# Dynamic adverse selection with the best and the worst in mind

Pascal Toquebeuf

► To cite this version:

Pascal Toquebeuf. Dynamic adverse selection with the best and the worst in mind. Mathematical Social Sciences, 2025, 139, pp.102490. 10.1016/j.mathsocsci.2025.102490. hal-05407714

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# Dynamic adverse selection with the best and the worst in mind ☆

Pascal Toquebeuf

University Grenoble-Alpes, INRAE, CNRS, Grenoble INP, GAEL, 38000, Grenoble, France

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

**JEL classification:**

D80

D81

**Keywords:**

Ambiguity

Optimism

Pessimism

Updating

Neo-additive

Adverse selection

## ABSTRACT

This paper analyzes a dynamic adverse selection market where buyers hold ambiguous beliefs about seller quality, modeled using neo-additive Choquet capacities and updated via optimistic, pessimistic, and Generalized Bayesian rules. First, we show that the choice of updating heuristic has a direct and systematic effect on the severity of adverse selection. While the optimistic and pessimistic rules invariably mitigate or amplify the problem, respectively, the Generalized Bayesian rule's impact is conditional, its trajectory toward collapse, efficiency, or a stable partial market depending on a persistent 'tug-of-war' between the buyer's static ambiguity attitude and the evolving probabilistic evidence. Our second main finding is that these immediate effects compound over time, leading to fundamentally different market trajectories. The pessimistic rule can drive the market to complete collapse, the optimistic rule can foster full participation, and the Generalized Bayesian path depends on the interplay between the buyer's attitude and the evolving evidence. We further analyze how baseline ambiguity and ambiguity aversion modulate these dynamics, uncovering a complex role for ambiguity in shaping the rate of market evolution.

## 1. Introduction

Economic analysis under uncertainty has traditionally relied heavily on the Subjective Expected Utility (SEU) model from Savage, where agents hold probabilistic beliefs over relevant states of the world (Savage, 1954). However, since the seminal work of Ellsberg (1961), extensive experimental evidence and theoretical considerations suggest that decision-makers often face situations characterized by ambiguity or Knightian uncertainty, where probabilities are unknown, unreliable, or vaguely perceived. Recognizing the empirical relevance of ambiguity, researchers have increasingly incorporated non-probabilistic beliefs into economic models across diverse fields, including strategic interactions (Eichberger et al., 2009), financial market behavior (Guerdjikova and Quiggin, 2014), asset pricing (Hansen and Miao, 2022), public good provision (Kishishita and Ozaki, 2020), job search (Blow and Faure-Grimaud, 2012), contracting (Mukerji, 1998), and insurance decisions (Alary et al., 2013).

A prominent and tractable approach for modeling choices under ambiguity is the Choquet Expected Utility (CEU) framework, initially characterized by Gilboa (1987) and Schmeidler (1989). An interesting particular case of CEU is the neo-additive model, introduced by Chateauneuf et al. (2007). This model captures ambiguity attitude through a simple structure: the value of an uncertain prospect is a weighted average of its expected value under a subjective probability measure and a combination of the worst and best possible outcomes.

While static models provide valuable insights, many economic interactions unfold dynamically. Standard Bayesian updating provides a canonical rule for revising probabilistic beliefs; however, updating ambiguous beliefs poses a significant challenge, because there is no unique way to update such beliefs. In the context of CEU preferences, the three most common updating rules for Choquet capacities, axiomatically characterized by Gilboa and Schmeidler (1993), Eichberger et al. (2007) and Horie (2013) are the naive Bayes rule, the Dempster-Shafer rule and Generalized Bayesian rule. When the capacity is neo-additive, Eichberger et al. (2010) show how these rules may be consistently applied to get conditional neo-additive capacities, each rule drawing unique feature in terms of conditional ambiguity and ambiguity attitude.

This paper applies these theoretical tools to a cornerstone problem in information economics: a market plagued by asymmetric information and adverse selection. Famously illustrated by Akerlof (1970) in the market for “lemons”, adverse selection describes how hidden information about product quality can lead to market unraveling as low prices offered by uninformed buyers drive out high-quality sellers. This fundamental issue has profound implications across various markets, including insurance (Rothschild and Stiglitz, 1976), credit (Stiglitz and Weiss, 1981), and labor markets (Spence, 1973). The dynamics of such markets have been studied extensively under standard

<footnote>☆ I am indebted to two anonymous referees from Mathematical Social Sciences for their extensive comments on a previous draft, which resulted in a substantial improvement in the quality of this paper.</footnote>

<footnote>E-mail address: pascal.toquebeuf@univ-grenoble-alpes.fr.</footnote>

<footnote>https://doi.org/10.1016/j.mathsocsci.2025.102490</footnote>

<footnote>Received 24 April 2025; Received in revised form 1 December 2025; Accepted 3 December 2025</footnote>

<footnote>Available online 3 December 2025</footnote>

<footnote>0165-4896/© 2025 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</footnote>
Bayesian assumptions (Guerrieri and Shimer, 2014; Hendel et al., 2005; Zhang, 2015), but the consequences of buyers holding and dynamically updating ambiguous beliefs remain less understood.

This paper bridges these streams of literature by investigating the impact of ambiguity and the choice of updating heuristic within a canonical dynamic adverse selection market. Our central findings show that the way buyers process ambiguous information is a critical determinant of market viability. We show that the optimistic and pessimistic rules generate stark, predetermined paths: the pessimistic rule consistently amplifies adverse selection and can lead to market collapse, whereas the optimistic rule consistently mitigates it and can foster full market participation. In contrast, the Generalized Bayesian rule's impact is conditional, with its trajectory toward collapse, efficiency, or a stable partial market depending on the interplay between the buyer's static disposition and the dynamic information they receive. We further analyze how baseline ambiguity, ambiguity aversion and time interact to shape these dynamics. In this framework, information is modeled as the observation of a market event or signal which allows buyers to update their probabilistic expectation about quality, and whose ex-ante likelihood, or specificity, shapes their updated perception of ambiguity.

The remainder of the paper is organized as follows. Section 2 details the static model setup. Section 3 introduces the dynamic framework, the ambiguity updating rules, and presents the core theorems on their immediate impact on adverse selection. Section 4 presents the main dynamic analysis, including the lemmas and theorems on long run market evolution. Section 5 covers welfare analysis and graphical illustrations. Section 6 discusses the paper's contribution in relation to the broader literature, and Section 7 concludes.

## 2. Static framework

To clarify the role of ambiguity preferences in a market context, this section introduces a static market model featuring a continuum of seller types and buyers. This initial step isolates the effects of valuation under ambiguity from the complexities associated with dynamic belief updates.

### 2.1. Model setup

We consider a static market populated by a continuum of potential sellers and a continuum of potential buyers. For normalization purposes, the mass of each group is set to unity. The state space is the set of possible seller types, represented by the interval $\Theta = [\theta, \bar{\theta}] \subset \mathbb{R}$. Let $\mathcal{E}$ denote the Borel sigma-algebra on $\Theta$, representing the set of relevant events. Seller types $\theta \in \Theta$ differentiate the quality of the product offered. The actual distribution of seller types is described by a probability density function (PDF) $f$ defined over the support $\Theta$. We assume $f(\theta) > 0$ for all $\theta \in \Theta$.

A seller of type $\theta$ provides a product with an inherent value $v(\theta)$. We assume $v: \Theta \to \mathbb{R}$ is continuous and non-decreasing in $\theta$. The minimum and maximum achievable values are $v_{min} = v(\theta)$ and $v_{max} = v(\bar{\theta})$, respectively. Each seller of type $\theta$ incurs a reservation price $S(\theta)$, where $S: \Theta \to \mathbb{R}$ is continuous and strictly increasing, $S' > 0$. To ensure a non-trivial market, we assume a non-degenerate type range, $\theta < \bar{\theta}$, with $v(\theta) < v(\bar{\theta})$ and $S(\theta) < S(\bar{\theta})$.

Buyers are homogeneous and face ambiguity regarding the true distribution $f(\theta)$. Their subjective belief about the distribution is represented by a PDF $g$ over $\Theta$, satisfying $g(\theta) > 0$ for all $\theta$. This density induces an additive probability measure $p$ on $(\Theta, \mathcal{E})$, where for any event $A \in \mathcal{E}$, $p(A) = \int_A g(\theta)d\theta$.

Buyers evaluate uncertain prospects according to the Choquet Expected Utility (CEU) framework, using preferences represented by a neo-additive capacity $v$ on $(\Theta, \mathcal{E})$. We follow Chateauneuf et al. (2007) in partitioning the event space $\mathcal{E}$ into null events $\mathcal{N}$ (those the agent deems impossible) and non-null events. It is important to distinguish the behavioral concept of a null event from the mathematical concept

of a Lebesgue-measure zero event. A non-empty set of states can have a Lebesgue measure of zero but still be considered possible by the decision-maker. In this model, consistent with the standard applications of neo-additive capacities, we adopt the simplest structure where the only null event is the impossible event, $\mathcal{N} = \{\emptyset\}$. The support of the capacity – the smallest closed set whose complement is a null event – is therefore the entire state space $\Theta$.

The neo-additive capacity $v$ is constructed from the additive probability measure $p$, a baseline degree of ambiguity $\delta \in [0, 1]$, and a baseline ambiguity aversion parameter $\alpha \in [0, 1]$. Here, $\alpha$ represents the weight placed on the worst outcome. The capacity $v: \mathcal{E} \to [0, 1]$ is defined as $v(A) = (1 - \delta)p(A) + \delta\mu_{1-\alpha}^{\mathcal{N}}(A)$ for all $A \in \mathcal{E}$, where $\mu_{1-\alpha}^{\mathcal{N}}$ is the Hurwicz capacity that assigns a value of $(1 - \alpha)$ to any non-empty, non-universal event. By Chateauneuf et al. (2007) lemma 3.1, the valuation for acquiring a product is then given by the Choquet integral, which for a neo-additive capacity simplifies to:

$$V(v, v) = \int_{\Theta} v(\theta)d\nu = (1 - \delta)\mathbb{E}_{p}[v(\theta)] + \delta[\alpha v_{min} + (1 - \alpha)v_{max}]$$

where $\mathbb{E}_{p}[v(\theta)]$ is the expected value under the additive measure $p$. We shall assume that buyers are risk-neutral, meaning that there is no utility function over outcomes $v(\theta)$.

### 2.2. Buyer's valuation

The buyer's Willingness-to-Pay (WTP) is determined by their subjective beliefs about seller quality. Therefore, their valuation, denoted $V$, is calculated using the Choquet integral with respect to their subjective distribution $g$. This valuation is a weighted average of an evidence-based component derived from their beliefs and a heuristic, ambiguity-driven component:

$$V = (1 - \delta)\mathbb{E}_{g}[\bar{v}] + \delta V_{amb} \quad (1)$$

where:

* The evidence-based component is the standard Bayesian valuation, calculated using the buyer's subjective distribution $g$:

$$\mathbb{E}_{g}[\bar{v}] = \int_{\emptyset}^{\bar{\theta}} v(\theta)g(\theta)d\theta. \quad (2)$$

* The degree of ambiguity, $\delta$, represents the buyer's lack of confidence in their subjective belief $g$.

