Trust, Reputation, and the Market for Lemons: A Free-Energy Model of Endogenous Belief Refinement

Abstract

This paper examines how internalized honesty and reputation can mitigate adverse selection when buyers lack an exogenous quality index. A privately informed seller probabilistically chooses whether to offer or withhold a product using a free-energy policy rule. Internalized norms enter as a prior policy bias, whereas reputation changes anticipated consequences. A buyer observing an offer updates beliefs about quality through Bayes’ rule and decides whether to purchase at an exogenous posted price. Seller payoffs specify reservation values, offering costs, and expected reputational losses. Seller policy incorporates the anticipated probability of purchase, and a behavioral consistency condition requires this anticipation to agree with the buyer’s optimal response. On the full-purchase branch, stronger honesty and reputation discourage lemon offers and increase the buyer’s posterior probability of high quality. I derive a threshold under which purchase after an offer is strategically consistent and high-quality trade occurs with positive probability. This is a result about posted-price trade rather than a complete Akerlof-type unraveling equilibrium. The framework shows how socially embedded behavior can generate informative seller actions without a physical quality index.

1. Introduction

This paper studies whether social trust, internalized norms, and reputation can mitigate adverse selection and support trade at a fixed posted price when buyers lack an index in the sense introduced by Spence (1973)—an observable and unalterable characteristic that cannot be strategically chosen by the seller but may convey information about hidden product quality. The central research question is whether a buyer can rationally refine a belief about hidden quality from the seller’s decision to enter the market itself. I develop a sequential Bayesian signaling environment with a behavioral consistency condition: the seller privately observes product quality and chooses whether to offer the product while anticipating the buyer’s response. The seller’s action is generated probabilistically by a free-energy policy rule, which assigns greater choice probability to courses of action with lower expected free energy while allowing prior dispositions toward particular actions to influence choice. An internalized disposition against exploiting another person shifts the seller’s prior tendency toward particular actions, while reputation changes the anticipated cost of opportunistic behavior. Because these forces affect low-quality sellers more strongly than high-quality sellers, an observed offer carries information about quality. The buyer then performs ordinary Bayesian updating and chooses whether to purchase; a consistency condition requires the anticipated and actual continuation responses to coincide. I characterize a full-purchase branch on which sufficiently strong social or reputational discipline raises the buyer’s posterior probability of high quality above the purchase threshold, thereby supporting positive high-quality trade at the posted price.

The starting point is the adverse-selection mechanism in Akerlof (1970). When the seller knows product quality but the buyer cannot distinguish high quality from low quality, the price buyers are willing to pay can become too low for high-quality sellers, causing good products to leave the market. Signaling and screening mechanisms provide important responses to this information problem. In particular, Spence (1973) distinguished manipulable signals from observable and unalterable indices.

To give the seller’s choices explicit economic foundations, I specify quality-dependent reservation values, costs of offering, and an expected reputational loss from knowingly selling a lemon. I map these consequences into the seller’s preferred-outcome distribution, so that expected-free-energy differences are derived rather than imposed.

Mamada (2022) develops an information-theoretic extension of the lemons model using such an index. In that model, a buyer observes a costly index correlated with automobile quality and refines the conditional probability that a car is good. Mutual information measures the reduction in uncertainty obtained from inspecting the index. If the information cost lies in an appropriate range, the market need not collapse. The contribution of the present paper is deliberately different. Here there need not be an exogenous physical characteristic, test result, or other index that the buyer can inspect. Instead, the seller’s own market-entry decision becomes an endogenous source of information.

This distinction is relevant when physical indices are expensive, weakly informative, or unavailable. Consider a transaction between long-time acquaintances. If an owner knows that a car is defective, an internalized norm against taking advantage of a friend may make the owner reluctant to offer the car to that person. The buyer, knowing something about the social relationship, can rationally interpret an offer as evidence about quality. A similar mechanism can operate in anonymous markets through reputation. A seller with valuable public feedback has more to lose from opportunistic behavior. Public ratings and feedback therefore affect not only buyers’ direct perceptions of sellers but also sellers’ incentives to place low-quality products on the market.

The paper uses the free-energy framework to formalize the seller’s stochastic policy choice. In active-inference models, policies can be selected according to expected free energy together with prior or habitual policy biases (Friston et al., 2016). Mamada (2026) applies this structure to social decision making and interprets an additive policy term as a cultural or habitual baseline bias. The present paper gives that term a narrower economic interpretation: an internalized norm of honest exchange changes the prior propensity of a seller to withhold a known lemon. This is distinct from the buyer’s Bayesian prior over product quality. The norm affects behavior first; the behavior then generates information; the buyer’s quality belief changes only through Bayes’ rule.

The analysis yields three principal results for the candidate full-purchase continuation. First, stronger internalized honesty and stronger reputational stakes reduce the probability that a lemon owner offers the product when the buyer’s response is held fixed. Second, because the offer probability is type dependent, the buyer’s posterior belief that an offered product is high quality rises monotonically with both forces on that branch. Third, an explicit threshold in the combined strength of the internalized norm and reputation characterizes when full purchase after an offer is consistent with the seller’s anticipated continuation response. The mechanism may be summarized as

$$\begin{array}{ccc}& & \mathrm{social}\mathrm{or}\mathrm{reputational}\mathrm{discipline}⟶\mathrm{type-dependent}\mathrm{seller}\mathrm{behavior}⟶\hfill \\ & & \mathrm{Bayesian}\mathrm{belief}\mathrm{refinement}⟶\mathrm{positive}\mathrm{trade}\mathrm{on}\mathrm{the}\mathrm{full-purchase}\mathrm{branch}.\hfill \end{array}$$

The remainder of the paper is organized as follows. Section 2 reviews the literature most directly related to social embeddedness, internalized norms, and reputation. Section 3 introduces the model. Section 4 derives comparative statics and a threshold supporting a consistent full-purchase continuation at the posted price. Section 5 discusses friendship and public reputation as applications. Section 6 relates the mechanism to information theory and to the earlier index model. Section 7 discusses the broader interpretation and implications of the results. Section 8 concludes.

2. Related Literature and Conceptual Framework

2.1. Social Relations and Internalized Norms

The idea that economic behavior is embedded in social relationships has a long history. Granovetter (1985) argues that economic action cannot be understood adequately by treating actors as socially isolated; ongoing social relations affect both behavior and institutions. In game theory, Kandori (1992) shows how community enforcement can sustain cooperation when information about conduct circulates through a social group and when reputation-like labels can be revised over time. These contributions emphasize external strategic consequences of social relations.

The present paper also allows a second channel: social rules may be internalized and thereby influence behavior even before an explicit external sanction is calculated. This idea has a long foundation in sociology and psychology. Berger (1963) emphasizes that socialization involves the internalization of the social world, so that social norms become incorporated into individuals’ identities, thoughts, and emotions rather than remaining purely external constraints. From a psychological perspective, Huffman (2004) discusses how ethical standards may operate partly through preconscious or unconscious processes, with conscience and anticipated guilt constraining behavior even in the absence of an immediately calculated external sanction. Consistent with these perspectives, Gintis (2014) gives socialization and the internalization of norms an explicit role in a broader behavioral account of strategic choice. Cross-cultural experimental evidence is also consistent with the proposition that economic behavior is shaped by socially learned expectations: Henrich et al. (2001) report substantial behavioral variation across small-scale societies in ultimatum, public-good, and dictator games, with preferences and expectations appearing to reflect group-specific institutions and cultural norms. Swidler (1986) similarly conceptualizes culture as a repertoire of habits, skills, and styles used to construct strategies of action.

The formal use of an internalized normative parameter also builds on Mamada (2024), which models the influence of societal norms on cooperation in a Prisoner’s Dilemma representation of the commons. That paper places norm-relevant dispositions directly into strategic choice. The present model differs in its economic mechanism: the norm does not directly make the buyer more trusting and does not directly change the true quality distribution. Instead, it changes the seller’s quality-contingent policy, which creates an informative likelihood for the buyer.

Observability and repeated social interaction can reinforce the same mechanism. For example, Mas and Moretti (2009) find that worker effort responds to visible coworkers and to coworkers with whom interaction is frequent, a pattern they interpret as evidence that social pressure can internalize free-riding externalities. In the current setting, a close social relationship can therefore affect seller behavior through both an internalized disposition and anticipated social consequences. The formal model separates these two channels.

2.2. Reputation and Market Discipline

The reputation literature provides a complementary economic mechanism. Klein and Leffler (1981) show how future rents can make contractual performance incentive compatible, while Shapiro (1983) models quality premiums as returns to reputation when buyers cannot observe quality before purchase. Reputation therefore supplies an intertemporal cost of opportunism.

Online marketplaces make such reputational information unusually visible. Resnick et al. (2006) conduct a controlled field experiment on eBay and find that an established high-reputation identity generated greater willingness to pay than newly created identities selling matched items. Cabral and Hortaçsu (2010) document substantial changes in sales and exit patterns following negative seller feedback. These studies do not imply that reputation is novel as a solution to adverse selection. Rather, they motivate the present paper’s specific question: how can internalized norms and reputation be represented within a common probabilistic policy model, and how does the resulting seller behavior become an endogenous information channel for the buyer?

