You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: |
    When ambiguous beliefs are represented by multiple priors, a decision maker (DM)’s updating rule may also include a step of refining the initial belief. Maximum Likelihood (ML) updating indicates one example of such a refinement, whereas Full Bayesian (FB) updating does not allow for any refinement. In fact, ML and FB are the two extremes of refining beliefs with likelihood in updating. To capture behaviors between these two extremes, the present paper proposes and axiomatizes a new updating rule, Relative Maximum Likelihood (RML), to represent conditional preferences that are not as extreme as either case. RML updates an intermediate set of priors, which can be expressed as a linear contraction of the initial set with respect to the set of maximum likelihood priors. The linear contraction parameter captures the DM’s relative inclination towards ML with respect to FB, it is also the threshold of a relative likelihood ratio test. Moreover, RML includes both FB and ML as the two extreme special cases.

    Additionally, the present paper provides an axiomatization of ML for preferences admit Maxmin Expected Utility (MEU) representation as a preliminary result for the exposition of RML.\
    *JEL: D80, D81*

    *Keywords: ambiguity, MEU, updating, full Bayesian, maximum likelihood, likelihood ratio test, contingent reasoning, ambiguous signals, robustness*
author:
- 'Xiaoyu Cheng[^1]'
bibliography:
- 'references.bib'
title: 'Relative Maximum Likelihood Updating of Ambiguous Beliefs[^2]'
---

Introduction
============

For decisions under uncertainty, when sufficient information to pin down a unique probability for uncertainty is lacking, the decision maker (DM)’s revealed preference sometimes is not consistent with any probabilistic belief but is consistent with a belief of multiple priors (Ellsberg 1961[@ellsberg1961risk], Machina and Schmeidler 1992[@machina1992more]). When this DM learns additional information, the updating of her multiple priors belief may actually involve two steps: first, she could use this information to make an inference about the plausibility of each prior and refine her initial belief by discarding the priors that are deemed to be implausible; then second, updates every prior in the refined belief using Bayes’ rule conditional on the information she receives.

The type of inference in the first step is widely used in non-Bayesian statistics. For example, maximum likelihood estimation makes an inference about the parameter values based on the likelihood of generating the observed data. Its counterpart in the ambiguity literature is **Maximum Likelihood (ML)** updating, in which the DM updates only the priors attain maximum likelihood of the observed event in her initial belief; i.e., only those priors are deemed to be plausible. In contrast, another well-known updating rule, **Full Bayesian (FB)**[^3], updates all the priors in an initial belief; in other words, it allows no such inference.

Hence, ML and FB, as two popular updating rules for multiple priors belief, are actually the two polar extremes in terms of making an inference with respect to the likelihood of observed information in updating. The conditional preferences that result from some intermediate inferences, for example, one in which the priors that attain almost maximum likelihood are also deemed to be plausible, cannot be captured by either of these two extremes. In fact, this type of intermediate behaviors are largely missing from the updating literature with axiomatization. In other words, the preference behaviors that corresponding to such an intermediate inference have not been extensively studied.

The present paper proposes an updating rule, **Relative Maximum Likelihood (RML)**, which is able to capture a full range of behaviors between FB and ML. Furthermore, its behavioral foundation is characterized when the decision criterion under multiple priors follows the Maxmin Expected Utility (MEU), in which the DM evaluates her decision according to the worst expected utility among all the priors (Gilboa and Schmeidler 1989[@gilboa1989maxmin]).

It is known in the literature that, FB is characterized by a behavioral axiom under MEU preferences (Pires 2002[@pires2002rule]), however ML is axiomatized only when preferences admit both MEU and Choquet Expected Utility (CEU) representations (Gilboa and Schmeidler 1993[@gilboa1993updating]), which is a strict special case of MEU preferences. As a preliminary result for the exposition of RML, Theorem \[thm1\] provides an axiomatization of ML under general MEU preferences[^4].

Relative Maximum Likelihood
---------------------------

RML captures an intermediate updating behavior between FB and ML in the sense that it updates a refined set of prior that is both the subset of the initial priors and also a superset of the maximum likelihood priors.

Formally, let a convex and closed set $C$ denote the initial set of priors, and for each conditional event $E$, let $C^{*}(E)$ denote the set of priors in $C$ that attain maximum likelihood of event $E$.[^5] For some parameter $\alpha \in [0,1]$, RML selects the following set of priors $C_{\alpha}(E)$ for updating when event $E$ occurs: $$C_{\alpha}(E) = (1-\alpha)C + \alpha C^{*}(E) =  \{(1-\alpha)p + \alpha q: p \in C \text{ and } q \in C^{*}(E) \}.$$ Namely, $C_{\alpha}(E)$ is a linear mixture of the two sets $C$ and $C^{*}(E)$, and the linear mixture is defined as an element-wise mixture.

Since $C^{*}(E) \subseteq C$, thus for all $\alpha \in [0,1]$ one has $C^{*}(E) \subseteq C_{\alpha}(E) \subseteq C$. Geometrically, the set $C_{\alpha}(E)$ is a linear contraction of the set $C$ with $C^{*}(E)$ being the center[^6]. Furthermore, it reduces to FB or ML when $\alpha$ equals to 0 or 1 respectively. Indeed, the parameter $\alpha$ captures a *relative* inclination towards ML with respect to FB, as $\alpha = 0$ and $\alpha = 1$ capture the two extreme inclinations.

To better illustrate how RML works and its relation with FB and ML, consider a version of the Ellsberg’s three-colored urn problem: an urn contains 30 red (R) balls, 60 black (B) and yellow (Y) balls, nature randomly draws a ball from the urn. Let $\{R, B, Y\}$ denote the color of the drawn ball, suppose the DM forms a multiple priors belief that coincides with the information of the urn: $$C = \{p \in \Delta(\{R,B,Y\}): p(R) = 1/3 \}$$ Then let $E = \{R,B\}$ be the observed event, i.e. the DM later learns that the drawn ball is not yellow. Under FB, the DM updates every prior in $C$ conditional on $E$ and results in the following set of posteriors: $$\Pi^{FB} = \{p \in \Delta(\{R,B\}): 1/3 \leq p(R) \leq 1 \}$$ Under ML, since the maximum likelihood of event $E$ is attained by the prior $p\in C$ such that $p(B) = 2/3$, then $C^{*}(E) = \{p \in \Delta(\{R,B, Y\}): p(R) = 1/3, p(B) = 2/3\}$ and its posterior is also: $$\Pi^{ML} = \{p \in \Delta(\{R,B\}): p(R) = 1/3\}$$ Under RML for some $\alpha \in [0,1]$, the updated set by definition is given by $$C_{\alpha}(E) = \{p \in \Delta(\{R,B, Y\}): p(R) = 1/3, p(B) \geq 2\alpha/3\}$$ and the set of posteriors is therefore: $$\Pi^{RML} = \{p \in \Delta(\{R,B\}): 1/3 \leq p(R) \leq 1/(1+2\alpha) \}$$ Consider a bet $f_{B}$ that pays 1 on state $B$ and nothing on the other states. With MEU preference, notice that the conditional evaluation of the bet under RML is given by $\frac{2\alpha}{1+2\alpha}$, which is increasing with respect to $\alpha$. Given the observed event $\{R,B\}$, the DM may infer that there cannot be too many yellow balls in the urn. A DM who is more willing to make such an inference would be more willing to conclude that there actually should be even fewer yellow balls, therefore the evaluation of the bet $f_{B}$ should also be higher. Hence, the value of parameter $\alpha$ calibrates exactly such an attitude. In section 3, it can be further shown that, $\alpha$ is also the threshold of a relative likelihood ratio test.

Furthermore, ranging over $\alpha \in [0,1]$, the conditional evaluation of the bet $f_{B}$ under RML traces out the full range of all possible values between $0$ and $2/3$, the conditional evaluations under FB and ML respectively. In this sense, RML is able to capture a full range of intermediate behaviors between FB and ML.

Representation Theorems
-----------------------

The main contribution of the present paper is identifying the behavioral axioms that characterize preferences represented by RML. Especially, two different representation theorems that feature in two different strength of identification of the parameter $\alpha$ are provided (Theorem \[thm2\], Theorem \[thm4\]).

The stronger one, Theorem \[thm4\], is the ultimate representation theorem that provides a characterization of RML with constant $\alpha$. Whereas, by dropping one of the axioms in Theorem \[thm4\], Theorem \[thm2\] characterizes the **Weak RML**, in which the parameter $\alpha[E]$ depends on $E$ such that it could be different across events.

Both representation theorems are important in their own right. A constant $\alpha$ would be more convenient for applications of RML and also provide sharper predictions for updating behaviors. On the other hand, an $\alpha[E]$ that depends on the conditional event $E$ would allow for more flexibility in terms of interpreting updating behaviors. For example, the experimental results shown by Table 4.2 in Liang (2019)[@liang2019] correspond to a preference under MEU with RML such that $\alpha[E] < 1/2$ when $E = $“good news” and $\alpha[E] > 1/2$ when $E = $“bad news”[^7].

The axiomatization of RML leverages on weakening of the axioms that characterize FB and ML, while the latter axioms are themselves weakening of a property so-called **Contingent Reasoning (CR)**.

In dynamic choice, CR relates a DM’s ex-ante preference to her conditional preferences in the same fashion as Savage’s P2 axiom (Sure-thing principle). Suppose a DM, conditional on an event $E$, is indifferent between an act[^8] $f$ and some constant act $x$: $f\sim_{E}x$. Consider then she compares between two acts ex-ante: $f_{E}g$ and $x_{E}g$, where $f_{E}g$ denotes an act that pays according to the act $f$ when event $E$ occurs, and pays according to $g$ otherwise.

CR requires that, her ex-ante preference could be recovered by evaluating her choices contingent on whether event $E$ occurs. In the above case, contingent on $E$ occurring, the DM is conditional indifferent between $f$ and $x$; while contingent on $E$ not occurring, she receives the same act $g$ in both acts. From this type of reasoning, the DM should further conclude that she is indifferent between $f_{E}g$ and $x_{E}g$ no matter what the act $g$ is[^9]. As a matter of fact, an expected utility maximizer with Bayesian updating always satisfies this CR condition.

Under MEU preferences, both FB and ML are well-known to sometimes violate CR (e.g. in the Ellsberg’s three-colored example). However, when the act $g$ pays on the event $E^{c}$ is restricted to be some specific act, it can be shown that FB and ML will satisfy CR in different special cases, and more importantly, those special cases are also sufficient to characterize FB and ML respectively.

Pires (2002)[@pires2002rule] characterizes FB by an axiom that imposes CR only when the act $g$ is restricted to be the **conditional certainty equivalence**, i.e. $f \sim_{E}x$ implies that $f_{E}x \sim x$. The present paper shows that ML would satisfy CR when the act $g$ is restricted to be **large consequences**, i.e. for all sufficiently large consequences $x^{*}$, $f \sim_{E}x$ implies that $f_{E}x^{*} \sim x_{E}x^{*}$. Furthermore, Theorem \[thm1\] establishes the necessary and sufficient relation between ML under MEU and this special CR condition.

The two axioms that characterize FB and ML are called **Contingent Reasoning for Conditional Certainty Equivalence (CR-CCE)** and **Contingent Reasoning for Large Consequences (CR-LC)** respectively, summarizing their contents.

As RML captures an intermediate behavior between FB and ML, the two axioms CR-CCE and CR-LC, will sometimes be violated under RML. By definition, RML deviates from FB and ML in two different directions (in the sense of subset and superset), which further implies that the two special CR conditions will also be violated in two different directions. Indeed, the **Undershooting and Overshooting (U-O)** axiom, which says that if $f\sim_{E}x$, then $f_{E}x \precsim x$ and $f_{E}x^{*} \succsim x_{E}x^{*}$ for all sufficiently large $x^{*}$, reflects the two directions of deviation from the CR conditions. Proposition \[prop\] suggest that the U-O axiom is necessary and sufficient for any updating rule that is intermediate between FB and ML just like the way in RML.

All intuitions about a DM not willing to be as extreme as either FB and ML are captured by the U-O axiom. However, Proposition \[prop\] further implies that this intuition per se is not sufficient to pin down any specific set that represents the DM’s conditional preferences. To achieve more concrete characterization of conditional preferences, more restrictions are needed to impose on the behaviors.

The **Weak CR-CCE** axiom features another direction of weakening the CR-CCE axiom, orthogonal to the direction of weakening in the U-O axiom. For any two acts $f$ and $g$, CR-CCE implies that if $f\sim_{E}g\sim_{E}x$, then $f_{E}x \sim g_{E}x$. Instead of imposing this CR condition on all acts that are conditionally indifferent, Weak CR-CCE imposes it only when these two acts are also indifference when sufficiently large consequences are paid on $E^{c}$, i.e. $f_{E}x^{*} \sim g_{E}x^{*}$ for all sufficiently large $x^{*}$.[^10]

Theorem \[thm2\] shows that the U-O and Weak CR-CCE axiom are necessary and sufficient for the preferences to be represented by the Weak RML with $\alpha[E]$ that may depend on the observed event $E$. To further characterize a constant $\alpha$ across different events, an additional axiom, **Event Consistency (EC)** will be shown to be sufficient. The EC axiom restricts the DM’s conditional preferences across different events by a similar requirement as in the Weak CR-CCE. Eventually, imposing U-O, Weak CR-CCE and EC axiom is necessary and sufficient for the preferences to be represented by RML with a constant $\alpha$ (Theorem \[thm4\]).

(Naive) Consequentialist Approach
---------------------------------

A fundamental issue for updating ambiguous beliefs is the fact that, for ambiguity sensitive choices, an updating rule cannot preserve both dynamic consistency and consequentialism at the same time (Hanany and Klibanoff 2007[@hanany2007updating], Siniscalchi 2009[@siniscalchi2009two]). By the definition in Hanany and Klibanoff (2007)[@hanany2007updating], dynamic consistency means that the ex-ante most preferred act should remain to be preferred after updating to any other acts that are both feasible and agree with it on the event not occurs; consequentialism means that the conditional preferences should not depend on event not occurs as well as the context of the decision problem (e.g. feasible acts etc.). Therefore, any updating rule for ambiguous beliefs needs to relax one of the two properties.

As RML is characterized by axioms that relaxing the contingent reasoning property which is related to dynamic consistency, and it also incorporates FB and ML as special cases, both of which are well-known to violate dynamic consistency, its violation of dynamic consistency thus can be expected. That being said, RML takes the consequentialist approach which keeps the consequentialism property but relaxes dynamic consistency.

For updating rules take the consequentialist approach, a common potential problem is that when dynamic consistency is violated, the DM would find that her ex-ante preferred course of action will not be carried out since she will prefer another action after updating. Concerning this problem, the naive approach considers the DM to be not sophisticated enough to anticipate that, so she would just follow her updating rule no matter what. In contrast, the sophisticated approach considers the DM to be able to anticipate that and uses consistent planning to commit herself to the ex-ante preferred action that is feasible under her updating.

In order to make the exposition of RML as clear as possible, the present paper takes the naive approach to highlight the updating rule per se. Using the framework developed in Siniscalchi (2011)[@siniscalchi2011dynamic], one will be able to apply RML in a sophisticated approach and consider its predictions.

Robust Applications of Ambiguity
--------------------------------

One important feature of RML is the fact that it unifies FB and ML to the same family of updating rules. The additional parameter $\alpha$ provides an abundant amount of freedom to model DMs’ different reactions with respect to information in the context of ambiguity. Especially, this freedom creates an additional lens to look at robustness questions related to applications of ambiguity.

Many recent applications of ambiguity to information economic models assume that players’ preferences are represented by MEU with FB updating (Bose and Renou 2014[@bose2014mechanism], Beauchene, Li and Li 2019[@beauchene2019ambiguous], Kellner and Le Quement 2018[@kellner2018endogenous]). Although all of these papers present results that the introduction of ambiguity is strictly beneficial for some player so that she can exploit the ambiguity aversion of the other players, their rather strong behavioral assumptions create an caveat that the strict benefits might be restricted only to this specific functional form instead of the general idea of ambiguity.

As FB is a special case of RML, relaxing the behavioral assumptions to MEU with RML updating for any $\alpha \in [0,1]$ provides one way to consider the robustness of the strict benefits by allowing the other players to have arbitrary attitude towards making an inference with respect to the likelihood of information.

In section 5, one example of such a type of robustness analysis is presented. I take the illustrative example in Beauchene, Li and Li (2019)[@beauchene2019ambiguous], in which they show that the sender can gain strictly more payoff than Bayesian persuasion by using ambiguous persuasion. Once one relaxes the behavioral assumption on the receiver from FB ($\alpha = 0$) to RML for any $\alpha \in [0,1]$, first of all, the device that approaching the sender’s optimal payoff[^11] will no longer induce the same action from a receiver with any $\alpha > 0$. Furthermore, the sender’s payoff when receiver’s $\alpha$ is non-zero will be strictly worse than Bayesian persuasion. Seemingly, the strict benefits do crucially depend on the FB assumption here.

Nonetheless, section 5 further shows that there still exists an ambiguous persuasion scheme in which the sender will be able to induce the same action from the receiver for any $\alpha \in [0,1]$, and more importantly, the sender’s payoff will be strictly higher than using Bayesian persuasion. It thus suggests that, the strict gain from using ambiguous signals in this example is robust to allowing for arbitrary attitudes toward inference with respect to the likelihood of information.\
The remainder of the paper is organized as follows: Section 2 sets up the environment and provides the characterization of ML under MEU preferences; Section 3 gives the formal definition of RML and Weak RML; Section 4 provides preference foundation for RML and Weak RML; Section 5 looks at a special case of the current setting and illustrates the predictions and interpretations that RML is able to offer; Section 6 shows an example of robustness analysis using RML in the context of information design; Section 7 talks about related literature; Section 8 concludes.

Preliminaries
=============

Set up
------

I adopt the Anscombe-Aumann framework. Let $\Omega$ be the set of *states of the world* with at least three states[^12], endowed with a sigma-algebra $\Sigma$ of *events* with generic element $E$. Let $X$ be the set of all simple (i.e. finite-support) lotteries over a set of consequences $Z$ and let $x$ denote a generic element of $X$. Let $\mathcal{F}$ denote the set of *bounded acts*, meaning that each $f \in \mathcal{F}$ is a bounded $\Sigma$-measurable function from $\Omega$ to $X$. With conventional abuse of notation, denote a constant act which maps all states $\omega \in \Omega$ to $x$ simply by $x$.

The primitive is a family of preferences $\{\succsim_{E}\}_{E\in \Sigma}$ over all acts $f\in \mathcal{F}$. Let $\succsim_{\Omega} \equiv \succsim$ denote the ex-ante preference and for all the other $E\in \Sigma$, let $\succsim_{E}$ denote the conditional preference when event $E$ occurs. ($\succsim_{\emptyset}$ is irrelevant.)