* The heuristic component is the buyer's attitudinal fallback, triggered by their lack of confidence:

$$V_{amb} = \alpha v(\theta) + (1 - \alpha)v(\bar{\theta}). \quad (3)$$

### 2.3. Adverse selection mechanism

Adverse selection arises because the price offered by buyers, $V$, may be too low to attract high-quality sellers. A seller of type $\theta$ participates only if $V \ge S(\theta)$. This defines a participation threshold $\theta^*$, which is the highest type willing to sell at price $V$:

$$\theta^*(V) = \sup\{\theta \in \Theta \mid V \ge S(\theta)\} \quad (4)$$

A lower threshold $\theta^*$ signifies a smaller, lower-quality pool of participating sellers, thus a more severe adverse selection problem. The severity can be measured by the quality shortfall between the unconditional average quality and the average quality of participating sellers:

$$AS_0(V) = \mathbb{E}_f[\bar{v}] - \mathbb{E}_f[\bar{v} \mid \theta \le \theta^*(V)] \ge 0 \quad (5)$$

A lower valuation $V$ leads to a lower $\theta^*$ and thus a weakly higher $AS_0$.
## 2.4. Ambiguity and ambiguity attitude

We now investigate how the ambiguity parameters $\alpha$ and $\delta$ affect market outcomes.

**Proposition 1.** Let $\delta > 0$. An increase in the baseline ambiguity aversion $\alpha$ weakly increases the adverse selection measure $AS_0$.

A more ambiguity averse buyer (higher $\alpha$) places more weight on the worst possible outcome, $v(\theta)$. This lowers their overall valuation. The lower price offered in the market excludes more high-quality sellers, thus shrinking the market and exacerbating adverse selection.

**Proposition 2.** The effect of an increase in the baseline degree of ambiguity $\delta$ on adverse selection depends on the sign of $(V_{amb} - E_g[\bar{v}])$:

(i) If $V_{amb} < E_g[\bar{v}]$, increasing $\delta$ increases $AS_0$.

(ii) If $V_{amb} > E_g[\bar{v}]$, increasing $\delta$ decreases $AS_0$.

An increase in $\delta$ means the buyer relies less on the rational expectation $E_g[\bar{v}]$ and more on the ambiguity component $V_{amb}$. If the buyer's attitude is relatively ambiguity averse, i.e. if $\alpha$ is high, such that the ambiguity component is lower than the rational expectation, then more ambiguity hurts their valuation, worsening adverse selection. Conversely, if they are relatively ambiguity loving, i.e. if $\alpha$ is low, more ambiguity can boost their valuation, mitigating adverse selection.

## 2.5. Market equilibrium

A market equilibrium $(P^*, Q^*, \theta^*)$ is determined by the intersection of aggregate supply and demand. Aggregate supply, $Q^S(P)$, is the mass of sellers for whom $S(\theta) \le P$, and is non-decreasing in price. Aggregate demand, $Q^D(P)$, is a step function: buyers demand 1 unit if the price is at or below their common valuation $V$, and 0 otherwise. We note that this is a posted-price mechanism where buyers' WTP is based on the unconditional distribution of seller types. This serves as a standard benchmark, following Akerlof (1970), to isolate the direct effect of ambiguity, in contrast to rational expectations models where the valuation must be consistent with the equilibrium quality of participating sellers.

This leads to three potential equilibrium regimes:

* Market Collapse: If $V < S(\theta)$, no trade occurs ($Q^* = 0$).

* Partial Market: If $S(\theta) \le V < S(\bar{\theta})$, the equilibrium price is $P^* = V$. The quantity traded is $Q^* = F(\theta^*)$, where $\theta^* = S^{-1}(V)$ is the marginal participating seller. This is the classic adverse selection outcome.

* Full Market: If $V \ge S(\bar{\theta})$, all sellers are willing to participate. The price is set to clear the market, e.g. $P^* = S(\bar{\theta})$, and $Q^* = 1$.

In a partial market equilibrium, the buyer's valuation $V$ directly determines all outcomes. The effects of ambiguity parameters $(\alpha, \delta)$ on $V$, as detailed in Propositions 1 and 2, translate directly into changes in the equilibrium price, quantity, and participation. For instance, higher ambiguity aversion, that is a larger $\alpha$, reduces $V$, leading to a lower price, lower quantity, and reduced total surplus, thereby increasing the inefficiency (deadweight loss) from adverse selection.

## 3. The dynamic model of adverse selection under ambiguity

In this section, we extend the static framework to a dynamic setting. The primary motivation for this extension is to analyze how the pool of available sellers evolves over time—a key feature of markets subject to adverse selection. Unlike in a one-shot model, sellers in a dynamic market are not captive participants; they can choose to exit if offered prices consistently fail to meet their reservation costs. By modeling this endogenous change in the seller pool, our framework can explicitly trace the process of market unraveling, where low prices drive out high-quality sellers, or, conversely, the process of market persistence, where

favorable prices encourage continued participation. This dynamic approach allows us to study not just the existence of an adverse selection problem, but the trajectory it follows as beliefs and market composition co-evolve.

### 3.1. Setup

The market operates over discrete time periods indexed by $t = 0, 1, 2, \dots$. The fundamental characteristics of sellers, namely their types $\theta$, the value function $v$, and the cost function $S$, remain as defined in the static model. However, the pool of available sellers may change over time. The true density of available sellers at the beginning of period $t$ is denoted by $f_t$, with the corresponding total mass being $\int f_t(\theta)d\theta \le 1$. Information relevant to the market arrives sequentially over time, represented by a filtration $(F_t)_{t \ge 0}$. The history of events up to time $t$ is denoted by $E_t \in \mathcal{F}_t$.

Buyers are endowed at time $t=0$ with a neo-additive capacity $v_0 = (g_0, \delta, \alpha)$, representing their initial subjective beliefs $g_0$ and baseline ambiguity preferences $(\delta, \alpha)$. The assumptions established in the static framework regarding seller characteristics and buyer preferences continue to hold in this dynamic setting.

### 3.2. Conditional capacities and belief updating

In a dynamic setting, buyers must have a way to revise their beliefs upon receiving new information, represented by the history $E_t$. For a standard Bayesian agent, where ambiguity is absent, that is $\delta = 0$, this is handled by the unique Bayesian update rule, yielding a posterior willingness-to-pay of $V_t^B(E_t) = E_{g_t}[\bar{v}|E_t]$. However, for non-additive Choquet beliefs, the generalization of conditioning is not unique. Several updating rules have been proposed, often arising from different ways of defining the relationship between unconditional and conditional expectations. This paper focuses on three prominent rules that are common in the literature.

When $\delta > 0$, buyers update their capacity based on the history $E_t$ using a rule $U \in \{O, P, GB\}$. Let $P(E_t|\theta)$ be the likelihood of history $E_t$ given state $\theta$, and let the prior probability of the history be $g_0(E_t) = \int P(E_t|\theta)g_0(\theta)d\theta$. Each updated capacity, $v_t^U$, uses the standard Bayesian posterior density $g_t(\theta|E_t)$ but, as shown by Eichberger et al. (2010), applies a distinct transformation to the baseline ambiguity parameters $(\delta, \alpha)$ to arrive at state-dependent values $(\delta_t^U, \alpha_t^U)$:

1. The Optimistic (O) rule, also known as the Naive Bayesian rule, sets the updated ambiguity attitude to pure optimism, $\alpha_t^O = 0$, with an updated ambiguity level of:

$$ \delta_t^O = \frac{\delta(1 - \alpha)}{(1 - \delta)g_0(E_t) + \delta(1 - \alpha)}. \qquad (6) $$

2. The Pessimistic (P) rule, equivalent to the Dempster-Shafer rule, sets the updated ambiguity attitude to pure pessimism, $\alpha_t^P = 1$, with an updated ambiguity level of:

$$ \delta_t^P = \frac{\delta\alpha}{(1 - \delta)g_0(E_t) + \delta\alpha}. \qquad (7) $$

3. The Generalized Bayesian (GB) rule, also known as the Full Bayesian rule, is unique in that it preserves the baseline ambiguity attitude, $\alpha_t^{GB} = \alpha$, with an updated ambiguity level of:

$$ \delta_t^{GB} = \frac{\delta}{(1 - \delta)g_0(E_t) + \delta}. \qquad (8) $$

The names “Optimistic” and “Pessimistic” have a deep behavioral foundation, as these updating rules have been axiomatically characterized by Gilboa and Schmeidler (1993). The updated belief depends on what outcome the agent assumes would have occurred had the conditioning event not happened. The optimistic rule corresponds to a decision-maker who assumes the worst possible outcome would
have occurred in the counterfactual, making them “happy” or optimistic about the information they received. Conversely, the pessimistic (Dempster-Shafer) rule corresponds to an agent who assumes the best possible outcome would have occurred, making them feel regretful or pessimistic about the actual event. In the specific context of neoaugitive capacities, these names gain further justification (Eichberger et al., 2010): the optimistic rule forces the updated attitude to pure optimism ($\alpha_t^O = 0$), the pessimistic rule forces it to pure pessimism ($\alpha_t^P = 1$), while only the Generalized Bayesian rule leaves the agent's baseline ambiguity attitude $\alpha$ unchanged (Eichberger et al., 2012). These three rules are frequently analyzed as they represent distinct, axiomatically-grounded approaches to the challenge of updating ambiguous beliefs.

### 3.3. Ex-post valuation and market evolution

Given the history $E_t$ and the application of updating rule $U \in \{O, P, GB\}$, the resulting ex-post willingness-to-pay of the buyer is denoted by $V_t^U(E_t)$. This valuation is calculated using the Choquet integral with respect to the updated capacity $v_{E_t}^U = (g_t(\theta|E_t), \delta_t^U, \alpha_t^U)$:

$$V_t^U(E_t) = (1 - \delta_t^U) \mathbb{E}_{g_t}[\bar{v}|E_t] + \delta_t^U (\alpha_t^U v(\underline{\theta}) + (1 - \alpha_t^U) v(\bar{\theta})) \quad (9)$$

Specifically, for the Optimistic rule ($\alpha_t^O = 0$):

$$V_t^O(E_t) = (1 - \delta_t^O) \mathbb{E}_{g_t}[\bar{v}|E_t] + \delta_t^O v(\bar{\theta}) \quad (10)$$

For the Pessimistic rule ($\alpha_t^P = 1$):

$$V_t^P(E_t) = (1 - \delta_t^P) \mathbb{E}_{g_t}[\bar{v}|E_t] + \delta_t^P v(\underline{\theta}) \quad (11)$$

And for the Generalized Bayesian rule ($\alpha_t^{GB} = \alpha$):

$$V_t^{GB}(E_t) = (1 - \delta_t^{GB}) \mathbb{E}_{g_t}[\bar{v}|E_t] + \delta_t^{GB} (\alpha v(\underline{\theta}) + (1 - \alpha) v(\bar{\theta})) \quad (12)$$

Seller participation in period $t$ depends on whether the prevailing willingness-to-pay $V_t^U(E_t)$ covers their cost $S(\theta)$. Consequently, the true distribution of seller types evolves over time based on this participation decision. The density of sellers available at the start of period $t+1$, conditional on history $E_t$, is proportional to the density at time $t$ restricted to those types who chose to participate:

$$f_{t+1}(\theta|E_t) \propto 1(V_t^U(E_t) \ge S(\theta)) \cdot f_t(\theta) \quad (13)$$

This density is then normalized to integrate to the new total mass. This dynamic process creates a feedback loop where buyer beliefs influence valuations, which determine seller participation, which in turn affects the composition of the seller pool and potentially future information signals, thereby influencing future beliefs.

