OSIPP Discussion Paper: DP-2021-E-006

Optimal Trade Mechanisms with Adverse Selection andInferential Mistakes
September 6, 2021
Takeshi Murooka*
Osaka School of International Public Policy (OSIPP), Osaka University

Takuro Yamashita†
Toulouse School of Economics, University of Toulouse

Abstract: We study an adverse selection environment, where a rational seller can trade a good of which
she privately knows its value to a buyer, and there are gains from trade. The buyer's types differ in theirdegree of inferential abilities: A rational type correctly infers the value of the good from the seller's offer,
whereas a naive type under-appreciates the correlation between the seller's private information and offer.
We characterize the optimal menu mechanism that maximizes the social surplus. Notably, no matter how
severe the adverse selection is (in particular, even when no trade is the unique possible outcome if allagents are rational), all types of buyers trade in the optimal mechanism. The rational buyer's trade occurs
at the expense of the naive buyer's losses. We also investigate a consumer-protection policy of limiting
the losses and discuss its implications.

JEL Codes: D82, D86, D90, D91
Keywords:

adverse selection, inferential naivety, mechanism design, behavioral contract theory,

consumer protection*

Email: murooka@osipp.osaka-u.ac.jp†

Email: takuro.yamashita@tse-fr.eu

Optimal Trade Mechanisms with
Adverse Selection and Inferential Mistakes∗
Takeshi Murooka†

Takuro Yamashita‡

September 4, 2021
Abstract
We study an adverse selection environment, where a rational seller
can trade a good of which she privately knows its value to a buyer, andthere are gains from trade. The buyer’s types differ in their degree ofinferential abilities: A rational type correctly infers the value of thegood from the seller’s offer, whereas a naive type under-appreciates
the correlation between the seller’s private information and offer. We
characterize the optimal menu mechanism that maximizes the social
surplus. Notably, no matter how severe the adverse selection is (inparticular, even when no trade is the unique possible outcome if allagents are rational), all types of buyers trade in the optimal mechanism. The rational buyer’s trade occurs at the expense of the naive
buyer’s losses. We also investigate a consumer-protection policy oflimiting the losses and discuss its implications.

∗

We are grateful to Paul Heidhues, Botond Kőszegi, Klaus Schmidt, and seminar andconference audiences for helpful comments. Takeshi acknowledges financial support from
JSPS KAKENHI (JP18H03640, JP19K01568, JP20K13451). Takuro acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (No. 714693), and ANR under grant ANR-17-EURE0010 (Investissements d’Avenir program). All remaining errors are our own.
†
Osaka School of International Public Policy, Osaka University. murooka@osipp.osakau.ac.jp‡
Toulouse
Schoolof
Economics,
UniversityofToulouse,
Capitole.
takuro.yamashita@tse-fr.eu

1

1

Introduction

From the seminal work of Akerlof (1970), adverse selection and resulting
market collapses have been extensively investigated. Samuelson (1984) andAttar, Mariotti, and Salanié (2011) show that, under a certain condition,a no-trade outcome is the only incentive-feasible allocation. These resultsbuild on a classical assumption that all parties are fully rational, in particular, an uninformed buyer can correctly infer the seller’s type from observing
her offer.1 However, recent experimental and empirical evidence suggeststhat some buyers are systematically biased in that they fail to recognize
the informational content of others’ actions.2 Eyster and Rabin (2005) propose a solution concept which incorporates such inferential mistakes (calleda cursed equilibrium) in which a naive agent fails to infer other agents’ private information from their actions.3 Under severe adverse selection (i.e., anenvironment where only no-trade outcome is possible by rational agents) andthe homogeneous type of buyer, Eyster and Rabin (2005) show that trade
can occur between such naive buyers and (relatively) low-value sellers.4
Building on, but beyond these studies, we analyze one-sided adverse selection problems with heterogeneous inferential abilities. In the model, arational seller has a single unit of a good v ∈ [0, 1], which is her private
information. Gains from trade is denoted by α > 1 in that a buyer’s value ofthe good is αv. Generalizing Eyster and Rabin (2005), we consider an envi1

