs to Subi_σ. For eachn∈Nwe let

xn≡(ei|σ) + (1/(n+1))1σ.

Since the preference relation≻i

σis monotone we havexn∈Prefiσ(ei|σ)implying that14

1

n+1¾ǫE

ξ_[ti_].

13_{The vector1}

σinRσis defined by1σ(ω) =1 for everyω∈σ.

14_{As usual, for any vectorz∈R}σ_{the notationξ·zis replaced byE}ξ_{[z]sinceξis a probability measure defined} onσ.

Passing to the limit this leads to the contradiction: 0¾ Eξ[ti]. We have thus proved that there existsǫ∈(0,ǫ]such that(ei|σ) +ǫti belongs to Prefi_σ(ei|σ). We claim that actually

(ei|σ) +ηti belongs to Pref_σi(ei|σ) for every η ∈ (0,ǫ]. Fix η ∈ (0,ǫ]. Following the previous argument we can prove that there existsη∈(0,η)such that(ei|σ) +ηtibelongs to Prefi_σ(ei|σ). Observe that(ei|σ)+ηtiis a convex combination of(ei|σ)+ηtiand(ei|σ)+ǫti. Since the set Prefi_σ(ei_{|σ)is convex, we get the desired result.}

We adapt the concepts of efficiency to this general framework.

Definition 13. The allocatione_is

• ex-ante efficientif there does not exist a feasible tradet_{such that each agentiprefers at}

the ex-ante stage the contingent consumptionei+titoeiin the sense thatei+ti≻i_Ωei;

• interim efficientif there does not exist a feasible tradet_{such that each agentiprefers}

at every interim stageπ∈Πi_{the contingent consumption(e}i_{|π) + (t}i_|π)to(ei_|π)in

the sense that(ei|π) + (ti|π)≻i

π(e

i_|π).

A direct consequence of Lemma 6 and the fundamental theorem of welfare economics is the following characterization of ex-ante efficiency due to Rigotti, Shannon, and Strzalecki (2008).

Theorem 2. The allocatione_{is ex-ante efficient if and only if}

\

i∈I

Subi_Ω6=;.

Following almost verbatim the arguments in the proof of Theorem 1 we obtain the fol-lowing characterization.15

Theorem 3. The allocatione_{is interim efficient if and only if}

\

i∈I

CPi”_(Subi

π)π∈Πi

—

6

=;.

Rigotti, Shannon, and Strzalecki (2008) studied the relationships between the notion of subjective belief and those arising in several common models of ambiguity. We propose to interpret the two previous characterization results for the two models of ambiguity studied in Kajii and Ui (2009). One could also do the same for the other models studied in Rigotti, Shannon, and Strzalecki (2008).16

15_{For every interim signalπ∈Π}i_{, a subjective beliefr}

π∈Subiπcan be interpreted as a probability measure in Prob(Ω)by posingr_π(ω) =0 for everyω6∈π. Therefore, we abuse notations and consider that Subi_πis a subset of Prob(Ω)implying that the formula

CPi”(Sub_πi)_π∈Πi

— . is well-defined.

16_{The Bewley’s incomplete preference model is not studied in Rigotti, Shannon, and Strzalecki (2008) since they}

restrict their attention to complete and transitive binary relations.

7.1. Bewley’s incomplete preferences

In this section we consider that each agenti’s preferences are defined as follows:

• ex-ante,

∀x∈RΩ

+, Pref

i

Ω(x)≡

¦

y∈RΩ

+ : ∀p∈P

i_{, E}p_[ui_(y)]>Ep_[ui_(x)]©

• for every interim signalπ∈Πi_,17

∀x∈Rπ

+, Pref

i

π(x)≡

¦

y∈Rπ

+ : ∀r∈Φ

i_{(π), E}r_[ui_(y)]>Er_[ui_(x)]©_.

For these specific convex preferences we can compute explicitly the set of subjective beliefs.

Lemma 7. For each agent i we have

Sub_Ωi =RAi(Pi) and Subi_π=RAi(Φi_{(π)), ∀π∈Π}i_.