### 3.4. Mitigation and amplification of dynamic adverse selection

This section compares the dynamic impact of the updating rules on the severity of adverse selection relative to the standard Bayesian benchmark (B), maintaining the assumption of risk neutrality for buyers. Let $g_t(\theta|E_t)$ represent the Bayesian posterior density derived from history $E_t$, and let $\mathbb{E}_{g_t}[\bar{v}|E_t]$ be the corresponding posterior mean value. The willingness-to-pay under updating rule $U \in \{O, P\}$ is $V_t^U(E_t)$, derived using the updated capacity parameters ($\delta_t^U, \alpha_t^U$), while the Bayesian willingness-to-pay is $V_t^B(E_t) = \mathbb{E}_{g_t}[\bar{v}|E_t]$.

The participation threshold under rule $U$ at time $t$ given history $E_t$, denoted $\theta^{*,U}(E_t)$, is the highest seller type willing to participate at the price $V_t^U(E_t)$:

$$\theta^{*,U}(E_t) = \sup\{\theta \in \Theta | V_t^U(E_t) \ge S(\theta)\} \quad (14)$$

Similarly, the Bayesian threshold $\theta^{*,B}(E_t)$ is determined using the Bayesian valuation $V_t^B(E_t)$. Let $f_t(\theta)$ be the true density of seller types present in the market at the beginning of period $t$. The dynamic measure of adverse selection under rule $U$ for period $t$, given history

$E_t$, compares the average quality of sellers available at time $t$ with the average quality of those who choose to participate for period $t+1$:

$$AS_t^U(E_t) = \mathbb{E}_{f_t}[\bar{v}] - \bar{v}_{t+1}^U(E_t) \quad (15)$$

Here, $\mathbb{E}_{f_t}[\bar{v}]$ represents the average quality among sellers present at time $t$, and $\bar{v}_{t+1}^U(E_t)$ denotes the average quality of those sellers who participate based on the valuation $V_t^U(E_t)$:

$$\bar{v}_{t+1}^U(E_t) = \mathbb{E}_{f_t}[\bar{v} | \theta \le \theta^{*,U}(E_t)] = \frac{\int_{\underline{\theta}}^{\theta^{*,U}(E_t)} v(\theta) f_t(\theta) d\theta}{\int_{\underline{\theta}}^{\theta^{*,U}(E_t)} f_t(\theta) d\theta} \quad (16)$$

This expression is valid provided the denominator, representing the mass of participating sellers, is positive. A lower participation threshold $\theta^{*,U}$ implies a potentially lower average quality of participating sellers $\bar{v}_{t+1}^U$, thus leading to a higher measure of adverse selection $AS_t^U$.

**Lemma 1.** For any history $E_t$, the valuations generated by the pessimistic, Bayesian, and optimistic rules rank as follows: $V_t^P(E_t) \le V_t^B(E_t) \le V_t^O(E_t)$.

The intuition for **Lemma 1** is that the pessimistic updating rule consistently yields the lowest willingness-to-pay by incorporating the worst possible outcome $v(\underline{\theta})$, while the optimistic rule yields the highest by incorporating the best possible outcome $v(\bar{\theta})$, effectively sandwiching the purely expectation-based Bayesian valuation. In simple terms, an optimist's higher valuation helps keep high-quality sellers in the market who would otherwise exit, while a pessimist's lower valuation actively drives them out.

**Lemma 2.** Let $V_{amb} = \alpha v(\underline{\theta}) + (1 - \alpha)v(\bar{\theta})$. Then:

(i) If $V_{amb} \le \mathbb{E}_{g_t}[\bar{v}|E_t]$, then $V_t^{GB}(E_t) \le V_t^B(E_t)$.

(ii) If $V_{amb} \ge \mathbb{E}_{g_t}[\bar{v}|E_t]$, then $V_t^{GB}(E_t) \ge V_t^B(E_t)$.

The intuition behind **Lemma 2** is straightforward. The Generalized Bayesian valuation is a weighted average of the evidence-based rational expectation $\mathbb{E}_{g_t}[\bar{v}|E_t]$ and the attitude-driven ambiguity component $V_{amb}$. Naturally, the final valuation is pulled toward the agent's attitudinal anchor. If the buyer has a cautious disposition, that is a low $V_{amb}$, their valuation will be below the Bayesian benchmark; if they have an adventurous disposition, that is a high $V_{amb}$, it will be above it.

Before analyzing how this ranking of valuations translates into market outcomes, we must first formally define how we measure the severity of adverse selection.

**Definition 1.** (Mitigation/Amplification of AS). Given history $E_t$, updating rule $U \in \{O, P\}$ exhibits relative:

* Mitigation of AS if $AS_t^U(E_t) \le AS_t^B(E_t)$.

* Amplification of AS if implying $AS_t^U(E_t) \ge AS_t^B(E_t)$.

Amplification and mitigation of adverse selection are relative because the entire analysis compares the outcomes under the three updating rules against the outcome of a Bayesian agent.

With this definition in hand, we can now state our first main theorems, which demonstrate that the WTP rankings established in the lemmas have a direct and systematic impact on whether adverse selection is mitigated or amplified.

**Theorem 1.** Let $\delta > 0$. For any given history $E_t$:

(i) The pessimistic rule relatively amplifies adverse selection.

(ii) The optimistic rule relatively mitigates adverse selection.

Furthermore, an increase in $\delta$ strengthens these respective effects, increasing the magnitude of both the amplification and the mitigation.

As stated in **Lemma 1**, the lower WTP resulting from the Pessimistic rule drives out more high-quality sellers compared to the
Bayesian benchmark, thus amplifying adverse selection. Conversely, the higher WTP associated with the Optimistic rule helps retain more high-quality sellers than the Bayesian benchmark, thereby mitigating adverse selection.

More baseline ambiguity $\delta$ increases the weight on the extreme outcomes in the updated valuations. Under the optimistic rule, this means more weight on the best outcome $v(\bar{\theta})$, which raises WTP. Under the pessimistic rule, it means more weight on the worst outcome $v(\theta)$, which lowers WTP. Therefore, ambiguity can be “good” for market participation under the optimistic updating rule and “bad” under the pessimistic one. Further, if the world is perceived as more ambiguous, that means a higher $\delta$, then behavioral rules that are sensitive to ambiguity, here O and P, will naturally produce outcomes that deviate more sharply from the ambiguity-neutral Bayesian benchmark.

**Theorem 2.** Let $\delta > 0$. For any given history $E_t$:

(i) If $V_{amb} \le E_{g_t}[\bar{v}|E_t]$, the GB rule relatively amplifies adverse selection.

(ii) If $V_{amb} \ge E_{g_t}[\bar{v}|E_t]$, the GB rule relatively mitigates adverse selection.

Furthermore, an increase in $\delta$ strengthens these conditional effects by increasing the magnitude of the deviation from the Bayesian benchmark.

The conditional nature of the Generalized Bayesian rule's behavior warrants further discussion, as it is governed by the inequality between the static ambiguity component, $V_{amb} = \alpha v(\bar{\theta}) + (1 - \alpha)v(\theta)$, and the dynamic posterior expectation, $E_{g_t}[\bar{v}|E_t]$. This inequality represents a tension between the buyer's fixed, attitude-driven valuation of the extreme outcomes and their evolving, evidence-based valuation of the average outcome. In terms of model parameters, a higher ambiguity aversion $\alpha$ directly lowers $V_{amb}$ and makes a downward deviation, $V_{amb} < E_{g_t}[\bar{v}|E_t]$, more likely. Furthermore, the information flow dynamically influences the posterior mean; for instance, good news that raises $E_{g_t}[\bar{v}|E_t]$ would also make a downward deviation more probable.

These parametric effects correspond to distinct real-world market characteristics. A downward deviation is more plausible in contexts defined by caution, such as insurance markets where the business model focuses on avoiding catastrophic loss, or in mature investment markets where a pessimistic investor might overweight the tail risk of a sudden crash. Conversely, an upward deviation, $V_{amb} > E_{g_t}[\bar{v}|E_t]$, is more characteristic of speculative settings with high potential upsides. This includes venture capital, where an optimistic focus on finding a single transformative success is necessary to justify investment in a portfolio where the average startup is expected to fail, or in pharmaceutical R&D, where the immense payoff from a blockbuster drug can outweigh a low probabilistic chance of success.

### 3.5. Ambiguity aversion

We now examine how the baseline ambiguity aversion parameter $\alpha$ influences the severity of adverse selection, depending on the updating rule. We define as $M_t^U = |V_t^U - V_t^B|$, with $V_t^B = E_{g_t}[\bar{v}|E_t]$, the magnitude of the deviation from expected utility implied by the updating rule $U \in \{O, P, GB\}$.

**Proposition 3.** Let $\delta > 0$. Baseline ambiguity aversion $\alpha$ has then the following effects:

(i) It weakly increases adverse selection for both rules:

$$ \frac{\partial AS_t^U}{\partial \alpha} \ge 0, \quad \text{for } U \in \{O, P\} $$

(ii) It has opposite effects on the magnitudes of the optimistic and pessimistic biases:

$$ \frac{\partial M_t^O}{\partial \alpha} < 0 \quad \text{and} \quad \frac{\partial M_t^P}{\partial \alpha} > 0 $$

A higher initial $\alpha$ reflects a more cautious disposition. Under pessimistic updating, this directly increases the updated ambiguity level, lowering WTP. Under optimistic updating, a higher $\alpha$ reduces the updated ambiguity level, but since the optimistic valuation is a mix of the high $v(\bar{\theta})$ and the lower expectation, reducing the weight on the former also lowers the overall WTP. In both cases, more baseline aversion leads to lower prices and worsens adverse selection.

This proposition states that a higher baseline ambiguity aversion $\alpha$ uniformly worsens adverse selection by depressing valuations. This can be seen in the context of insurance pricing after a novel catastrophe. Imagine a new, specific type of natural disaster emerges, creating significant ambiguity for insurers. An insurer with low ambiguity aversion might set premiums based on its best-guess models, attracting a broad mix of businesses. In contrast, an insurer with high ambiguity aversion is acutely sensitive to the unknown, worst-case losses. To protect itself, it will set extremely high premiums. Such a price ensures that only the businesses that know they are located in the most vulnerable spots will find it worthwhile to purchase a policy. The insurer's pool therefore becomes saturated with the highest-risk clients, making the adverse selection problem severe and potentially rendering the insurance line unprofitable.

**Proposition 4.** Let $\delta > 0$. Baseline ambiguity aversion $\alpha$ has then the following effects:

(i) It weakly increases adverse selection:

$$ \frac{\partial AS_t^{GB}}{\partial \alpha} \ge 0 $$

(ii) It has the following conditional effects on $M_t^{GB}$:

$$ V_t^{GB} < V_t^B \Rightarrow \frac{\partial M_t^{GB}}{\partial \alpha} > 0 \quad \text{and} \quad V_t^{GB} > V_t^B \Rightarrow \frac{\partial M_t^{GB}}{\partial \alpha} < 0 $$

Statement (ii) says that, under the GB rule, an increase in ambiguity aversion strictly increases the magnitude of a downward deviation and strictly decreases the magnitude of an upward deviation from the Bayesian benchmark.

The result for the GB rule in Proposition 4 stems from a direct mechanism: greater ambiguity aversion, that is a higher $\alpha$, makes the buyer's attitudinal anchor, $V_{amb}$, more pessimistic by placing more weight on the worst outcome. This directly depresses their overall willingness-to-pay, thus worsening adverse selection. This explains the proposition's finding on the deviation's magnitude: a more pessimistic anchor naturally pulls the valuation further down from the benchmark, increasing a downward deviation's size, or reins it in from above, decreasing an upward deviation's size.

### 4. Dynamic market evolution

This section explores the influence of other model parameters, focusing on the dynamic evolution of the market.