We use female pronouns to refer to a seller and male pronouns to refer to a buyer.
See, for instance, Brown, Camerer, and Lovallo (2012), Enke and Zimmermann (2019),
Enke (2020), and Jin, Luca, and Martin (2021). There is also accumulating evidence that,in some industries, a fraction of buyers make suboptimal inferences and purchase decisions;see Heidhues and Kőszegi (2018) on consumer behavior in markets and Beshears, Choi,
Laibson, and Madrian (2018) on household decision makings.
3
For theoretical works applying the concept of cursed equilibrium, see Eyster, Rabin,and Vayanos (2019), Kondor and Kőszegi (2017), and Ispano and Schwardmann (2017).
4
As related theoretical studies, Jehiel (2005) and Jehiel and Koessler (2008) propose
an analogy-based expectation equilibrium in which an agent bundles unobserved states into
analogy classes. Spiegler (2016) develops a model of non-rational expectations based on a
Bayesian-network approach. For reviews, see Eyster (2019) and Jehiel (2020).
22

ronment where the buyer is either a rational type (i.e., a standard Bayesiantype) or a naive type (i.e., he under-appreciates the correlation between theseller’s private information and action).
It turns out that environments with the co-existence of rational and naive
types of buyers are fundamentally different from those with either a rationalor naive type only. Notably, even with severe adverse selection where no
trade would be the unique possible outcome if only rational agents existed,
we show that, with the co-existence of rational and naive types, there exists
a mechanism in which both types of buyers trade with positive probability. Intuitively, the trade between the naive buyer and the low-value seller
generates positive externalities to other types: given that the (relatively)
low-value seller trades with the naive buyer, there exists a mutually agreeable offer between the (relatively) high-value seller and the rational buyer,
which generates a positive trade surplus.
We characterize optimal menu mechanisms when the buyer’s type of inferential ability is heterogeneous. We show that the optimal mechanism always
involves the rational buyer’s trade with positive probability. The mechanism has the double-separation property: The seller types are separated sothat the (relatively) low-value seller offers a low-price option that is tradedwith the naive buyer, and the higher-value seller takes a higher-price optionthat is traded with the rational buyer. When the buyer’s value follows either the binary distribution or the power distribution (including the uniform
distribution), the optimal menu mechanism has only two trading options.
With a modest condition, the mechanism achieves a strictly higher expected
total surplus than the one-option mechanism, and with cross-subsidization
between different types of the buyer, ex ante Pareto improvement may also
be possible.
Nevertheless, we show that it is impossible for both types of buyers tomake non-negative expected payoffs. It implies that the naive buyer’s loss
is necessary for achieving a positive trade surplus.5 Hence, as a potential
5

In a companion paper (Murooka and Yamashita, 2020), we show that this impossibility

3consumer-protection policy which can go together with the positive trade surplus, we investigate partial protection of the buyer in that the naive buyer’s
loss in terms of his (actual) expected payoff is limited to a certain amount.
We characterize the optimal menu mechanism under the consumer protection. The optimal mechanism still exhibits the double-separation property,
and it highlights a new trade-off between the naive buyer’s protection andthe social surplus.
This paper belongs to the growing literature on behavioral mechanism
design and behavioral contract theory.6 As most closely related to ours,
Eliaz and Spiegler (2006, 2008) and Heidhues and Kőszegi (2010) investigate
optimal contract designs with different degrees of agent naivety. In contrast
to these studies, we focus on the market for lemons and shed light on the
new positive externality to rational agents, as well as discussing its economic
implications.
Our results are also related to the literature on behavioral industrial organization, in particular, interplays between rational and naive consumers.
Most studies have focused on negative externalities imposed by rational consumers, e.g., cross-subsidization from naive to rational consumers (Gabaix
and Laibson, 2006; Heidhues, Kőszegi, and Murooka, 2017; Jehiel, 2018).7
For positive externalities from naive to rational consumers, de Clippel, Eliaz,
and Rozen (2014) and Johnen (2020) show that equilibrium prices can belower if there are more naive consumers in the market. Our result highlights a positive externality from naive to rational consumers too, but its
logic is quite different: We show that the trade between a rational buyer
and a (relatively) high-type seller occurs in the market for lemons, becausea (relatively) low-type seller trades with a naive buyer. Based on this novel
separation property, we investigate the welfare implications of the optimal
result holds for any kind of behavioral errors.
6
See Kőszegi (2014) for a review.
7
Herweg and Müller (2016) analyze the market for lemons with an overconfident buyer.
Different from our results, they show that a high-quality good is traded with the overconfident buyer in equilibrium, whereas a rational buyer is crowded out of the market.