The arguments of the proof are standard: the result follows from the concavity of the utility indexui. As a direct consequence of Theorem 2, Theorem 3 and the previous lemma, we obtain the following necessary and sufficient conditions for ex-ante and interim efficiency.

Proposition 8. Assume that all agents have Bewley’s incomplete preferences. The allocation

e_{is ex-ante efficient if and only if}

\

i∈I

RAi_(Pi_)6=;.

Proposition 8 corresponds to Proposition 1 in Kajii and Ui (2009) which is due to Bewley (2001) and Rigotti and Shannon (2005).

Proposition 9. Assume that all agents have Bewley’s incomplete preferences. The allocation

e_{is interim efficient if and only if}

\

i∈I

CPi◦RAi(Φi)6=;.

Proposition 9 corresponds to Proposition 2 in Kajii and Ui (2009).18

17_{Ifxis a vector inR}π

+andris a probability in Prob(π)we letEr[ui(x)]≡ P

ω∈πr(ω)ui(x(ω)). 18_{Actually in Kajii and Ui (2009) the necessary and sufficient condition for interim efficiency is}

\

i∈I

RAi◦CPi(Φi)6=;.

We proved (see Lemma 3) that this condition is equivalent to the one proposed in Proposition 9.

7.2. Gilboa–Schmeidler’s MaxMin expected utility preferences

In this section we consider that each agenti’s preferences are defined as follows:

• ex-ante,

∀x∈RΩ

+, Pref

i

Ω(x)≡

¨

y∈RΩ

+ : min

p∈Pi

Ep[ui(y)]>min

p∈Pi

Ep[ui(x)]

«

• for every interim signalπ∈Πi_,

∀x∈Rπ

+, Pref

i

π(x)≡

y∈Rπ

+ : min

r∈Φi_(π)

Er[ui(y)]> min

r∈Φi_(π)

Er[ui(x)]

.

For these specific convex preferences Rigotti, Shannon, and Strzalecki (2008) have computed explicitly the set of subjective belies.19

Lemma 8. For each agent i we have

Sub_Ωi =RAi◦Acti(Pi) and Subi_π=RAi◦Acti(Φi(π)), ∀π∈Πi.

As a direct consequence of Theorem 2, Theorem 3 and the previous lemma, we obtain the necessary and sufficient conditions for ex-ante and interim efficiency presented in Propo-sition 1 and Theorem 1.

8. Interim efficiency and common knowledge

In this section we consider the framework of the previous section where each agent i

is endowed with an ex-ante preference relation≻i

Ω and an interim preference relation≻iπ

for every private signalπ ∈ Πi_{. We assume that Assumption 3 is satisfied, i.e., for each} σ∈ {Ω} ∪Πi, the preference relation≻_σi is irreflexive, convex, monotone and continuous.

The no-trade theorem of Milgrom and Stokey (1982) says that, upon the new arrival of information to the agents, it cannot be commonly known among them that there are some trading opportunities which can make them mutually beneficial, provided that the previous allocation (before the arrival of the new information) is ex-ante efficient. To illustrate this we consider that the allocatione_{is the outcome of an ex-ante trade process and assume it is}

ex-ante efficient. If the state of nature iss∈Ω, each agentiknows (and only knows) at the interim stage that the true state belongs toπi_≡Πi_{(s). One should first define which objects}

agents may have incentive to trade. If agentiproposes to trade a consumption bundle x_πi :

πi_→R

+he is revealing to the other agents his private signalπi. We follow Wilson (1978) by considering that agents do not want to communicate their private information. If we assume that the family(Πi₎

i∈I of private partitions is common knowledge then agents can trade

consumption bundles contingent to the common knowledge eventE≡Πc_(s)whereΠc_(s)is

the unique atom of the common knowledge partitionΠc_.20 _{Let (y}i

E)i∈I be an allocation of

consumption bundlesyi

E ∈R+E contingent to the eventEand feasible, i.e.,

∀ω∈E, X

i∈I

y_Ei(ω) =X i∈I

ei(ω).

19_{See Lemma 1 in Rigotti, Shannon, and Strzalecki (2008) and Appendix A in Kajii and Ui (2009).}

20_Πc_{is the finest partition among those which are coarser than all partitions{Π}i_{:i∈I}. See Aumann (1976).}