#### 4.1. Long-run market dynamics

The core of the dynamic analysis lies in how the market evolves as information accumulates over time. As time passes, the history of events $E_t$ becomes more precise, implying its prior probability $g_0(E_t)$ decreases. This drives the dynamics of WTP and adverse selection.

**Lemma 3.** If $\delta \in (0, 1)$, as time $t$ increases:

(i) $V_{t+1}^P \le V_t^P$ and $AS_{t+1}^P \ge AS_t^P$.

(ii) $V_{t+1}^O \ge V_t^O$ and $AS_{t+1}^O \le AS_t^O$.
As time increases, information becomes more precise and has a lower prior probability. Both the pessimistic and the optimistic updating rules interpret these news as a reason to increase the weight on ambiguity. With the pessimistic updating rule, more ambiguity means a greater focus on the worst outcome, depressing the buyer's WTP over time and worsening adverse selection. Under the optimistic updating rule, it means a greater focus on the best outcome, boosting the buyer's WTP and weakening adverse selection. This creates two starkly different paths over time: pessimism leads to a downward spiral, while optimism leads to a virtuous cycle.

**Proposition 5.** The sequences {$M_t^O$}$_{t \ge 0}$ and {$M_t^P$}$_{t \ge 0}$ are non-decreasing.

In words, as time increases, the magnitudes of both the optimistic bias and the pessimistic bias weakly increase.

**Proposition 6.** The sequence {$M_t^{GB}$}$_{t \ge 0}$ is non-decreasing.

**Proposition 6** states that, as time increases, the magnitude of the deviation of the GB valuation from the Bayesian benchmark weakly increases.

These results show that as uncertainty is resolved, the behavioral models based on neo-additive beliefs diverge from the classical one. A Bayesian agent incorporates new information and their valuation converges based on that evidence. In contrast, an ambiguity-sensitive agent sees new information as a reason to rely more on their behavioral bias: optimism, pessimism, or their initial attitude $\alpha$. This causes the gap between their valuation and the rational Bayesian benchmark to widen over time, highlighting a fundamental difference in how they learn.

**Theorem 3.** For any $\alpha \in [0, 1]$, any $\delta \in (0, 1)$ and any $t \ge 0$:

$$ (i) \quad \theta_{t+1}^{*,P} \le \theta_t^{*,P} \quad \text{and} \quad \lim_{t \to \infty} V_t^P < S(\theta) \quad \Rightarrow \quad \lim_{t \to \infty} Q_t^* = 0. $$

$$ (ii) \quad \theta_{t+1}^{*,O} \ge \theta_t^{*,O} \quad \text{and} \quad \lim_{t \to \infty} V_t^O \ge S(\theta) \quad \Rightarrow \quad \lim_{t \to \infty} Q_t^* = 1. $$

Statement (i) means that under the pessimistic rule, the market follows a path of non-increasing participation. If the limiting WTP falls below the lowest seller cost, the market is destined for complete collapse. Statement (ii) means that under the optimistic rule, the market follows a path of non-decreasing participation. A partial market may evolve toward full efficiency.

This theorem formalizes the long-run consequences of the dynamics identified in **Lemma 3**. The pessimistic rule fuels a “death spiral”: falling prices over time drive out good sellers, which justifies even lower prices, potentially unraveling the market entirely. The optimistic rule fuels market persistence and growth: rising prices over time attract better sellers, which justifies higher prices, a process that can potentially eliminate adverse selection.

The immediate amplification of adverse selection under the pessimistic rule, established in **Theorem 1**, is not a one-time effect. It initiates a dynamic feedback loop where lower valuations progressively drive out higher-quality sellers. This self-reinforcing mechanism sets the market on a path of non-increasing participation, which, as we formalize in **Theorem 3**, can culminate in a complete market collapse. Conversely, by **Theorem 1**, the mitigating effect of the optimistic rule fuels a virtuous cycle. By sustaining higher valuations, it encourages participation from better sellers, which in turn justifies the buyer's optimism. **Theorem 3** demonstrates how this process compounds over time, potentially guiding a partial market toward full efficiency.

The model's core dynamic prediction is that pessimistic updating can lead to market unraveling, while optimistic updating can foster persistence. This is well-illustrated by real-world examples. Consider the nascent market for used electric vehicles, where ambiguity about battery health is high. Early negative news can trigger a pessimistic spiral as described by **Theorem 3**, as falling prices drive out good cars, confirming the pessimism and leading to market collapse. Conversely,

in the early days of a new cryptocurrency, early positive returns can trigger an optimistic spiral. Focusing on the best-case scenario, buyers' WTP soars, attracting more participants and creating a self-sustaining market, just as the optimistic trajectory suggests. We now turn to the Generalized Bayesian rule, which exhibits a fundamentally different asymptotic behavior.

**Lemma 4.** For any $\alpha \in [0, 1]$ and any $\delta \in (0, 1)$, the valuation under the Generalized Bayesian rule converges asymptotically to the static ambiguity component:

$$ \lim_{t \to \infty} V_t^{GB} = V_{amb} = \alpha \nu(\theta) + (1 - \alpha) \nu(\bar{\theta}) \quad (17) $$

This lemma establishes that, unlike the Bayesian agent whose valuation converges to the true expected value, the Generalized Bayesian agent's valuation eventually becomes dominated by their baseline ambiguity attitude. This convergence leads to the following characterization of long-run market outcomes.

**Theorem 4.** The long-run market equilibrium under the Generalized Bayesian rule is determined solely by the buyer's baseline ambiguity attitude relative to the seller's cost structure:

$$ (i) \quad \text{Collapse:} \quad \text{If } V_{amb} < S(\theta), \text{ the market collapses, } \lim_{t \to \infty} Q_t^* = 0. $$

$$ (ii) \quad \text{Full Efficiency:} \quad \text{If } V_{amb} \ge S(\bar{\theta}), \text{ the market achieves full participation, } \lim_{t \to \infty} Q_t^* = 1. $$

$$ (iii) \quad \text{Partial Market:} \quad \text{If } S(\theta) \le V_{amb} < S(\bar{\theta}), \text{ the market converges to a stable partial equilibrium with participation } \theta_{\infty}^* = S^{-1}(V_{amb}) \text{ and traded quantity } Q_{\infty}^* \in (0, 1). $$

This theorem highlights a fundamental distinction between the Generalized Bayesian rule and its Optimistic and Pessimistic counterparts. While the latter drive the market toward extreme outcomes, either full efficiency or total collapse, the Generalized Bayesian rule allows for a stable “middle ground”. As information accumulates, the buyer's valuation becomes increasingly anchored by their static ambiguity attitude, $V_{amb}$. Consequently, if this baseline attitude is moderate, falling strictly between the minimum and maximum seller costs, the market will neither unravel nor clear completely, but instead converge to a stable partial equilibrium (case iii). In the short to medium run, the trajectory may be volatile as the “tug-of-war” between the evolving evidence $E_{g_t}[\bar{v}|E_t]$ and the static anchor $V_{amb}$ plays out. However, as the ambiguity weight $\delta_t^{GB}$ approaches unity, the influence of probabilistic fluctuations fades, ensuring asymptotic stability.

These theoretical limiting cases align with distinct real-world behaviors. Case (ii), where a high $V_{amb}$ drives efficiency, mirrors a pharmaceutical company evaluating a novel drug; despite poor probabilistic evidence of success, a strong optimistic focus on the massive potential payoff, that is a high $V_{amb}$, sustains funding until project completion. Conversely, Case (i) reflects a cautious fund manager evaluating a volatile asset class; a low baseline $V_{amb}$ means that even if early returns are decent, the persistent weight on worst-case scenarios eventually drives the manager to exit, collapsing that market segment.

## 4.2. The impact of ambiguity and ambiguity aversion on market evolution

Having established the long-run direction of the market trajectories, we now turn to the rate of this evolution. We analyze how the baseline parameters, ambiguity $\delta$ and aversion $\alpha$, modulate the speed at which the market pricing changes over time.

The short run corresponds to the initial period of the market, where information is still generic and uncertainty is high. This is a phase of discovery, where agents' beliefs are volatile and highly sensitive to foundational news. Their reactions to initial signals are strong, leading to rapid adjustments in their willingness-to-pay.

The long run, in contrast, represents a more mature phase of the market. After a significant amount of information has been processed,
a clear trend in valuations has been established. New information tends to be more incremental, leading to minor refinements rather than major shifts. In this phase, behavior is more stable, and valuations begin to converge toward a final equilibrium.

The model captures this distinction formally through a transition time, denoted $t^*$. This is the point in time that separates the short-run discovery phase from the long-run convergence phase. Before this time, for $t < t^*$, the market exhibits its short-run characteristics, and after this time, for $t > t^*$, it exhibits its long-run characteristics. The precise value of this transition time is determined by the agent's ambiguity preferences, as detailed in the following propositions.

Let $g_U^* = g_0(E_U^*)$, where $t_U^*$ is the transition time ensuring the passage from short run to long run under the updating rule $U \in \{O, P\}$.

**Definition 2 (Critical Information Threshold for an Optimist).** For an optimist, the critical information threshold is the prior probability value $g_O^* = \delta(1 - \alpha)/(1 - \delta)$.

Based on this definition, the market is in its short-run phase when $g_0(E_t) > g_O^*$ and enters its long-run phase when information becomes more precise, $g_0(E_t) < g_O^*$. Since $g_0(E_t)$ is non-increasing in time, such a $t_O^*$ exists. Note that this transition time $t_O^*$ is weakly decreased by a greater $\delta$ and weakly increased by a greater $\alpha$.

**Proposition 7.** Under the optimistic rule, a greater $\delta$:

(i) Accelerates the rate of decrease in adverse selection for $t < t_O^*$.

(ii) Dampens the rate of decrease in adverse selection for $t > t_O^*$.

Early in the market's life, when $t < t_O^*$, a greater baseline ambiguity $\delta$ makes an optimist more responsive to news, accelerating the rise in their WTP. Later in time, for $t > t_O^*$, when much is already known, a high baseline $\delta$ acts as a buffer, making the optimist's high valuation less sensitive to further incremental news, thus stabilizing the market.

**Definition 3 (Critical Information Threshold for a Pessimist).** Under the pessimistic rule, the critical information threshold is the prior probability value $g_P^* = \delta\alpha/(1 - \delta)$.

The market is then in its short-run phase when $g_0(E_t) > g_P^*$ and enters its long-run phase when $g_0(E_t) < g_P^*$. The transition time $t_P^*$ is weakly decreased by an increase in either $\delta$ or $\alpha$.

**Proposition 8.** Under the pessimistic rule, a greater $\delta$:

(i) Accelerates the rate of increase in adverse selection for $t < t_P^*$.

(ii) Dampens the rate of increase in adverse selection for $t > t_P^*$.

This shows that initial ambiguity $\delta$ can either accelerate or dampen market trends depending on the time elapsed. The evolution of a trendy restaurant scene illustrates this well. Early on, high ambiguity accelerates the initial hype following a good review. Years later, after the transition time, the same ambiguity dampens the effect of further reviews, leading to more stable prices.

The baseline ambiguity aversion, $\alpha$, also modulates the rate of evolution over time, with its effect changing before and after the transition times $t_O^*$ and $t_P^*$.

**Proposition 9.** Under the optimistic rule, a greater $\alpha$:

(i) Dampens the rate of decrease in adverse selection for $t < t_O^*$.

(ii) Accelerates the rate of decrease in adverse selection for $t > t_O^*$.

A greater ambiguity aversion makes an optimist more cautious. This caution has two effects. First, as noted after Definition 2, it increases the transition time $t_O^*$, meaning the initial phase of slow, dampened WTP growth lasts longer. The agent requires more accumulated evidence before their valuation begins to rise rapidly. Second, it governs

the reaction speed within these phases: the agent is less reactive to initial news (dampening) but more reactive to later, confirming news (acceleration).