4menu mechanism, the impossibility of full consumer protection, and the effectof partial consumer protection.
This paper is organized as follows. In Section 2, we set up a model of
one-sided adverse selection and introduce each type of agents. In Section 3,
we illustrate the properties of the optimal trade mechanism when the buyer’svalue follows the binary distribution. In Section 4, we analyze the optimal
trade mechanism under general distributions and its properties. In Section
5, we investigate partial protection of the (naive) buyer. Section 6 discusses
other extensions and concludes.

2

Model

A seller has a single unit of a good, whose value to her is v ∈ [0, 1] as her
private information. A buyer’s value is αv. The buyer does not know v, but
only knows that it follows a distribution with cdf F . Let µ = E[v] denote
its mean, and in case F admits a density, let f denote its density. They arerisk neutral in monetary transfers. If the good is traded with probabilityq ∈ [0, 1] with monetary transfer p ∈ R, then the seller’s payoff is p − vq, andthe buyer’s payoff is αvq − p. Hence, the social surplus from trade is given
by S = (α − 1)vq. We assume α > 1 so that it is socially optimal to always
make them trade. We also assume that α is not too large:
αE[v|v ≤ p] − p < 0,(1)

for all p > 0.8 As is well-known in the literature, Condition (1) implies severe
adverse selection in the sense that only no-trade outcome can be incentivefeasible (if the agents are rational). To provide some intuition, note that, if
the reverse of this inequality holds for some p > 0, then there is a simple
posted-price scheme that induces some trading: Any seller type v ≤ p would
wish to trade with price p, and the buyer would accept it as his expected
8

For example, if v follows a uniform distribution, then the condition is equivalent to
α < 2.

5

payoff is E[αv − p|v ≤ p] ≥ 0. Condition (1) says that, except for p = 0,such a mutually-agreeable price does not exist.
In the main part of the paper, we consider the following class of “menumechanisms” as possible trading protocols. A menu mechanism is denotedby (qm , pm )M
m=0 , where M ∈ N, (q0 , p0 ) = (0, 0), and for each m = 1, . . . , M ,qm ∈ [0, 1] and pm ∈ R. In the mechanism, the seller (knowing his v) chooses
one option in the menu, say m. Then, the buyer, without knowing v but
observing the seller’s choice m, either accepts it or rejects it. If accepted, then
the trade happens with probability qm with monetary transfer pm ; otherwise,
no trade occurs.9 Our goal is to obtain insights about the mechanisms thatare optimal in terms of the expected social surplus.
The key element of the analysis is the buyer’s belief updating given theseller’s choice in the menu. We assume that the buyer has one of the two
types: a rational type, or a naive type. The rational type is a standard“Bayesian rational” type: Once he sees the seller’s choice of (qm , pm ), hecorrectly updates his belief about the seller’s value v, and then makes anacceptance decision. The naive type, on the other hand, (incorrectly) believes
that the seller’s choice of (qm , pm ) does not convey any information about v
to him, and hence, he makes an acceptance decision without updating his
belief. Let ψ ∈ [0, 1] denote the fraction of the rational type among thebuyers, and 1 − ψ denote the fraction of the naive type. Our formulationof the naive type coincides with the fully cursed agent in Eyster and Rabin(2005), as well as the coarsest analogy partition and the private information
analogy partition in Jehiel and Koessler (2008). We discuss several kinds ofextensions incorporating a partially-naive type in Section 6.1.
Given each v, let σ(v) ∈ {0, . . . , M } denote the seller’s choice. Given
the seller’s choice m ∈ {0, . . . , M }, let dr (m), dn (m) ∈ {0, 1} denote the9

The class of mechanisms may appear restrictive; for example, the buyer cannot input
any message other than his acceptance decision. Nevertheless, as long as a mechanism
follows the posterior implementability developed by Green and Laffont (1987), the qualitative feature of the paper would not change with a more general class of mechanisms.
We discuss this point in Section 6.2.

6

rational and naive buyer’s acceptance decisions, respectively. A strategy
profile (σ(·), dr (·), dn (·)), or simply (σ, d), is a (Bayesian) equilibrium if itsatisfies:• (IC) Incentive compatibility for the seller: For each v and m ∈ {0, . . . , M },pσ(v) − vqσ(v) [ψdr (σ(v)) + (1 − ψ)dn (σ(v))]≥ pm − vqm [ψdr (m) + (1 − ψ)dn (m)];• (IR-r) Individual rationality for the rational type of buyer: For eachm ∈ {0, . . . , M },Z 1
dr (m)(αvqm − pm )1{σ(v)=m} dF
0
Z 1≥ (1 − dr (m))(αvqm − pm )1{σ(v)=m} dF ;0