First of all, assume that the ex-ante preference $\succsim$ admits a Maxmin Expected Utility representation and is represented by a set $C \subseteq \Delta(\Omega)$ and an affine utility function $u$ such that for all $f,g \in \mathcal{F}$: $$f \succsim  g  \Leftrightarrow \min\limits_{p \in C} \int_{\Omega} u(f)dp \geq  \min\limits_{p \in C}  \int_{\Omega} u(g) dp$$ where $u(f)$ denote a function $Y : \Omega \rightarrow {\mathbb{R}}$ such that $Y(\omega) = u(f(\omega))$ for all $\omega \in \Omega$ and call $u(f)$ the *utility profile* of act $f$. In addition, assume that $\succsim$ has finitely many plausible priors[^13].

The finitely many plausible priors assumption is extremely useful since it helps simplify the statements of the axioms to a great extent while conveying the same intuitions. All characterizations can be achieved without this assumption by different axioms as shown in the appendix. On the other hand, this assumption is also arguably reasonable as it still covers a tremendous number of popular cases in applications of multiple priors, for example the $\epsilon$-contamination.

Assume that $X$ is unbounded under the ex-ante preference in the sense of the following: there exists $x \succ y$ in $X$ such that, for all $\alpha \in (0,1)$, there exists $z \in X$ that satisfies either $y \succ \alpha z + (1-\alpha)x$ or $\alpha z + (1-\alpha)y \succ x$. By Lemma 29 in Maccheroni et al. (2006)[@maccheroni2006ambiguity], it is equivalent to say that the set $u(X)$ is unbounded.

For each $E\in \Sigma$, for any $f,g \in \mathcal{F}$, let $f_{E}g$ denote an act that maps all $\omega \in E$ to $f(\omega)$ and maps all $\omega \in E^{c}$ to $g(\omega)$. An event $E$ is $\succsim$-null if for all $f,g,h \in \mathcal{F}$, one has $f_{E}h \sim g_{E}h$. Under MEU, an event $E$ is $\succsim$-null if and only if $p(E) = 0$ for all $p \in C$. Thus, $E$ is $\succsim$-nonnull if there exists $p\in C$ such that $p(E) > 0$.

Furthermore, define an event $E$ to be strict $\succsim$-nonnull if for all $x,x' \in X$ such that $x \succ x'$, one has $x_{E}x' \succ x'$. Under MEU, an event $E$ is strict $\succsim$-nonnull if and only if $p(E) > 0$ for all $p \in C$. Therefore, an event $E$ is $\succsim$-nonnull but is also strict $\succsim$-null if there exists $p, p'\in C$ such that $p(E) =0$ and $p'(E) >0$.

For each $\succsim$-nonnull $E\in \Sigma$, assume that the conditional preference $\succsim_{E}$ also admits a Maxmin Expected Utility representation and is represented by a set $C_{E} \subseteq \Delta(\Omega)$ and the same utility function $u$ such that for all $f,g \in \mathcal{F}$: $$f \succsim_{E}  g  \Leftrightarrow \min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp\geq  \min\limits_{p \in C_{E}}\int_{\Omega} u(g)dp$$ Meanwhile, the conditional preference $\succsim_{E}$ for $\succsim$-null $E$ is unrestricted.

Finally, assume that the conditional preferences satisfy *consequentialism*: first, for all $p \in C_{E}$, $p(E) = 1$, i.e. the complement of the conditional event is irrelevant for conditional preference; and second, ex-ante preference and conditional event $E$ completely determine the conditional preference $\succsim_{E}$, which rules out the possibility that the updating rule may depend on the context of the decision problem.\
All properties assumed for the primitive have axiomatize foundations:

-   Maxmin Expected Utility representation: axioms from Gilboa and Schmeidler (1989)[@gilboa1989maxmin].

-   Finitely many plausible priors: no local hedging axiom from Siniscalchi (2006)[@siniscalchi2006behavioral].

-   $u$ is independent of $E$: state independence axiom from Pires (2002)[@pires2002rule] or unchanged tastes axiom from Hanany and Klibanoff (2007)[@hanany2007updating].

-   Consequentialism: null complement axiom and independence from feasible sets axiom from Hanany and Klibanoff (2007)[@hanany2007updating].

Setting up in this way enables the establishing of a clean if and only if connection between the key axioms and the updating rules. Especially, the most important behavioral foundations that characterize different updating rules will be highlighted in the representation theorems.

For example, FB updating is defined in Pires (2002)[@pires2002rule] by the following:

The primitive $\{ \succsim_{E}\}_{E\in \Sigma}$ is represented by FB updating if the following holds for all $f\in\mathcal{F}$: for all $\succsim$-nonnull $E \in \Sigma$, $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp \leq \min\limits_{p \in C: p(E) \neq 0} \int_{E} u(f)\frac{dp}{p(E)}$$ and especially for all strict $\succsim$-nonnull event $E\in \Sigma$, $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp = \min\limits_{p \in C} \int_{E} u(f)\frac{dp}{p(E)}$$

Her axiomatization result can be simplified, in this framework, to the statement that FB is necessary and sufficient to the following axiom (which is exactly her A9 axiom):\
**Axiom CR-CCE** (Contingent Reasoning for Conditional Certainty Equivalence). For all $\succsim$-nonnull event $E\in \Sigma$, for all $f \in \mathcal{F}$ and $x\in X$, if $f \sim_{E}x$ then $f_{E}x \sim x$.

$\{ \succsim_{E}\}_{E\in \Sigma}$ is represented by FB updating if and only if the CR-CCE axiom holds.

This statement of the theorem highlights that the behavioral foundation of FB is the relaxation of the following Contingent Reasoning (CR) property:\
**Contingent Reasoning** For all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$ and $x\in X$, if $f\sim_{E}x$ then $f_{E}g \sim x_{E}g$ for all $g\in \mathcal{F}$.\
CR-CCE relaxes CR in a way that only requires ex-ante indifference when the act $g$ pays on $E^{c}$ is restricted to be the conditional certainty equivalence $x$.

Maximum Likelihood Updating
---------------------------

The primitive $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by ML updating if the following holds for all $\succsim$-nonnull $E \in \Sigma$ and for all $f\in \mathcal{F}$: $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp = \min\limits_{p \in C^{*}(E)} \int_{E} u(f)\frac{dp}{p(E)}$$ where $C^{*}(E) = \arg\max\limits_{p \in C} p(E)$.

In other words, the conditional preference $\succsim_{E}$ for $\succsim$-nonnull $E$ is represented by the set of posteriors that updated from the priors attain maximum likelihood of event $E$.

When the ex-ante preference $\succsim$ admits both MEU and CEU representations and the set $X$ is bounded such that there exists a maximal element $x^{*}$: $x^{*} \succsim x$ for all $x\in X$, Gilboa and Schmeidler (1993)[@gilboa1993updating] show that, ML is characterized by **Contingent Reasoning for Best Consequence**:\
**Axiom CR-BC** (Contingent Reasoning for Best Consequence). For all $\succsim$-nonnull event $E\in \Sigma$, for all $f \in \mathcal{F}$ and $x\in X$, if $f \sim_{E}x$ then $f_{E}x^{*} \sim x_{E}x^{*}$.\
Notice that CR-BC also relaxes CR in a similar manner as CR-CCE. CR-BC claims that, the DM’s ex-ante evaluation of the act $f_{E}x^{*}$ is equivalently given by her contingent reasoning with respect to the event $E$. Gilboa and Schmeidler (1993)[@gilboa1993updating] offer a “pessimistic” interpretation of this behavior: the DM’s conditional preference of an act $f$ comes from the consideration that the best consequence would have been received had the complement event happened. In other words, her conditional evaluation of the act $f$ reflects a disappointment that event $E$ actually occurs. Moreover, such a disappointment reflected in her conditional preference is triggered by the fact that she assigns maximum probability to the event $E$ under her ex-ante preference.

Although their intuition about the behavioral foundation of ML updating is qualitatively correct, yet the intuition captured by CR-BC would become imprecise when one enlarges the domain of preferences to the more general MEU case.

The following example illustrates exactly a case where the ex-ante preference admits MEU but not CEU representation, and the CR-BC axiom is violated under ML updating.

\[exp3\] Consider a state space $\Omega = \{\omega_{1}, \omega_{2}, \omega_{3}\}$, denote any $p \in \Delta(\Omega)$ by a vector with three coordinates $p = (p_{1}, p_{2}, p_{3})$ such that $p_{1} + p_{2} + p_{3} = 1$. Consider a set $C \subseteq \Delta(\Omega)$ that is a convex hull of the following three points: $(1/2, 0, 1/2), (0, 1/2, 1/2)$ and $(1/3, 1/3, 1/3)$.

Let $X = [0,1]$ with utility function $u(x) = x$. An act $f$ pays $1$ at $\omega_{1}$, pays $0$ at $\omega_{2}$ and is undetermined on state $\omega_{3}$. Let the conditional event $E$ be the two states $\{\omega_{1}, \omega_{2}\}$.

First of all, it is straightforward to verify that a MEU preference with respect to this $C$ does not admit a CEU representation[^14].

If the conditional preference is given by ML updating, as $C^{*}(E)$ contains only the extreme point $(1/3,1/3,1/3)$ then $$f \sim_{E} 1/2$$ The CR-BC axiom implies that $f_{E}x^{*}$ should be indifferent to $1/2 _{E} x^{*}$ where $x^{*} = 1$. However, it is actually the case that $$f_{E}1 \prec 1/2_{E}1$$ i.e. the CR-BC axiom is false.

One might notice that the breakdown of CR-BC axiom results from the fact that $f_{E}1$ and $1/2_{E}1$ are evaluated at different extreme points, especially, the act $f_{E}1$ is not evaluated at the extreme point $(1/3, 1/3, 1/3)$. In other words, her ex-ante evaluation of the act $f_{E}1$ **does not** trigger the disappointment reflected in her conditional evaluation of $f$ that event $E$ occurs. Moreover, the graphical illustration given in figure \[fig:2\] suggests that it is exactly because the consequence $x^{*} = 1$ is not good enough to trigger such a disappointment for this act $f$.

![Graphical Illustration of Example \[exp3\][]{data-label="fig:2"}](2){width="1.0\linewidth"}

In figure \[fig:2\], the blue arrow indicates how the act $f_{\omega_{1}, \omega_{2}}x$ would be changing (in angle) if one increases the consequence $x$. When $x = 1$, the act $f_{\omega_{1}, \omega_{2}}1$ is evaluated at the extreme point $(0, 1/2, 1/2)$ according to MEU and when $x = 2$, the act $f_{\omega_{1}, \omega_{2}}2$ is indifferent between evaluating at the two extreme points $(1/3, 1/3, 1/3)$ and $(0, 1/2, 1/2)$. If one keeps increasing $x$, apparently for all $x \geq 2$, the act $f_{\omega_{1}, \omega_{2}}x$ will be evaluated at the extreme point $(1/3, 1/3, 1/3)$ (When $x$ goes to infinity, the act eventually will be parallel to the bottom line of the triangle).

Therefore, for this act $f$ and conditional event $E = \{\omega_{1}, \omega_{2}\}$, the consequence $x$ paid on the event $E^{c}$ needs to be at least better than $x = 2$ in order to trigger the disappointment reflected in her conditional preference. Consequently, the assumption that $X$ is bounded by $x^{*} = 1$ would prevent that from happening. In fact, one can also verify that $f_{E}2 \sim 1/2_{E}2$ in this example.

At a higher level, Gilboa and Schmeidler (1993)[@gilboa1993updating]’s intuition about the behavioral foundation of ML updating is qualitatively correct, i.e. the conditional evaluation of an act $f$ under ML reflects a disappointment that event $E$ occurs while some sufficiently good consequence would have been paid had event $E$ not occur. However under general MEU, the best consequence $x^{*}$ sometimes is not good enough to trigger such a disappointment in the ex-ante preference. Example \[exp3\] shows exactly a case where the threshold of sufficiently good consequence $(x=2)$ is strictly greater than the best consequence $(x^{*} = 1)$.

Hence, in the current setting where $X$ is assumed to be unbounded, the behavioral intuition for ML updating under MEU is more precisely given by the following axiom:\
**Axiom CR-LC** (Contingent Reasoning for Large Consequences). For all $\succsim$-nonnull event $E \in \Sigma$, for all $f \in \mathcal{F}$ and $x\in X$, there exists $\bar{x}_{E, f} \in X $ such that if $f \sim_{E}x$ then $f_{E}x^{*} \sim x_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E, f}$.\
The CR-LC axiom explicitly states that the threshold for large consequences $\bar{x}_{E,f}$ would depend on the conditional event $E$ and act $f$. More importantly, the existence of such a threshold for every $E$ and $f$ is guaranteed by assumptions on the primitives: $X$ is unbounded and $C$ has finitely many extreme points.

The reason why $X$ being unbounded is important for the existence of $\bar{x}_{E,f}$ has been explained above. For the other assumption, when $C$ contains infinitely many extreme points, for example, is a circle in the three-dimensional simplex, for the act $f$ in example $\ref{exp3}$, $f_{E}x$ is evaluated at the extreme point that maximizes the likelihood of event $E$ only if $x$ goes to infinity. In that case, a threshold $\bar{x}_{E,f}$ will not exist. In contrast, when $C$ contains finitely many extreme points, such a threshold always exists[^15].

The following theorem shows that CR-LC axiom is necessary and sufficient for ML updating under MEU.

\[thm1\] $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by ML updating if and only if the CR-LC axiom holds.

**Remark.** If one drops the finitely many extreme points assumption, as the above discussion suggested, the threshold $\bar{x}_{E,f}$ would not always exist. When it does not exist, one can find a sequence of thresholds such that the difference between the evaluations of $f_{E}x^{*}$ and $x_{E}x^{*}$ for all $x^{*}$ greater than the threshold vanishes as the threshold increases. Formal statement of the axiom and characterization of ML in this case can be found in appendix \[apx1\].

Relative Maximum Likelihood
===========================

The present paper proposes a new updating rule for multiple priors beliefs called Relative Maximum Likelihood (RML) in order to capture the intermediate behaviors between FB and ML. It features a parameter $\alpha \in [0,1]$ that is able to calibrate an extent of *relative* inclination towards ML with respect to FB, and it is also the threshold of a *relative likelihood ratio test*, related to the (absolute) likelihood ratio test used in statistics.

Formally, let $C$ denote the initial set of priors and for some event $E \in \Sigma$ let $C^{*}(E)$ denote the subset of $C$ assigning maximum likelihood of event $E$. For some $\alpha \in [0,1]$, RML updates the following set of priors: $$C_{\alpha}(E) = (1-\alpha)C + \alpha C^{*}(E) =  \{(1-\alpha)p + \alpha q: p \in C \text{ and } q \in C^{*}(E) \}$$ Geometrically, as illustrated in Figure \[fig:1\], the set $C_{\alpha}(E)$ is a linear contraction of the set $C$ with $C^{*}(E)$ being the center. In Figure \[fig:1\], the triangle represents the simplex of probability distributions over the three states $\omega_{1}, \omega_{2}$ and $\omega_{3}$. The larger hexagon represents the set of priors $C$ and when $E = \{\omega_{1}, \omega_{2}\}$ the bottom line of it represents $C^{*}(E)$. Then the blue shaded area represents $C_{\alpha}(E)$ for some $\alpha \in [0,1]$. More specifically, $C_{\alpha}(E)$ is the closed convex hull of points that are linear mixtures of extreme points in $C$ and extreme points in $C^{*}(E)$.

![image](3){width="1.1\linewidth"} \[fig:1\]

![image](1){width="1.1\linewidth"} \[fig:3\]

Observe that when $\alpha = 0$, $C_{\alpha}(E)$ coincides with $C$ and RML reduces to FB, which means that the corresponding conditional preference does not express any inclination towards ML. On the contrary, when $\alpha = 1$, $C_{\alpha}(E)$ becomes $C^{*}(E)$ and RML reduces to ML. In this case, the corresponding conditional preference shows an extreme inclination towards ML.

All the other $\alpha \in (0,1)$ capture an intermediate behavior between FB and ML where the conditional preference has *some* inclination towards ML. Moreover, the set $C_{\alpha}(E)$ shrinks as $\alpha$ increases reflects the fact that larger $\alpha$ corresponds to a DM being more willing to make an inference with respect to likelihood and thus discards more priors from her initial belief.

This fact further suggests a connection between the parameter $\alpha$ and some threshold for refining priors that depends on likelihood. (Absolute) Likelihood ratio test is commonly used in statistics to determine whether a statistical model with fewer parameters is “good enough” compared with a model with maximum number of parameters. The former model is good enough if the likelihood ratio of generating the observed data compared with the latter model exceeds some threshold. In other words, a model is plausible only if it generates the observed data with some likelihood that its ratio with respect to the maximum likelihood exceeds the threshold.

Accordingly, in updating multiple priors, a DM may find a prior to be plausible only if it generates the observed event with some likelihood that its ratio with respect to the maximum likelihood exceed a threshold, for example $$\frac{p(E)}{\max\limits_{p \in C}p(E)} \geq \lambda$$ In this sense, the level of this threshold ($\lambda$) also reflects the extent of willingness to make an inference using likelihood. In fact, the updating rule proposed in Epstein and Schneider (2007)[@epstein2007learning] explicitly uses such a likelihood ratio test to determine whether a prior is plausible or not.

For RML, its functional form suggests a related but different test that is used for selecting priors. Notice that, for all $p \in C_{\alpha}(E)$, $$\frac{p(E)}{\max\limits_{p \in C}p(E)} \geq \alpha + (1-\alpha) \frac{\min\limits_{p \in C}p(E)}{\max\limits_{p \in C}p(E)}$$ or equivalently $$\frac{p(E)-\min\limits_{p \in C}p(E) }{\max\limits_{p \in C}p(E) - \min\limits_{p \in C}p(E)} \geq \alpha$$ from the second formula it is obvious that $\min\limits_{p\in C}p(E)$ becomes an additional benchmark for conducting likelihood ratio test. That being said, a prior is plausible only if it generates the information with some likelihood that its relative difference with respect to the minimum likelihood compared with the difference between maximum likelihood and minimum likelihood exceeds the threshold. Therefore, it is called a **relative likelihood ratio test**. Apparently, when the minimum likelihood of event $E$ is zero, the two likelihood ratio tests coincide.

However, RML does not require only the relative likelihood ratio test for prior selection. Passing the relative likelihood ratio test is only necessary for a prior to be updated under RML, the linear contraction suggests that the shape of the set of initial priors $C$ also plays a crucial role in determining whether a prior will be updated. In fact, the set $C_{\alpha}(E)$ sometimes is a strict subset of the set of priors passing the relative likelihood ratio test.

Let $\hat{C}_{\alpha}(E)$ denote the latter set of priors: $$\hat{C}_{\alpha}(E) = \left\{p\in C: \frac{p(E)-\min\limits_{p \in C}p(E) }{\max\limits_{p \in C}p(E) - \min\limits_{p \in C}p(E)} \geq \alpha\right\}$$ Figure \[fig:3\] shows a case where $C_{\alpha}(E) \subsetneqq \hat{C}_{\alpha}(E)$. For the same scenario as in figure \[fig:1\], notice that the area below the red dashed line and in the set $C$ is the set $\hat{C}_{\alpha}(E)$ with the same $\alpha$ as in $C_{\alpha}(E)$. Hence the area below the red dashed line yet is not in $C_{\alpha}(E)$ represents the priors that pass the relative likelihood ratio test with some $\alpha$, but are not updated under RML with this $\alpha$.\
Finally, as all the discussions above look at a single conditional event $E$, and the DM’s updating across different events is actually not restricted. It is different from FB or ML, where as the DM’s attitudes towards ML is fixed to be a constant: $\alpha = 0$ or $1$ respectively, her updating across different events also needs to be consistent with this constant such that will always be FB or ML.