**Proposition 10.** Under the pessimistic rule, a greater $\alpha$:

(i) Accelerates the rate of increase in adverse selection for $t < t_P^*$.

(ii) Dampens the rate of increase in adverse selection for $t > t_P^*$.

Under the pessimistic rule, greater ambiguity aversion makes buyers more sensitive to initial negative signals and more entrenched in their pessimistic outlook. This causes the transition time $t_P^*$ to decrease, meaning the market enters the later phase of dampened decline sooner. Before this earlier transition, their deep-seated pessimism causes them to react strongly to initial bad news, accelerating the WTP decline. After this point, with their valuation already low, their reaction to further news is dampened as there is less room for their WTP to fall.

In essence, greater ambiguity aversion makes agents more entrenched. Early in time, this appears as inertia, slowing their reaction to initial news. Later in time, it appears as an overreaction to news that confirms the established trend, accelerating the market's path toward its final state.

This analysis of a “transition time” separating distinct dynamic phases is specific to the optimistic and pessimistic rules. The concept arises from a specific mathematical property where the interaction effect between a baseline ambiguity parameter, like $\alpha$ or $\delta$, and information specificity changes character at an identifiable threshold. For the Generalized Bayesian rule, this property is absent. Its interaction effects are either monotonic or do not yield a simple threshold. Consequently, the dynamic evolution under the GB rule cannot be neatly divided into the same short-run and long-run phases of accelerating or dampening trends.

### 4.3. Limit cases

We examine here two special cases: pure ambiguity loving versus pure ambiguity aversion, and pure ignorance about the state probabilities.

#### 4.3.1. Extreme ambiguity loving vs. extreme ambiguity aversion

When we consider a purely ambiguity-loving agent with $\alpha = 0$ and a purely ambiguity-averse agent with $\alpha = 1$, their transition times become identical. Let this common transition time be $t^*$, corresponding to the threshold $g_0^* = \delta/(1 - \delta)$.

**Corollary 1.** For pure types with a common transition time $t^*$, a higher $\delta$ has the following effects for both the optimistic and the pessimistic rule:

(i) For $t < t^*$, a higher $\delta$ accelerates the rate of change.

(ii) For $t > t^*$, a higher $\delta$ dampens the rate of change.

This reveals a remarkable symmetry. Early in time, ambiguity acts as an amplifier of divergence, making both pure types more extreme in their reactions. Later in time, its role reverses to that of a stabilizer, making both agents less sensitive to further information.

#### 4.3.2. Pure ignorance

A final point of interest is the boundary case of pure ignorance, where the agent's baseline ambiguity level is maximal, i.e., $\delta = 1$. In this scenario, the model's dynamics cease to exist, and the equilibrium becomes static.

When $\delta = 1$, the buyer completely disregards their subjective probability distribution $g$ and, by extension, any new information that arrives. The initial valuation formula from Eq. (1) simplifies to:

$$V = (1 - 1)\mathbb{E}_g[\bar{v}] + 1 \cdot [\alpha v(\theta) + (1 - \alpha)v(\bar{\theta})] = \alpha v(\theta) + (1 - \alpha)v(\bar{\theta})$$

The buyer's willingness-to-pay is determined solely by their ambiguity attitude $\alpha$ and the extreme possible outcomes, $v(\theta)$ and $v(\bar{\theta})$.
Crucially, the updating rules also become static. The updated ambiguity levels, $\delta_t^O$, $\delta_t^P$, and $\delta_t^{GB}$, are always equal to 1, regardless of the history $E_t$.

* Under the optimistic rule, the WTP is constant for all time: $V_t^O = v(\bar{\theta})$.

* Under the pessimistic rule, the WTP is constant for all time: $V_t^P = v(\bar{\theta})$.

* Under the Generalized Bayesian rule, the WTP is constant for all time and equals the initial ambiguity component: $V_t^{GB} = \alpha v(\bar{\theta}) + (1-\alpha)v(\bar{\theta}) = V_{amb}$.

Because the WTP never changes, the market equilibrium is locked in from $t=0$. There is no evolution, and the concepts of transition times and dynamic phases become ill-defined. The case of $\delta = 1$ thus represents a degenerate form of decision-making where the agent is so overwhelmed by ambiguity that they ignore all evidence. This highlights that the dynamic results of the paper are predicated on the condition $\delta < 1$, which allows for at least some weight to be placed on observed information.

# 5. Welfare analysis

This section analyzes the welfare implications of ambiguity in both the static and dynamic market settings. Standard welfare measures are calculated based on the equilibrium outcome ($P^*, \theta^*$) using the true value function $v$, cost function $S$, and density of seller types $f$. Key welfare components include Producer Surplus (PS), Consumer Surplus (CS), Total Surplus (TS), and Deadweight Loss (DWL).

To illustrate these welfare effects and provide analytical tractability for our graphical analysis, we often adopt a specific uniform-linear specification. In this context, we assume seller types are uniformly distributed, $\theta \sim U[0, 1]$, and that the value and cost functions are linear: $v(\theta) = v_0 + v_1\theta$ and $S(\theta) = c_0 + c_1\theta$. The buyer's willingness-to-pay is then given by $V(\alpha, \delta) = v_0 + v_1(1/2 + \delta(1/2 - \alpha))$. The market outcome – be it collapse, a partial equilibrium, or full participation – depends directly on how this valuation compares to the range of seller costs, $[c_0, c_0 + c_1]$.

## 5.1. Static welfare analysis

In a static partial market equilibrium, the buyer's willingness-to-pay $V$ directly determines the market outcome, as the price is $P^* = V$ and the marginal seller is $\theta^* = S^{-1}(V)$. An increase in ambiguity aversion, represented by a higher $\alpha$, reduces the buyer's WTP, as shown in **Proposition 1**. This leads to a lower equilibrium price and reduced seller participation. Consequently, producer surplus decreases as fewer sellers participate and receive a lower price. At the same time, the deadweight loss increases as the gap between the actual and efficient level of trade widens. The effect on consumer surplus remains ambiguous due to the trade-off between the benefit of a lower price and the loss from a smaller, lower-quality pool of sellers. Overall, greater pessimism exacerbates adverse selection and shrinks the market, reducing total welfare. These static trade-offs are illustrated in **Fig. 1**.

The effect of an increase in the baseline ambiguity level, $\delta$, depends on the buyer's disposition, as detailed in **Proposition 2**. If the buyer is relatively pessimistic, such that their ambiguity component is lower than the rational expectation ($V_{amb} < E_g[\bar{v}]$), more ambiguity lowers their WTP. This worsens all market outcomes similarly to an increase in $\alpha$. Conversely, if the buyer is relatively optimistic ($V_{amb} > E_g[\bar{v}]$), more ambiguity boosts their WTP and mitigates adverse selection. **Fig. 2** visually summarizes these comparative statics for both ambiguity aversion and the conditional effect of the ambiguity level.

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Static welfare comparison. The buyer's WTP determines the participation threshold ($\theta^*$) and welfare. The GB valuation is shown for an illustrative case where it is below the Bayesian benchmark ($V_{amb} < E_g[\bar{v}]$).</figcaption></figure>

## 5.2. Dynamic welfare analysis

In the dynamic framework, welfare outcomes diverge significantly depending on the updating rule used by the buyer. The choice of updating rule creates a clear ranking of outcomes. The pessimistic rule consistently yields a lower WTP than the Bayesian benchmark, which is in turn lower than the WTP from the optimistic rule. The Generalized Bayesian rule's valuation is positioned relative to the benchmark based on the buyer's initial attitude. This WTP ranking translates directly to welfare: the pessimistic rule leads to lower participation and lower total surplus, while the optimistic rule fosters higher participation and improves overall welfare. The GB rule's effect is similarly conditional. Over time, these effects compound. As established in our main theorems, the pessimistic rule can drive a “death spiral” of declining WTP, while the optimistic rule can fuel a virtuous cycle toward full efficiency. The GB rule follows a path toward collapse, efficiency, or a stable partial equilibrium, depending on the relationship between the buyer's baseline attitude and the market costs. These divergent trajectories are depicted in **Fig. 3**.

The initial ambiguity parameters also influence the entire trajectory of dynamic welfare. A higher baseline ambiguity aversion $\alpha$ weakly decreases WTP under all three updating rules, uniformly worsening adverse selection. The effect of a greater baseline ambiguity level $\delta$ is more nuanced. For the optimistic rule, it increases WTP, whereas for the pessimistic rule, it decreases WTP. For the Generalized Bayesian rule, a higher $\delta$ is detrimental to welfare if the buyer's initial assessment is a downward deviation but is beneficial if it is an upward deviation.

# 6. Discussion

Adverse selection, the market distortion arising from asymmetric information, has been a central theme in economic theory since the seminal work of Akerlof (1970). While foundational static models laid out the core problem (Rothschild and Stiglitz, 1976; Wilson, 1977), much of the modern literature has focused on dynamic settings. This research explores how outcomes are shaped by evolving information and participation, but often through different lenses. For instance, some models focus on structural frictions; Guerrieri et al. (2010) integrate competitive search to show how bilateral matching can discipline markets, while Guerrieri and Shimer (2014) link these frictions to illiquidity
<figure><img src="image_4.png" /><figcaption>Fig. 2. Comparative statics of ambiguity. (a) Higher baseline ambiguity aversion $\alpha$ shrinks the market for all rules (Propositions 1, 3, 4). (b) Higher ambiguity $\delta$ can expand or shrink the market, illustrating the conditional effect for the GB rule and the static model (Proposition 2, Theorem 2).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Divergent Market Trajectories. O and P rules follow monotonic paths (Lemma 3). The GB rule's trajectory converges asymptotically to the baseline ambiguity attitude (Lemma 4), leading to distinct long-run states (Theorems 3 & 4).</figcaption></figure>

and fire sales. Other models emphasize market structure; Hendel and Lizzeri (1999) show that the interaction with a new goods market endogenizes ownership and ensures the used market never fully collapses. A third strand investigates the subtleties of rational learning and information aggregation in search environments (Lauermann and Wolinsky, 2016) or the conditions for achieving efficiency over time under standard learning (Hendel et al., 2005; Zhang, 2015).

A common thread in this literature is the assumption that agents process information according to standard Bayesian probability theory—they face risk but not ambiguity. This paper contributes by introducing a behavioral friction into this framework. We depart from the standard setup by assuming buyers perceive ambiguity and update their beliefs using the optimistic, pessimistic, and Generalized Bayesian rules for neo-additive capacities. To provide a consolidated overview of our central dynamic findings, Table 1 summarizes the comparative statics for each updating rule.