2.3. Free Energy, Policy Priors, and Bayesian Belief Refinement

Expected free energy provides a way to evaluate alternative courses of action when their consequences are uncertain. In the present model, a policy a simply means a course of action available to the seller, such as offering the product for sale or withholding it from the market. For each privately known quality type q, the seller evaluates the policies available to them and attempts to minimize expected free energy. Thus, a policy with a lower value of $G_q\left(a\right)$ is more attractive to the seller. Because policy selection will subsequently be modeled probabilistically, the seller need not choose the policy with the lowest expected free energy with certainty; rather, lower expected free energy makes a policy more likely to be selected. A general active-inference policy score can be written as

$$G_q\left(a\right)=D_{\mathrm{KL}}\left(Q_q(o\mid a)\parallel P_q^{\mathrm{pref}}\left(o\right)\right)-\beta {\mathcal{I}}_q\left(a\right),$$

(1)

where q denotes the seller’s privately known quality type, a denotes a policy, $Q_q(o\mid a)$ is the seller’s predictive probability distribution over possible outcomes o if policy a is chosen, and $P_q^{\mathrm{pref}}\left(o\right)$ is a probability distribution representing how desirable those outcomes are to the seller (Friston et al., 2016).

The first term in Equation (1), $D_{\mathrm{KL}}\left(Q_q(o\mid a)\parallel P_q^{\mathrm{pref}}\left(o\right)\right)$, is the Kullback–Leibler (KL) divergence between the outcomes the seller expects under policy a and the outcomes the seller prefers. For discrete outcomes it is defined as

$$D_{\mathrm{KL}}\left(Q_q(o\mid a)\parallel P_q^{\mathrm{pref}}\left(o\right)\right)=\sum _o{Q}_q(o\mid a){\log }_2{\displaystyle \frac{Q_q(o\mid a)}{P_q^{\mathrm{pref}}\left(o\right)}}.$$

(2)

Throughout this paper, logarithms used in information-theoretic quantities are taken to base 2. Accordingly, KL divergence, expected information gain, and mutual information are measured in bits.

The KL divergence is nonnegative and becomes zero when the predicted and preferred distributions coincide. Accordingly, a smaller KL divergence means that the outcomes anticipated from a policy are more closely aligned with the seller’s preferred outcomes. The second term, ${\mathcal{I}}_q\left(a\right)$, is the expected information gain associated with policy a. It measures how much the seller expects to learn by choosing that policy and observing its consequences. More formally, if x denotes a latent state about which the seller is uncertain, the expected information gain can be written as

$${\mathcal{I}}_q\left(a\right)=\sum _o{Q}_q(o\mid a)D_{\mathrm{KL}}\left(Q_q(x\mid o,a)\parallel Q_q(x\mid a)\right).$$

(3)

This expression compares the seller’s belief about the latent state before and after observing an outcome and then averages that change over possible outcomes. A larger value therefore means that the policy is expected to resolve more uncertainty. Because ${\mathcal{I}}_q\left(a\right)$ enters Equation (1) with a negative sign, greater expected information gain lowers expected free energy and, other things equal, makes an informative policy more attractive. This is the epistemic component of expected free energy, whereas the KL term captures how closely anticipated outcomes conform to preferred outcomes (Friston et al., 2016). In the one-step seller decision studied in this paper, however, the seller already knows whether the product is of high or low quality, and no additional information about product quality needs to be acquired. Consequently, the epistemic term does not drive the comparative statics developed below. The central role of $G_q\left(a\right)$ in this model is therefore to evaluate the seller’s possible actions after material, social, and reputational consequences have been incorporated into the preferred-outcome structure.

Following the policy form used by Mamada (2026), the conditional probability that a seller of quality type q selects policy a is

$$P(a\mid q)={\displaystyle \frac{\exp \{E_q\left(a\right)-\gamma G_q\left(a\right)\}}{{\sum }_{a^{\prime }}\exp \{E_q\left(a^{\prime }\right)-\gamma G_q\left(a^{\prime }\right)\}}},\gamma >0,$$

(4)

where $E_q\left(a\right)$ is an additive log-policy bias and $\gamma$ is the precision parameter governing how strongly differences in expected free energy affect choice. Equation (4) gives a stochastic implementation of free-energy minimization: holding the policy bias fixed, a lower value of $G_q\left(a\right)$ increases the probability that policy a is selected. The parameter $\gamma$ determines how strongly the seller responds to differences in expected free energy. When $\gamma$ is larger, policies with lower expected free energy are favored more strongly; when $\gamma$ is smaller, policy selection is more stochastic. Holding expected free energy fixed, a larger value of $E_q\left(a\right)$ represents a stronger baseline disposition toward that policy.

A key conceptual distinction is important. The term $E_q\left(a\right)$ is not the buyer’s prior probability that the product is of high quality. Nor is $E_q\left(a\right)$ itself a belief about the hidden quality state. Instead, it represents a prior tendency of a seller of type q to choose one policy rather than another. A change in $E_q\left(a\right)$ therefore changes the seller’s behavior, not the buyer’s belief directly. The buyer observes the seller’s action and only then updates the belief about the unobserved quality type using Bayes’ rule. Thus, policy priors affect the buyer’s posterior belief indirectly by changing the type-dependent likelihood of observing a particular seller action. This separation between the seller’s prior tendency over policies and the buyer’s belief over quality states is central to the model.

3. Model

3.1. Players, Quality, and Timing

There are two players, a seller S and a buyer B. Nature chooses product quality

$$q\in \{G,L\},$$

where G denotes high quality and L denotes a lemon. The common prior is

$$P(q=G)=\pi ,P(q=L)=1-\pi ,0<\pi <1.$$

(5)

The seller observes q, while the buyer does not.

The seller chooses

$$a\in \{O,W\},$$

where O means offer the product to the buyer and W means withhold it. If the seller chooses W, the game ends. If the seller chooses O, the buyer observes the offer but not q and chooses whether to buy the product at an exogenously given posted price P. Before choosing a policy, the seller anticipates a conditional purchase probability $b\in [0,1]$. The variable b is the buyer’s probability of purchasing after an offer, not the unconditional probability of a sale. The anticipated value of b enters the seller’s evaluation of the consequences of offering; Section 3.5 requires that this anticipation agree with the buyer’s optimal continuation response.

The buyer values the two qualities at $v_G$ and $v_L$, where

$$v_G>P>v_L.$$

(6)

Thus, a known high-quality product is worth purchasing at P, whereas a known lemon is not. The buyer is risk neutral in the baseline model.

Let

$$\mu =P(G\mid O)$$

denote the buyer’s posterior probability that the product is of high quality after observing an offer. Given this posterior belief, the buyer’s reservation price is

$$P_B^R\left(\mu \right)=\mu v_G+(1-\mu )v_L.$$

(7)

Because the buyer is risk neutral, this reservation price is the expected value of the product conditional on observing an offer. Purchasing is optimal whenever

$$P\le P_B^R\left(\mu \right).$$

(8)

It is strictly optimal when the inequality is strict, whereas the buyer may mix between purchasing and declining when equality holds.

All parameters governing seller behavior, including the social relationship and publicly visible reputation, are common knowledge. The buyer therefore knows the likelihood with which each seller type offers the product even though the buyer does not observe the type itself.

Seller Payoffs and Reservation Values

Let $s_q\ge 0$ be the type-dependent reservation value of retaining the product, and let $c_q\ge 0$ be the cost of making an offer, paid even when the buyer declines. For a knowingly low-quality sale, let $\lambda R$ denote an expected reputational loss, where $\lambda >0$ and $R\ge 0$. This loss can summarize the present value of forfeited future business or social sanctions and is incurred only when a lemon is sold. If an offer is rejected, the seller retains the product. The resulting payoff structure is shown in Table 1.

Table 1. Seller payoffs conditional on quality and the buyer’s response.

Outcome	High Quality (G)	Lemon (L)
Offer; sale (B)	$P-c_G$	$P-c_L-\lambda R$
Offer; rejection (N)	$s_G-c_G$	$s_L-c_L$
Withhold (W)	$s_G$	$s_L$

These are seller payoffs, not buyer valuations: $s_q$ need not equal $v_q$. In particular, $s_G>s_L$ may capture the higher reservation value conventionally associated with good-quality products. At the posted price, the seller’s expected monetary and reputational payoff from offering, given an anticipated purchase probability b, is

$$\begin{array}{cc}\hfill {\overline{U}}_G(O;b)& =b(P-c_G)+(1-b)(s_G-c_G),\hfill \\ \hfill {\overline{U}}_L(O;b,R)& =b(P-c_L-\lambda R)+(1-b)(s_L-c_L),\hfill \end{array}$$

(9)

whereas withholding gives $U_q\left(W\right)=s_q$. Hence the payoff gains from offering are

$$\begin{array}{cc}\hfill \Delta U_G\left(b\right)& =b(P-s_G)-c_G,\hfill \\ \hfill \Delta U_L(b;R)& =b(P-s_L-\lambda R)-c_L.\hfill \end{array}$$

(10)

This makes the seller’s response to anticipated buyer behavior explicit: no sale means no price revenue and no reputational loss, but an offer still incurs its cost.