• (IR-n) Individual rationality for the naive type of buyer: For eachm ∈ {0, . . . , M },dn (m)(αµqm − pm ) ≥ (1 − dn (m))(αµqm − pm ).
Given an equilibrium, the expected social surplus is:
Z 1(α − 1)vqσ(v) [ψdr (σ(v)) + (1 − ψ)dn (σ(v))]dF,
0and therefore, an optimal mechanism is given by solving the following problem:
Z 1max(α − 1)vqσ(v) [ψdr (σ(v)) + (1 − ψ)dn (σ(v))]dF
0sub. to(IC), (IR-r), (IR-n).

Two benchmark results are well-known in the literature. First, if only
the rational type exists (ψ = 1), then as in Samuelson (1984), a no-trademechanism is optimal under Condition (1): q(v) = p(v) = 0 for all v ∈ [0, 1].
7

Second, if only the naive type exists (ψ = 0), then as in Eyster and Rabin(2005), an optimal mechanism exhibits M = 1 (i.e., one-price mechanism)with (q1 , p1 ) = (1, αµ): In the mechanism, the seller with v > αµ chooses(q0 , p0 ) = (0, 0), and hence no trade happens regardless of the buyer’s choice;the seller with v < αµ chooses (q1 , p1 ) = (1, αµ), and the (naive) buyer
accepts it. The expected total surplus is:
Z αµ(α − 1)vdF ≥ 0.
0

Despite its trade possibility, note that the buyer’s expected payoff is necessarily negative due to Condition (1):Z αµ(αv − αµ)dF
0= αE[v|v ≤ αµ] − αµ < 0.

3

Illustration with Binary-Value Distribution

Some of the main results of the paper (though not all) can be simply illustrated with a binary-value distribution: v = 1 with probability µ ∈ (0, 1)
and v = 0 with probability 1 − µ (and hence, E[v] = µ). In this case,
Condition (1) is equivalent to αµ < 1.
In the binary case, both benchmarks yield zero expected surplus: Even if
only the naive type exists (ψ = 0), only the seller with v = 0 wishes to trade.
So the resulting surplus is zero and the buyer earns a negative payoff −αµ.
Now we illustrate that, if ψ ∈ (0, 1), then there exists a mechanism that
achieves strictly positive expected surplus. In this sense, environments withco-existence of rational and naive types are fundamentally different fromthose with either a rational or naive type only.
αµ. Consider the menu mechanism
We first analyze the case with ψ < 1+αµ
where the seller chooses (q, p) ∈ {(0, 0), (1, 1), (1, αµ)} (and the buyer accepts
or rejects what the seller chooses). It is easy to verify that the following
constitutes an equilibrium: The seller chooses (q, p) = (1, 1) if v = 1, while8(q, p) = (1, αµ) if v = 0; and the rational buyer accepts only (q, p) = (1, 1),
while the naive buyer accepts only (q, p) = (1, αµ).10
The trading possibilities are summarized in the following table:
naive
v=0
X
v=1

rational
X

The expected surplus is now strictly positive: ψ(α−1) > 0. The intuitionis simple. In this mechanism, the seller with v = 0 is matched with the naive
buyer with price αµ. Given that they are out, now only the seller with v = 1
remains. Therefore, we can find a mutually agreeable price (e.g., p = 1) for
the seller with v = 1 and the rational buyer, and their trade creates positive
trade surplus. In this sense, the existence of the naive type generates a“positive externality” to the high-value seller and the rational buyer. On
the other hand, note that this positive externality is at the cost of the naive
buyer’s loss.
αµThe expected surplus is increasing in ψ as long as ψ < 1+αµ, but thisαµmechanism no longer works once ψ becomes higher than 1+αµ . This is beαµcause, with ψ > 1+αµ, even the low-value seller prefers (q, p) = (1, 1) to(1, αµ). In this case, we can modify the mechanism by partially suppressing
the trade with price p = 1 in order to restore the incentive feasibility. Morespecifically, consider the modified menu mechanism where the seller chooses(q, p) ∈ {(0, 0), (q ∗ , 1), (1, αµ)}, where q ∗ = (1−ψ)αµ< 1. In this mechanism,
ψthe same “separating” strategy profile as above continues to be an equilibrium: The seller with v = 0 chooses (q, p) = (1, αµ) which is only acceptedby the naive buyer; while the seller with v = 1 chooses (q, p) = (q ∗ , 1) which
10