As the table illustrates, our central findings show that the choice of ambiguity updating heuristic fundamentally alters market outcomes. Theorems 1 and 3 show that the optimistic and pessimistic rules create stark, predetermined paths: pessimistic updating consistently amplifies adverse selection and leads toward market collapse, whereas optimistic updating consistently mitigates it and fosters market persistence. In contrast, the Generalized Bayesian rule's impact is conditional, its trajectory depending on the interplay between the buyer's initial ambiguity attitude and the evolving probabilistic evidence, as stated by Theorems 2 and 4. The table further details how market dynamics are shaped by the model's core parameters. We show that a greater baseline level of ambiguity strengthens the amplifying or mitigating effects of

<table><caption>**Table 1**
Summary of dynamic comparative statics results.</caption><thead><tr><th>Comparative static</th><th>Optimistic (O) rule</th><th>Pessimistic (P) rule</th><th>Generalized Bayesian (GB) rule</th></tr></thead><tbody><tr><td colspan="4"><strong>Effect of an increase in ambiguity aversion (α)</strong></td></tr><tr><td>Willingness-to-pay (V<sub>t</sub>)</td><td>Decreases (Proposition 3)</td><td>Decreases (Proposition 3)</td><td>Decreases (Proposition 4)</td></tr><tr><td>Adverse selection (AS<sub>t</sub>)</td><td>Increases (Proposition 3)</td><td>Increases (Proposition 3)</td><td>Increases (Proposition 4)</td></tr><tr><td>Deviation magnitude (|V<sub>t</sub> - V<sub>t</sub><sup>B</sup>|)</td><td>Decreases (Proposition 3)</td><td>Increases (Proposition 3)</td><td>Conditional (Proposition 4)</td></tr><tr><td colspan="4"><strong>Effect of an increase in ambiguity level (δ)</strong></td></tr><tr><td>Willingness-to-pay (V<sub>t</sub>)</td><td>Increases (Theorem 1)</td><td>Decreases (Theorem 1)</td><td>Conditional (Theorem 2)</td></tr><tr><td>Adverse selection (AS<sub>t</sub>)</td><td>Decreases (Theorem 1)</td><td>Increases (Theorem 1)</td><td>Conditional (Theorem 2)</td></tr><tr><td>Deviation magnitude (|V<sub>t</sub> - V<sub>t</sub><sup>B</sup>|)</td><td>Increases (Theorem 1)</td><td>Increases (Theorem 1)</td><td>Increases (Theorem 2)</td></tr><tr><td colspan="4"><strong>Effect of time (Time ↑, g<sub>0</sub>(E<sub>t</sub>) ↓)</strong></td></tr><tr><td>WTP trajectory (V<sub>t</sub>)</td><td>Non-decreasing (Lemma 3)</td><td>Non-increasing (Lemma 3)</td><td>Converge to V<sub>amb</sub> (Lemma 4)</td></tr><tr><td>AS trajectory</td><td>Non-increasing (Lemma 3)</td><td>Non-decreasing (Lemma 3)</td><td>Conditional (Lemma 4)</td></tr><tr><td>Deviation magnitude (|V<sub>t</sub> - V<sub>t</sub><sup>B</sup>|)</td><td>Increases (Proposition 5)</td><td>Increases (Proposition 5)</td><td>Increases (Proposition 6)</td></tr></tbody></table>

Note: The conditional effects for the GB rule depend on the sign of the deviation $V_{amb} - E_g[\tilde{v}]E_t$.
the updating rules. Similarly, our Propositions 3 and 4 demonstrate that greater ambiguity aversion not only uniformly worsens adverse selection but also systematically alters the magnitude of the deviation from the Bayesian benchmark. Finally, we find that as information accumulates over time, these behavioral effects compound, and that the initial ambiguity parameters play a complex, state-dependent role in accelerating or dampening the market's evolution.

Our results also contribute to a growing literature that integrates ambiguity into canonical adverse selection models. Our work is particularly close to Giraud and Thomas (2017), who study a static principal-agent model using the same neo-additive framework with a continuum of types. They find that ambiguity leads to a contract with no distortion for the most efficient type but pooling for the least efficient types. Our paper can be seen as a dynamic extension of their setting, demonstrating how these initial contract distortions evolve over time as buyers update their ambiguous beliefs, leading to the divergent market trajectories we identify. Our findings also resonate with Koufopoulos and Kozhan (2016), who, in a competitive insurance model with discrete types and maxmin preferences, show that ambiguity can support a full-insurance pooling equilibrium and lead to more generous contracts for low-risk types. While their setting is different, their results and ours both highlight a key theme: ambiguity can fundamentally alter the classic separation result in adverse selection, often by making pooling equilibria more likely.

A crucial distinction in our framework arises from dynamic consistency. Bayesian agents satisfy the law of iterated expectations, but this is a significant challenge for non-expected utility models (Machina, 1989; Epstein and LeBreton, 1993; Sarin and Wakker, 1998; Eichberger et al., 2005). The optimistic and pessimistic updating rules, like many non-Bayesian approaches including the Generalized Bayesian rule, generally violate this law, leading to dynamically inconsistent valuations (cf. Lapied and Toquebeuf, 2010, 2012). This lack of dynamic consistency is a key driver of the distinct market trajectories we observe. The current analysis assumes agents are myopic regarding their future preference shifts. Exploring the behavior of sophisticated agents aware of their inconsistency is a valuable extension, though likely at the cost of tractability. Our approach arguably captures a form of bounded rationality where agents follow specific heuristics without full foresight.

The distinction between myopic and sophisticated agents offers a rich direction for future research. A sophisticated agent, in the sense of Strotz (1955), anticipates their future preference changes and chooses the best plan of action available to them, given their future selves' anticipated behavior. In the context of ambiguity, Eichberger et al. (2018) propose a solution concept of consistent planning where agents take account of possible changes in their ambiguity-driven preferences. They show that in strategic settings like the centipede game, sophisticated and optimistic agents can sustain cooperation, a result that starkly contrasts with standard backward induction logic. This raises a compelling question for our model: could sophistication alter the market collapse trajectory under pessimism? Our finding of a pessimistic death spiral relies on myopic agents repeatedly reacting negatively to information, which lowers prices and confirms their pessimistic outlook. A sophisticated agent, foreseeing this self-defeating spiral, might attempt to pre-commit to a different strategy. Conversely, if pre-commitment is not possible, a sophisticated pessimist might anticipate their future excessive caution and exit the market even earlier. Investigating whether consistent planning can rescue a market from a pessimistic collapse, or whether it simply accelerates the exit of rational agents who foresee the inevitable, remains a key extension.

By integrating this richer set of ambiguity updating rules, this paper offers behavioral mechanisms for market persistence or unraveling, complementing the literature that focuses on other causes. For example, where Guerrieri et al. (2010) explain outcomes through structural search frictions, and where Hendel and Lizzeri (1999) find persistence through the interaction of new and used goods markets, our model

shows that the buyer's psychological attitude toward ambiguity can, on its own, generate similar or even more extreme outcomes. Furthermore, while sophisticated search models like Lauermann and Wolinsky (2016) delve into the nuances of rational information aggregation, our work tackles the complementary question of how *departures* from the standard Bayesian framework affect the more fundamental outcome of market existence and viability.

The assumption of a homogeneous buyer population, while useful for isolating the model's core mechanisms, abstracts from the rich heterogeneity of ambiguity attitudes in the real world. The starkly different market trajectories predicted by the different updating rules gain practical relevance when we consider in which contexts these attitudes are likely to prevail.

Pessimistic ambiguity attitudes are often observed in markets where downside risk is particularly salient. The insurance market is a prime example; individuals and firms purchase insurance precisely to guard against worst-case scenarios, making them naturally sensitive to ambiguity surrounding novel or hard-to-predict risks. Similarly, financial markets during a crisis are often dominated by pessimistic ambiguity, leading to a flight to quality that drains liquidity from risky assets, an outcome consistent with the market collapse trajectory driven by the pessimistic updating rule.

Conversely, optimistic ambiguity attitudes are more likely to characterize markets with high-upside potential and speculative elements. Venture capital and technology startup ecosystems, for instance, operate on an optimistic focus on finding the next transformative company. A pessimistic or even purely Bayesian approach would likely deem most ventures un-investable. This aligns with our model's optimistic trajectory, where a high WTP is maintained, mitigating adverse selection and fostering market participation. Similar dynamics can be seen in speculative markets for novel assets like cryptocurrencies or collectibles.

Beyond these contextual examples, a growing body of field and experimental evidence confirms that ambiguity attitudes are indeed heterogeneous across real-world populations and have tangible economic consequences. Field experiments with investors in both the Netherlands and China, for instance, find a wide spectrum of attitudes, from strong ambiguity aversion to ambiguity seeking (Anantanasuwong et al., 2024; Potamites and Zhang, 2012). This heterogeneity is not random; it correlates with demographic factors, with some studies finding that younger and male investors are less ambiguity averse, and that higher financial literacy is linked to lower *perceived* ambiguity (Anantanasuwong et al., 2024; Balafoutas and Sutter, 2019). This has direct market-level effects: as Bossaerts et al. (2010) show, ambiguity-averse individuals may opt out of holding ambiguous assets entirely, which affects risk-sharing and asset prices for the whole market. This phenomenon is also observed in other domains; studies with farmers in both Ghana and France find that ambiguity aversion is common and directly reduces their willingness to purchase crop insurance (Ali et al., 2021; Bougherara et al., 2017). This empirical evidence provides some external validity for theoretical models, like ours, that explore the consequences of moving beyond a representative agent with a single, known attitude toward ambiguity.

Understanding these dynamics provides a behavioral lens relevant to interventions (Fuchs and Skrzypacz, 2015). Policies focused on providing information might have counterintuitive effects, as an information-based intervention could have vastly different marginal impacts depending on the market's history and the agents' baseline ambiguity. While beyond the scope of this paper, these differing attitudes could also be systematically linked to regional or demographic factors. Recognizing this heterogeneity provides a natural bridge to future work. A model incorporating buyers with different ambiguity attitudes could explore more complex market dynamics, such as market segmentation, the persistence of niche optimistic markets amid general pessimism, and the overall resilience of a market populated by diverse decision-makers.
## 7. Conclusion

This paper investigated how ambiguity preferences and dynamic belief updating affect a market with adverse selection. Using the neo-additive capacity model, we analyzed how buyer behavior under ambiguity drives market outcomes away from the standard Bayesian benchmark when beliefs are revised using the optimistic, pessimistic, and Generalized Bayesian updating rules.

Our findings demonstrate that the way buyers update ambiguous beliefs is a critical determinant of market viability. We show that the pessimistic rule consistently amplifies adverse selection, creating a dynamic that can lead to market unraveling, while the optimistic rule consistently mitigates it, fostering market persistence and efficiency. In contrast, the Generalized Bayesian rule has a conditional impact, with its trajectory toward collapse, efficiency, or a stable partial market depending on the interplay between the buyer's static attitude and the evolving probabilistic evidence. We further detailed how initial ambiguity aversion, the initial level of ambiguity and time interact to shape these dynamics, revealing that the baseline level of ambiguity plays a complex, state-dependent role in accelerating or dampening these trends.

In sum, this paper's central contribution is the demonstration that the choice of updating heuristic is a critical determinant of market viability under ambiguity. We have shown that pessimistic and optimistic updating create predetermined paths toward market collapse and persistence, respectively, while the Generalized Bayesian path is endogenously shaped by the interplay of attitude and evidence.

### Declaration of Generative AI and AI-assisted technologies in the writing process

During the preparation of this work the author used Gemini 2.0 Flash in order to help for generating the .tex of the model setup in a previous version of the paper. After using this tool, the author reviewed and edited the content as needed and takes full responsibility for the content of the published article.

### Funding sources

This work has used resources from University Grenoble-Alpes and Grenoble Applied Economics Laboratory.