3.2. Seller Policy Under Internalized Norms and Reputation

To relate the economic payoffs in Table 1 to Equation (1), use the three mutually exclusive outcomes $o\in \{B,N,W\}$: a completed sale, an offer rejected by the buyer, and withholding. Write $U_q(o;R)$ for the payoff of each outcome in the table and impose the explicit exponential preference encoding

$$P_q^{\mathrm{pref}}(o;R)={\displaystyle \frac{\exp \{U_q(o;R)/\tau \}}{{\displaystyle \sum _{o^{\prime }\in \{B,N,W\}}\exp \{U_q(o^{\prime };R)/\tau \}}}},\tau >0.$$

(11)

The parameter $\tau$ converts monetary payoffs into dimensionless preference weights; this exponential mapping is an explicit behavioral assumption, not an identity implied by Bayes’ rule or the original free-energy principle. In this one-shot specification, there is no decision-relevant latent state to learn, so I set ${\mathcal{I}}_q\left(O\right)={\mathcal{I}}_q\left(W\right)=0$. Using the payoff table, the predictive outcome distributions are

$$\begin{array}{cc}\hfill Q_q(B\mid O;b)& =b,Q_q(N\mid O;b)=1-b,Q_q(W\mid O;b)=0,\hfill \\ \hfill Q_q(B\mid W)& =0,Q_q(N\mid W)=0,Q_q(W\mid W)=1.\hfill \end{array}$$

(12)

Under this specification, substitution in the base-2 KL divergence yields

$$\begin{array}{cc}\hfill \Delta G_q(b;R)& =G_q(O;b,R)-G_q(W;R)\hfill \\ \hfill & =-H_2\left(b\right)-{\displaystyle \frac{\Delta U_q(b;R)}{\tau \ln 2}},\hfill \end{array}$$

(13)

where $H_2\left(b\right)=-b{\log }_{2}b-(1-b){\log }_2(1-b)$ is the binary entropy, continuously extended by $H_2\left(0\right)=H_2\left(1\right)=0$. For example, writing $Z_q={\sum }_o\exp \{U_q(o;R)/\tau \}$, the offer divergence equals $-H_2\left(b\right)-{\overline{U}}_q(O;b,R)/\left(\tau \ln 2\right)+{\log }_2{Z}_q$, while the withhold divergence equals $-s_q/\left(\tau \ln 2\right)+{\log }_2{Z}_q$; their common normalization cancels. The entropy term represents uncertainty about whether an offer results in a sale or rejection, not an additional monetary benefit.

For clarity, the intermediate KL-divergence calculations underlying Equation (13) are given explicitly. The exponential preference distribution in Equation (11) assigns strictly positive probability to each outcome. With the common normalizing constant $Z_q={\sum }_{o\in \{B,N,W\}}{e}^{U_q(o;R)/\tau }$, its three probabilities are

$$\begin{array}{cc}\hfill P_q^{\mathrm{pref}}(B;R)& ={\displaystyle \frac{e^{U_q(B;R)/\tau }}{Z_q}},\hfill \\ \hfill P_q^{\mathrm{pref}}(N;R)& ={\displaystyle \frac{e^{U_q(N;R)/\tau }}{Z_q}},\hfill \\ \hfill P_q^{\mathrm{pref}}(W;R)& ={\displaystyle \frac{e^{U_q(W;R)/\tau }}{Z_q}}.\hfill \end{array}$$

The same $Z_q$ applies to both policies because the seller’s preferred-outcome distribution is defined over the same outcome set. By Equation (12), offering produces the predictive distribution $Q_q(·\mid O;b)=(b,1-b,0)$. Using the base-2 definition of KL divergence and the convention that a zero-probability term contributes zero gives

$$\begin{array}{cc}\hfill G_q(O;b,R)& =b{\log }_2{\displaystyle \frac{b}{P_q^{\mathrm{pref}}(B;R)}}+(1-b){\log }_2{\displaystyle \frac{1-b}{P_q^{\mathrm{pref}}(N;R)}}\hfill \\ \hfill & =b{\log }_2\left({\displaystyle \frac{b{Z}_q}{e^{U_q(B;R)/\tau }}}\right)+(1-b){\log }_2\left({\displaystyle \frac{(1-b)Z_q}{e^{U_q(N;R)/\tau }}}\right).\hfill \end{array}$$

Apply ${\log }_2(x/y)={\log }_{2}x-{\log }_{2}y$ and ${\log }_2\left(e^x\right)=x/\ln 2$. Expanding both terms and collecting coefficients of the common normalization yields

$$\begin{array}{cc}\hfill G_q(O;b,R)& =b{\log }_{2}b+(1-b){\log }_2(1-b)+[b+(1-b)]{\log }_2{Z}_q\hfill \\ \hfill & -{\displaystyle \frac{b{U}_q(B;R)+(1-b)U_q(N;R)}{\tau \ln 2}}\hfill \\ \hfill & =-H_2\left(b\right)-{\displaystyle \frac{{\overline{U}}_q(O;b,R)}{\tau \ln 2}}+{\log }_2{Z}_q.\hfill \end{array}$$

The second equality uses both the binary-entropy definition and the expected payoff from offering, ${\overline{U}}_q(O;b,R)=b{U}_q(B;R)+(1-b)U_q(N;R)$, as defined in Equation (9). Thus, the negative binary-entropy term arises from the two possible buyer responses, while the other nonconstant term is proportional to the seller’s expected payoff.

For withholding, Equation (12) implies $Q_q(·\mid W)=(0,0,1)$. Hence its KL divergence contains a single nonzero term:

$$\begin{array}{cc}\hfill G_q(W;R)& ={\log }_2{\displaystyle \frac{1}{P_q^{\mathrm{pref}}(W;R)}}\hfill \\ \hfill & ={\log }_2\left({\displaystyle \frac{Z_q}{e^{U_q(W;R)/\tau }}}\right)\hfill \\ \hfill & =-{\displaystyle \frac{s_q}{\tau \ln 2}}+{\log }_2{Z}_q,\hfill \end{array}$$

because the payoff from withholding is $U_q(W;R)=s_q$. Subtracting the withholding divergence from the offering divergence therefore gives

$$\begin{array}{cc}\hfill G_q(O;b,R)-G_q(W;R)& =-H_2\left(b\right)-{\displaystyle \frac{{\overline{U}}_q(O;b,R)-s_q}{\tau \ln 2}}\hfill \\ \hfill & =-H_2\left(b\right)-{\displaystyle \frac{\Delta U_q(b;R)}{\tau \ln 2}}.\hfill \end{array}$$

In particular, the two copies of ${\log }_2{Z}_q$ cancel exactly. This completes the derivation of Equation (13). The entropy contribution vanishes under certain rejection or certain purchase, $H_2\left(0\right)=H_2\left(1\right)=0$, and reaches its maximum of one bit at $b=1/2$. Under the specified KL preference encoding, it lowers the free-energy difference at intermediate purchase probabilities. This is an implication of the representation, not an additional monetary payoff or an assumed preference for uncertainty.

Finally, Equation (10) makes the economic components of Equation (13) explicit for each seller type:

$$\begin{array}{cc}\hfill \Delta G_G\left(b\right)& =-H_2\left(b\right)+{\displaystyle \frac{c_G-b(P-s_G)}{\tau \ln 2}},\hfill \\ \hfill \Delta G_L(b;R)& =-H_2\left(b\right)+{\displaystyle \frac{c_L-b(P-s_L-\lambda R)}{\tau \ln 2}}.\hfill \end{array}$$

On the full-purchase continuation $b=1$, the entropy term disappears and the expected payoff advantages reduce to

$$\begin{array}{cc}\hfill \Delta U_G\left(1\right)& =P-s_G-c_G,\hfill \\ \hfill \Delta U_L(1;R)& =P-s_L-c_L-\lambda R.\hfill \end{array}$$

Substituting these two expressions into Equation (13) yields the free-energy differences on the full-purchase branch. In particular, the lemon seller’s difference separates into a baseline component and a component proportional to the reputational stake:

$$\Delta G_L(1;R)={\displaystyle \frac{s_L+c_L-P}{\tau \ln 2}}+{\displaystyle \frac{\lambda R}{\tau \ln 2}}.$$

For the seller-policy calculations below, $G_q\left(O\right)$ abbreviates $G_q(O;1)$ (or $G_L(O;1,R)$ when reputation is explicit). Let $\rho >0$ denote the increase in the lemon seller’s expected-free-energy difference per unit increase in the reputational stake R. From Equations (10) and (13), the full-purchase free-energy differences and reputational coefficient are

$$\begin{array}{cc}\hfill \Delta G_G& ={\displaystyle \frac{s_G+c_G-P}{\tau \ln 2}},\hfill \\ \hfill \Delta G_L^0& ={\displaystyle \frac{s_L+c_L-P}{\tau \ln 2}},\hfill \\ \hfill \rho & ={\displaystyle \frac{\lambda }{\tau \ln 2}}>0.\hfill \end{array}$$

(14)

With these definitions, the lemon seller’s free-energy difference can be written as $\Delta G_L(1;R)=\Delta G_L^0+\rho R$. Thus, $\Delta G_G<0$ if and only if $P>s_G+c_G$, and $\Delta G_L^0<0$ if and only if $P>s_L+c_L$. The original free-energy differences are therefore no longer unrestricted constants: they represent net economic incentives on the full-purchase branch. Section 3.5 allows b to vary and imposes behavioral consistency.

Let

$$\Delta G_G\equiv G_G\left(O\right)-G_G\left(W\right)$$

(15)

be the difference in expected free energy between offering and withholding for a high-quality seller. For the high-quality seller, I assume that there is no prior or habitual policy bias favoring either offering or withholding, so that

$$E_G\left(O\right)=E_G\left(W\right).$$

Because only differences in the policy-bias terms affect the softmax probabilities, any common value assigned to $E_G\left(O\right)$ and $E_G\left(W\right)$ cancels from the numerator and denominator of the policy rule. Their common value can therefore be normalized to zero without loss of generality:

$$E_G\left(O\right)=E_G\left(W\right)=0.$$

This baseline assumption leaves the high-quality seller’s choice governed solely by the expected-free-energy difference between offering and withholding.