For RML, two different definitions will be provided (Weak RML, RML) differs exactly in the sense that whether the updating rule, or more specifically the parameter $\alpha$, depends on the conditional event $E$.

The primitive $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by Weak RML if for all $\succsim$-nonnull $E\in \Sigma$, either there exists $\alpha[E] \in (0,1]$ such that for all $f\in \mathcal{F}$: $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp = \min\limits_{p \in C_{\alpha[E]}(E)} \int_{E} u(f)\frac{dp}{p(E)}$$ or $\succsim_{E}$ is represented by FB ($\alpha[E] = 0$).

It should be noted that this definition separates the case $\alpha[E] \in (0,1]$ and FB ($\alpha[E] = 0$) only because the definition of FB treats events that are both $\succsim$-nonnull and strict $\succsim$-null differently. Whereas, for any $\succsim$-nonnull event $E$, when $\alpha[E] \in (0,1]$ every $p \in C_{\alpha[E]}(E)$ satisfies $p(E) > 0$.

The definition of Weak RML allows for the possibility that $\alpha[E]$ may be different across events. For example, it is able to capture an updating behavior in which the DM updates with FB ($\alpha[E] = 0$) for some event $E$ and updates with ML ($\alpha[E'] = 1$) for another event $E'$. Such inconsistency across events may come from different likelihood of events, different context (good news, bad news), or even just different labels. Although one can argue whether this inconsistency is desirable or not, by offering such flexibility, Weak RML is able to accommodate a broader class of updating behaviors. Especially, some experimental evidence also reflects such inconsistency (Liang 2019[@liang2019]).

On the other hand, Weak RML may not be strong enough in applications when sharper predictions of the updating behaviors are needed. Therefore accordingly, RML will be defined to restrict that $\alpha$ needs to be a constant across events.

The primitive $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by RML if either there exists $\alpha\in (0,1]$ such that for all $\succsim$-nonnull $E\in \Sigma$ and for all $f\in \mathcal{F}$: $$\min\limits_{p \in C_{E}} \int_{\Omega} u(f)dp = \min\limits_{p \in C_{\alpha}(E)} \int_{E} u(f)\frac{dp}{p(E)}$$ or $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by FB ($\alpha[E] = 0$).

Preference Foundation
=====================

For a DM who finds both FB and ML to be too extreme in terms of updating beliefs, she will sometimes deviate from the behaviors that characterize FB or ML. Recall that FB and ML correspond to two different relaxations of CR: if $f \sim_{E}x$, FB implies CR-CCE, i.e. $f_{E}x \sim x$; whereas ML implies CR-LC, i.e. $f_{E}x^{*} \sim x_{E}x^{*}$ for all sufficiently large $x^{*}$.

In order to be not as extreme as either FB or ML, the DM would necessarily update some set $C(E)$ such that $C^{*}(E) \subseteq C(E) \subseteq C$. It further implies her conditional evaluation of any act $f$ would be weakly higher than under FB and weakly lower than under ML. Let $x, x^{FB}$ and $x^{ML}$ denote the conditional certainty equivalence of an act $f$ under different updating rules such that $x^{ML} \succsim x \succsim x^{FB}$.

First notice that CR-CCE implies $f_{E}x^{FB} \sim x^{FB}$. As one replaces $x^{FB}$ by $x$ on both sides of this indifference, since $x \succsim x^{FB}$ and the better consequence is received only on event $E^{c}$ for the left hand side, yet it will be received on all states for the right hand side. Thus it would imply that the increase in terms of utility will be higher for the right hand side, i.e. $f_{E}x \precsim x$.

Intuitively, this DM’s ex-ante comparison between the acts $f_{E}x$ and $x$ can no longer be recovered exactly from contingent reasoning. Instead, for the reason that she discards some priors for conditional evaluation of the act $f$, her ex-ante evaluation of $f_{E}x$ would be **undershooting**, meaning that she is now only confident in the fact that $f_{E}x$ should be weakly worse than $x$.

On the other hand, as CR-LC implies $f_{E}x^{*} \sim x^{ML}_{E}x^{*}$ for all sufficiently large $x^{*}$, replacing $x^{ML}$ by the worse consequence $x$ would imply that $f_{E}x^{*} \succsim x_{E}x^{*}$ for all sufficiently large $x^{*}$. Similarly, the DM’s ex-ante comparison between $f_{E}x^{*}$ and $x_{E}x^{*}$ cannot be recovered exactly by contingent reasoning as well. Since now her conditional evaluation of the act $f$ depends on more priors than under ML, her ex-ante evaluation of $f_{E}x^{*}$ would be **overshooting**. It means that instead of knowing she should be indifferent, now she is only confident in that $f_{E}x^{*}$ should be weakly better than $x_{E}x^{*}$.

The following axiom summarizes exactly the above intuitions:\
**Axiom U-O** (Undershooting and Overshooting) For all $\succsim$-nonnull event $E \in \Sigma$, for all $f \in \mathcal{F}$ and $x\in X$, there exists $\bar{x}_{E,f} \in X $ such that if $f \sim_{E}x$, then $f_{E}x \precsim x$ and $f_{E}x^{*} \succsim x_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E,f}$.\
The previous discussions suggest that the U-O axiom is necessary for a DM who updates some intermediate set between FB and ML. Moreover, this intuition is also sufficient such that the U-O axiom is able to characterize such an updating rule.

\[prop\] For each $\succsim$-nonnull $E$, the conditional preference $\succsim_{E}$ is represented by the set of posteriors of the set $C(E)$ such that $C^{*}(E) \subseteq C(E) \subseteq C$ if and only if the U-O axiom holds.

The proof of this proposition relies on separating hyperplane arguments, similar to the proof of Proposition 12 in Hanany and Klibanoff (2007)[@hanany2007updating] thus omitted in the present paper.\
The U-O axiom captures exactly the intuitions about a DM who is not willing to be as extreme as either FB or ML. However, Proposition \[prop\] claims that this intuition is not strong enough to further pin down some specific set to represent the DM’s conditional preference. In order to get more concrete characterization of conditional preferences, consider a different direction of weakening the CR-CCE axiom:

For any two acts $f,g \in \mathcal{F}$, CR-CCE implies that if $f \sim_{E}g \sim_{E}x$, then $f_{E}x \sim g_{E}x$, i.e. contingent reasoning is also true for comparison between the two acts $f_{E}x$ and $g_{E}x$. However, when the conditional preference is not as extreme as FB, the U-O axiom implies that the DM now can only conclude $f_{E}x \precsim x$ and $g_{E}x \precsim x$, but the comparison between $f_{E}x$ and $g_{E}x$ is undetermined.

To impose some restrictions on the DM’s ex-ante preference between $f_{E}x$ and $g_{E}x$, suppose that a DM would conclude $f_{E}x \sim g_{E}x$ only when the two acts $f$ and $g$ are sufficiently “similar”. In particular, suppose these two acts are “pessimistically similar” such that when sufficiently large consequences is paid on event $E^{c}$, it would trigger the same extent of disappointment reflected in the conditional evaluations of the two acts. To be more specific, it means that $f_{E}x^{*} \sim g_{E}x^{*}$ for all sufficiently large $x^{*}$.

Therefore the next axiom, **Weak CR-CCE**, is a relaxation of CR-CCE such that only when both $f \sim_{E} g \sim_{E} x$ and $f_{E}x^{*} \sim g_{E}x^{*}$ are true then it would imply that $f_{E}x \sim g_{E}x$. However, notice that there is a special case for RML where the second quantifier is redundant: when $\alpha = 1$, i.e. RML reduces to ML, $f\sim_{E} g$ would necessarily imply that $f_{E}x^{*} \sim g_{E}x^{*}$ by the CR-LC axiom. In this case, the additional restriction power from the second quantifier is no longer meaningful, and thus the implication should not always be true as well. Therefore, in the formal statement of Weak CR-CCE, I separate this special case from the general statement:\
**Axiom Weak CR-CCE.** For all $\succsim$-nonnull event $E\in \Sigma$, at least one of the following scenarios hold:

(i) For all $f, g \in \mathcal{F}$, there exists $\bar{x}_{E,f,g} \in X$ such that if $f \sim_{E} g$, then $f_{E}x^{*} \sim g_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E,f,g}$.

(ii) For all $f, g \in \mathcal{F}$ and $x\in X$, there exists $\bar{x}_{E,f,g} \in X$ such that if $f \sim_{E} g \sim_{E}x $ and $f_{E}x^{*} \sim g_{E}x^{*} $ for all $x^{*} \succsim \bar{x}_{E,f,g}$, then $f_{E}x \sim g_{E}x$.

The following theorem shows that the additional behavioral restriction imposed by the Weak CR-CCE axiom together with the U-O axiom are sufficient to characterize conditional preferences that are represented by Weak RML.

\[thm2\] $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by Weak RML if and only if the U-O and Weak CR-CCE axioms hold. Furthermore, for each $\succsim$-nonnull $E \in \Sigma$, $\alpha[E]$ is unique if there exists $p, p' \in C$ s.t. $p(E) \neq p'(E)$.

For sufficiency of the U-O and Weak CR-CCE axiom, fix any $\succsim$-nonnull $E\in \Sigma$, the proof proceeds by the following three steps:

For all $f\in \mathcal{F}$, there exists $\alpha[E,f]$ such that for all sufficiently large consequences $x^{*} \in X$ one has $$\label{eq}
(1-\alpha[E,f])\text{CE}(f_{E}x) + \alpha[E,f] \text{CE}(f_{E}x^{*}) \sim (1-\alpha[E,f])x + \alpha[E,f] \text{CE}(x_{E}x^{*})$$ where CE$(f)$ denotes the certainty equivalence of the act $f$. The quantifier “sufficiently large” means that $x^{*}$ needs to be greater than the following three thresholds: (i)$x$ (ii) $\bar{x}_{E,f}$ and (iii) $\hat{x}_{E,f}$, where $\hat{x}_{E,f}$ denotes the consequence such that for all $x^{*} \succsim \hat{x}_{E,f}$ one has $$\min\limits_{p\in C} \int_{\Omega} u(f_{E}x^{*}) dp = \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*}) dp$$

Once $x^{*}$ is sufficiently large, the existence of an $\alpha[E,f]$ such that does not depend on $x^{*}$ is guaranteed. Furthermore, this $\alpha[E,f]$ is unique if either $f_{E}x \prec x$ or $f_{E}x^{*} \succ x_{E}x^{*}$ holds.

Equation (\[eq\]) implies that the DM’s conditional evaluation of any $f \in \mathcal{F}$ can be represented by $$\min\limits_{p\in C_{\alpha[E, f]}(E)} \int_{E} u(f) \frac{dp}{p(E)}$$

By plugging into the expression of each certainty equivalence in equation (\[eq\]), one is able to show that under another MEU preference $\succsim'$, which is represented by the set $C_{\alpha[E,f]}(E)$, the following indifference holds: $f_{E}x \sim' x$. Then it further implies that the conditional evaluation of the act $f$, equals to $u(x)$, also can be represented by a FB updating of preference $\succsim'$. Notice that when $\alpha \in (0,1]$, $C_{\alpha}(E)$ is always strict $\succsim'$-nonnull.

The Weak CR-CCE axiom implies that $\alpha[E,f]$ needs to be the same across all $f \in \mathcal{F}$ and it is unique if $C \neq C^{*}(E)$, i.e. there exists $p, p' \in C$ such that $p(E) \neq p'(E)$.

First, if there does not exist any $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique, then by step 1, it is the case both $f_{E}x \sim x$ and $f_{E}x^{*} \sim x_{E}x^{*}$ hold. In other words, the conditional preference are given by both FB and ML, which further implies $C = C^{*}(E)$. Thus, if $C \neq C^{*}(E)$, then there exists at least one $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique.

Fix any $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique. For any $l \in {\mathbb{R}}_{++}$, let $f_{l}$ denote an act such that $u(f_{l}(\omega)) = l \cdot u(f(\omega))$ for all $\omega \in E$. For any $\lambda \in [0,1]$ and $y\in X$, let $f_{\lambda}y$ denote the mixture $\lambda f + (1-\lambda)y$ of an act $f$ and the consequence $y$. Equation (\[eq\]) implies that for all $l\in {\mathbb{R}}_{++}$, $\lambda \in [0,1]$ and $y \in X$, $\alpha[E, f_{l\lambda}y] = \alpha[E,f]$. Namely, these two operations on $f$ preserves $\alpha[E,f]$.

If the first scenario of the Weak CR-CCE axiom holds, since it is equivalent to CR-LC axiom, it would imply that $\alpha[E,f] = 1$ for all $f \in \mathcal{F}$.

If the first scenario is false, then the second scenario must be true. For any $g\in \mathcal{F}$ that cannot be obtained from $f$ by the two operations that preserves $\alpha[E,f]$, one can always construct an $g_{l\lambda}y$ such that the following two conditions hold: $$f \sim_{E} g_{l\lambda}y \text{ and } f_{E}x^{*} \sim [g_{l\lambda}y]_{E}x^{*}$$ By the implication of the second scenario of Weak CR-CCE, it is necessarily the case that: first, $\alpha[E,f] \neq 1$ and second, $\alpha[E, g_{l\lambda}y] = \alpha[E,g] = \alpha[E,f]$. This argument applies to arbitrary $g\in \mathcal{F}$, thus $\alpha[E,f]$ needs to be a constant in this case as well.

To conclude, combining both scenarios ($\alpha[E,f] = 1$ and $\alpha[E,f]\neq 1$) implies that $\alpha[E,f]$ is a constant across all $f\in \mathcal{F}$ for each $\succsim$-nonnull $E\in \Sigma$.

Theorem \[thm2\] shows that the U-O and Weak CR-CCE axioms characterize the Weak RML updating rule where the parameter $\alpha[E]$ may be different across events. It is obviously because of the fact that both U-O and Weak CR-CCE only restrict behaviors within each conditional event $E$, yet does not explicitly restrict behaviors across events. To impose such an additional restriction, again consider another different direction of weakening the CR-CCE axiom:

For any two $\succsim$-nonnull events $E_{1}$ and $E_{2}$, if there are two acts $f,g$ such that $f \sim_{E_{1}}x$ and $g\sim_{E_{2}}x$, i.e. the conditional evaluation of the two acts under the two events $E_{1}$ and $E_{2}$ respectively are the same, then CR-CCE further implies that $f_{E_{1}}x \sim g_{E_{2}}x$. Thus CR-CCE actually implicitly imposes such a restriction on behaviors across different events, and therefore characterizes FB where the conditional preference is consistent across events.

For conditional preferences that are not as extreme as FB, only $f_{E_{1}}x \precsim x$ and $g_{E_{2}}x \precsim x$ are known. Hence to impose some restrictions on behaviors across events in the similar manner as in Weak CR-CCE, one can also restrict that $f_{E_{1}}x \sim g_{E_{2}}x $ only when the two acts $f$ and $g$ are in addition “pessimistically similar”. However notice that, since the conditional events are different here, the sufficiently large consequences need to satisfy some additional restriction to be able to calibrate the exact same extent of disappointment that event $E_{1}$ or $E_{2}$ occurs. To be more specific, in order to claim that $f_{E_{1}}x_{1}^{*} \sim g_{E_{2}}x_{2}^{*}$, one needs to find $x_{1}^{*}$ and $x_{2}^{*}$ that are not only sufficiently large, but also satisfies $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$.

Furthermore, the same special case ($\alpha = 1$) needs to be handled separately. Thus the axiom that further restricts behaviors across different events is given by the following:\
**Axiom EC** (Event Consistency). For all $\succsim$-nonnull events $E_{1}$ and $E_{2}$, at least one of the following scenarios hold:

(i) For all $f, g\in \mathcal{F}$ and for all $x\in X$, there exists $\bar{x}_{E_{1}, E_{2}, f,g} \in X$ such that if $f\sim_{E_{1}}x$ and $g \sim_{E_{2}} x$, then $ f_{E_{1}}x_{1}^{*} \sim g_{E_{2}}x_{2}^{*}$ for all $x_{1}^{*}, x_{2}^{*} \succsim \bar{x}_{E_{1}, E_{2}, f,g} $ whenever $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$.

(ii) For all $f, g\in \mathcal{F}$ and for all $x\in X$, there exists $\bar{x}_{E_{1}, E_{2}, f,g} \in X$ such that if $f\sim_{E_{1}}x$, $g \sim_{E_{2}} x$ and$ f_{E_{1}}x_{1}^{*} \sim g_{E_{2}}x_{2}^{*}$ for all $x_{1}^{*}, x_{2}^{*} \succsim \bar{x}_{E_{1}, E_{2}, f,g} $ whenever $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$, then $f_{E_{1}}x \sim g_{E_{2}}x$.

The following theorem shows that the additional restriction imposed by the EC axiom exactly characterizes a constant $\alpha$ for Weak RML.

\[thm4\] $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by RML if and only if the U-O, AC and EC axioms hold. Furthermore, $\alpha$ is unique if there exists $\succsim$-nonnull $E \in \Sigma$ such that $p(E) \neq p'(E)$ for some $p,p' \in C$.

Given Theorem \[thm2\], the only remaining proof is to show EC axiom is necessary and sufficient to characterize a constant $\alpha$ across all events. While a detailed proof can be found in the appendix, from the similarity between Weak CR-CCE and EC, the argument here is similar to the step 3 in the proof of Theorem \[thm2\].

Updating Ambiguous Signals
==========================

One special case in the current setting is when the state space $\Omega$ can be divided into payoff-relevant states $\Theta$ and payoff-irrelevant signals $S$. Namely, the grand state space $\Omega$ is given by the Cartesian product $\Theta \times S$ and a generic element is denoted by $\theta \times s$. The conditional event in which signal $s$ realizes is given by $E = \Theta \times s$. The set of available acts denoted by $\mathcal{F}_{\Theta}$ contains acts that depend only on $\Theta$: for all $f\in \mathcal{F}_{\Theta}$, $f(\theta\times s) = f(\theta \times s')$ for all $\theta \in \Theta$ and $s, s' \in S$.

Any environment involves states and signals can be modeled using this special framework. In general, a DM’s ambiguous belief may express ambiguity both on the marginal belief over states[^16] $\Theta$ and signals $S$. This section focuses on a further special case where the DM has an unambiguous marginal belief over the states. Formally, $p( A \times S) = p'(A \times S)$ for all $A\times S \in \Sigma$ and $p, p' \in C$.

It is the case where the DM has probabilistic belief about the states whereas the signals are ambiguous in the sense that the exact correlation between states and signals is unknown. In reality, one can imagine a situation in which a decision problem may have been encountered repeatedly such that the distribution over the states could be learned from historical data. Meanwhile for the current problem, an additional new data set (signal) is available to the DM. It is potentially informative for the true state from her knowledge about the data; however, the exact correlation between the data and the states is unknown.