It is immediate that the naive buyer accepts only (q, p) = (1, αµ). The rational buyer
accepts only (q, p) = (1, 1) because (q, p) = (1, 1) means that the seller’s type is v = 1 (andhence trading yields the rational buyer’s expected payoff of α − 1 > 0). Given the buyer’s
behavior, the seller with v = 1 earns zero payoff by choosing (1, 1), while choosing (1, αµ)yields a negative expected payoff; the seller with v = 0 prefers (1, αµ) to (1, 1) despite its
αµlower price (recall αµ < 1), because ψ < 1+αµ
is equivalent to (1 − ψ)αµ > ψ and theseller with v = 0 prefers to sell the good to the naive buyer who only accepts (1, αµ).

9is only accepted by the rational buyer. In particular, note that the seller
v = 0 (weakly) prefers (q, p) = (1, αµ) to (q ∗ , 1) because (1 − ψ)αµ ≥ ψq ∗ .
The expected surplus is again strictly positive:ψq ∗ (α − 1) = (1 − ψ)αµ(α − 1) > 0.
Note that this is decreasing in ψ, and in particular, converging to zero asψ → 1, consistent with Samuelson (1984).
In both of the cases, the mechanisms are doubly-separating: The seller
types are separated so that the low-value seller takes a low-price option thatthe high-value seller does not take, and the high-value seller takes a highprice option that the low-value seller does not take. The buyer types are
separated too so that the naive type only accepts the low-price option, while
the rational type only accepts the high-price option. The low-value seller
is matched with the naive type, and the high-value seller is matched withthe rational type. In the next section, we show that this double-separation
property is a necessary feature of the optimal mechanism with a general classof distributions.
In terms of the induced trade surplus, there are several observations thatdo (or do not) continue to hold with more general distributions. First, because of the positive externality made by the naive buyer, the rational buyer
earns a strictly higher expected payoff than under the “one-option” menu
mechanism, {(qm , pm )}M
m=0 = {(0, 0), (1, αµ)}. On the other hand, the naive
buyer and the seller continue to earn the same expected payoff. In this sense,
relative to the optimal mechanisms in the two benchmarks, our mechanism
attains Pareto improvement. With a general class of distributions, it is notnecessarily the case, because the existence of the rational type may cause anegative externality to the naive type, which does not occur in the binary
case.
Second, the expected surplus is non-monotone in ψ, and more importantly, the peak is with an interior ψ. We think it has an important implication in consumer education: The peak with ψ > 0 means that someconsumer education could be socially beneficial, depending on the “current”
10ψ. With a general class of distributions, the peak may or may not be withan interior ψ.
Finally, in the optimal mechanism above, the naive buyer earns a strictly
negative payoff by matching with the low-value seller, while the rational buyer
earns a strictly positive payoff. Thus, there might seem to be a room for crosssubsidization to make both buyer types better off (in the actual expected
payoff).11 However, we show that such cross-subsidization is impossible,
regardless of the (binary or non-binary) distribution. That is, if a mechanism
achieves strictly positive expected surplus, it necessarily implies that thenaive buyer earns a strictly negative expected payoff.

4

Doubly-Separating Menu and Its Welfare
Properties

In this section, we first show that the double-separation property is a necessary feature of the optimal mechanism with a general class of distributions.
We then investigate welfare properties of the optimal mechanism.

4.1

Main theorem

We first state our main theorem that characterizes the optimal menu mechanism {(qm , pm )}M
m=0 , where (q0 , p0 ) = (0, 0) is (if any) chosen by the seller
with the highest values and (qM , pM ) = (1, αµ) is chosen by the seller withthe lowest values.
Theorem 1. An optimal menu mechanism {(qm , pm )}M
m=0 has (i) M ≥ 1,pm+1pm(ii) (qM , pM ) = (1, αµ), (iii) qm > qm+1 for m = 1, . . . , M − 1, and is (iv)associated with the following equilibrium: There exists v ∗ ≤ αµ such that theseller with v < v ∗ chooses m = M , and the seller with v > v ∗ chooses m 6= M ;11

Recall that each type of the buyer’s perceived expected payoff is always non-negative,
although it is not correct for the naive type ex post.