Substituting $q=G$ and $a=O$ into the general policy-selection rule in Equation (4) gives

$$\begin{array}{cc}\hfill {\sigma }_G\equiv P(O\mid G)& ={\displaystyle \frac{\exp \{-\gamma G_G\left(O\right)\}}{\exp \{-\gamma G_G\left(O\right)\}+\exp \{-\gamma G_G\left(W\right)\}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{1+\exp \left\{\gamma [G_G\left(O\right)-G_G\left(W\right)]\right\}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{1+\exp \{\gamma \Delta G_G\}}}.\hfill \end{array}$$

(16)

The second equality is obtained by dividing both the numerator and the denominator by $\exp \{-\gamma G_G\left(O\right)\}$. Thus, Equation (16) is the two-policy specialization of Equation (4). In the economically relevant case in which offering a high-quality product has lower expected free energy than withholding it,

$$G_G\left(O\right)<G_G\left(W\right),$$

so that $\Delta G_G<0$. It follows that $\exp \{\gamma \Delta G_G\}<1$ and therefore

$${\sigma }_G>{\displaystyle \frac{1}{2}}.$$

Hence, a high-quality seller is more likely to offer the product than to withhold it.

For a seller who owns a lemon, introduce an internalized honesty parameter $\eta \ge 0$. The interpretation is that a seller who knows that the product is of low quality may have an internalized disposition against exploiting the buyer by deliberately offering the lemon. This disposition is represented as a prior policy bias favoring withholding over offering.

Because only differences in the policy-bias terms affect the softmax probabilities in Equation (4), one of the two terms can be normalized to zero without loss of generality. I therefore take offering the lemon as the reference policy and set

$$E_L\left(O\right)=0,E_L\left(W\right)=\eta .$$

(17)

Thus, $\eta$ measures the relative prior policy bias in favor of withholding the lemon:

$$E_L\left(W\right)-E_L\left(O\right)=\eta .$$

The parameter $\eta$ should not be interpreted as a probability or as a monetary payoff. Rather, it is an additive log-policy bias in the stochastic policy rule. When $\eta =0$, the lemon seller has no prior policy disposition favoring withholding over offering, so the choice between the two policies is governed solely by their expected-free-energy consequences. When $\eta >0$, withholding receives greater prior weight, and, holding expected free energy fixed, the probability of withholding increases relative to the probability of offering. A larger $\eta$ therefore represents a stronger internalized honesty disposition against knowingly offering a lemon to the buyer. Indeed, holding the expected-free-energy terms fixed, increasing $\eta$ by one unit multiplies the relative policy weight attached to withholding rather than offering by a factor of e.

Reputation affects the seller through a different channel. Let $R\ge 0$ denote the seller’s reputational stake. As defined in Equation (14), $\rho >0$ measures how strongly the reputational stake increases the expected free-energy cost of offering a lemon on the full-purchase branch. Define

$$\Delta G_L\left(R\right)\equiv G_L(O;R)-G_L(W;R)=\Delta G_L^0+\rho R,\rho >0,$$

(18)

where $\Delta G_L^0$ is the expected-free-energy difference between offering and withholding in the absence of reputational stakes, evaluated on the candidate full-purchase continuation $b=1$. Equation (14) identifies this baseline with the seller’s net monetary incentive and derives $\rho =\lambda /\left(\tau \ln 2\right)$. A larger R raises the expected free-energy cost of selling a lemon because it represents an expected loss of future rents, feedback-related business, or social standing.

Substituting $q=L$ and $a=O$ into the general policy-selection rule in Equation (4), together with Equation (17), gives

$$\begin{array}{cc}\hfill {\sigma }_L(\eta ,R)\equiv P(O\mid L)& ={\displaystyle \frac{\exp \{-\gamma G_L(O;R)\}}{\exp \{-\gamma G_L(O;R)\}+\exp \{\eta -\gamma G_L(W;R)\}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{1+\exp \{\eta +\gamma [G_L(O;R)-G_L(W;R)]\}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{1+\exp \{\eta +\gamma \Delta G_L\left(R\right)\}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{1+\exp \left\{\eta +\gamma \left(\Delta G_L^0+\rho R\right)\right\}}}.\hfill \end{array}$$

(19)

The second equality is obtained by dividing both the numerator and the denominator by $\exp \{-\gamma G_L(O;R)\}$. Thus, Equation (19) is the lemon-seller counterpart of Equation (16): the probability of offering depends jointly on the seller’s prior policy disposition, represented by $\eta$, and on the expected-free-energy difference between offering and withholding.

The model allows

$$\Delta G_L^0<0.$$

In this case, absent internalized honesty and reputation, offering the lemon has lower expected free energy than withholding it and is therefore more attractive under the policy rule. In particular, when $\eta =0$ and $R=0$,

$${\sigma }_L(0,0)={\displaystyle \frac{1}{1+\exp \{\gamma \Delta G_L^0\}}},$$

which exceeds $1/2$ whenever $\Delta G_L^0<0$. Internalized honesty and reputation counteract this baseline incentive. From Equation (19),

$${\sigma }_L(\eta ,R)<{\displaystyle \frac{1}{2}}$$

if and only if

$$\eta +\gamma \left(\Delta G_L^0+\rho R\right)>0,$$

or equivalently,

$$\eta +\gamma \rho R>-\gamma \Delta G_L^0.$$

(20)

Thus, sufficiently strong internalized honesty, sufficiently valuable reputation, or a combination of the two can make offering a lemon less likely than withholding it.

3.3. The Buyer’s Posterior Belief

On the candidate full-purchase continuation $b=1$, an offer is informative because the two quality types generally use different mixed policies. Bayes’ rule gives

$$\mu (\eta ,R)\equiv P(G\mid O)={\displaystyle \frac{\pi {\sigma }_G}{\pi {\sigma }_G+(1-\pi ){\sigma }_L(\eta ,R)}}.$$

(21)

This is the paper’s endogenous belief-refinement mechanism. No physical index needs to be inspected. The observed seller action plays the role of a signal because its likelihood depends on hidden quality.

The distinction from Mamada (2022) is summarized in Table 2.

Table 2. Exogenous index information and endogenous social information.

	Index Model	Present Model
Hidden variable	Product quality	Product quality
Observed information	Physical index	Seller’s offer decision
Source of correlation	Joint quality–index distribution	Type-dependent seller policy
Behavioral mechanism	Costly index inspection	Internalized norm and reputation
Belief refinement	$P(G\mid \mathrm{index})$	$P(G\mid O)$
Information source	Exogenous to seller’s action	Endogenous to strategic behavior

3.4. The Buyer’s Purchase Rule

On the candidate full-purchase continuation $b=1$, the buyer’s expected value conditional on an offer is

$$V_B(\eta ,R)=\mu (\eta ,R)v_G+\left[1-\mu (\eta ,R)\right]v_L.$$

(22)

A purchase is optimal if

$$V_B(\eta ,R)\ge P.$$

(23)

At equality the buyer is indifferent and may choose either action or mix; this possibility is incorporated into the continuation-response correspondence below. Define the critical posterior

$${\mu }^{\ast }\equiv {\displaystyle \frac{P-v_L}{v_G-v_L}}.$$

(24)

By Equation (6), $0<{\mu }^{\ast }<1$. Moreover, the buyer’s purchase condition in Equation (23),

$$\mu (\eta ,R)v_G+\left[1-\mu (\eta ,R)\right]v_L\ge P,$$

can be rearranged as

$$\mu (\eta ,R)(v_G-v_L)\ge P-v_L.$$

Since $v_G-v_L>0$, this is equivalent to

$$\mu (\eta ,R)\ge {\displaystyle \frac{P-v_L}{v_G-v_L}}={\mu }^{\ast }.$$

Thus, ${\mu }^{\ast }$ is the minimum posterior probability of high quality required for the buyer to purchase the offered product.

3.5. Strategic Consistency and the Buyer’s Continuation Response

The preceding seller-policy and posterior expressions evaluate the candidate full-purchase continuation $b=1$. The explicit seller payoffs now determine the free-energy evaluation for every anticipated buyer response $b\in [0,1]$. Following the sale/rejection outcome probabilities in Equation (12), an offer generates the predictive distribution

$$Q_q(o\mid O;b,R)=b{Q}_q(o\mid O,B;R)+(1-b)Q_q(o\mid O,N;R),$$

(25)

where the conditional distributions place unit mass on the mutually exclusive sale and rejection outcomes. Withholding places unit mass on W. For the high-quality seller, the reputation argument can be suppressed. Equations (10) and (13) then imply the explicit continuation-dependent free-energy differences

$$\begin{array}{cc}\hfill \Delta G_G\left(b\right)& =-H_2\left(b\right)+{\displaystyle \frac{c_G-b(P-s_G)}{\tau \ln 2}},\hfill \\ \hfill \Delta G_L(b;R)& =-H_2\left(b\right)+{\displaystyle \frac{c_L-b(P-s_L-\lambda R)}{\tau \ln 2}}.\hfill \end{array}$$

(26)

The nonlinear binary-entropy term follows from the KL divergence rather than being appended to the payoff function. At the endpoints $b=0$ and $b=1$, it vanishes. In particular, the full-purchase case is recovered exactly:

$$\Delta G_G\left(1\right)=\Delta G_G,\Delta G_L(1;R)=\Delta G_L^0+\rho R.$$

(27)

Thus the economic payoff assumptions complete the feedback structure introduced in response to buyer behavior; the free-energy functions are no longer left unspecified for $b<1$.