This special case has been adopted in various applications: Bose and Renou (2014)[@bose2014mechanism], Beauchene, Li and Li (2019)[@beauchene2019ambiguous], Kellner and Le Quement (2018)[@kellner2018endogenous]. In addition, several recent experimental papers also focus on understanding subjects’ response to ambiguous signals in this case: Liang (2019)[@liang2019], Shishkin Ortoleva (2019)[@shishkin2019ambiguous], Kellner, Quement and Riener (2019)[@kellner2019reacting]. In the following, I will use a stylized example to illustrate the theoretical predictions RML is able to offer here.

\[exp1\] There are two payoff-relevant states: $\Theta = \{ \theta_{1},\theta_2\}$ with unambiguous marginal prior $p(\theta_{1}) = \beta \in [0,1]$. The DM is evaluating a bet $f$ that pays one at $\theta_{1}$ and nothing at $\theta_{2}$. There are two signals: $S = \{s_{1}, s_{2}\}$ and the signaling structure is ambiguous, means that there are two possible correlations between signals and states: $$\begin{aligned}
p(s_{i}| \theta_{i}) = \lambda_{1}  \\
p(s_{i}| \theta_{i}) = \lambda_{2} \end{aligned}$$ and which correlation generates the signal is unknown. Without loss of generality, assume $\lambda_{1} \geq \lambda_{2}$ and $\frac{\lambda_{1} + \lambda_{2}}{2} \geq \frac{1}{2}$. Namely, the signal $s_{1}$ is “on average” an informative signal for the state $\theta_{1}$. Moreover the first signaling device governed by $\lambda_{1}$ is more accurate than $\lambda_{2}$.

Suppose a DM forms a multiple priors belief coincides exactly to the information given in this example. Her set of priors $C$ can be easily parameterized by a parameter $\mu \in [0,1]$ which denotes the probability that the first signaling device governed by $\lambda_{1}$ is the one generates the signals. Namely, $C = \{p_{\mu} \in \Delta(\Theta \times S) : \mu \in [0,1]\}$ where $p_{\mu}$ is defined as in the following: $$\begin{aligned}
&p_{\mu} (\theta_{1} \times \{s_{1}, s_{2}\}) = \beta \\
&p_{\mu} (\theta_{2} \times \{s_{1}, s_{2}\}) = 1-\beta\\
&p_{\mu}(\theta_{1}\times s_{1}) = \beta[\mu\cdot \lambda_{1} + (1-\mu) \cdot \lambda_{2}]\\
&p_{\mu}(\theta_{2}\times s_{1}) = (1-\beta)[\mu\cdot (1-\lambda_{1}) +(1-\mu)\cdot (1-\lambda_{2}) ]\\
&p_{\mu} (\{\theta_{1}, \theta_{2} \}\times s_{1}) = p_{\mu}(\theta_{1}\times s_{1}) + p_{\mu}(\theta_{2}\times s_{1})\end{aligned}$$

To fully characterize the DM’s behavior in this example, there are in total six different cases: $\beta > 1/2, \beta = 1/2$ and $\beta < 1/2$ combined with signal realization $s_{1}$ or $s_{2}$. Only one of the cases will be derived here in detail and a summary of behaviors in all these cases will be provided afterwards in Table \[table1\].

Consider the case $\beta > 1/2$ and signal $s_{1}$ is realized. Notice that when $\beta > 1/2$, the likelihood of signal $s_{1}$, given by $p_{\mu} (\{\theta_{1}, \theta_{2} \}\times s_{1})$ is maximized when $\mu = 1$. Moreover, thanks to this simple parametrization of the set $C$, for any $\alpha \in [0,1]$ the set $C_{\alpha}(\{\theta_{1}, \theta_{2} \}\times s_{1}) $ is simply given by $$C_{\alpha}(\{\theta_{1}, \theta_{2} \}\times s_{1}) = \{p_{\mu} \in C: \mu \in [\alpha, 1] \}$$ Under RML, the DM updates only the priors $p_{\mu}$ for $\mu \in [\alpha, 1]$ and for each $p_{\mu}$ the posterior about states after observing $s_{1}$ is given by $$\pi_{\mu}(\theta_{1} | s_{1}) = \frac{p_{\mu}(\theta_{1}\times s_{1})}{p_{\mu} (\{\theta_{1}, \theta_{2} \}\times s_{1})}$$

Therefore the DM’s set of posteriors will be the following interval[^17]: $\left[ \pi_{\alpha}(\theta_{1} | s_{1}), \pi_{1}(\theta_{1} | s_{1})\right]$. Apparently, under FB, the DM’s set of posteriors will be $\left[ \pi_{0}(\theta_{1} | s_{1}), \pi_{1}(\theta_{1} | s_{1})\right]$; and under ML, the DM will end up with a singleton posterior: $\pi_{1}(\theta_{1} | s_{1})$.

Furthermore under MEU, the DM’s conditional evaluation of the bet $f$ will be given by the lowest posterior of state $s_{1}$. Let $x_{f}^{FB}, x_{f}^{RML}$ and $x_{f}^{ML}$ denote the conditional certainty equivalence of the bet $f$ under FB, RML and ML updating, then one has $x_{f}^{FB} \leq x_{f}^{RML} \leq x_{f}^{ML}$ for all $\alpha \in [0,1]$. Especially, $x_{f}^{RML}$ is increasing with respect to $\alpha$ and coincides with $x_{f}^{FB}$ and $x_{f}^{ML}$ when $\alpha = 0$ and $\alpha = 1$ respectively.

Table \[table1\] summarizes the predictions of behaviors under RML in all six cases. The first two rows are the only cases where the DM’s evaluation of $f$ depends on the value of $\alpha$. In other words, those are the cases where different inferences with respect to likelihood might affect one’s conditional evaluation of $f$ after updating, and thus RML is able to provide richer predictions than FB and ML.

The 3rd and 4th row are the cases where the MEU evaluation of $f$ are given by the posteriors updated from the maximum likelihood priors, thus even the updated beliefs are different under FB and ML, MEU makes their evaluations the same. For this reason, although the updated belief under RML is also different from FB and ML for any $\alpha \in (0,1)$, the MEU evaluation under RML is still the same.

Finally, the 5th and 6th rows represent the same type of special case where every prior agrees with the likelihood of the signal. It is the case where FB coincides with ML, and therefore RML does not have an extra bite. All the priors will be updated under RML for all $\alpha \in [0,1]$, thus the updated belief will always be the same.

  --------- --------- ----------- ---------------------------------- -----------------------
   $\beta$   Signal    ML prior               Evaluation                 Comparison with
                                                of $f$                probabilistic signal
                                                                      $\alpha <1/2$: lower
   $>1/2$    $s_{1}$   $\mu = 1$   $\pi_{\alpha}(\theta_{1}|s_{1})$   $\alpha = 1/2$: equal
                                                                      $\alpha >1/2$: higher
                                                                      $\alpha <1/2$: lower
   $>1/2 $   $s_{2}$   $\mu = 0$   $\pi_{\alpha}(\theta_{1}|s_{2})$   $\alpha = 1/2$: equal
                                                                      $\alpha >1/2$: higher
   $<1/2 $   $s_{1}$   $\mu = 0$     $\pi_{0}(\theta_{1}|s_{1})$       All $\alpha$: lower
   $<1/2 $   $s_{2}$   $\mu = 1$     $\pi_{1}(\theta_{1}|s_{2})$       All $\alpha$: lower
   $=1/2 $   $s_{1}$   All $\mu$     $\pi_{0}(\theta_{1}|s_{1})$       All $\alpha$: lower
   $=1/2 $   $s_{2}$   All $\mu$     $\pi_{1}(\theta_{1}|s_{2})$       All $\alpha$: lower
  --------- --------- ----------- ---------------------------------- -----------------------

  : Summary of Example \[exp1\] under RML[]{data-label="table1"}

The last column of Table \[table1\] presents a comparison that is highlighted in experiments by Liang (2019)[@liang2019]. It compares the DM’s s conditional evaluations of $f$ under this ambiguous signaling structure with a probabilistic signaling structure where the correlation equals to $\frac{\lambda_{1} + \lambda_{2}}{2}$. The latter is designed to reflect an average accuracy of the two signaling devices in the ambiguous signaling structure. For probabilistic signals, assume that the DM follows Bayesian updating.

In the case where $s_{1}$ is realized, notice that the DM’s posterior under the probabilistic signaling structure is exactly given by $\pi_{1/2}(\theta_{1} | s_{1})$. Therefore, for $\beta \leq 1/2$, the DM’s conditional evaluation under ambiguous signal is always **lower** than her conditional evaluation under probabilistic signal. On the other hand, when $\beta > 1/2$, this comparison would depend on the value of $\alpha$. More specifically, $\alpha < 1/2$ implies that the conditional evaluation under ambiguous signal is lower, $\alpha > 1/2$ implies that under ambiguous signal is higher and $\alpha = 1/2$ implies they are the same. Therefore, according to RML, this type of comparison sometimes actually reflect the DM’s attitudes towards making an inference with respect to likelihood captured by $\alpha$.

For the experimental results shown by the Table 4.2 in Liang (2019)[@liang2019], each row there can be categorized into some case in the Table \[table1\] here, as $s_{1}$ is “good news” and $s_{2}$ is “bad news” in his terminology. A rather interesting pattern emerged from the choice data is that, when $\beta > 1/2$, the subjects’ comparison is lower after receives good news ($s_{1}$) and higher after receives bad news ($s_{2}$). Notice that from Table \[table1\], the two directions of the comparisons are aligned with $\alpha < 1/2$ and $\alpha > 1/2$ respectively. It implies that the subjects may have different attitudes towards making inferences when the signal realization is different, especially when they are associates with meanings such as “good news” and “bad news”. This pattern can be captured by the **Weak RML**, which is characterized exactly to reflect such a flexibility in behaviors.

Moreover, the choice data shown in the other rows corresponding to cases where $\alpha$ is irrelevant for this comparison, is also aligned with behaviors under RML such that the DM finds her evaluation of $f$ is lower under ambiguous signal[^18]. Therefore, the Weak RML with $\alpha[\{\theta_{1}, \theta_{2} \}\times s_{1}]  <1/2$ and $\alpha[\{\theta_{1}, \theta_{2} \}\times s_{2}]  >1/2$, not only offers an interpretation for the different directions of comparison across signals, but in fact also provides an approach to accommodate almost all the behavioral patterns in this experiment[^19].\
In summary, for this special case where DM has probabilistic belief over states and ambiguous belief over signals, RML is able to provide richer predictions when inferences with respect to likelihoods affects behaviors. Moreover, such richness on the other hand would also be useful for considering the robustness of behaviors under ambiguous signals when the DM can potentially make an inference with respect to likelihoods.

Robust Ambiguous Persuasion: An Example
=======================================

This section elaborates the last comment from the previous section in the context of information design, or more specifically, persuasion. The scope here is restricted to an illustrative example, more general results are left for future work.

Consider a persuasion environment where the sender can commit to some signaling device to induce actions from the receiver and suppose the sender and receiver have a common probabilistic belief about the states.

Bayesian persuasion describes the case where the available signaling devices for the sender are only probabilistic. If the sender also has access to ambiguous devices, which specifies a set of probabilistic devices and the probability of using any one of them is unknown[^20], Beauchene, Li and Li (2019)[@beauchene2019ambiguous] show that when the receiver’s preference is represented by MEU with FB, the sender is able to gain strictly more payoff by sending ambiguous signals compared with the optimal probabilistic device.

Notice that once the sender commits to an ambiguous device, from the receiver’s point of view, it is exactly the updating ambiguous signals situation extensively discussed in the previous section. Then apparently the FB assumption excludes the possibility that the receiver may use the likelihood of signals to further infer about the plausibility of each probabilistic device. This fact thus suggests a caveat that the strict gain from using ambiguous signals may only come from the FB assumption instead of the general idea of ambiguity. In other words, it might seem to be that the ambiguous signals could help the sender exploit ambiguity aversion of the receiver only when inferences about the initial belief are not allowed.

RML provides exactly the tool to address such an issue. In the following, I take the illustrative example from Beauchene, Li and Li (2019) [@beauchene2019ambiguous] and relax their FB assumption to RML with any $\alpha \in [0,1]$. Then for this particular example, first of all, the ambiguous device that approaches the sender’s optimal payoff under FB assumption is not robust. In particular, for RML with any $\alpha > 0$, their ambiguous device cannot induce the same action from the receiver as under FB ($\alpha = 0$), and the sender’s payoff is strictly worse than using the optimal probabilistic device.

However on the other hand, for this example there still exists a *robust* ambiguous device such that induce the same action from the receiver and generates the same payoff for the sender for all $\alpha \in [0,1]$ under RML. Moreover, the sender gains strictly more payoff from this robust ambiguous device compared with the optimal probabilistic device. In other words, the strict benefits from using ambiguous signals in this example is robust to any possible inferences with respect to the likelihood of signals in updating.

\[exp4\] There are two states $\{ \omega_{l}, \omega_{h}\}$ with a uniform marginal prior, and the receiver has three feasible actions: $\{a_{l}, a_{m}, a_{h}\}$. The payoff of sender and receiver for each state and action is given as follows:

             $\omega_{l}$   $\omega_{h}$
  --------- -------------- --------------
   $a_{l}$      (-1,3)        (-1,-1)
   $a_{m}$      (0,2)          (0,2)
   $a_{h}$      (1,-1)         (1,3)

where in each cell, the first number is sender’s payoff and the second is receiver’s.

The payoff structure here is standard for persuasion, where the sender always prefers the receiver to take higher action yet the receiver prefers to choose the action that matches the state.

First consider the FB assumption, the following replicates the discussions from Beauchene, Li and Li (2019)[@beauchene2019ambiguous]. For more in depth explanations, readers are advised to refer to the original paper.

The ambiguous device they offer under FB assumption can be constructed by the following two steps: first, identify the *base probabilistic devices* that generate the desired set of posteriors, and then construct the probabilistic devices from these base devices in a way that is able to hedge against the sender’s own ambiguity.

In this example, let $\{m_{l}, m_{h}\}$ be the set of signals and then the base probabilistic devices $\pi_{1}$ and $\pi_{2}$ are:

   $\pi_{1}(m|\omega)$   $\omega_{l}$   $\omega_{h}$
  --------------------- -------------- --------------
         $m_{l}$            $2/3$            0
         $m_{h}$            $1/3$           $1$

   $\pi_{2}(m|\omega)$   $\omega_{l}$   $\omega_{h}$
  --------------------- -------------- --------------
         $m_{l}$            $3/4$          $1/4$
         $m_{h}$            $1/4$          $3/4$

If the sender’s ambiguous device consists of these two base devices, then the receiver would form the following set of posteriors: $$\begin{split}
&p(\omega_{h}|m_{l}) = \{0, 1/4\} \\
&p(\omega_{h}|m_{h}) = \{3/4, 3/4\}
\end{split}$$ where the first and second posterior in each set is updated from the first and second base device respectively. Given these posteriors, the receiver with MEU preference would take action $a_{m}$ when signal $m_{l}$ is realized and takes action $a_{h}$ when signal $m_{h}$ is realized. The posterior $p(\omega_{h}|m_{l}) = 1/4$ is crucial since any posterior that assigns less probability on $\omega_{h}$ would induce the receiver to take action $a_{l}$ and makes the sender worse off.

Given the receiver’s action for each signal, the sender’s evaluation of the ambiguous device may also be affected by the existence of ambiguity. If the sender uses an ambiguous device which contains exactly these two base devices, then under MEU her ex-ante payoff is given by $$\frac{1}{2} u_{s}(a_{m}) + \frac{1}{2} u_{s}(a_{h})$$ where $u_{s}$ stands for the sender’s payoff function and it is exactly the payoff from using the optimal probabilistic device. Thus the sender needs to hedge against this ambiguity to get a higher payoff.

Consider the following construction: first increase the number of signals to four such that now the signals are $\{m_{l}, m_{h}, m_{l}', m_{h}'\}$. Then consider a probabilistic device generated by a mixture of the two base devices: $\pi_{1}' = \lambda \pi_{1} \oplus (1-\lambda)\pi_{2}$, which represents a device sending signals $\{m_{l}, m_{h}\}$ with probability $\lambda$ according to the base device $\pi_{1}$ and sending signals $\{m_{l}', m_{h}'\}$ with probability $(1-\lambda)$ according to $\pi_{2}$. One can verify that given this device, the receiver’s posteriors coincide with $\pi_{1}$ when $m \in \{m_{l}, m_{h}\}$ and coincides with $\pi_{2}$ when $m \in \{m_{l}', m_{h}'\}$.

Moreover, consider the following two probabilistic devices constructed in the same manner $\pi_{1}' = \lambda \pi_{1} \oplus (1-\lambda)\pi_{2}$ and $\pi_{2}' = (1-\lambda)\pi_{2}\oplus \lambda \pi_{1} $:

   $\pi_{1}'(m|\omega)$        $\omega_{l}$             $\omega_{h}$
  ---------------------- ------------------------ ------------------------
         $m_{l}$           $\lambda \cdot 2/3$               0
         $m_{h}$           $\lambda \cdot 1/3$           $\lambda$
         $m_{l}'$         $(1-\lambda)\cdot 3/4$   $(1-\lambda)\cdot 1/4$
         $m_{h}'$         $(1-\lambda)\cdot 1/4$   $(1-\lambda)\cdot 3/4$

   $\pi_{2}'(m|\omega)$        $\omega_{l}$             $\omega_{h}$
  ---------------------- ------------------------ ------------------------
         $m_{l}$          $(1-\lambda)\cdot 3/4$   $(1-\lambda)\cdot 1/4$
         $m_{h}$          $(1-\lambda)\cdot 1/4$   $(1-\lambda)\cdot 3/4$
         $m_{l}'$          $\lambda \cdot 2/3$               0
         $m_{h}'$          $\lambda \cdot 1/3$           $\lambda$

When $m \in \{m_{l}, m_{h}\}$, the posterior of $\pi_{1}'$ coincides with $\pi_{1}$ and the posterior of $\pi_{2}'$ coincides with $\pi_{2}$, so that the set of posteriors generated by the ambiguous device $\Pi' = \{\pi_{1}', \pi_{2}'\}$ remains the same as the base devices. Then signals $m_{l}$ and $m_{l}'$ would induce the receiver to take action $a_{m}$ and signals $m_{h}$ and $m_{h}'$ would induce action $a_{h}$.

Furthermore notice that, the difference between these two probabilistic devices is **only** the label of the signals. Thus, given the receiver’s action is the same across signal $m_{l}$ and $m_{l}'$ as well as across signals $m_{h}$ and $m_{h}'$. The two probabilistic devices induce each action with the same frequency, hence the sender’s ex-ante payoff will be the same across these two devices. Therefore using an ambiguous device that contains these two probabilistic devices will not generate any ambiguity for sender’s ex-ante payoffs.

Especially, the sender’s ex-ante payoff from this ambiguous device is given by $$\label{ap}
\left[\frac{1}{2}(1-\lambda) + \frac{1}{3}\lambda\right]u_{s}(a_{m}) + \left[\frac{1}{2}(1-\lambda) + \frac{2}{3}\lambda \right]u_{s}(a_{h})$$ which is strictly higher than using the optimal probabilistic device when $\lambda > 0$ and it is also increasing in $\lambda$.