## Appendix. Proofs

### A.1. Proofs for Section 2

**Proof of Proposition 1.** We analyze the effect of the ambiguity aversion parameter $\alpha$ on WTP $V$, participation $\theta^*$, and adverse selection $AS_0$. To find the effect on WTP, we differentiate the valuation $V$ from Eq. (1) with respect to $\alpha$:

$$ \frac{\partial V}{\partial \alpha} = \delta(\nu(\theta) - \nu(\bar{\theta})) \le 0 \qquad (18) $$

since $\nu(\theta) < \nu(\bar{\theta})$ and $\delta > 0$. A lower $V$ leads to a non-increasing participation threshold $\theta^*(V)$ and thus a non-decreasing adverse selection measure $AS_0$. $\square$

**Proof of Proposition 2.** We analyze the effect of the ambiguity level $\delta$ by differentiating $V$ from Eq. (1):

$$ \frac{\partial V}{\partial \delta} = V_{amb} - E_g[\bar{v}] \qquad (19) $$

The sign is determined by the sign of $(V_{amb} - E_g[\bar{v}])$. A higher (lower) $V$ leads to a weakly higher (lower) $\theta^*$ and a weakly lower (higher) $AS_0$. $\square$

### A.2. Proofs for Section 3

**Proof of Lemma 1.** Let $\mathbb{E} = E_{g_t}[\bar{v}|E_t]$. Note $\nu(\bar{\theta}) \le \mathbb{E} \le \nu(\bar{\theta})$. $V_t^O$ from Eq. (10) is a convex combination of $\mathbb{E}$ and $\nu(\bar{\theta})$, so $V_t^O \ge \mathbb{E} = V_t^B$. $V_t^P$ from Eq. (11) is a convex combination of $\mathbb{E}$ and $\nu(\bar{\theta})$, so $V_t^P \le \mathbb{E} = V_t^B$. The ranking $V_t^P \le V_t^B \le V_t^O$ follows. $\square$

**Proof of Lemma 2.** The GB valuation $V_t^{GB}$ (Eq. (12)) is a convex combination of $E_{g_t}[\bar{v}|E_t]$ and $V_{amb}$. It must therefore lie weakly between these two values. $\square$

**Proof of Theorem 1.** We first prove statement (i). From Lemma 1, $V_t^P(E_t) \le V_t^B(E_t) \le V_t^O(E_t)$. $\theta^{*,U}(E_t)$, is the supremum of the set of seller types for whom the valuation covers the cost: $\theta^{*,U}(E_t) = \sup\{\theta \in \Theta | V_t^U(E_t) \ge S(\theta)\}$. Since $S'(\theta) > 0$, the mapping from $V_t^U$ to $\theta^{*,U}$ is non-decreasing.

This monotonicity implies that the ordering of valuations directly translates to an ordering of participation thresholds:

$$ \theta^{*,P}(E_t) \le \theta^{*,B}(E_t) \le \theta^{*,O}(E_t) $$

Adverse selection is defined as $AS_t^U(E_t) = E_{f_t}[\bar{v}] - E_{f_t}[\bar{v}]\theta \le \theta^{*,U}(E_t)$]. Because the value function $\nu(\bar{\theta})$ is non-decreasing, the conditional expectation $E_{f_t}[\bar{v}|\theta \le x]$ is a non-decreasing function of its upper bound $x$. Therefore, the inequalities for the participation thresholds yield an inverse relationship for $AS_t^U$.

For the optimistic rule, $\theta^{*,O}(E_t) \ge \theta^{*,B}(E_t)$ implies $E_{f_t}[\bar{v}|\theta \le \theta^{*,O}(E_t)] \ge E_{f_t}[\bar{v}|\theta \le \theta^{*,B}(E_t)]$. This leads to $AS_t^O(E_t) \le AS_t^B(E_t)$, satisfying the definition of mitigation.

For the pessimistic rule, $\theta^{*,P}(E_t) \le \theta^{*,B}(E_t)$ implies $E_{f_t}[\bar{v}|\theta \le \theta^{*,P}(E_t)] \le E_{f_t}[\bar{v}|\theta \le \theta^{*,B}(E_t)]$. This leads to $AS_t^P(E_t) \ge AS_t^B(E_t)$, satisfying the definition of amplification.

To prove statement (ii), we show that $\partial M_t^U / \partial \delta > 0$ for $U \in \{O, P\}$.

For $U = O$, $M_t^O = V_t^O - V_t^B \ge 0$. Its derivative with respect to $\delta$ is:

$$ \frac{\partial M_t^O}{\partial \delta} = \frac{\partial(V_t^O - V_t^B)}{\partial \delta} = \frac{\partial V_t^O}{\partial \delta} $$

We have $\partial V_t^O / \partial \delta = (\nu(\bar{\theta}) - E_{g_t}[\bar{v}|E_t])(\partial \delta_t^O / \partial \delta) \ge 0$ as both terms are non-negative. Therefore, an increase in $\delta$ increases the deviation magnitude, meaning $V_t^O$ increases. A higher WTP leads to a higher participation threshold $\theta_t^{*,O}$, which strengthens the mitigation of adverse selection.

For $U = P$, $M_t^P = V_t^B - V_t^P \ge 0$. Its derivative with respect to $\delta$ is:

$$ \frac{\partial M_t^P}{\partial \delta} = \frac{\partial(V_t^B - V_t^P)}{\partial \delta} = -\frac{\partial V_t^P}{\partial \delta} $$

We have $\partial V_t^P / \partial \delta = (\nu(\bar{\theta}) - E_{g_t}[\bar{v}|E_t])(\partial \delta_t^P / \partial \delta) \le 0$, since the first term is non-positive and the second is non-negative. Therefore, $-\partial V_t^P / \partial \delta \ge 0$, and the $M_t^P$ increases with $\delta$. This means $V_t^P$ decreases, leading to a lower $\theta_t^{*,P}$, which strengthens the amplification of adverse selection.

This completes the proof. $\square$

**Proof of Theorem 2.** The first part of the theorem follows directly from the WTP rankings in Lemma 2. A lower WTP leads to a lower participation threshold and worsens adverse selection, while a higher WTP lessens it.

To prove statement (ii), we show that an increase in $\delta$ increases $M_t^{GB} = |V_t^{GB} - V_t^B|$. The Bayesian WTP, $V_t^B$, is independent of $\delta$. We consider the two cases:

(a) $V_{amb} > E_{g_t}[\bar{v}|E_t]$ implies $M_t^{GB} = V_t^{GB} - V_t^B$. Its derivative is $\partial M_t^{GB} / \partial \delta = \partial V_t^{GB} / \partial \delta$. An increase in $\delta$ places more weight on the higher value, $V_{amb}$, so $\partial V_t^{GB} / \partial \delta \ge 0$. Thus, the magnitude of the positive deviation increases, strengthening the mitigation.

(b) $V_{amb} < E_{g_t}[\bar{v}|E_t]$ implies $M_t^{GB} = V_t^B - V_t^{GB}$. Its derivative is $\partial M_t^{GB} / \partial \delta = -\partial V_t^{GB} / \partial \delta$. An increase in $\delta$ places more weight on the lower value, $V_{amb}$, so $\partial V_t^{GB} / \partial \delta \le 0$. Therefore, $-\partial V_t^{GB} / \partial \delta \ge 0$, and the magnitude of the negative deviation increases, strengthening the amplification.
In both scenarios, a greater $\delta$ increases $M_t^U$. $\square$

**Proof of Proposition 3.** Let $\mathbb{E} = \mathbb{E}_{g_t}[\bar{v}|E_t]$. The effect of $\alpha$ on WTP operates through $\delta_t^U$. For the optimistic rule, an increase in $\alpha$ lowers $\delta_t^O$ ($\partial\delta_t^O/\partial\alpha \le 0$), which means $\partial V_t^O/\partial\alpha = (v(\bar{\theta}) - \mathbb{E})(\partial\delta_t^O/\partial\alpha) \le 0$. For the pessimistic rule, an increase in $\alpha$ raises $\delta_t^P$ ($\partial\delta_t^P/\partial\alpha \ge 0$), which means $\partial V_t^P/\partial\alpha = (v(\theta) - \mathbb{E})(\partial\delta_t^P/\partial\alpha) \le 0$. In both cases, WTP is non-increasing in $\alpha$, implying weakly higher $AS_t^U$, proving the first statement.

Furthermore, these changes in WTP determine the bias magnitudes. $M_t^O$ decreases as its derivative is $\partial V_t^O/\partial\alpha \le 0$. $M_t^P$ increases as its derivative is $-\partial V_t^P/\partial\alpha \ge 0$. $\square$

**Proof of Proposition 4.** Under the GB rule, $\delta_t^{GB}$ and $\mathbb{E}_{g_t}[\bar{v}|E_t]$ are independent of $\alpha$. Differentiating $V_t^{GB}$ gives: $\partial V_t^{GB}/\partial\alpha = \delta_t^{GB}(v(\theta) - v(\bar{\theta})) \le 0$. Since WTP is non-increasing in $\alpha$, $AS_t^{GB}$ is non-decreasing, proving the first statement.

To prove the second statement, we analyze the effect of $\alpha$ on the deviation magnitude, $|V_t^{GB} - V_t^B|$.

(i) **Downward Deviation ($V_t^{GB} < V_t^B$):** The magnitude is $M_{down} = V_t^B - V_t^{GB}$. Its derivative is $\partial M_{down}/\partial\alpha = -\partial V_t^{GB}/\partial\alpha \ge 0$. The magnitude increases.

(ii) **Upward Deviation ($V_t^{GB} > V_t^B$):** The magnitude is $M_{up} = V_t^{GB} - V_t^B$. Its derivative is $\partial M_{up}/\partial\alpha = \partial V_t^{GB}/\partial\alpha \le 0$. The magnitude decreases.

This completes the proof. $\square$

### A.3. Proofs for Section 4

**Proof of Lemma 3.** We analyze the sensitivity of $V_t^U$ to $g_0 \equiv g_0(E_t)$, noting that $g_0$ is non-increasing in time. Let $\mathbb{E} = \mathbb{E}_{g_t}[\bar{v}|E_t]$. For both rules, a higher $g_0$ leads to a weakly lower updated ambiguity level $\delta_t^U$ ($\partial\delta_t^U/\partial g_0 \le 0$). We use the chain rule: $\partial V_t^U/\partial g_0 = (\partial V_t^U/\partial\delta_t^U)(\partial\delta_t^U/\partial g_0)$.

(i) For the pessimistic rule, $\partial V_t^P/\partial\delta_t^P = v(\bar{\theta}) - \mathbb{E} \le 0$. Thus, $\partial V_t^P/\partial g_0 = (\le 0) \cdot (\le 0) \ge 0$. Since $g_0$ is non-increasing over time, the sequence $\{V_t^P\}_{t \ge 0}$ is non-increasing. Lower WTP implies weakly higher $AS_t^P$.

(ii) For the optimistic rule, $\partial V_t^O/\partial\delta_t^O = v(\bar{\theta}) - \mathbb{E} \ge 0$. Thus, $\partial V_t^O/\partial g_0 = (\ge 0) \cdot (\le 0) \le 0$. Since $g_0$ is non-increasing over time, $\{V_t^O\}_{t \ge 0}$ is non-decreasing. Higher WTP implies weakly lower $AS_t^O$. $\square$

**Proof of Proposition 5.** The magnitude of the bias for rule U is $M_t^U(g_0)$. We analyze its derivative with respect to $g_0$.

(i) $M_t^O(g_0) = V_t^O - V_t^B$. From the proof of Lemma 3, $\partial V_t^O/\partial g_0 \le 0$. Thus $\partial M_t^O/\partial g_0 \le 0$. Since $g_0$ is non-increasing in time, $M_t^O$ is non-decreasing.