Keeping the policy priors in Equation (17) and the zero differential bias for the high-quality type, the seller’s response to an anticipated b is

$$\begin{array}{cc}\hfill {\sigma }_G\left(b\right)& ={\displaystyle \frac{1}{1+\exp \{\gamma \Delta G_G\left(b\right)\}}},\hfill \\ \hfill {\sigma }_L(b;\eta ,R)& ={\displaystyle \frac{1}{1+\exp \{\eta +\gamma \Delta G_L(b;R)\}}}.\hfill \end{array}$$

(28)

These probabilities are strictly between zero and one whenever the free-energy differences are finite. The buyer therefore reaches the offer information set with positive probability and computes the posterior

$$\mu (b;\eta ,R)={\displaystyle \frac{\pi {\sigma }_G\left(b\right)}{\pi {\sigma }_G\left(b\right)+(1-\pi ){\sigma }_L(b;\eta ,R)}}.$$

(29)

The buyer’s net expected value from purchasing is

$$D_B(b;\eta ,R)=\mu (b;\eta ,R)v_G+\left[1-\mu (b;\eta ,R)\right]v_L-P.$$

(30)

Given the seller policies induced by the anticipated response b, the buyer’s optimal purchase probabilities form the correspondence

$$\mathcal{B}(b;\eta ,R)=\left\{\begin{array}{cc}\left\{1\right\},\hfill & D_B(b;\eta ,R)>0,\hfill \\ ,\hfill & D_B(b;\eta ,R)=0,\hfill \\ \left\{0\right\},\hfill & D_B(b;\eta ,R)<0.\hfill \end{array}\right.$$

(31)

3.5.1. Behavioral Bayesian Signaling Equilibrium

For given parameters $(\pi ,P,v_G,v_L,\eta ,R,\gamma )$ and specified continuation-dependent expected-free-energy functions, I call $(b^{\ast },{\sigma }_G^{\ast },{\sigma }_L^{\ast }$, ${\mu }^{\mathrm{eq}}$) a behavioral Bayesian signaling equilibrium if

$$\begin{array}{cc}\hfill {\sigma }_G^{\ast }& ={\sigma }_G\left(b^{\ast }\right),\hfill \\ \hfill {\sigma }_L^{\ast }& ={\sigma }_L(b^{\ast };\eta ,R),\hfill \\ \hfill {\mu }^{\mathrm{eq}}& =\mu (b^{\ast };\eta ,R),\hfill \\ \hfill b^{\ast }& \in \mathcal{B}(b^{\ast };\eta ,R).\hfill \end{array}$$

(32)

The equilibrium posterior is the buyer’s posterior under the equilibrium seller policies, whereas the critical posterior is the minimum posterior required for purchase. These quantities coincide if and only if the buyer is indifferent between purchasing and declining. The final condition closes the feedback loop: the purchase probability anticipated in the seller’s policy evaluation equals the buyer’s optimal continuation probability under the resulting Bayesian posterior. This is a behavioral equilibrium concept with a stochastic seller policy, not a claim that the seller’s choices satisfy conventional expected-utility sequential rationality or that the outcome is a standard perfect Bayesian equilibrium.

3.5.2. Existence and Possible Continuation Responses

Assume the continuation-dependent free-energy differences are finite and continuous in $b\in [0,1]$. Then $D_B(b;\eta ,R)$ is continuous and at least one solution of Equation (32) exists. To see this, if $D_B(0;\eta ,R)\le 0$, then $b^{\ast }=0$ is consistent; otherwise, if $D_B(1;\eta ,R)\ge 0$, then $b^{\ast }=1$ is consistent. In the remaining case $D_B(0;\eta ,R)>0>D_B(1;\eta ,R)$, the intermediate value theorem provides an interior $b^{\ast }\in (0,1)$ with $D_B(b^{\ast };\eta ,R)=0$, and the buyer may mix. This argument ensures existence but not uniqueness.

In particular, the full-purchase branch $b^{\ast }=1$ is an equilibrium continuation if and only if

$$\mu (1;\eta ,R)\ge {\mu }^{\ast }.$$

(33)

By Equation (27), the probabilities ${\sigma }_G\left(1\right)$ and ${\sigma }_L(1;\eta ,R)$ coincide with the baseline expressions in Equations (16) and (19). The following comparative statics and threshold results characterize this candidate full-purchase branch. When that branch is not consistent, the actual continuation must be obtained from Equation (32) using the explicit payoff-based free-energy differences in Equation (26).

4. Results

The results in this section first characterize the seller-policy and belief-refinement effects conditional on the candidate full-purchase continuation $b=1$. Holding b fixed is essential: the derivative formulas below do not in general describe comparative statics across equilibria when $b^{\ast }$ changes with the parameters.

4.1. Social Discipline and Lemon Entry

Proposition 1.

On the candidate full-purchase branch $b=1$, the lemon seller’s probability of offering is strictly decreasing in both the internalized honesty parameter η and the reputational stake R:

$${\displaystyle \frac{\partial {\sigma }_L}{\partial \eta }}=-{\sigma }_L(1-{\sigma }_L)<0,{\displaystyle \frac{\partial {\sigma }_L}{\partial R}}=-\gamma \rho {\sigma }_L(1-{\sigma }_L)<0.$$

(34)

Proof. 

Let

$$x=\eta +\gamma (\Delta G_L^0+\rho R).$$

From Equation (19), ${\sigma }_L={(1+e^x)}^{-1}$. Hence

$${\displaystyle \frac{d{\sigma }_L}{dx}}=-{\displaystyle \frac{e^x}{{(1+e^x)}^2}}=-{\sigma }_L(1-{\sigma }_L).$$

Since $\partial x/\partial \eta =1$ and $\partial x/\partial R=\gamma \rho >0$, the two derivatives in Equation (34) follow. □

Proposition 1 separates two mechanisms that may look similar observationally. The term $\eta$ represents a disposition already incorporated into the seller’s policy prior, whereas R changes the expected consequence of opportunism. Both make lemons less likely to be offered.

4.2. Endogenous Belief Refinement

Proposition 2.

Let $\pi =P(q=G)$ denote the common prior probability that the product is of high quality, with $0<\pi <1$, and suppose that $0<{\sigma }_G\le 1$, where ${\sigma }_G=P(O\mid G)$ is independent of η and R. Then, holding the buyer’s response fixed at $b=1$, the posterior probability that an offered product is of high quality is strictly increasing in both the internalized honesty parameter η and the reputational stake R:

$${\displaystyle \frac{\partial \mu }{\partial \eta }}>0,{\displaystyle \frac{\partial \mu }{\partial R}}>0.$$

(35)

Proof. 

Bayes’ rule in Equation (21) implies the posterior-odds identity

$$\ln {\displaystyle \frac{\mu }{1-\mu }}=\ln {\displaystyle \frac{\pi }{1-\pi }}+\ln {\displaystyle \frac{{\sigma }_G}{{\sigma }_L(\eta ,R)}}.$$

(36)

Here and in the following derivatives, ln denotes the natural logarithm. Holding the buyer’s continuation fixed at $b=1$, the prior $\pi$ and the high-quality offer probability ${\sigma }_G$ are independent of $\eta$ and R. Proposition 1 therefore gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial \eta }}\ln {\displaystyle \frac{\mu }{1-\mu }}& =-{\displaystyle \frac{1}{{\sigma }_L}}{\displaystyle \frac{\partial {\sigma }_L}{\partial \eta }}=1-{\sigma }_L>0,\hfill \\ \hfill {\displaystyle \frac{\partial }{\partial R}}\ln {\displaystyle \frac{\mu }{1-\mu }}& =-{\displaystyle \frac{1}{{\sigma }_L}}{\displaystyle \frac{\partial {\sigma }_L}{\partial R}}=\gamma \rho (1-{\sigma }_L)>0.\hfill \end{array}$$

(37)

Since $0<\mu <1$, differentiation of the log-odds transformation yields $\partial \mu /\partial x=\mu (1-\mu )\partial \ln [\mu /(1-\mu \left)\right]/\partial x$ for $x\in \{\eta ,R\}$. Hence

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \mu }{\partial \eta }}& =\mu (1-\mu )(1-{\sigma }_L)>0,\hfill \\ \hfill {\displaystyle \frac{\partial \mu }{\partial R}}& =\gamma \rho \mu (1-\mu )(1-{\sigma }_L)>0.\hfill \end{array}$$

(38)

These identities prove the proposition. The equivalent quotient-rule calculation is recorded in Appendix A. □

The result formalizes a particular meaning of trust. The buyer does not simply choose to believe that a friend or a highly rated seller is honest. Instead, the social environment changes the likelihood of observing an offer from a lemon owner. Once the likelihood changes, a more favorable posterior follows from Bayes’ rule.

Equation (36) also distinguishes the information supplied by the offer from the buyer’s prior. The posterior exceeds the prior precisely when ${\sigma }_G>{\sigma }_L$. An increase in honesty or reputation raises the likelihood ratio and thus improves the evidence in favor of high quality, but need not make an offer favorable evidence relative to the prior at every parameter value.

Corollary 1.

The likelihood ratio associated with observing an offer,

$${\Lambda }_O(\eta ,R)\equiv {\displaystyle \frac{P(O\mid G)}{P(O\mid L)}}={\displaystyle \frac{{\sigma }_G}{{\sigma }_L(\eta ,R)}},$$

(39)

is strictly increasing in both η and R, conditional on $b=1$.