Therefore, the optimal ambiguous persuasion can be approached by letting $\lambda \rightarrow 1$, notice that $\lambda$ cannot be exactly one.\
Above is the Beauchene, Li and Li (2019)[@beauchene2019ambiguous]’s construction of the ambiguous device under FB assumption. The fact that their optimal device relies on letting $\lambda$ approaches one but cannot be exactly one already suggests its fragility when FB assumption is slightly deviated.

More formally, such a concern is exactly originated from the fact that the likelihood of generating the signals by each device also depends on $\lambda$. If letting $\lambda \rightarrow 1$, the likelihood of generating the same signal by each device might be severely different.

Let $l_{i}(m)$ denote the likelihood of generating signal $m$ under device $\pi_{i}'$:

                    $l_{1}(m)$         $p(\omega_{h}|m)$
  ---------- ------------------------ -------------------
   $m_{l}$     $\lambda \cdot 1/3$             0
   $m_{h}$     $\lambda \cdot 2/3$            3/4
   $m_{l}'$   $(1-\lambda)\cdot 1/2$          1/4
   $m_{h}'$   $(1-\lambda)\cdot 1/2$          3/4

                    $l_{2}(m)$         $p(\omega_{h}|m)$
  ---------- ------------------------ -------------------
   $m_{l}$    $(1-\lambda)\cdot 1/2$          1/4
   $m_{h}$    $(1-\lambda)\cdot 1/2$          3/4
   $m_{l}'$    $\lambda \cdot 1/3$             0
   $m_{h}'$    $\lambda \cdot 2/3$            3/4

When $\lambda \rightarrow 1$, notice that the likelihood of generating signal $m_{l}$ by device $\pi_{2}'$ goes to $0$, whereas the likelihood of $m_{l}$ by device $\pi_{1}'$ goes to $1/3$. Thus intuitively, knowing $\lambda \rightarrow 1$, whenever signal $m_{l}$ is observed, the receiver should be almost sure that it is generated by device $\pi_{1}'$. However, notice that the crucial posterior $1/4$ is in fact generated by the other device $\pi_{2}'$.

Indeed, for RML with any $\alpha > 0$, since the device $\pi_{2}'$ has the minimum likelihood of generating signal $m_{l}$, the crucial posterior $1/4$ is always excluded from the set of posteriors. Then the receiver would find $a_{l}$ is strictly better than $a_{m}$, thus the desired action $a_{m}$ can no longer induced. Furthermore, one can also verify that the sender’s payoff is equivalent to some non-optimal probabilistic device, thus she instead becomes strict worse off using ambiguous signals compared with using the optimal probabilistic device when the receiver slightly deviates from FB in the direction of ML.

Therefore the first observation is that, ambiguous devices that are strictly profitable than the optimal probabilistic device under FB may sharply depend on this assumption. Any slight deviation from FB in the direction of ML could actually result in a complete reversal of the comparison.

Nonetheless notice that, when $\lambda = 3/5$, the likelihood of generating signal $m_{l}$ by device $\pi_{1}'$ and $\pi_{2}'$ becomes the same and it is also true for the signal $m_{l}'$. Consequently, the receiver cannot use likelihood to make any inference about the devices. It is the case where FB and ML coincides and the receiver will always update with respect to both devices under RML no matter what the $\alpha$ is. As the crucial posterior $1/4$ will always be updated, the sender is able to induce the action $a_{m}$ from the receiver when signal $m_{l}$ or $m_{l}'$ realizes.

Furthermore, even though the likelihood of generating $m_{h}$ and $m_{h}'$ is not the same across the two devices, as the corresponding posteriors are the same, these signals can always induce the same action $a_{h}$ from the receiver as well. That being said, the ambiguous device $\pi'$ with $\lambda = 3/5$ is able to always induce action $a_{m}$ and $a_{h}$ with corresponding signals regardless of the receiver’s attitudes towards making inferences, i.e. for any RML receiver with any $\alpha \in [0,1]$. In addition, the sender’s payoff will also be given by equation (\[ap\]) after plugging into $\lambda = 3/5$, which is strictly higher than using the optimal probabilistic device.

Therefore, this ambiguous device is *robustly better* than probabilistic devices in the sense that it is able to generate strictly more payoff for the sender regardless of the receiver’s attitudes between FB and ML. More importantly, it further suggests that the strict gain from using ambiguous signals is also robust to the concern that the receiver may use likelihoods to make inferences.\
This section presents an example of considering the robustness question for applications of ambiguity, especially it illustrates how RML can be applied as a tool to look at robustness with respect to updating ambiguity. Although the scope here is restricted only to this example, more results along this line hopefully may help resolve some potential doubts and concerns about applications of ambiguity. Undoubtedly, RML considers only one dimension of robustness, the other dimensions captured by other updating models such as dynamic consistency (Hanany and Klibanoff 2007[@hanany2007updating]) are also equally important.

Related Literature
==================

The present paper adds to the literature on dynamic choice under ambiguity by characterizing the RML updating rule. As mentioned in the introduction, for ambiguity sensitive choice, one cannot preserve both dynamic consistency and consequentialism at the same time (Hanany and Klibanoff 2007[@hanany2007updating], Siniscalchi 2009[@siniscalchi2009two]). RML takes the consequentialist approach such that relaxes dynamic consistency. Pires (2002)[@pires2002rule] and Gilboa and Schmeidler (1993)[@gilboa1993updating] are the two most closely related papers that also take the consequentialist approach. Along another route, Hanany and Klibanoff (2007, 2009)[@hanany2007updating][@hanany2009updating] axiomatize updating rules that preserve dynamic consistency yet do not require consequentialism.

The idea of refining initial belief as new information arrives is essential in many different non-Bayesian updating rules, and it is also not an exclusive feature for beliefs with multiple priors. When the initial belief is a singleton, for example, Ortoleva (2012)[@ortoleva2012modeling] characterizes a hypothesis testing updating rule such that if the likelihood of information received is too low under the initial belief, then the DM will revise that initial belief and find a different prior for updating. Since the initial belief is probabilistic, the hypothesis testing updating rule emphasizes on dealing with unexpected events such as those with a probability of 0.

Zhao (2017)[@zhao2017] also considers the probabilistic belief and focuses on unexpected information, especially when the information takes the form “event $A$ is more likely than event $B$” such that it contradicts to the DM’s initial belief. He characterizes the Pseudo-Bayesian updating rule in which the DM updates another prior that is closest to the initial belief in terms of Kullback-Leibler divergence and is subject to the constraint specified in the unexpected information.

When the information is ambiguous yet the DM is ambiguity neutral and forms a single prior belief, Suleymanov (2018)[@suleymanov2018a] characterizes the Robust Maximum Likelihood updating rule in which the DM revises her initial belief according to the maximum likelihood of the observed event. Namely, the DM is an expected utility maximizer both ex-ante and conditionally, yet the posterior is not updated from the prior. Thus the main difference between the Robust Maximum Likelihood and ML as well as RML (Relative Maximum Likelihood) is that the latter updating rules require that the posteriors have to be updated from the subset of those priors that represent the DM’s ex-ante preference. In other words, RML necessarily reduces to Bayesian updating when the DM’s ex-ante preference is represented by expected utility, which is not true for Robust Maximum Likelihood updating.

In cases where the initial belief is a set of priors, one way of refining is to rule out priors from the initial set. Epstein and Schneider (2007)[@epstein2007learning] proposes an updating rule without characterization, such that the refining is done according to the likelihood ratio test. RML belongs to this category of ruling out priors, and more importantly, relative likelihood ratio test is the necessary criterion for selecting priors under RML. The dynamic consistent updating rule characterized in Hanany and Klibanoff (2007)[@hanany2007updating] also features ruling out priors, and the criteria there is to maintain the optimality of the ex-ante preferred act.

Yet another way of refining is to consider a different set of priors, where it is possible that some priors are not included in the initial belief. Ortoleva (2014)[@Ortoleva2014] characterizes the hypothesis testing updating rule for multiple priors: if the likelihood of the observed event is too low under some prior in the initial belief, then the DM will revise her initial belief and change to another set of priors for updating.

When the initial belief further involves confidence ranking proposed by Hill (2013)[@hill2013confidence], based on a similar motivation, Hill (2019)[@hill2019] characterizes an updating rule under this framework which also features in refining initial belief by the information received, and it include FB and ML as special cases as well.

Except for the theoretical development on this idea of revising and refining initial belief, De Filippis et al. (2019)[@de2018non] also identifies such behavior in a social learning experiment. Their finding suggests that the non-Bayesian behavior observed in the experiment is consistent with a generalized ML updating rule where subjects revise their initial belief according to the information they receive. As mentioned in section 5, there are several recent experimental papers, Liang (2019)[@liang2019], Shishkin Ortoleva (2019)[@shishkin2019ambiguous] and Kellner, Quement and Riener (2019)[@kellner2019reacting] testing updating behaviors under ambiguity in a special environment.

Gul and Pesendorfer (2019)[@gul2017evaluating] observe a common unintuitive feature of both FB and ML that is “all news is bad news”. Namely, a DM sometimes find that the ex-ante preferred alternative is dominated by another alternative no matter what the realization of the signal is. Imposing the “not all news can be bad news” axiom, they characterize an updating by proxy rule for preferences admitting CEU representation with capacities that are totally monotone, which is a strict subset of the preferences considered in the present paper.\

Concluding Remarks
==================

Recall the updating procedure for multiple priors beliefs proposed at the beginning of introduction. Bayesian updating of single prior beliefs is a special case where the first step is absent. In the case where the true probability law that governs the uncertainty is known, such absence is reasonable since one cannot further refine the belief but can only update it conditional on the information received.

However, in scenarios where the underlying probability law is unknown, the DM needs to form a conjecture about the uncertainty for decision making. Whether the conjecture is a singleton or a set of probabilities, it all seems too stringent to require the DM to always stick with her initial conjecture despite new information she might receive. Thus Bayesian updating of the conjecture belief actually reflects a confidence about her initial belief for updating. Accordingly, updating rules that do not reflect such a confidence and allow for refining initial belief should also be reasonable.

As mentioned in the related literature, several different updating rules have been proposed to capture the situation in which initial conjecture is a singleton and it may be revised after seeing new information. For when the initial beliefs are multiple priors, the present paper proposes RML updating rule, in which the initial beliefs are revised based on likelihood of the information observed. More importantly, the present paper pinpoints the behaviors that are equivalent to such an updating rule, which provides a preference foundation for using likelihood as a criterion for updating.

One application of RML highlighted in the present paper is using it for robustness analysis in applications involve ambiguity. As only one example is illustrated here, more general results are left for future research.

Proofs of the results
=====================

Throughout all the proofs, let $C$ denote the set of priors representing the ex-ante preference; $C^{*}(E)$ denote the subset of $C$ assigning maximum likelihood of event $E$: $C^{*}(E) \equiv \{p \in C: p(E) \geq p'(E) \forall p' \in C\}$ and $p^{*}(E)$ denote the maximum likelihood of event $E$: $p^{*}(E) \equiv \max\limits_{p \in C}p(E)$.

Proof of Theorem \[thm1\]
-------------------------

**Necessity.** The necessity of CR-LC axiom for ML updating is proved via the following three lemmas:

\[lem1\] Suppose the conditional preferences are represented by ML. For all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$ and for all $x, x^{*} \in X$, if $f \sim_{E} x $, $x^{*} \succsim x$ and $$\min\limits_{p \in C} \int_{\Omega} u(f_{E}x^{*})dp = \min\limits_{p \in C^{*}(E)}  \int_{\Omega} u(f_{E}x^{*})dp$$ then $f_{E}x^{*} \sim x_{E}x^{*}$.

Suppose the conditional preferences are represented by ML. For any $\succsim$-nonnull $E$, $f \sim_{E}x$ implies that $\min\limits_{p \in C^{*}(E)} \int_{E} u(f)\frac{dp}{p(E)}= u(x)$, then one can further derive $$\begin{aligned}
\min\limits_{p \in C} \int_{\Omega} u(f_{E}x^{*}) dp &= \min\limits_{p \in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*}) dp \\
&= \min\limits_{p\in C^{*}(E)}  \left[ \int_{E}  u(f_{E}x^{*}) \frac{dp}{p(E)} \cdot p(E) + (1-p(E))u(x^{*}) \right] \\
& = p^{*}(E)\cdot \min\limits_{p\in C^{*}(E)} \int_{E} u(f)\frac{dp}{p^{*}(E)} +  (1-p^{*}(E))u(x^{*}) \\
&= p^{*}(E) u(x) + (1-p^{*}(E))u(x^{*}) \\
&= \min\limits_{p \in C} \int_{\Omega}u(x_{E}x^{*})dp\end{aligned}$$ where the third equality follows from $p(E) = p^{*}(E)$ for all $p \in C^{*}(E)$, the last equality follows from the fact that $u(x^{*}) \geq u(x)$ as $x^{*} \succsim x$.

\[lem2\] For all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$, if there exists $\bar{x}\in X$ such that $$\min\limits_{p \in C} \int_{\Omega}u(f_{E}\bar{x})dp = \min\limits_{p \in C^{*}(E)}  \int_{\Omega} u(f_{E}\bar{x})dp$$ then for all $x^{*} \succsim \bar{x}$ one has $$\min\limits_{p \in C}  \int_{\Omega}u(f_{E}x^{*})dp = \min\limits_{p \in C^{*}(E)} \int_{\Omega}u(f_{E}x^{*})dp$$

For any $\succsim$-nonnull $E \in \Sigma$ and any $f\in \mathcal{F}$. Suppose there exists $x\in X$ such that $\min\limits_{p \in C} \int_{\Omega}u(f_{E}x)dp = \min\limits_{p \in C^{*}(E)}  \int_{\Omega} u(f_{E}x)dp$. Towards a contradiction suppose there also exists $x'$ such that $x' \succsim x$ as well as $$\min\limits_{p\in C}\int_{\Omega}u(f_{E}x')dp < \min\limits_{p\in C^{*}(E)} \int_{\Omega}u(f_{E}x')dp$$ Then this strict inequality further implies: $$\begin{split}
&\min\limits_{p\in C} \left[ \int_{E} u(f)dp + u(x')(1-p(E)) \right]< \min\limits_{p\in C^{*}(E)} \left[\int_{E} u(f)dp + u(x') (1-p(E)) \right] \\
&\min\limits_{p\in C}  \left[ \int_{E} u(f)dp + u(x')(1-p(E)) \right]<  \min\limits_{p\in C^{*}(E)}\left[ \int_{E} u(f)dp+ u(x)(1-p^{*}(E)) + (u(x')-u(x))(1-p^{*}(E)) \right]\\
&\min\limits_{p\in C} \left[ \int_{E} u(f)dp + u(x')(1-p(E)) \right] - (u(x')-u(x))(1-p^{*}(E)) <  \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp\\
&\min\limits_{p\in C} \left[\int_{E} u(f)dp + u(x)(1-p(E)) + (u(x')-u(x))(p^{*}(E)- p(E)) \right] <  \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp\\
&\min\limits_{p\in C} \left[ \int_{\Omega} u(f_{E}x) dp + (u(x')-u(x))(p^{*}(E)- p(E)) \right]<  \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp
\end{split}$$ Notice that, for the LHS of the last inequality, minimum of $\int_{\Omega} u(f_{E}x) dp $ can be obtained at some $p \in C^{*}(E)$ which also minimizes the second term $(u(x')-u(x))(p^{*}(E)- p(E))$, since for all $p\in C$, $p^{*}(E) - p(E) \geq 0$. Thus the minimum of LHS is obtained at some p with $p(E) = p^{*}(E)$ and it implies $$\min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp = \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp < \min\limits_{p\in C^{*}(E)}\int_{\Omega} u(f_{E}x) dp$$ which is a contradiction.

\[lem4\] If $\succsim$ admits MEU representation with finitely many plausible priors, then for all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$, an $\bar{x}_{E,f} \in X$ such that both $$\min\limits_{p \in C} \int_{\Omega}u(f_{E}\bar{x}_{E,f})dp = \min\limits_{p \in C^{*}(E)}  \int_{\Omega} u(f_{E}\bar{x}_{E,f})dp$$ and $\bar{x}_{E,f} \succsim_{E} f$ always exists.

First show that when $C$ contains only finitely many extreme points, an $\bar{x}_{E,f}$ such that $f_{E}\bar{x}_{E,f}$ is evaluated at some extreme point in $C^{*}(E)$ always exists.

Let $q$ be any extreme point in $C^{*}(E)$ and let $p$ be any extreme point in $C$. The act $f_{E}\bar{x}_{E,f}$ is evaluated at $q$ if for all $p \in C$, $$\int_{\Omega} u(f_{E}\bar{x}_{E,f}) dq \leq \int_{\Omega} u(f_{E}\bar{x}_{E,f}) dp$$ It can be further derived as $$\label{eq1}
\int_{E} u(f) dq + (1-q(E)) u(\bar{x}_{E,f}) \leq \int_{E} u(f) dp + (1-p(E)) u(\bar{x}_{E,f})$$ Notice that the first term of both LHS and RHS does not depend on $\bar{x}_{E,f}$, furthermore, $(1-q(E)) \leq (1-p(E))$ as $q \in C^{*}(E)$. When $p$ is also in $C^{*}(E)$, the value of $\bar{x}_{E,f}$ does not matter and one can pin down the $q \in C^{*}(E)$ such that minimizes the evaluation of $f_{E}\bar{x}_{E,f}$ among all extreme points in $C^{*}(E)$.

Fix that $q$, and then for extreme points not in $C^{*}(E)$, the following inequality becomes strict: $(1-q(E)) < (1-p(E))$. Then for all $f, p, E$, since $X$ is unbounded under the ex-ante preference, there always exists an $\bar{x}_{E,f,p}$ such that the inequality (\[eq1\]) holds. As inequality (\[eq1\]) also holds for any $x \succsim \bar{x}_{E,f,p}$, it suffice to let $\bar{x}_{E,f}$ be $ \max \{ \max\limits_{p} \bar{x}_{E,f,p}, x\}$ for $x \sim_{E} f$ where the “max” is according to the ex-ante preference. The existence of such an $\bar{x}_{E,f}$ is guaranteed by $X$ being unbounded under the ex-ante preference and there are only finitely many extreme points in $C$

In summary, for all $\succsim$-nonnull $E\in \Sigma$, for all $f \in \mathcal{F}$ and $x \in X$, Lemma \[lem2\] and \[lem4\] together show the existence of a threshold $\bar{x}_{E,f}$ such that for all $x^{*} \succsim \bar{x}_{E,f}$, the evaluation of the act $f_{E}x^{*}$ is given by some prior in $C^{*}(E)$, meanwhile $x^{*} \succsim x$. Then Lemma \[lem1\] shows that, when $\succsim_{E}$ is given by ML updating, for all $x^{*} \succsim \bar{x}_{E,f}$ one has $f_{E}x^{*} \sim x_{E}x^{*}$.\
**Sufficiency.** For sufficiency of CR-LC axiom, fix any $\succsim$-nonnull $E\in \Sigma$, consider the contra positive statement: not ML updating implies not CR-LC.