(ii) $M_t^P(g_0) = V_t^B - V_t^P$. From the proof of Lemma 3, $\partial V_t^P/\partial g_0 \ge 0$. Thus $\partial M_t^P/\partial g_0 = -\partial V_t^P/\partial g_0 \le 0$. Since $g_0$ is non-increasing in time, $M_t^P$ is non-decreasing. $\square$

**Proof of Proposition 6.** We show that $\partial M_t^{GB}/\partial g_0(E_t) \le 0$, where $M_t^{GB} = |V_t^{GB} - V_t^B|$. Recall that $V_t^{GB}$ is a convex combination of $V_t^B$ and $V_{amb}$ weighted by $\delta_t^{GB}$. The derivative of the valuation with respect to the prior probability is:

$$ \frac{\partial V_t^{GB}}{\partial g_0} = \frac{\partial V_t^{GB} \partial \delta_t^{GB}}{\partial \delta_t^{GB} \partial g_0} = (V_{amb} - \mathbb{E}_{g_t}[\bar{v}|E_t]) \cdot \frac{\partial \delta_t^{GB}}{\partial g_0} $$

From Eq. (8), $\delta_t^{GB}$ is decreasing in $g_0$, so $\partial\delta_t^{GB}/\partial g_0 \le 0$.

(i) If $V_{amb} > \mathbb{E}_{g_t}[\bar{v}|E_t]$, then $M_t^{GB} = V_t^{GB} - V_t^B$. The derivative $\partial V_t^{GB}/\partial g_0$ is the product of a positive term ($V_{amb}-\mathbb{E}$) and a non-positive term, so it is $\le 0$. Thus, $\partial M_t^{GB}/\partial g_0 \le 0$.

(ii) If $V_{amb} < \mathbb{E}_{g_t}[\bar{v}|E_t]$, then $M_t^{GB} = V_t^B - V_t^{GB}$. The derivative $\partial V_t^{GB}/\partial g_0$ is the product of a negative term and a non-positive term, so

it is $\ge 0$. Consequently, the derivative of the magnitude is $\partial M_t^{GB}/\partial g_0 = -\partial V_t^{GB}/\partial g_0 \le 0$.

In both cases, the deviation magnitude decreases as $g_0$ increases. Since $g_0(E_t)$ is non-increasing over time $t$, the magnitude $M_t^{GB}$ is non-decreasing in $t$. $\square$

### Proof of Theorem 3.

(i) From Lemma 3, the sequence $\{V_t^P\}_{t \ge 0}$ is non-increasing and bounded below by $v(\theta)$. By the Monotone Convergence Theorem, it converges to a limit $V_\infty^P$. If $V_\infty^P < S(\theta)$, then for some $T$, for all $t \ge T$, $Q_t^* = 0$.

(ii) From Lemma 3, the sequence $\{V_t^O\}_{t \ge 0}$ is non-decreasing and bounded above by $v(\bar{\theta})$. It converges to a limit $V_\infty^O$. If $V_\infty^O \ge S(\bar{\theta})$, then for some $T$, for all $t \ge T$, $Q_t^* = 1$. $\square$

**Proof of Lemma 4.** Recall that the updated ambiguity level for the Generalized Bayesian rule is given by Eq. (8). Since $\Theta$ is continuous, $\lim_{t \to \infty} g_0(E_t) = 0$. Taking the limit of $\delta_t^{GB}$:

$$ \lim_{t \to \infty} \delta_t^{GB} = \frac{\delta}{(1 - \delta) \cdot 0 + \delta} = 1 $$

The valuation $V_t^{GB}$ is defined as $V_t^{GB} = (1 - \delta_t^{GB})\mathbb{E}_{g_t}[\bar{v}|E_t] + \delta_t^{GB}V_{amb}$. Since the conditional expectation $\mathbb{E}_{g_t}[\bar{v}|E_t]$ is bounded for all $t$, and $(1 - \delta_t^{GB}) \to 0$, the weight on the expectation vanishes. Thus, $\lim_{t \to \infty} V_t^{GB} = V_{amb}$. $\square$

**Proof of Theorem 4.** From Lemma 4, we know that for sufficiently large $t$, the valuation $V_t^{GB}$ approaches $V_{amb}$. The market outcome is determined by the participation condition $V_t^{GB} \ge S(\theta)$.

(i) If $V_{amb} < S(\theta)$, the limiting valuation is below the minimum cost. Thus, $\theta_t^* \to \theta$ and $Q_t^* \to 0$.

(ii) If $V_{amb} \ge S(\bar{\theta})$, the limiting valuation exceeds the maximum cost. Thus, $\theta_t^* \to \bar{\theta}$ and $Q_t^* \to 1$.

(iii) If $S(\bar{\theta}) \le V_{amb} < S(\bar{\theta})$, the valuation converges to a value that strictly intersects the cost curve. Since $S(\bar{\theta})$ is continuous and strictly increasing, the participation threshold converges to a unique interior point $\theta_\infty^* = S^{-1}(V_{amb})$, resulting in a stable partial market. $\square$

The following two lemmas will be used in order to prove the remaining propositions.

**Lemma 5.** Let $AS_t$ be the measure of adverse selection and $V_t$ be WTP at time $t$.

(i) The rate of change of adverse selection, $d(AS_t)/dt$, has the opposite sign of the rate of change of WTP, $d(V_t)/dt$.

(ii) The speed of change of adverse selection, $|d(AS_t)/dt|$, is proportional to the speed of change of WTP, $|d(V_t)/dt|$.

**Proof.** By the chain rule, $d(AS_t)/dt = (d(AS_t)/dV_t) \cdot (dV_t/dt)$. Since $AS_t$ is a non-increasing function of $V_t$, we have $d(AS_t)/dV_t \le 0$. Part (i) follows directly. Part (ii) follows because the factor of proportionality is $|d(AS_t)/dV_t| \ge 0$. $\square$

**Lemma 6.** The second-order cross-partial derivatives of $\delta_t^O$ and $\delta_t^P$ are given by:

$$ \frac{\partial^2 \delta_t^O}{\partial g_0 \partial \delta} = \frac{(1 - \alpha)[\delta(1 - \alpha) - g_0(1 - \delta)]}{((1 - \delta)g_0 + \delta(1 - \alpha))^3} \quad (20a) $$

$$ \frac{\partial^2 \delta_t^P}{\partial g_0 \partial \delta} = \frac{\alpha[\delta\alpha - g_0(1 - \delta)]}{((1 - \delta)g_0 + \delta\alpha)^3} \quad (20b) $$

$$ \frac{\partial^2 \delta_t^O}{\partial \alpha \partial g_0} = \frac{\delta(1 - \delta)^2(g_0 - g_O^*)}{((1 - \delta)g_0 + \delta(1 - \alpha))^3} \quad (20c) $$

$$ \frac{\partial^2 \delta_t^P}{\partial \alpha \partial g_0} = \frac{\delta(1 - \delta)[\delta\alpha - g_0(1 - \delta)]}{((1 - \delta)g_0 + \delta\alpha)^3} \quad (20d) $$
**Proof.** The results follow from direct differentiation. For (20a), we first find $\partial\delta_t^O/\partial\delta = (1-\alpha)g_0/D_O^2$, where $D_O = ((1-\delta)g_0 + \delta(1-\alpha))$. Differentiating with respect to $g_0$ yields the result. The other derivations are analogous. ☐

**Proof of Proposition 7.** By Lemma 5, the rate of decrease in $AS_t^O$ is proportional to the rate of increase of $V_t^O$. We analyze the sign of $\partial/\partial\delta(dV_t^O/dt)$. Since $dg_0/dt < 0$, the sign of this term is the opposite of the sign of $\partial^2V_t^O/(\partial\delta\partial g_0)$, which has the same sign as $\partial^2\delta_t^O/(\partial g_0\partial\delta)$ from Lemma 6. This sign is determined by $(g_O^* - g_0)$.

(i) For $t < t_O^* \Rightarrow g_0 > g_O^*$, we have $(g_O^* - g_0) < 0$. By Lemma 6, $\partial^2V_t^O/(\partial\delta\partial g_0) \le 0$. The effect on the rate is non-negative, accelerating the decrease in adverse selection.

(ii) For $t > t_O^* \Rightarrow g_0 < g_O^*$, we have $(g_O^* - g_0) > 0$. By Lemma 6, $\partial^2V_t^O/(\partial\delta\partial g_0) \ge 0$. The effect on the rate is non-positive, dampening the decrease in adverse selection. ☐

**Proof of Proposition 8.** By Lemma 5, the rate of increase of $AS_t^P$ is proportional to the speed of decrease of $V_t^P$, $|d(V_t^P)/dt|$. We analyze the sign of $\partial/\partial\delta|dV_t^P/dt|$. Since $dg_0/dt < 0$, this has the same sign as $\partial^2V_t^P/(\partial\delta\partial g_0)$, which has the opposite sign of $\partial^2\delta_t^P/(\partial g_0\partial\delta)$ from Lemma 6. This sign is determined by $(g_P^* - g_0)$.

(i) For $t < t_P^* \Rightarrow g_0 > g_P^*$, we have $(g_P^* - g_0) < 0$. By Lemma 6, $\partial^2\delta_t^P/(\partial\delta_0\partial\delta) \le 0$, so $\partial^2V_t^P/(\partial\delta\partial g_0) \ge 0$. The effect on the speed is non-negative, accelerating the increase in adverse selection.

(ii) For $t > t_P^* \Rightarrow g_0 < g_P^*$, we have $(g_P^* - g_0) > 0$. By Lemma 6, $\partial^2\delta_t^P/(\partial\delta_0\partial\delta) \ge 0$, so $\partial^2V_t^P/(\partial\delta\partial g_0) \le 0$. The effect on the speed is non-positive, dampening the increase in adverse selection. ☐

**Proof of Proposition 9.** Following the logic for Proposition 7, we analyze the sign of $\partial/\partial\alpha(dV_t^O/dt)$. This has the opposite sign of $\partial^2V_t^O/(\partial\alpha\partial g_0)$, which has the same sign as $\partial^2\delta_t^O/(\partial\alpha\partial g_0)$ from Lemma 6. This is determined by $(g_0 - g_O^*)$.

(i) For $t < t_O^* \Rightarrow g_0 > g_O^*$, we have $(g_0 - g_O^*) > 0$. By Lemma 6, $\partial^2V_t^O/(\partial\alpha\partial g_0) \ge 0$. The effect on the rate is non-positive, dampening the decrease in adverse selection.

(ii) For $t > t_O^* \Rightarrow g_0 < g_O^*$, we have $(g_0 - g_O^*) < 0$. By Lemma 6, $\partial^2V_t^O/(\partial\alpha\partial g_0) \le 0$. The effect on the rate is non-negative, accelerating the decrease in adverse selection. ☐

**Proof of Proposition 10.** Following the logic for Proposition 8, we analyze the sign of $\partial/\partial\alpha|dV_t^P/dt|$. This has the same sign as $\partial^2V_t^P/(\partial\alpha\partial g_0)$, which has the opposite sign of $\partial^2\delta_t^P/(\partial\alpha\partial g_0)$ from Lemma 6. This is determined by $(g_P^* - g_0)$.

(i) For $t < t_P^* \Rightarrow g_0 > g_P^*$, we have $(g_P^* - g_0) < 0$. By Lemma 6, $\partial^2\delta_t^P/(\partial\alpha\partial g_0) \le 0$, so $\partial^2V_t^P/(\partial\alpha\partial g_0) \ge 0$. The effect on the speed is non-negative, accelerating the increase in adverse selection.

(ii) For $t > t_P^* \Rightarrow g_0 < g_P^*$, we have $(g_P^* - g_0) > 0$. By Lemma 6, $\partial^2\delta_t^P/(\partial\alpha\partial g_0) \ge 0$, so $\partial^2V_t^P/(\partial\alpha\partial g_0) \le 0$. The effect on the speed is non-positive, dampening the increase in adverse selection. ☐

## Data availability

No data was used for the research described in the article.

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