Proof. 

By Equation (36), the log-likelihood ratio is the difference between posterior and prior log odds:

$$\ln {\Lambda }_O=\ln {\displaystyle \frac{\mu }{1-\mu }}-\ln {\displaystyle \frac{\pi }{1-\pi }}.$$

(40)

The prior is fixed, so Equation (37) implies

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\Lambda }_O}{\partial \eta }}& ={\Lambda }_O(1-{\sigma }_L)>0,\hfill \\ \hfill {\displaystyle \frac{\partial {\Lambda }_O}{\partial R}}& =\gamma \rho {\Lambda }_O(1-{\sigma }_L)>0.\hfill \end{array}$$

(41)

Both signs follow from ${\Lambda }_O>0$, $0<{\sigma }_L<1$, and $\gamma \rho >0$. No additional quotient-rule calculation is required. □

Thus, social discipline makes an offer more diagnostic of quality. This is the exact informational link between the seller’s internal state and the buyer’s posterior belief.

4.3. A Threshold for a Consistent Full-Purchase Continuation

Theorem 1

(Full-Purchase Consistency Threshold). Let

$${\overline{\sigma }}_L\equiv {\displaystyle \frac{\pi {\sigma }_G(1-{\mu }^{\ast })}{(1-\pi ){\mu }^{\ast }}}.$$

(42)

On the candidate branch $b=1$, if ${\overline{\sigma }}_L\ge 1$, then full purchase is a self-consistent buyer response for every $\eta \ge 0$ and $R\ge 0$. If $0<{\overline{\sigma }}_L<1$, the full-purchase branch is consistent if and only if

$$\eta +\gamma \rho R\ge \underset{{\displaystyle {\Theta }^{\ast }}}{\underbrace{\ln \left({\displaystyle \frac{1-{\overline{\sigma }}_L}{{\overline{\sigma }}_L}}\right)-\gamma \Delta G_L^0}}.$$

(43)

Here, ln denotes the natural logarithm. Thus, when ${\Theta }^{\ast }>0$, sufficiently strong internalized honesty, a sufficiently valuable reputational stake, or a combination of the two supports the full-purchase continuation. High-quality trade then occurs with positive probability at the exogenously given price P; this statement alone does not establish prevention of Akerlof-type price–participation unraveling. If $0<{\overline{\sigma }}_L<1$ and ${\Theta }^{\ast }\le 0$, the full-purchase condition is satisfied for every admissible $\eta \ge 0$ and $R\ge 0$, including $\eta =R=0$.

Proof. 

By Equation (33), the full-purchase continuation is consistent if and only if

$$\mu (1;\eta ,R)\ge {\mu }^{\ast }.$$

Using Equation (21), this condition becomes

$${\displaystyle \frac{\pi {\sigma }_G}{\pi {\sigma }_G+(1-\pi ){\sigma }_L}}\ge {\mu }^{\ast }.$$

Because the denominator is strictly positive, multiplying both sides by it gives

$$\pi {\sigma }_G\ge {\mu }^{\ast }\left[\pi {\sigma }_G+(1-\pi ){\sigma }_L\right].$$

Expanding the right-hand side,

$$\pi {\sigma }_G\ge {\mu }^{\ast }\pi {\sigma }_G+{\mu }^{\ast }(1-\pi ){\sigma }_L.$$

Rearranging,

$$\pi {\sigma }_G(1-{\mu }^{\ast })\ge {\mu }^{\ast }(1-\pi ){\sigma }_L.$$

Since $(1-\pi ){\mu }^{\ast }>0$, division by this positive quantity gives

$${\sigma }_L\le {\displaystyle \frac{\pi {\sigma }_G(1-{\mu }^{\ast })}{(1-\pi ){\mu }^{\ast }}}={\overline{\sigma }}_L.$$

Hence, the buyer’s purchase condition is equivalent to

$${\sigma }_L\le {\overline{\sigma }}_L.$$

(44)

If ${\overline{\sigma }}_L\ge 1$, Equation (44) always holds because $0<{\sigma }_L<1$. Therefore, the buyer purchases after an offer for every $\eta \ge 0$ and $R\ge 0$.

Now suppose $0<{\overline{\sigma }}_L<1$. Substituting Equation (19) into Equation (44) gives

$${\displaystyle \frac{1}{1+\exp \{\eta +\gamma (\Delta G_L^0+\rho R)\}}}\le {\overline{\sigma }}_L.$$

Multiplying both sides by the positive denominator yields

$$1\le {\overline{\sigma }}_L\left[1+\exp \{\eta +\gamma (\Delta G_L^0+\rho R)\}\right].$$

Subtracting ${\overline{\sigma }}_L$ from both sides gives

$$1-{\overline{\sigma }}_L\le {\overline{\sigma }}_L\exp \{\eta +\gamma (\Delta G_L^0+\rho R)\}.$$

Because ${\overline{\sigma }}_L>0$, division by ${\overline{\sigma }}_L$ yields

$${\displaystyle \frac{1-{\overline{\sigma }}_L}{{\overline{\sigma }}_L}}\le \exp \{\eta +\gamma (\Delta G_L^0+\rho R)\}.$$

Equivalently,

$$\exp \{\eta +\gamma (\Delta G_L^0+\rho R)\}\ge {\displaystyle \frac{1-{\overline{\sigma }}_L}{{\overline{\sigma }}_L}}.$$

Since $0<{\overline{\sigma }}_L<1$, the quantity $(1-{\overline{\sigma }}_L)/{\overline{\sigma }}_L$ is strictly positive. Taking the natural logarithm of both sides therefore preserves the inequality. Because $\ln \left(\exp z\right)=z$, it follows that

$$\eta +\gamma (\Delta G_L^0+\rho R)\ge \ln \left({\displaystyle \frac{1-{\overline{\sigma }}_L}{{\overline{\sigma }}_L}}\right).$$

Rearranging gives

$$\eta +\gamma \rho R\ge \ln \left({\displaystyle \frac{1-{\overline{\sigma }}_L}{{\overline{\sigma }}_L}}\right)-\gamma \Delta G_L^0={\Theta }^{\ast },$$

which is precisely Equation (43).

The natural logarithm appears here because the seller’s policy rule in Equation (19) is expressed using the exponential function $\exp (·)$, whose inverse is $\ln (·)$. This is distinct from the base-2 logarithms used in the information-theoretic quantities of the model, such as KL divergence and mutual information.

Finally, ${\sigma }_G>0$ by Equation (16). Since $0<\pi <1$, high quality occurs with positive prior probability. Therefore, when the buyer accepts an observed offer, high-quality trade occurs with unconditional probability

$$\pi {\sigma }_G>0.$$

 □

4.3.1. Economic Interpretation: Participation at a Fixed Posted Price

The payoff foundations in Section 3 make the high-quality seller’s offer probability sensitive to the posted price and the reservation value, even though the price itself is not determined in equilibrium. On the candidate full-purchase branch, Equations (14) and (16) imply

$${\sigma }_G\left(1\right)={\displaystyle \frac{1}{1+\exp \left\{\frac{\gamma }{\tau \ln 2}(s_G+c_G-P)\right\}}}.$$

(45)

Consequently, holding the other primitives fixed,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\sigma }_G\left(1\right)}{\partial s_G}}& =-{\displaystyle \frac{\gamma }{\tau \ln 2}}{\sigma }_G\left(1\right)[1-{\sigma }_G\left(1\right)]<0,\hfill \\ \hfill {\displaystyle \frac{\partial {\sigma }_G\left(1\right)}{\partial P}}& ={\displaystyle \frac{\gamma }{\tau \ln 2}}{\sigma }_G\left(1\right)[1-{\sigma }_G\left(1\right)]>0.\hfill \end{array}$$

(46)

A higher high-quality reservation value therefore discourages offering, while a higher posted price encourages it. Unlike a deterministic participation cutoff, however, the stochastic free-energy policy assigns positive offering probability to both types at all finite parameter values; it does not force all high-quality sellers to withdraw when their reservation values exceed the price.

The distinction between an offer and a completed sale is equally important. In any behavioral Bayesian signaling equilibrium from Equation (32), the unconditional probabilities of a sale and of a high-quality sale are, respectively,

$$\begin{array}{cc}\hfill \mathrm{Pr}\left(\mathrm{sale}\right)& =b^{\ast }\left[\pi {\sigma }_G\left(b^{\ast }\right)+(1-\pi ){\sigma }_L(b^{\ast };\eta ,R)\right],\hfill \\ \hfill \mathrm{Pr}(G,\mathrm{sale})& =\pi b^{\ast }{\sigma }_G\left(b^{\ast }\right).\hfill \end{array}$$

(47)

Thus $b^{\ast }=1$ implies positive high-quality trade, whereas $b^{\ast }=0$ implies no completed trade. If the threshold in Theorem 1 fails, full purchase is not a consistent continuation, but this does not rule out an interior solution with $0<b^{\ast }<1$ and positive trade. Conversely, satisfying that threshold does not establish that the full-purchase continuation is unique.

This result is deliberately narrower than prevention of classical Akerlof-type unraveling. In the latter mechanism, buyers’ willingness to pay, the endogenous market price, and quality-dependent seller participation interact. In the present model, P is held fixed and no price-clearing or price-setting condition closes that feedback; moreover, participation is stochastic rather than a deterministic reservation-value cutoff. Theorem 1 therefore characterizes conditions sustaining trade at the specified posted price, not an equilibrium price path or a general absence of market unraveling in the sense of Akerlof (1970).