Let $C_{E}$ be the closed and convex set of posteriors represents the conditional preference $\succsim_{E}$. Not ML updating implies that $C_{E} \neq \{\frac{p}{p(E)}: p \in C^{*}(E)\}$. In other words, either there exists $\tilde{p} \in C^{*}(E)$ such that $\frac{\tilde{p}}{\tilde{p}(E)} \notin C_{E}$, or there exists $q \in C_{E}$ such that $q \notin \{\frac{p}{p(E)}: p \in C^{*}(E)\}$ or both.

Not CR-LC means that there exists $f\in \mathcal{F}$ and $x\in X$ such that $f\sim_{E}x$ and for all $\bar{x} \in X$, there exists $x^{*} \in X$ such that $x^{*} \succsim \bar{x}$ and it is not the case $f_{E}x^{*} \sim x_{E}x^{*}$.

For the two different cases of not ML, since both $C_{E}$ and $\{\frac{p}{p(E)}: p \in C^{*}(E)\}$ are convex and closed set, the same type of separating hyperplane argument can be applied to both cases. Thus the proof here only shows the implication of the first case, while the same argument applies to the other case.

Formally, in the first case, there exists $\tilde{p} \in C^{*}(E)$ such that $\frac{\tilde{p}}{\tilde{p}(E)} \notin C_{E}$, strong separating hyperplane theorem implies that there exists an act $f\in \mathcal{F}$ such that $$\int_{E} u(f) \frac{d\tilde{p}}{\tilde{p}(E)} < \min\limits_{p\in C_{E}} \int_{E} u(f)dp$$ Then first as $\tilde{p} \in C^{*}(E)$, one has $\min\limits_{p\in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)} \leq \int_{E} u(f) \frac{d\tilde{p}}{\tilde{p}(E)}$. Second, for any $x\in X$, $f \sim_{E} x$ implies that $\min\limits_{p\in C_{E}} \int_{E} u(f)dp = u(x)$. These inequalities and equality together imply that $$\min\limits_{p\in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)} < u(x)$$

On the other hand by Lemma \[lem4\], there always exists an $\bar{x}_{E,f}$ such that $\bar{x}_{E,f} \succsim x$ and for all $x^{*} \succsim \bar{x}_{E,f}$, $$\min\limits_{p \in C} \int_{\Omega} u(f_{E}x^{*}) dp = \min\limits_{p \in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp$$ Then for any $x^{*} \succsim \bar{x}_{E,f}$, the following is true: $$\begin{split}
\min\limits_{p \in C} \int_{\Omega} u(f_{E}x^{*}) dp & = \min\limits_{p \in C^{*}(E)}  \left[ \int_{E} u(f) \frac{dp}{p(E)} \cdot p(E) + x^{*}(1-p(E)) \right]\\
                    & =p^{*}(E) \min\limits_{p \in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)} +  x^{*}(1-p^{*}(E))\\
                    & < p^{*}(E) \cdot u(x) + u(x^{*})(1-p^{*}(E)) \\
                    &= \min\limits_{p \in C} \int_{\Omega} u(x_{E}x^{*}) dp
\end{split}$$ It means that, if $f\sim_{E}x$ then $f_{E}x^{*} \prec x_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E,f}$. That is, for this $f\in \mathcal{F}$, for any $\bar{x} \in X$, there always exists $x^{*} \succsim \bar{x}$ such that it is not the case $f_{E}x^{*} \sim x_{E}x^{*}$, i.e. the CR-LC axiom is not true.

The argument for the second case is analogously the same and combining both cases shows that not ML updating implies not CR-LC.

Proof of Theorem \[thm2\]
-------------------------

The necessity of the U-O axiom is given before introducing the axiom. For the Weak CR-CCE axiom, $\alpha[E] = 1$ implies the first scenario, $\alpha[E] \neq 1$ implies the second, and both scenarios are true when $\alpha[E]$ is not unique. This is immediate by noticing that equation (\[equ1\]) holds with $\alpha[E]$ is necessary under Weak RML, which is further given by reversing the arguments in step 2 for sufficiency.\
For sufficiency, fix any $\succsim$-nonnull $E\in \Sigma$, the proof proceeds by the following steps:

: Show that for all $f\in \mathcal{F}$, there exists $\alpha[E,f] \in [0,1]$ such that for all sufficiently large consequence $x^{*} \in X$ one has $$\label{equ1}
(1-\alpha[E,f])\text{CE}(f_{E}x) + \alpha[E,f] \text{CE}(f_{E}x^{*}) \sim (1-\alpha[E,f])x + \alpha[E,f] \text{CE}(x_{E}x^{*})$$ where CE$(f)$ denote the certainty equivalence of the act $f$. Furthermore, $\alpha[E,f]$ is unique if either $f_{E}x \prec x$ or $f_{E}x^{*} \succ x_{E}x^{*}$ hold.\
When the U-O axiom is true, $f \sim_{E} x$ implies that $f_{E}x \precsim x$ and there exists $\bar{x}_{E,f}\in X$ such that $f_{E}x^{*} \succsim x_{E}x^{*}$ for all $x^{*} \succsim \bar{x}_{E,f}$.

First consider all $x^{*} \in X$ such that $x^{*} \succsim \bar{x}_{E,f}$ and $x^{*} \succsim x$, then the following inequalities hold: $f_{E}x^{*} \succsim x_{E}x^{*} \succsim x \succsim f_{E}x$, which further implies that, **for each $x^{*}$**, there always exists an $\alpha[E,f] \in [0,1]$ such that the following equation holds: $$(1-\alpha[E,f]) u(\text{CE}(f_{E}x)) +  \alpha[E,f] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E,f]) u(x) +  \alpha[E,f] u(\text{CE}(x_{E}x^{*}))$$ Notice that, $\alpha[E,f]$ here may depend on the value of $x^{*}$ because the act $f_{E}x^{*}$ could be evaluated at different extreme points for different $x^{*}$. However, in the case where $f_{E}x^{*}$ is always evaluated at the extreme points in $C^{*}(E)$, as both $u(\text{CE}(f_{E}x^{*}))$ and $u(\text{CE}(x_{E}x^{*}))$ have the common term $u(x^{*})(1-\max\limits_{p\in C} p(E))$ which cancels out, $\alpha[E,f]$ does not depend on the value of $x^{*}$ any more.

By Lemma \[lem2\] and \[lem4\], for all $f\in \mathcal{F}$, there exists another threshold $\hat{x}_{E,f}$ such that for all $x^{*} \succsim \hat{x}_{E,f}$, one has $$\min\limits_{p \in C} \int_{\Omega}u(f_{E}x^{*})dp = \min\limits_{p \in C^{*}(E)}  \int_{\Omega} u(f_{E}x^{*})dp$$

Therefore, there exists $\alpha[E,f]$ such that the following is true for all $x^{*} \succsim \max\{\bar{x}_{E,f}, x, \hat{x}_{E,f}\}$:[^21] $$\label{equ}
(1-\alpha[E,f]) u(\text{CE}(f_{E}x)) +  \alpha[E,f] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E,f]) u(x) +  \alpha[E,f] u(\text{CE}(x_{E}x^{*}))$$ Furthermore, this $\alpha[E,f]$ is unique if either $f_{E}x \prec x$ or $f_{E}x^{*} \succ x_{E}x^{*}$ hold.\
Equation (\[equ\]) implies that the DM’s conditional evaluation of any $f\in \mathcal{F}$ can be represented by $$\min\limits_{p\in C_{\alpha[E, f]}(E)} \int_{E} u(f) \frac{dp}{p(E)} = u(x)$$ for some $\alpha[E,f] \in (0,1]$ or is represented by FB.

For all sufficiently large $x^{*}$, the LHS of equation (\[equ\]) can further be derived as: $$\begin{aligned}
&\quad (1-\alpha[E, f]) u(CE(f_{E}x)) + \alpha[E, f] u(\text{CE}(f_{E}x^{*}))\\
& =(1-\alpha[E, f])  \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp + \alpha[E, f] \min\limits_{p\in C} \int_{\Omega} u(f_{E}x^{*})dp \\
& = (1-\alpha[E, f])  \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp + \alpha[E, f] \min\limits_{q\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dq \\
& = (1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp+ \alpha[E, f]\left[ \min\limits_{q\in C^{*}(E)}  \int_{E} u(f) \frac{dq}{p^{*}(E)}\cdot p^{*}(E)  +\right. \\ 
&\left.\quad (1-p^{*}(E)) u(x^{*})  \right] \\
& = (1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp+ \alpha[E, f]\left[ \min\limits_{q\in C^{*}(E)}  \int_{E} u(f) \frac{dq}{p^{*}(E)}\cdot p^{*}(E)  +\right. \\ 
&\left.\quad (1-p^{*}(E)) u(x)  \right]  + \alpha[E, f] [u(x^{*}) - u(x)][1-p^{*}(E)]\end{aligned}$$ where the second equality follows from $f_{E}x^{*}$ is evaluated at some $p\in C^{*}(E)$.

On the other hand, the RHS of equation (\[equ\]) can also be derived as $$\begin{aligned}
&\quad (1-\alpha[E, f]) u(x) + \alpha[E, f] u( \text{CE}(x_{E}x^{*}))\\
& = (1-\alpha[E, f])u(x) + \alpha[E, f]\min\limits_{p \in C} \int_{\Omega} u(x_{E}x^{*})dp\\
& = (1-\alpha[E, f])u(x) +  \alpha[E, f][u(x)p^{*}(E) + u(x^{*})(1-p^{*}(E))] \\
& = u(x) + \alpha[E, f] [u(x^{*}) - u(x)][1-p^{*}(E)]\end{aligned}$$ Observe that now equalizing the LHS and RHS of equation (\[equ\]) implies $$\begin{aligned}
u(x) & =  (1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp+ \alpha[E, f]\left[ \min\limits_{q\in C^{*}(E)}  \int_{E} u(f) \frac{dq}{p^{*}(E)}\cdot p^{*}(E)  +(1-p^{*}(E)) u(x)  \right] \\
& = (1-\alpha[E, f]) \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp + \alpha[E, f]  \min\limits_{q\in C^{*}(E)}  u(f_{E}x)dq \\
& = \min\limits_{p\in C}  \min\limits_{q\in C^{*}(E)} \int_{\Omega} u(f_{E}x) d((1-\alpha[E, f]) p + (\alpha[E, f])q)\\
& = \min\limits_{p \in C_{\alpha[E, f]}(E)} \int_{\Omega} u(f_{E}x)dp\end{aligned}$$ where $C_{\alpha[E,f]}(E) = (1-\alpha[E,f])C + \alpha[E,f]C^{*}(E)$.

As $\alpha[E,f] \in [0,1]$, if $\alpha[E,f] = 0$, then by definition the CR-CCE axiom holds, thus the conditional evaluation of $f$ will be represented by FB. If $\alpha \in (0,1]$, notice that from the last equation one can further derive: $$\begin{aligned}
0  & = \min\limits_{p \in C_{\alpha[E, f]}(E)} \int_{\Omega} u(f_{E}x)dp - u(x) \\
    & =  \min\limits_{p \in C_{\alpha[E, f]}(E)} \int_{\Omega} [u(f_{E}x) - u(x)]dp \\
    & = \min\limits_{p \in C_{\alpha[E, f]}(E)}  \int_{E} [u(f) - u(x)]dp \\
    & = \min\limits_{p \in C_{\alpha[E, f]}(E)} \left[ \int_{E} u(f)dp - u(x)p(E) \right]\\\end{aligned}$$

When $E$ is $\succsim$-nonnull, $p^{*} (E) > 0$ thus $p(E) > 0$ for all $p \in C_{\alpha[E,f]}(E)$, then the last equality further implies $$0 = \min\limits_{p \in C_{\alpha[E, f]}(E)} \left[ \int_{E} u(f)\frac{dp}{p(E)} - u(x) \right]$$ i.e. $$\min\limits_{p\in C_{\alpha[E, f]}(E)} \int_{E} u(f) \frac{dp}{p(E)} = u(x)$$ which represents the conditional evaluation of $f$ under $\succsim_{E}$ since $f \sim_{E}x$.\
In summary, the conditional evaluation of $f$ is either given by FB ($\alpha[E,f] = 0$) or is given by $$\min\limits_{p\in C_{\alpha[E, f]}(E)} \int_{E} u(f) \frac{dp}{p(E)} = u(x)$$ for some $\alpha \in (0,1]$.\
The Weak CR-CCE axiom implies $\alpha[E, f]$ is constant across all $f\in \mathcal{F}$, and it is unique if $C \neq C^{*}(E)$.\
**Equivalent Class.** First, for any two acts $f, f'$, denote them by $f \equiv_{E} f'$ if $f(\omega) = f'(\omega)$ for all $\omega \in E$. Then an equivalence class of acts can be accordingly defined: $$[f] = \{ f' \in \mathcal{F}: f' \equiv_{E} f \}$$ By step 1, $\alpha [E, f] = \alpha[E, f']$ whenever $f' \in [f]$. Thus in the following, I abuse notation to use $f$ denote the whole class of acts $[f]$.\
**When $\alpha[E]$ is not unique.** If there does not exists any $f\in \mathcal{F}$ such that $\alpha[E, f]$ is unique. Then by step 1, for all $f\in \mathcal{F}$ and $x\in X$ such that $f \sim_{E}x$, it implies that both $f_{E}x \sim x$ and $f_{E}x^{*} \sim x_{E}x^{*}$ for all sufficiently large $x^{*}$ hold.

Then it is the case in which both the CR-CCE axiom and the CR-LC axiom hold at the same time. Namely, the conditional preference $\succsim_{E}$ can be represented by both FB and ML. When $E$ is strict $\succsim$-nonnull, it implies that $C = C^{*}(E)$; when $E$ is $\succsim$-nonnull but not strict, it is impossible since $C^{*}(E) \subsetneqq C$ implies that the conditional preference under ML is represented by a strict subset of the set under FB. In other words, $C \neq C^{*}(E)$ implies that there exists at least an $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique.\
When unique $\alpha[E,f]$ exists, fix some $f\in \mathcal{F}$ such that $\alpha[E, f]$ is unique.

**Two operations on acts that preserve $\alpha[E,f]$.** For any $l\in {\mathbb{R}}_{++}$, let $f_{l}$ denote an act such that $u(f_{l}(\omega)) = l \cdot u(f(\omega))$ for all $\omega \in \Omega$. i.e. the utility profile of $f_{l}$ is a positive linear transformation of the utility profile of $f$.

Then $f\sim_{E}x$ implies that $f_{l} \sim_{E} x_{l}$ for $u(x_{l}) = l \cdot u(x)$. For all $x^{*}$ that are sufficiently large for both $f$ and $f_{l}$, equation (\[equ\]) for $f_{l}$ implies that $$\begin{aligned}
&~~~~~~~~(1-\alpha[E, f_{l}]) u(\text{CE}(f_{lE}x_{l})) + \alpha[E, f_{l}] u(\text{CE}(f_{lE}x^{*}))  = (1-\alpha[E, f_{l}]) u(x_{l}) + \alpha[E, f_{l}] u(\text{CE}(x_{lE}x^{*})) \\
&\Rightarrow (1-\alpha[E, f_{l}]) l \cdot u(\text{CE}(f_{E}x)) + \alpha[E, f_{l}] \min\limits_{p\in C^{*}(E)} \int_{E} l \cdot u(f) dp  =(1-\alpha[E, f_{l}]) l \cdot u(x) \\
& + \alpha[E, f_{l}] l \cdot u(x)p^{*}(E)\\
&\Rightarrow(1-\alpha[E, f_{l}])  u(\text{CE}(f_{E}x)) + \alpha[E, f_{l}] \min\limits_{p\in C^{*}(E)} \int_{E}u(f) dp  = (1-\alpha[E, f_{l}])  u(x) + \alpha[E, f_{l}] u(x)p^{*}(E)\\
&\Rightarrow(1-\alpha[E, f_{l}]) u(\text{CE}(f_{E}x)) + \alpha[E, f_{l}] u(\text{CE}(f_{E}x^{*}))  = (1-\alpha[E, f_{l}]) u(x) + \alpha[E, f_{l}] u(\text{CE}(x_{E}x^{*}))\end{aligned}$$ on the other hand, $f\sim_{E} x$ also implies that $$(1-\alpha[E, f]) u(\text{CE}(f_{E}x)) + \alpha[E, f] u(\text{CE}(f_{E}x^{*}))  = (1-\alpha[E, f]) u(x) + \alpha[E, f] u(\text{CE}(x_{E}x^{*}))$$ Therefore, $\alpha[E,f_{l}] = \alpha[E,f]$ for all $l \in {\mathbb{R}}_{++}$ since $\alpha[E,f]$ is unique.

Next, for any $y \in X$, for any $\lambda \in [0,1]$ consider the act $f_{\lambda}y = \lambda f + (1-\lambda)y $, which is an Anscombe-Aumann mixture of acts. By certainty independence, $f\sim_{E}x$ implies that $f_{\lambda} y \sim_{E}  \lambda x + (1-\lambda)y$. Let $x_{\lambda}y\in X$ denote the consequence on the RHS. Then for all $x^{*}$ that are sufficiently large for $f$ and $f_{\lambda}y$, equation (\[equ\]) for $f_{\lambda}y$ implies that $$\begin{aligned}
&(1-\alpha[E, f_{\lambda}y]) u(\text{CE}([f_{\lambda}y]_{E}x_{\lambda}y) + \alpha[E, f_{\lambda}y] u(\text{CE}([f_{\lambda}y]_{E}x^{*})) \\
&= (1-\alpha[E, f_{\lambda}y]) u(x_{\lambda}y) + \alpha[E, f_{\lambda}y] u(\text{CE}([x_{\lambda}y]_{E}x^{*}))\end{aligned}$$ Notice that $$\begin{aligned}
u(\text{CE}([f_{\lambda}y]_{E}x_{\lambda}y) = u(\text{CE}([f_{E}x]_{\lambda}y)) =  \lambda u(\text{CE}(f_{E}x)) + (1-\lambda)u(y)\end{aligned}$$ where the first equality follows from Anscombe-Aumann mixture, the second equality also follows from certainty independence.

Then one can further derive $$\begin{aligned}
&(1-\alpha[E, f_{\lambda}y]) [\lambda u(\text{CE}(f_{E}x)) + (1-\lambda)u(y)] + \alpha[E, f_{\lambda}y] 
\min\limits_{p\in C^{*}(E)} \int_{E} u(f_{\lambda}y)dp \\
&= (1-\alpha[E, f_{\lambda}y]) u(x_{\lambda}y) + \alpha[E, f_{\lambda}y] u(x_{\lambda}y) p^{*}(E)\end{aligned}$$ which further implies that $$(1-\alpha[E, f_{\lambda}y]) u(\text{CE}(f_{E}x) + \alpha[E, f_{\lambda}y] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E, f_{\lambda}y]) u(x) + \alpha[E, f_{\lambda}y] u(\text{CE}(x_{E}x^{*}))$$ Then since $f\sim_{E}x$ and $\alpha[E,f]$ is unique, one can also conclude that $\alpha[E,f_{\lambda}y] = \alpha[E,f]$ for all $\lambda\in [0,1]$ and $y\in X$.