The threshold has an intuitive interpretation. Define effective social discipline as

$$S\equiv \eta +\gamma \rho R.$$

(48)

Then the full-purchase branch is consistent whenever $S\ge {\Theta }^{\ast }$ in the case $0<{\overline{\sigma }}_L<1$. On this branch, internalized honesty and reputation are therefore substitutes in generating trustworthy seller behavior. An increase $dR$ can be offset by a decrease

$$d\eta =-\gamma \rho dR$$

without changing ${\sigma }_L(1;\eta ,R)$ or the associated buyer posterior on the full-purchase branch. This equivalence need not extend to equilibria with different $b^{\ast }$.

4.3.2. Comparative Statics of the Full-Purchase Threshold

The buyer’s critical posterior in Equation (24) allows the critical lemon-offer probability in Equation (42) to be expressed directly in terms of economic primitives:

$$K\equiv {\overline{\sigma }}_L={\displaystyle \frac{\pi {\sigma }_G(v_G-P)}{(1-\pi )(P-v_L)}},v_G>P>v_L.$$

(49)

For $K\ge 1$, full purchase is a consistent continuation for all $\eta ,R\ge 0$, irrespective of the baseline free-energy difference. In the interior case $0<K<1$, Theorem 1 gives ${\Theta }^{\ast }=\ln [(1-K)/K]-\gamma \Delta G_L^0$. The derivatives below treat $\pi ,{\sigma }_G,P,v_G,v_L$ as separate threshold arguments, holding $\Delta G_L^0$ fixed and, when differentiating with respect to any one argument, holding the others fixed:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\Theta }^{\ast }}{\partial \pi }}& =-{\displaystyle \frac{1}{(1-K)\pi (1-\pi )}}<0,\hfill \\ \hfill {\displaystyle \frac{\partial {\Theta }^{\ast }}{\partial {\sigma }_G}}& =-{\displaystyle \frac{1}{(1-K){\sigma }_G}}<0,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\Theta }^{\ast }}{\partial v_G}}& =-{\displaystyle \frac{1}{(1-K)(v_G-P)}}<0,\hfill \\ \hfill {\displaystyle \frac{\partial {\Theta }^{\ast }}{\partial v_L}}& =-{\displaystyle \frac{1}{(1-K)(P-v_L)}}<0,\hfill \\ \hfill {\left.{\displaystyle \frac{\partial {\Theta }^{\ast }}{\partial P}}\right|}_{{\sigma }_G,\Delta G_L^0}& ={\displaystyle \frac{1}{1-K}}\left({\displaystyle \frac{1}{v_G-P}}+{\displaystyle \frac{1}{P-v_L}}\right)>0.\hfill \end{array}$$

(51)

A more favorable prior or a larger high-quality offer probability makes an observed offer more reassuring and reduces the minimum effective social discipline needed for full purchase. A higher valuation for either quality type similarly relaxes the buyer’s purchase condition. With the seller’s offer probabilities and baseline free-energy difference held fixed, raising the posted price increases the buyer’s required posterior and tightens the full-purchase condition. These are local results for $0<K<1$; when $K\ge 1$, no positive threshold is needed, and when $0<K<1$ but ${\Theta }^{\ast }\le 0$, the condition $\eta +\gamma \rho R\ge {\Theta }^{\ast }$ holds for all admissible $\eta ,R\ge 0$.

The payoff microfoundations introduced above also imply that a change in P affects both seller types’ offer incentives. It would therefore be incorrect to interpret the positive partial derivative in Equation (51) as the total effect of the posted price. Set $\kappa \equiv \gamma /\left(\tau \ln 2\right)>0$. Using Equation (14) and the full-purchase high-quality offer probability in Equation (45), I obtain

$${\displaystyle \frac{d{\Theta }^{\ast }}{dP}}={\displaystyle \frac{1}{1-K}}\left[{\displaystyle \frac{1}{v_G-P}}+{\displaystyle \frac{1}{P-v_L}}+\kappa ({\sigma }_G-K)\right],0<K<1.$$

(52)

To derive this expression, note that $d{\sigma }_G/dP=\kappa {\sigma }_G(1-{\sigma }_G)$ and $d\Delta G_L^0/dP=-1/\left(\tau \ln 2\right)$. Thus the total price effect includes changes in the information content of an offer and in the lemon seller’s monetary incentive. It is positive when ${\sigma }_G\ge K$, but its sign need not be positive for all admissible parameters when ${\sigma }_G<K$. The structural effect of a higher high-quality reservation value has an unambiguous sign:

$${\displaystyle \frac{d{\Theta }^{\ast }}{d{s}_G}}={\displaystyle \frac{\kappa (1-{\sigma }_G)}{1-K}}>0.$$

(53)

This follows because raising $s_G$ reduces ${\sigma }_G$, thereby requiring stronger social discipline to support full purchase. The derivatives in this paragraph characterize the candidate $b=1$ continuation; they do not assert that the realized equilibrium purchase probability varies smoothly or uniquely with the primitives.

4.4. Numerical Illustration

The numerical illustration now uses the explicit seller payoff structure rather than imposing unrelated preferred-outcome and predictive probabilities. Because the expected information gain vanishes for both seller actions in this specification, the coefficient $\beta$ drops out of the free-energy difference and no numerical value for it is needed. Let

$$\begin{array}{c}P=70,s_G=65,s_L=64,\\ c_G=2.8,c_L=4.6,\tau =1/\ln 2,\gamma =1,\lambda =1.\end{array}$$

(54)

The high-quality seller has the higher reservation value $s_G>s_L$, and the lemon seller faces the higher offering cost. Since $\tau \ln 2=1$, the payoff-based full-purchase differences are

$$\Delta G_G=65+2.8-70=-2.2,\Delta G_L^0=64+4.6-70=-1.4,\rho =1.$$

(55)

Thus, absent norms and reputation, both types have incentives to sell at a completed-sale continuation, though the strength of those incentives differs. The full-purchase offer probabilities follow from the same free-energy policy rule as before:

$$\begin{array}{cc}\hfill {\sigma }_G\left(1\right)& ={\displaystyle \frac{1}{1+e^{-2.2}}}\approx 0.90025,\hfill \\ \hfill {\sigma }_L(1;\eta ,0)& ={\displaystyle \frac{1}{1+e^{\eta -1.4}}}.\hfill \end{array}$$

(56)

For the buyer, take

$$\pi =0.5,v_G=100,v_L=30.$$

(57)

At $P=70$, the critical posterior is ${\mu }^{\ast }=(70-30)/(100-30)=4/7\approx 0.57143$. Consequently,

$$\begin{array}{cc}\hfill {\overline{\sigma }}_L& ={\displaystyle \frac{0.5\left(0.90025\right)(1-4/7)}{0.5(4/7)}}\approx 0.67519,\hfill \\ \hfill {\Theta }^{\ast }& =\ln \left({\displaystyle \frac{1-0.67519}{0.67519}}\right)+1.4\approx 0.66826.\hfill \end{array}$$

(58)

At these illustrative values, $K\approx 0.67519$ and $\kappa =1$. Equation (52) therefore gives $d{\Theta }^{\ast }/dP\approx 0.87249>0$. This is the total local price sensitivity of the full-purchase threshold in the specified payoff model, not merely the effect of increasing the buyer’s reservation-price requirement.

First consider $\eta =R=0$. On the hypothetical full-purchase branch,

$${\sigma }_L(1;0,0)\approx 0.80218,\mu (1;0,0)\approx 0.52880,V_B(1;0,0)\approx 67.02<70.$$

(59)

Hence $b=1$ is not a consistent continuation at these parameter values. Unlike the preceding reduced-form illustration, the payoff specification now also permits the actual continuation response to be computed. With $R=0$ and $\tau \ln 2=1$, Equation (26) becomes

$$\begin{array}{cc}\hfill \Delta G_G\left(b\right)& =-H_2\left(b\right)+2.8-5b,\hfill \\ \hfill \Delta G_L(b;0)& =-H_2\left(b\right)+4.6-6b.\hfill \end{array}$$

(60)

Substitution into the seller policies and Bayes’ rule gives $D_B(0;0,0)\approx 19.65>0$ and $D_B(1;0,0)\approx -2.98<0$. An interior solution therefore exists. Numerically solving $D_B(b;0,0)=0$ gives the following self-consistent mixed-purchase continuation:

$$\begin{array}{c}{b}^{\ast }\approx 0.70551,{\sigma }_G\left(b^{\ast }\right)\approx 0.83230,\\ {\sigma }_L(b^{\ast };0,0)\approx 0.62422,{\mu }^{\mathrm{eq}}=\mu (b^{\ast };0,0)=4/7.\end{array}$$

(61)

The buyer is indifferent at the stated posterior and can optimally purchase with probability $b^{\ast }$. These values describe a consistent continuation, without implying that it is unique. In this mixed-purchase continuation, the equilibrium posterior equals the critical purchase threshold.

Now increase internalized honesty to $\eta =1$ while keeping $R=0$. On the full-purchase branch,

$${\sigma }_L(1;1,0)\approx 0.59869,{\mu }^{\mathrm{eq}}=\mu (1;1,0)\approx 0.60059,V_B(1;1,0)\approx 72.04>70.$$

(62)

In this case the equilibrium posterior is strictly above the critical purchase threshold, so purchasing is strictly optimal for the buyer. The threshold $\eta =1>{\Theta }^{\ast }\approx 0.66826$ is satisfied, so $b^{\ast }=1$ is now a self-consistent continuation. High-quality trade occurs with unconditional probability $\pi {\sigma }_G\left(1\right)\approx 0.45012>0$. The comparison illustrates how an internalized honesty disposition can support full purchase while leaving the posted price, seller reservation values, offering costs, and product valuations unchanged. It does not establish uniqueness or a full endogenous-price Akerlof equilibrium.

5. Applications: Friendship and Public Reputation

5.1. Friendship and Long-Term Acquaintance

Suppose the same physical car can be offered either to a stranger or to a long-time acquaintance. Let ${\eta }_F$ denote the internalized honesty parameter when the buyer is a friend and ${\eta }_S$ the corresponding parameter for a stranger. If

$${\eta }_F>{\eta }_S,$$

then Proposition 1, conditional on a common full-purchase continuation, implies

$$P(O\mid L,\mathrm{friend})<P(O\mid L,\mathrm{stranger}).$$

Consequently, Proposition 2 yields

$$P(G\mid O,\mathrm{friend})>P(G\mid O,\mathrm{stranger}),$$

(63)

provided the good seller’s offer probability is otherwise unchanged. Equation (63) is a rational form of relational trust. Friendship does not mechanically change the quality of the car; it changes the probability that an informed seller would expose the buyer to a lemon, and the buyer rationally conditions on that behavior.

This interpretation is consistent with the embeddedness perspective of Granovetter (1985). It is also compatible with community-enforcement models such as Kandori (1992), although the present paper distinguishes internalized behavior from externally enforced sanctions. A long-term relationship may raise both $\eta$ and R: the seller may feel a stronger internalized obligation toward the buyer and may also expect greater future social costs from opportunism.

5.2. Public Reputation and Online Marketplaces

In an online marketplace, direct friendship may be absent, but public feedback can create a reputational stake. On the full-purchase continuation, a higher R increases $\Delta G_L\left(R\right)$ and therefore lowers $P(O\mid L)$. Buyers who understand the incentive effect can rationally assign a higher posterior probability of quality to an offer from a seller with more valuable reputation.

The empirical reputation literature supports the economic importance of this mechanism. Resnick et al. (2006) show experimentally that an established eBay identity commanded greater willingness to pay than new identities selling matched items. Cabral and Hortaçsu (2010) find that negative feedback is associated with changes in sales dynamics and seller exit. These findings are consistent with reputation having economic value that can be lost through poor performance. The model’s claim is not that review systems eliminate asymmetric information. Rather, a reputational stake changes the endogenous likelihood that a privately informed seller chooses opportunistic market entry.

Public reputation may also interact with internalized norms. In a highly embedded social group, $\eta$ may be large even when formal reputation scores do not exist. In a relatively anonymous online market, $\eta$ may be smaller but R may be large because public feedback affects future business. Equation (48) shows that these institutional environments can generate similar buyer posteriors through different behavioral channels.

6. Information-Theoretic Interpretation

The connection to information theory can be stated directly. Let the hidden variable be quality $Q\in \{G,L\}$ and the observed variable be seller action $A\in \{O,W\}$. Conditional on a specified purchase continuation, the seller’s policy induces a binary channel. On the full-purchase branch, the channel is

$$P(O\mid G)={\sigma }_G,P(O\mid L)={\sigma }_L(\eta ,R).$$

(64)

An observed offer changes the buyer’s prior odds according to

$${\displaystyle \frac{P(G\mid O)}{P(L\mid O)}}={\displaystyle \frac{\pi }{1-\pi }}{\displaystyle \frac{{\sigma }_G}{{\sigma }_L(\eta ,R)}}.$$

(65)

The second factor is the Bayes factor generated by seller behavior. By the corollary to Proposition 2, stronger internalized honesty or reputation raises this factor monotonically when $b=1$ is held fixed.

For comparison, Mamada (2022) treats an exogenous index as the information source and measures the average uncertainty reduction by mutual information. In the present model, one can similarly define the mutual information between Q and A as

$$I(Q;A)=\sum _{q\in \{G,L\}}\sum _{a\in \{O,W\}}P\left(q\right)P(a\mid q){\log }_2{\displaystyle \frac{P(a\mid q)}{P\left(a\right)}}.$$

(66)

The important distinction is that the conditional distribution $P(a\mid q)$ is not produced by a physical inspection technology. It is produced endogenously by the seller’s free-energy policy. Social norms and reputation therefore alter the information structure of the game itself.

I do not impose a global monotonicity claim for $I(Q;A)$ with respect to $\eta$ or R without additional restrictions, because mutual information depends on the entire distribution of both O and W. The conditional result is narrower and stronger for the observed transaction node: the likelihood ratio associated with an offer and the posterior $P(G\mid O)$ both rise monotonically on the full-purchase branch. Thus, the paper’s information-theoretic claim does not depend on treating every increase in social discipline as a global increase in mutual information.

7. Discussion

The model deliberately combines conventional Bayesian inference with a nonstandard behavioral specification for policy choice. This separation is useful. If the internalized-norm parameter were inserted directly into the buyer’s prior probability of quality, trust would be assumed rather than explained. Here, the buyer remains a conventional Bayesian updater and takes an optimal continuation action. The behavioral departure occurs on the seller side, where a norm or habit affects the probability of actions through the policy prior E. The seller’s anticipated buyer response enters the expected-free-energy calculation, and a consistency condition links the resulting action frequencies to the buyer’s optimal behavior and posterior.

A second advantage is that internalized norms and reputation are not conflated. Internalized honesty enters $E_L\left(W\right)$ because it represents a baseline policy disposition. Reputation enters expected free energy because it changes anticipated future consequences. This distinction parallels the difference between behavior sustained by internalized socialization and behavior sustained by external incentives. Both are economically relevant, but they need not have the same origin. The literature review motivating this paper emphasizes the internalization channel, while the classical reputation literature emphasizes the external-incentive channel (Gintis, 2014; Klein & Leffler, 1981; Shapiro, 1983).

The framework is intentionally parsimonious. The seller payoff specification explicitly identifies completed-sale revenue, type-dependent reservation values, offering costs, and the expected reputational cost of selling a lemon. It also supplies the outcomes following buyer rejection, so the consistency condition permits quantitative evaluation of full purchase, no purchase, or mixed purchase. No general uniqueness claim is imposed. The exponential payoff-to-preference mapping and zero epistemic term are maintained behavioral assumptions, not consequences of payoff maximization. The result is deliberately limited to trade at an exogenous posted price; a full Akerlof-type model would additionally determine price jointly with quality-dependent participation and buyers’ willingness to pay. Furthermore, the stochastic offering rule does not impose a sharp seller reservation-value participation cutoff. Several other extensions remain for later work. First, the price P is exogenous within the present subgame. An endogenous price-setting stage could allow the seller’s price itself to become informative. Second, public reputation can be modeled dynamically, with current conduct changing a future reputation state. Third, the buyer could also use a probabilistic free-energy policy rather than the deterministic expected-value purchase rule in Equation (23). Fourth, sellers may differ heterogeneously in $\eta$, in which case the buyer would infer both product quality and seller disposition. These extensions would enrich the model but are not required for the central belief-refinement result.

The interpretation should also remain precise about terminology. Unlike a Spence-type index, the seller’s offer is manipulable and therefore functions as a signal. The contribution is consequently not that adverse selection can be mitigated without any signal whatsoever. Rather, it is that an expensive exogenous index need not exist: socially disciplined strategic behavior itself can generate a signal whose likelihood is informative about hidden quality.

The threshold comparative statics identify the separate roles of the quality prior, the high-quality seller’s offering behavior, and the buyer’s valuations. The structural payoff specification additionally shows why changes in the posted price must account for both types’ endogenous offering responses, rather than only the buyer’s willingness to pay.

8. Conclusions

This paper develops a free-energy extension of the market for lemons in which social trust is generated endogenously. A privately informed seller chooses whether to offer a product. The seller’s payoff structure identifies type-dependent reservation values, offering costs, and expected reputation losses, which are mapped into a preferred-outcome distribution to derive the free-energy differences. Internalized honesty changes the seller’s prior policy bias, while reputation changes the expected free-energy cost of opportunistic market entry. On the full-purchase branch, these mechanisms reduce the probability that lemon owners offer their products. Because the seller’s offer decision is correlated with hidden quality, the buyer rationally refines the belief about quality after observing an offer.

On a candidate full-purchase continuation, stronger internalized honesty and stronger reputational stakes increase the posterior probability that an offered product is high quality. The model also yields an explicit threshold determining when full purchase is a self-consistent continuation; its dependence on prior quality, high-quality offering, valuations, and price has been characterized by comparative statics. If the threshold is satisfied, high-quality trade occurs with positive probability at the exogenous posted price even when the buyer cannot purchase a costly physical index. These findings characterize a mechanism for sustaining trade at a specified posted price. They neither determine an equilibrium price nor establish the absence of Akerlof-type price–participation unraveling, and the full-purchase equilibrium need not be unique.

The mechanism provides a bridge between two forms of information. In the earlier index model, information is produced by inspecting a characteristic correlated with quality. In the present model, information is produced by strategic behavior whose type dependence is itself shaped by norms and reputation. This suggests a broader research program in Bayesian games: institutions, culture, and social relationships may matter not only because they change preferences or payoffs, but also because they endogenously change the information content of observable actions.