In summary, for all $l \in {\mathbb{R}}_{++}$, $y \in X$ and $\lambda \in [0,1]$, $\alpha[E, f_{l\lambda} y] = \alpha[E,f]$.\
**The Weak CR-CCE axiom implies $\alpha[E,f]$ is constant across acts.** First consider the case where the first scenario of Weak CR-CCE is true. Since the fist scenario is equivalent to the CR-LC axiom. Thus it would imply that $\alpha[E,f] = 1 $ for all $f\in \mathcal{F}$. Then $\alpha[E,f]$ is indeed a constant in this case.\
Next, suppose the first scenario of Weak CR-CCE is false, then it further implies that the second scenario must be true.

For any $g\in \mathcal{F}$ such that cannot be obtained from $f$ by the two operations preserve $\alpha[E,f]$. Without loss of generality, I normalize its utility profile such that $u(g(\omega)) \geq 0$ for all $\omega \in E$. Consider the act $g_{l\lambda}y$ for any $l \in {\mathbb{R}}_{++}$, $\lambda \in (0,1]$ and $y \in X$. By previous result, $\alpha[E, g_{l\lambda}y] = \alpha[E,g]$, thus it suffice to show that $\alpha[E,f] = \alpha[E,g_{l\lambda}y]$ for some $l \in {\mathbb{R}}_{++}$, $\lambda \in (0,1]$ [^22]and $y \in X$.

As the conditional preference $\succsim_{E}$ admits MEU representation, let it be represented by the set $C_{E}$. Let $p_{f}$ and $p_{g}$ denote the extreme points in $C_{E}$ that evaluate the act $f$ and $g$ respectively. By certainty independence, the act $g_{l\lambda}y$ is also evaluated at the same extreme point as $g$.

Then $f \sim_{E} g_{l\lambda}y$ when $$u(g_{l\lambda}y) \cdot p_{g}  =  \lambda u(g_{l})\cdot p_{g} + (1-\lambda) u(y) = u(f) \cdot p_{f}$$

Next, for all $x^{*}$ that are sufficiently large for $f$ and $g_{l\lambda}y$, consider the condition $g_{l\lambda E}x^{*} \sim f_{E}x^{*}$, which is equivalent to $$\begin{aligned}
&~~~~~~~~\min\limits_{p\in C} \int_{\Omega} u([g_{l\lambda}y]_{E}x^{*}) = \min\limits_{p\in C} \int_{\Omega} u(f_{E}x^{*})\\
&\Rightarrow \min\limits_{p\in C^{*}(E)} \int_{E} u(g_{l\lambda}y) dp + (1-p^{*}(E))u(x^{*}) = \min\limits_{p\in C^{*}(E)} \int_{E} u(f) dp + (1-p^{*}(E))u(x^{*})\\
&\Rightarrow \min\limits_{p\in C^{*}(E)} \int_{E} u(g_{l\lambda}y) dp = \min\limits_{p\in C^{*}(E)} \int_{E} u(f) dp \\
& \Rightarrow \min\limits_{p\in C^{*}(E)} \int_{E} u(g_{l\lambda}y) \frac{dp}{p(E)} = \min\limits_{p\in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)} \end{aligned}$$ where the last equality follows from $p(E) = p^{*}(E)$ for all $p \in C^{*}(E)$.

For the set of posteriors of $C^{*}(E)$, let $q_{f}$ and $q_{g}$ denote the two extreme points that evaluate the act $f$ and $g$ respectively. Again, the act $g_{l\lambda}y$ is also evaluated at the same extreme point as $g$. Thus, the above condition can also be written as

$$u(g_{l\lambda}y) \cdot q_{g}  = \lambda u(g_{l})\cdot q_{g}+ (1-\lambda) u(y) = u(f) \cdot q_{f}$$

Consider the following construction:

In a two dimensional space, draw the two points $(1, u(f) \cdot q_{f})$ and $(1,u(f) \cdot p_{f})$ as well as the two points $(2, u(g_{l}) \cdot q_{g})$ and $(2,u(g_{l}) \cdot p_{g})$. Notice that, by the U-O axiom, the set $C_{E}$ is a superset of the set of posteriors of $C^{*}(E)$, thus it is necessarily the case that $u(f) \cdot q_{f} \geq u(f) \cdot p_{f}$ and $u(g_{l}) \cdot q_{g} \geq u(g_{l}) \cdot p_{g}$. Now for different possibilities of the two inequalities:

-   $u(f) \cdot q_{f} =  u(f) \cdot p_{f}$ and $u(g_{l}) \cdot q_{g} \geq  u(g_{l}) \cdot p_{g}$.

    In this case, consider when $g=x$ in the second scenario of the Weak CR-CCE axiom. The first equality implies that $u(f) \cdot q_{f} =  u(f) \cdot p_{f} = u(x)$, i.e. $f_{E}x \sim x_{E}x^{*}$ as well. Then the second scenario of the Weak CR-CCE axiom would imply that $f_{E}x \sim x$. By step 1, both $f_{E}x \sim x_{E}x^{*}$ and $f_{E}x \sim x$ holds imply that $\alpha[E,f]$ is not unique, a contradiction to the current assumption.

-   $u(f) \cdot q_{f} > u(f) \cdot p_{f}$ and $u(g_{k}) \cdot q_{g} = u(g_{k}) \cdot p_{g}$.

    This case is symmetry to the first case, and it necessarily implies that $\alpha[E,g]$ is not unique. Then it suffice to let $\alpha[E, g] = \alpha[E, f]$ for this case.

-   $u(f) \cdot q_{f} > u(f) \cdot p_{f}$ and $u(g_{k}) \cdot q_{g} > u(g_{k}) \cdot p_{g}$.

    In this case, $\alpha[E,f] \neq 1$ because $p_{f} \neq q_{f}$. In the following one can further show that the second scenario of the Weak CR-CCE axiom implies that $\alpha[E,f] = \alpha[E,g]$.

Consider in the Case 3, find $l \in {\mathbb{R}}_{++}$ such that the following two strict inequalities hold: $$\begin{aligned}
&u(g_{l})\cdot p_{g} > u(f) \cdot p_{f} \\
&u(g_{l})\cdot q_{g} - u(g_{l})\cdot p_{g}  > u(f) \cdot q_{f} - u(f) \cdot p_{f} \end{aligned}$$

Since both LHS of the inequalities are increasing with respect to $l$, thus one can always find such $l$ (recall our normalization $u(g(\omega)) \geq 0$ for all $\omega \in E$). Once $l$ is identified, draw two straight lines, one connects $(1, u(f) \cdot q_{f}), (2, u(g_{k}) \cdot q_{g})$ and the other one connects $(1,u(f) \cdot p_{f}), (2,u(g_{k}) \cdot p_{g})$. By construction, these two lines must intersect at some point on the southwest of the point $(1,u(f) \cdot p_{f})$, and denote it by $(z, u(y))$. Then the desired $\lambda$ is given by $\frac{1-z}{2-z}$ and this $g_{l\lambda}y$ satisfies the two conditions: $$f \sim_{E} g_{l\lambda}y \text{ and } f_{E}x^{*} \sim [g_{l\lambda}y]_{E}x^{*}$$

In the following, I abuse notation to use $g$ denote the act $g_{l\lambda}y$ such that two conditions above hold.

By step 1, $f\sim_{E} x$ implies that $$\label{equ11}
(1-\alpha[E,f]) u(\text{CE}(f_{E}x)) + \alpha[E,f] u(\text{CE}(f_{E}x^{*})) = (1-\alpha[E,f]) u(x) + \alpha[E,f] u(\text{CE}(x_{E}x^{*}))$$ and $g \sim_{E} x$ also implies that $$\label{equ10}
(1-\alpha[E,g]) u(\text{CE}(g_{E}x)) + \alpha[E,g] u(\text{CE}(g_{E}x^{*})) = (1-\alpha[E,g]) u(x) + \alpha[E,g] u(\text{CE}(x_{E}x^{*}))$$

Consider the LHS of equation (\[equ10\]) and denote it by $L$: $$\begin{aligned}
L & = (1-\alpha[E, g])  u(\text{CE}(g_{E}x))  + \alpha[E, g] u(\text{CE}(g_{E}x^{*})) \\
& = (1-\alpha[E, f]) u(\text{CE}(g_{E}x)) + \alpha[E, f] u(\text{CE}(g_{E}x^{*})) + [\alpha[E, g] - \alpha[E, f]](u(\text{CE}(g_{E}x^{*})) - u(\text{CE}(g_{E}x)))\\
& = L' +[\alpha[E, g] - \alpha[E, f]] M_{1} \end{aligned}$$ Meanwhile the RHS of equation (\[equ10\]) denoted by $R$ can be further derived to $$\begin{aligned}
R & =  (1-\alpha[E, g]) u(x) + \alpha[E, g] u(\text{CE}(x_{E}x^{*})) \\
& = (1-\alpha[E, f]) u(x) + \alpha[E, f] u(\text{CE}(x_{E}x^{*})) + [\alpha[E, g] - \alpha[E, f]](u(\text{CE}(x_{E}x^{*})) - u(x))\\
& = R' + [\alpha[E, g] - \alpha[E, f]] M_{2} \end{aligned}$$ Notice that by equation (\[equ11\]), $R'$ also equals to $(1-\alpha[E, f])\text{CE}(f_{E}x) + \alpha[E, f] \text{CE}(f_{E}x^{*})$. Then as $\alpha[E, f] \neq 1$, the second scenario of Weak CR-CCE implies that $L' = R'$ as $f_{E}x \sim g_{E}x$ and $f_{E}x^{*} \sim g_{E}x^{*}$ hold.

Then as $L = R$, one has $$\begin{aligned}
&L - R  =  L' +[\alpha[E, g] - \alpha[E, f]] M_{1} - R' - [\alpha[E, g] - \alpha[E, f]] M_{2} = 0 \\ 
&\Rightarrow L' - R' =  [\alpha[E, f] - \alpha[E, g]] [M_{1} - M_{2}]\end{aligned}$$ Further notice that $$\begin{aligned}
M_{1} - M_{2}& =  [u(\text{CE}(g_{E}x^{*})) - u(\text{CE}(g_{E}x))] - [ u(\text{CE}(x_{E}x^{*})) - u(x)] \\
& = [\min\limits_{p\in C} \int_{\Omega}u(g_{E}x^{*}) - \min\limits_{p\in C} \int_{\Omega}u(g_{E}x)] - [u(x)p^{*}(E) + u(x^{*})(1-p^{*}(E)) - u(x)] \\
& = [\min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x^{*}) - \min\limits_{p\in C} \int_{\Omega}u(g_{E}x)] - [u(x^{*}) - u(x)](1-p^{*}(E)) \\
& = [\min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x^{*}) - \min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x)] - [u(x^{*}) - u(x)](1-p^{*}(E)) \\
& + [\min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x) - \min\limits_{p\in C} \int_{\Omega}u(g_{E}x) ] \\
& = \min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g_{E}x) - \min\limits_{p\in C} \int_{\Omega}u(g_{E}x)\end{aligned}$$ where the last equality follows from $$\begin{aligned}
\min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g) - \min\limits_{p\in C^{*}(E)} \int_{\Omega}u(g) &= [\min\limits_{p\in C^{*}(E)} \int_{E}u(g_{E}x^{*}) - \min\limits_{p\in C^{*}(E)} \int_{E}u(g_{E}x)]p^{*}(E) \\
&\quad+ [u(x^{*}) - u(x)](1-p^{*}(E)) \end{aligned}$$

\[lem5\] For any $f\in \mathcal{F}$ such that $\alpha[E, f]$ is unique, if $f\sim_{E}x$ then $$\min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp < \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp$$

Since if $\alpha[E, f]$ is unique, it implies that for $f\sim_{E}x$, one of the inequalities $f_{E}x^{*} \succsim x_{E}x^{*}$ and $x \succsim f_{E}x$ is strict. Notice that the first inequality implies $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f)dp \geq u(x)p^{*}(E)$$ and the second implies that $$u(x) \geq \min\limits_{p\in C} \int_{\Omega} u(f_{E}x)dp$$

Add the term $u(x)(1-p^{*}(E))$ to both sides of the first inequality and combine both inequalities and recall that one of them has to be strict yield: $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f)dp + u(x)(1-p^{*}(E)) >  \min\limits_{p\in C} \int_{\Omega} u(f_{E}x)dp$$ which is equivalent to $\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x) dp > \min\limits_{p\in C} \int_{\Omega} u(f_{E}x) dp $.

Denote it by $$\Delta f_{E}x \equiv  \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x)dp - \min\limits_{p\in C} \int_{\Omega} u(f_{E}x)dp$$

If $\alpha[E,g]$ is not unique, then it suffice to let $\alpha[E,g] = \alpha[E,f]$. If it is also unique, given Lemma \[lem5\], $g\sim_{E}x$ implies that $$\Delta g_{E}x > 0$$

Therefore the difference $$L' - R' =  [\alpha[E, f] - \alpha[E, g]] [M_{1} - M_{2}] = [\alpha[E, f] - \alpha[E, g]]\cdot \Delta g_{E}x$$ is 0 if and only if $\alpha[E, f] = \alpha[E, g]$.

To conclude, when the first scenario of the Weak CR-CCE axiom is false, it implies that the second scenario must be true. Fix $f\in \mathcal{F}$ such that $\alpha[E,f]$ is unique, for any $g\in \mathcal{F}$, when $\alpha[E,g]$ is not unique, it suffice to let $\alpha[E,g] = \alpha[E,f]$. If $\alpha[E,g]$ is unique, the second scenario implies that it cannot be the case that $\alpha[E,f] = 1$. Then when $\alpha[E,f] \neq 1$, by construction of $g_{l\lambda}y$, the second scenario of the Weak CR-CCE axiom further suggests that it is necessarily the case $\alpha[E, f] = \alpha[E,g]$.

Finally, combining the two cases ($\alpha[E, f] = 1$ and $\alpha[E, f] \neq 1$) yields that $\alpha[E,f]$ needs to be a constant across all $f\in \mathcal{F}$ for each $\succsim$-nonnull $E\in \Sigma$.

Proof of Theorem \[thm4\]
-------------------------

Given Theorem \[thm2\], the only remaining proof here is to show $\alpha[E]$ is a constant across all events if and only if the EC axiom holds.

The necessity of the EC axiom is also immediate when one plug a constant $\alpha[E]$ into equation (\[equ1\]), $\alpha[E] = 1$ implies the first scenario, and $\alpha[E] \neq 1$ implies the second. Both scenarios are true when $\alpha[E]$ is not unique.\
In the following I show that, EC implies $\alpha[E]$ needs to be a constant across all $\succsim$-nonnull events.

First of all, consider the case there does not exists any $\succsim$-nonnull $E\in \Sigma$ such that $C \neq C^{*}(E)$. Then it implies that all $p \in C$ agree with the probability of all $\succsim$-nonnull event $E$. Namely, all $p\in C$ agree on the probability of all $\succsim$-nonnull events.

Thus, if there exists one $\succsim$-nonnull event $E$ such that $C \neq C^{*}(E)$, i.e. $\alpha[E]$ is unique, then it suffice to let this $\alpha[E]$ to be the constant $\alpha$ across all events.

Next, when there exists at least two $\succsim$-nonnull events, $E_{1}$ and $E_{2}$, such that both $\alpha[E_{1}]$ and $\alpha[E_{2}]$ are unique. Find any $f,g\in \mathcal{F}$ and $x\in X$ such that $f \sim_{E_{1}} x$ and $g \sim_{E_{2}} x$. For all $x^{*}\in X$ that are sufficiently large for act $f$ under event $E_{1}$ and for act $g$ under event $E_{2}$. (The same definition as in the proof of Theorem \[thm2\]), identify two consequences $x_{1}^{*}$ and $x_{2}^{*}$ such that $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$.

By Theorem \[thm2\], $f \sim_{E_{1}} x$ implies that $$\label{eq2}
(1-\alpha[E_{1}]) u(\text{CE}(f_{E_{1}}x)) + \alpha[E_{1}] u(\text{CE}(f_{E_{1}}x_{1}^{*})) =  (1-\alpha[E_{1}]) u(x) + \alpha[E_{1}] u(\text{CE}(x_{E_{1}}x_{1}^{*}))$$

Meanwhile $g \sim_{E_{2}} x$ also implies $$\label{eq3}
(1-\alpha[E_{2}]) u(\text{CE}(g_{E_{2}}x)) + \alpha[E_{2}] u(\text{CE}(g_{E_{2}}x_{2}^{*})) =  (1-\alpha[E_{2}]) u(x) + \alpha[E_{2}] u(\text{CE}(x_{E_{2}}x_{2}^{*}))$$

If the first scenario of the EC axiom is true, equation (\[eq2\]) and (\[eq3\]) always imply $f_{E_{1}}x_{1}^{*} \sim g_{E_{2}}x_{2}^{*}$ for all such $f,g$ only when $\alpha[E_{1}] = \alpha[E_{2}] = 1$. The same applies to all $\succsim$-nonnull $E$ with unique $\alpha[E]$, thus in this case $\alpha[E] = 1$ for all $\succsim$-nonnull $E$.

If the first scenario is false, the EC axiom further implies that the second scenario must be true. Consider a similar construction as in the proof of Theorem \[thm2\]. Notice that since $\alpha[E]$ is a constant across all $f\in \mathcal{F}$ for each $\succsim$-nonnull $E$, thus all operations on acts preserve $\alpha[E]$.

Fix any $f\in \mathcal{F}$ such that $f\sim_{E_{1}}x$. For any $g \in \mathcal{F}$, let $g_{l}$ denote the act such that $u(g_{l}(\omega)) = l \cdot u(g(\omega))$ where normalize $u$ such that $u(g(\omega))$ is non-negative for all $\omega \in E_{2}$. Then for any consequence $y\in X$, for any $\lambda \in [0,1]$, let the act $g_{l\lambda}y$ denote the mixture $\lambda g_{l} + (1-\lambda)y $ of the act $g_{l}$ and consequence $y$. It is important to notice that the act $g_{l\lambda}y$ is always evaluated at the same extreme point as $g$.

Then let $p_{f}$ denote the extreme point in $C_{E_{1}}$ that evaluates the act $f$, and let $p_{g}$ denote the extreme point in $C_{E_{2}}$ that evaluates the act $g$, where $C_{E_{1}}$ and $C_{E_{2}}$ are the two sets representing the conditional preference $\succsim_{E_{1}}$ and $\succsim_{E_{2}}$ respectively.

Then $g_{l\lambda}y \sim_{E_{2}} x$ when $$u(g_{l\lambda}y)\cdot p_{g} = \lambda u(g_{l})\cdot p_{g} + (1-\lambda) u(y) = u(x)$$

Further, since $f \sim_{E_{1}} x$, it is also suffice to let $$\lambda u(g_{l})\cdot p_{g} + (1-\lambda) u(y) = u(f)\cdot p_{f}$$

Next, for all $x^{*}$ that are sufficiently large for $f$ under $E_{1}$ and $g_{l\lambda}y$ under $E_{2}$, consider the condition $[g_{l\lambda}y]_{E}x^{*}_{2} \sim f_{E_{1}}x^{*}_{1}$, which is equivalent to $$\begin{aligned}
&~~~~~~~~\min\limits_{p\in C} \int_{\Omega} u([g_{l\lambda}y]_{E}x_{2}^{*}) dp = \min\limits_{p\in C} \int_{\Omega} u(f_{E_{1}}x^{*}_{1}) dp \\
&\Rightarrow \min\limits_{p\in C^{*}(E_{2})}  \int_{E_{2}} u(g_{l\lambda}y) dp + u(x_{2}^{*}) (1- p^{*}(E_{2})) = \min\limits_{p\in C^{*}(E_{1})}  \int_{E_{1}} u(f) dp + u(x^{*}_{1}) (1- p^{*}(E_{1}))\end{aligned}$$ where $p^{*}(E_{1})$ denotes $\max\limits_{p\in C} p(E_{1})$ and $p^{*}(E_{2})$ is defined accordingly.

Let $q_{f}$ and $q_{g}$ be the two extreme points in $C^{*}(E_{1})$ and $C^{*}(E_{2})$ that evaluates $f$ and $g_{l\lambda}y$ respectively, then the last equality can be written as: $$u(g_{l\lambda}y) \cdot q_{g} + u(x_{2}^{*}) (1- p^{*}(E_{2})) = u(f)\cdot q_{f} + u(x^{*}_{1}) (1- p^{*}(E_{1}))$$

Furthermore as the condition $x_{E_{1}}x_{1}^{*} \sim x_{E_{2}}x_{2}^{*}$ implies that $$u(x) p^{*}(E_{1}) + u(x_{1}^{*})(1-p^{*}(E_{1}))  = u(x) p^{*}(E_{2}) + u(x_{2}^{*}) (1-p^{*}(E_{2}))$$ i.e. $$u(x_{1}^{*})(1-p^{*}(E_{1}))   -  u(x_{2}^{*}) (1-p^{*}(E_{2}))  = u(x)[p^{*}(E_{2}) - p^{*}(E_{1})]$$ As $E_{1}$ and $E_{2}$ are chosen arbitrarily, without loss of generality to let $p^{*}(E_{2}) - p^{*}(E_{1}) \geq 0$ and also normalize $u$ such that $u(x) \geq 0$.

Then $[g_{l\lambda}y]_{E}x^{*}_{2} \sim f_{E_{1}}x^{*}_{1}$ is equivalent to $$\lambda u(g_{l})\cdot q_{g} + (1-\lambda)u(y)  = u(f)\cdot q_{f} + u(x)[p^{*}(E_{2}) - p^{*}(E_{1})]$$

Again, since both $\alpha[E_{1}]$ and $\alpha[E_{2}]$ are unique, the set represents the conditional preference is a superset of the corresponding set of posteriors of maximum likelihood priors. Thus the following two inequality are still necessarily true: $u(f) \cdot q_{f} \geq u(f) \cdot p_{f}$ and $u(g_{k}) \cdot q_{g} \geq u(g_{k}) \cdot p_{g}$. For different possibilities of the two inequalities:

-   Either $u(f) \cdot q_{f} = u(f) \cdot p_{f}$ or $u(g_{k}) \cdot q_{g} = u(g_{k}) \cdot p_{g}$.

    Under this case, without loss of generality assume the first equality is true. Consider $g=x$ in the second scenario of the EC axiom. The equality implies that when $f \sim_{E_{1}}x$, one has $f_{E_{1}}x_{2}^{*} \sim x_{E_{2}}x_{2}^{*}$ as well. Then the second scenario of EC would imply that $f_{E_{1}}x \sim x$. By step 1 in the proof of Theorem \[thm2\], it further implies that $\alpha[E_{1}]$ is not unique, a contradiction to the current assumption.

-   $u(f) \cdot q_{f} > u(f) \cdot p_{f}$ and $u(g_{k}) \cdot q_{g} > u(g_{k}) \cdot p_{g}$.

    In this case, both $\alpha[E_{1}]$ and $\alpha[E_{2}]$ cannot be 1.

Consider in the Case 2, given our normalization, it is also the case that $u(f) \cdot q_{f}+ u(x)[p^{*}(E_{2}) - p^{*}(E_{1})] > u(f) \cdot p_{f}$ and $u(g_{l}) \cdot q_{g} > u(g_{l}) \cdot p_{g}$. Now for a slight different construction from the one in proof of Theorem \[thm2\]:

In a two dimensional space, draw the two points $(1, u(f) \cdot q_{f} + u(x)[p^{*}(E_{2}) - p^{*}(E_{1})])$ and $(1,u(f) \cdot p_{f})$ as well as the two points $(2, u(g_{l}) \cdot q_{g})$ and $(2,u(g_{l}) \cdot p_{g})$.

First find $l \in {\mathbb{R}}_{++}$ such that the following two strict inequalities hold: $$\begin{aligned}
&u(g_{l})\cdot p_{g} > u(f) \cdot p_{f} \\
&u(g_{l})\cdot q_{g} - u(g_{l})\cdot p_{g}  > u(f) \cdot q_{f} + u(x)[p^{*}(E_{2}) - p^{*}(E_{1})]- u(f) \cdot p_{f} \end{aligned}$$

Since both LHS of the inequalities are increasing with respect to $l$, thus one can always find such $l$. Once $l$ is identified, draw two straight lines, one cross $(1, u(f) \cdot q_{f}), (2, u(g_{l}) \cdot q_{g})$ and the other one cross $(1,u(f) \cdot p_{f} + u(x)[p^{*}(E_{2}) - p^{*}(E_{1})]), (2,u(g_{l}) \cdot p_{g})$. These two lines must intersect at some point on the southwest of the point $(1,u(f) \cdot p_{f})$, and denote it by $(z, u(y))$. Then the desired $\lambda$ is given by $\frac{1-z}{2-z}$, and such $g_{l\lambda}y$ satisfies the two conditions in the second scenario of EC.

From this point on, apply exactly the same argument in step 3 of the proof of Theorem \[thm2\] would imply that $\alpha[E_{1}] = \alpha[E_{2}]$, and furthermore $\alpha[E]$ needs to be a constant across all $\succsim$-nonnull $E\in \Sigma$.

To conclude, combining the two cases ($\alpha[E_{1}] = 1$ and $\alpha[E_{1}] \neq 1$) imply that $\alpha$ is a constant across all $\succsim$-nonnull event $E$. Furthermore, $\alpha$ is unique if there exists $\succsim$-nonnull $E\in \Sigma$ such that not all $p\in C$ agree on $p(E)$.

ML under general MEU {#apx1}
====================

The characterization results in the main text rely on the finitely many plausible priors assumption. As mentioned, such an assumption helps simplify the axioms and conveys the same intuition. This section in the appendix provides characterization results without this assumption.

For the ex-ante preference $\succsim$, this section assumes only that it admits MEU representation with some closed and convex set $C$, which may have infinitely many plausible priors. In addition, endow the space $\Delta(\Omega)$ with weak topology. Except for these, every other assumption about the primitive $\{\succsim_{E}\}_{E\in \Sigma}$ is the same as in Section 2.

The definition is the same as before, $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by ML updating if for all $\succsim$-nonnull $E\in \Sigma$ and for all $f\in \mathcal{F}$: $$\min\limits_{p\in C_{E}} \int_{\Omega}u(f) dp = \min\limits_{p\in C^{*}(E)} \int_{E} u(f) \frac{dp}{p(E)}$$

For the general MEU case where $C$ may contain infinitely many plausible priors, consider the following axiom:\
**Axiom Approximate CR-LC**. For all $\succsim$-nonnull $E\in \Sigma$, for all $f\in \mathcal{F}$ and for all $x, z, w \in X$ such that $z \succ w$, there exists $\bar{x}_{E,f,z,w}$ such that if $f \sim_{E}x$ then $$\frac{1}{2} f_{E}x^{*} + \frac{1}{2} w \prec \frac{1}{2} x_{E}x^{*} + \frac{1}{2} z$$ and $$\frac{1}{2} f_{E}x^{*} + \frac{1}{2}z \succ \frac{1}{2} x_{E}x^{*} + \frac{1}{2}w$$ for all $x^{*} \succsim \bar{x}_{E,f,z,w}$.\
Notice that when $\bar{x}_{E,f}$ exists, CR-LC implies Approximate CR-LC. Whereas, in the case where $\bar{x}_{E,f}$ does not exist such that CR-LC is silent, Approximate CR-LC imposes additional restriction on behaviors. It restricts that the difference between $f_{E}x^{*}$ and $x_{E}x^{*}$ should be arbitrarily small when $x^{*}$ is sufficiently large.

Hence, Approximate CR-LC conveys essentially the same intuition as CR-LC. The following representation theorem claims that Approximate CR-LC is equivalent to ML in the current setting.

\[thm5\] $\{\succsim_{E}\}_{E\in \Sigma}$ is represented by ML updating if and only if the Approximate CR-LC\* axiom holds.

First consider the following lemma:

\[alem\] For any $\succsim$-nonnull $E\in \Sigma$ and $f\in \mathcal{F}$, for any $\epsilon > 0$ there exists $\bar{x}_{E,f,\epsilon} \in X$ such that $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp -  \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*}) dp  < \epsilon$$ for all $x^{*} \succsim \bar{x}_{E,f,\epsilon}$.

For any $\succsim$-nonnull $E\in \Sigma$ and $f\in \mathcal{F}$, either there exists $\bar{x}_{E,f}$ such that $$\min\limits_{p\in C} \int_{\Omega} u(f_{E}x^{*}) dp = \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*}) dp$$ for all $x^{*} \succsim \bar{x}_{E,f}$ or not. If it is the first case, then this lemma is trivially true.

Consider the case there does not exist $\bar{x}_{E,f}$ for some $E$ and $f$. For each $x \in X$, let $p_{x}$ be the probability measure in $C$ that evaluates the act $f_{E}x$, i.e. $p_{x} \equiv \arg\min\limits_{p\in C} \int_{\Omega} u(f_{E}x)dp $. Let $q$ denote the probability measure in $C^{*}(E)$ that evaluates the act $f_{E}x$ and notice that it does not depend on the value of $x$.

Then one has $$\begin{aligned}
            &\quad \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x)dp -  \min\limits_{p\in C}\int_{\Omega} u(f_{E}x) dp \\
            & = \int_{E} u(f)dq + u(x)(1-p^{*}(E)) - \int_{E} u(f)dp_{x} - u(x)(1-p_{x}(E))
            \end{aligned}$$

Take derivative with respect to $u(x)$ and apply envelope theorem yields $$\frac{d}{du(x)} \left[ \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x)dp -  \min\limits_{p\in C}\int_{\Omega} u(f_{E}x) dp \right] = p_{x}(E) - p^{*}(E)$$

The current assumption $p_{x} \notin C^{*}(E)$ implies that $p_{x}(E) - p^{*}(E) < 0$, i.e. the difference is decreasing with respect to $u(x)$. Furthermore, since the difference is bounded below by zero, monotone convergence theorem implies that $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x)dp -  \min\limits_{p\in C}\int_{\Omega} u(f_{E}x) dp \rightarrow 0$$ for $u(x) \rightarrow \infty$, i.e. the lemma holds.

For necessity of the Approximate CR-LC axiom, recall Lemma \[lem1\] implies that if $f\sim_{E}x$ then under ML updating $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp = \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp$$ Thus for any $\epsilon > 0$ there exists $\bar{x}_{E,f,\epsilon}$ such that $$\min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*}) dp  < \epsilon$$ for all $x^{*} \succsim \bar{x}_{E,f,\epsilon}$. Then for each $z \succ w$, it suffice to let $\epsilon = u(z) - u(w)$ and Approximate CR-LC axiom will hold.\
For sufficiency, fix any $\succsim$-nonnull $E$ and consider the contra positive statement: not ML updating implies not Approximate CR-LC axiom. Not ML means that $C_{E} \neq \{ \frac{p}{p(E)}: p \in C^{*}(E) \}$. In other words, either thre exists $\tilde{p} \in C^{*}(E)$ such that $\frac{\tilde{p}}{\tilde{p}(E)} \notin C_{E}$ or there exists $q \in C_{E}$ such that $q \notin \{\frac{p}{p(E)}: p \in C^{*}(E)\}$ or both.

Consider the first case, by the same strong separating hyperplane argument in the proof of Theorem \[thm1\], there exists $f\in \mathcal{F}$ and $x\in X$ such that $f\sim_{E}x$ and $$\min\limits_{p\in C^{*}(E)} \int_{E}u(f)\frac{dp}{p(E)} < u(x)$$ Then for all $x^{*} \succsim x$, $$\begin{aligned}
        \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp -  \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp = u(x)p^{*}(E) - \min\limits_{p\in C^{*}(E)} \int_{E}u(f)dp> 0
        \end{aligned}$$ where $p^{*}(E) = \max\limits_{p\in C} p(E)$. That is, there exists $\delta_{E,f} > 0 $ such that $$\begin{aligned}
        \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp -  \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp > \delta_{E,f} > 0
        \end{aligned}$$ Now for all $x^{*} \succsim x$, $$\begin{aligned}
        &\quad \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp \\
        &=  \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp  + \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp  - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp \\
        & > \delta_{E,f}  > 0
        \end{aligned}$$ the last inequality comes from the fact that $\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp  - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp \geq 0$. Then it suffice to find $z \succ w$ such that $u(z) - u(w) < \delta_{E,f}$ and it will imply that for all $x^{*} \succsim x$: $$\frac{1}{2} x_{E}x^{*} + \frac{1}{2} w \succ \frac{1}{2}f_{E}x^{*} + \frac{1}{2}z$$ i.e. the Approximate CR-LC axiom fails.

Now consider the second case, there exists $q \in C_{E}$ such that $q \notin \{\frac{p}{p(E)}: p \in C^{*}(E)\}$. By strong separating hyperplane, there exists $f \sim_{E}x$ such that $$u(x) = \min\limits_{p\in C_{E}} \int_{\Omega} u(f) dp \leq \int_{\Omega} u(f) dq < \min\limits_{p\in C^{*}(E)} \int_{E}u(f)\frac{dp}{p(E)}$$ Then it further implies for all $x^{*} \succsim x$, $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp  - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp  =  \min\limits_{p\in C^{*}(E)} \int_{E}u(f)dp - u(x)p^{*}(E) > 0$$ i.e. there exists $\delta_{E,f}$ such that $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp  - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp  > \delta_{E,f} > 0$$ On the other hand, by Lemma \[alem\], for any $\epsilon >0$ there exists $\bar{x}_{E,f,\epsilon}$ such that $$\min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*}) dp < \epsilon$$ for all $x^{*} \succsim \bar{x}_{E,f,\epsilon}$. Now for all $x^{*} \succsim \max\{x, \bar{x}_{E,f,\epsilon}\}$, $$\begin{aligned}
        &\quad  \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp \\
        &=  \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp  + \min\limits_{p\in C^{*}(E)} \int_{\Omega} u(f_{E}x^{*})dp  - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp\\
        & > -\epsilon + \delta_{E,f} 
        \end{aligned}$$ Now find any $z \succ w$ such that $u(z) - u(w) = \eta < \delta_{E,f}$ and let $\epsilon = \delta_{E,f} - \eta > 0$. Then for all $x^{*} \succsim \max\{x, \bar{x}_{E,f,\epsilon}\}$ the previous result implies that $$\min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp > -\epsilon + \delta_{E,f} = \eta > 0$$ Then $$\begin{aligned}
        \frac{1}{2}  \min\limits_{p\in C}\int_{\Omega} u(f_{E}x^{*})dp + \frac{1}{2}u(w) - \min\limits_{p\in C} \int_{\Omega} u(x_{E}x^{*})dp - \frac{1}{2}u(z) > \frac{1}{2}[\eta - \eta] = 0 
        \end{aligned}$$ i.e. $\frac{1}{2} f_{E}x^{*} + \frac{1}{2} w \succ \frac{1}{2}x_{E}x^{*} + \frac{1}{2}z$ for all $x^{*} \succsim \bar{x}_{E,f,\epsilon}$. Thus the Approximate CR-LC axiom fails in this case as well. Combine both cases shows that, not ML implies not Approximate CR-LC.

**Remark.** The characterization results for Weak RML and RML can be extended similarly to this general case. The statement of the axioms and also the additional steps in the proof will be analogously the same as in ML, thus omitted in the present paper.

[^1]: Department of Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University, Evanston, IL, USA. E-mail: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>

[^2]: I am grateful to Peter Klibanoff and Marciano Siniscalchi for invaluable guidance and discussions throughout the completion of this paper. I thank Eddie Dekel, Eran Hanany, Jingyi Xue, and especially Lorenzo Stanca for insightful discussions. I thank all participants at Northwestern Strategy Bag Lunch and Theory Lunch for comments. All remaining errors are my own.

[^3]: Some may also refer to it as prior-by-prior updating

[^4]: With additional technical assumption, the characterization of ML without this assumption is given by Theorem \[thm5\].

[^5]: These notations will be used throughout the present paper.

[^6]: A graphical illustration can be found in section 3.

[^7]: Details will be provided in section 4.

[^8]: An act $f$ is a function maps from the state space to the consequence space.

[^9]: Some may refer to this condition also as **Dynamic Consistency**. The present paper follows the terminology in Hanany and Klibanoff (2007)[@hanany2007updating], in which dynamic consistency is defined to require the other direction of implication: ex-ante indifference implies conditional indifference.

[^10]: In the formal statement of the axiom, a special case needs to be stated separately, which is the case $f\sim_{E}g$ always implies $f_{E}x^{*} \sim g_{E}x^{*}$, i.e. when RML reduces to ML ($\alpha = 1$).

[^11]: The optimal payoff cannot be achieved but can only be approached arbitrarily close.

[^12]: When there are only two states, the conditional preferences are trivial.

[^13]: Plausible priors are defined in Siniscalchi (2005), under MEU, it equivalently means that the set $C$ is a polytope in $\Delta(\Omega)$, i.e. it is a convex and closed set with finitely many extreme points.

[^14]: This $C$ is not a core of any convex capacity.

[^15]: See Lemma \[lem4\].

[^16]: I abuse terminology to use states to denote payoff-relevant states thereafter.

[^17]: Notice that $\pi_{\mu}(\theta_{1} | s_{1})$ is increasing in $\mu$.

[^18]: Only row 8 of Table 4.2 in Liang (2019) is different from the prediction of RML. The last two rows do not specify the signals thus are ignored.

[^19]: The same conclusion can be drawn also from a within-subject comparison shown in Table B.4 of Liang (2019).

[^20]: The sender can choose a set of probabilistic devices and delegate the choice from this set to a third party or to the draw from an Ellsberg urn to make the signals ambiguous.

[^21]: Throughout this proof, whenever I say “for all $x^{*}$ that are sufficiently large for some acts”, it means that for each one of those acts, say $f$, $x^{*} \succsim \max\{\bar{x}_{E,f}, x, \hat{x}_{E,f}\}$ holds.

[^22]: The case $\lambda = 0$ is excluded since when $\lambda = 0$, $g_{l\lambda}y$ coincides with $y$, and $\alpha[E,y]$ is not unique.