11

the naive buyer accepts only (qM , pM ), and the rational buyer accepts any(qm , pm ) except (qM , pM ).
As in the binary case, the mechanism separates the seller types based onsome threshold type v ∗ . The optimal v ∗ depends on specific distributions.
It also separates the naive buyer and the rational buyer. The fundamental
intuition for this mechanism to achieve a positive surplus is similar to thebinary case: The seller with v < v ∗ is matched with the naive type bychoosing a lower-price option, (qM , pM ) = (1, αµ). Given this, the seller withv > v ∗ can trade with the rational buyer with higher-price options (qm , pm )where m = 1, . . . , M − 1.
In general, different from the binary distribution case, there may exist
multiple options that further separate the high-value seller types. For acertain class of environments, however, at most two options (in addition to
the no-trade option) would be enough.
Example 1. Let F be a power distribution, F (v) = v γ with γ > 0.12 Then,γ
and Condition (1) becomes αµ < 1.
µ = 1+γ
In the power-distribution case, the optimal mechanism is a doubly-separating
menu mechanism with only two trading options, as summarized in Proposition 1.
Proposition 1. Let F be a power distribution, F (v) = v γ , with γ > 0 and
αµ < 1. Then, the optimal menu mechanism is {(qm , pm )}M
m=0 with M = 2,
(q0 , p0 ) = (0, 0), p1 > αµ, and (q2 , p2 ) = (1, αµ).
There are three properties in the optimal mechanism with the binary
distribution: (i) The optimal mechanism (a two-option mechanism in thebinary case) Pareto dominates the one-option mechanism; (ii) the expected
total surplus is maximized at an interior ψ; and (iii) the naive-type buyer
necessarily makes a loss. In the following subsections, we reexamine those
properties in more general non-binary environments.
12

A uniform distribution is a special case with γ = 1.

12

4.2

Suboptimality of one-price mechanism

The above theorem does not exclude a possibility that the one-option mechanism is optimal. The next proposition provides a sufficient condition withwhich a strictly better mechanism than the one-price mechanism exists.
Proposition 2. If vf (v) is (locally) strictly increasing at v = αµ, then the
optimal mechanism achieves strictly higher expected social surplus than the
one-option mechanism.
Note that for any differentiable f (·), (vf (v))0 = f (v) + vf 0 (v), and thesecond term approaches 0 as v approaches 0. Hence, many popular distributions satisfy the condition if αµ is sufficiently small. Proposition 2 implies
that inducing the trade between a (relatively) high-value seller and a rationalbuyer is surplus-enhancing. To provide some intuition, consider the following two-option mechanism: {(0, 0), (q, p), (1, αµ)}. The optimal one-option
mechanism does not have the option (q, p), and thus, under the one-option
mechanism, any seller with type v < αµ trades with the naive buyer, while
any seller type with v > αµ does not trade. With the additional option(q, p) with p > αµq, any seller type with v < v ∗ trades with the naive buyer,
and any seller type with v ∈ (v ∗ , p) trades with the rational buyer. Thatv ∗ < αµ may be interpreted as a “skimming” phenomenon: The seller withv ∈ (v ∗ , αµ) corresponds to the high-end of those types who would tradewith the naive buyer without the new option (q, p). With the new option,these sellers now trade with the rational buyer, and both the seller and therational buyer are better off with this new option.

4.3

Pareto improvement relative to the one-option mechanism

Recall that, with a binary distribution, the two-option mechanism achieves“ex post” Pareto improvement relative to the one-option mechanism: The
rational buyer is better off, and the naive buyer and the seller are indifferent.
13With continuous distributions, it is impossible for the optimal mechanism to attain ex post Pareto improvement relative to the one-option mechanism, because the above skimming logic clearly implies that the naive buyer
is worse off without cross-subsidization. However, it may be possible toachieve “ex ante” Pareto improvement: The buyer’s average expected payoff of both types (with these types’ probability weights) may be improved.
And of course, if the buyer is ex ante Pareto improved, then appropriate
cross-subsidization can make all the buyer types ex post Pareto improved.
Example 2.
1. By a continuity argument, even if F is continuous, if F
is not too far from a binary distribution (e.g., a continuous bi-modal
distribution with peaks close to v = 0 and v = 1), the optimal mechanism can achieve ex ante Pareto improvement relative to the one-option
mechanism, and hence ex post Pareto improvement after appropriate
cross-subsidization.
2. Let F be a power distribution, where F (v) = v γ with γ > 0. Then, under the optimal mechanism with any incentive-feasible cross-subsidization,
the payoff of the naive type is strictly lower than the one-price mechanism.13 Intuitively, this is because the seller receives most of the gainsof the social surplus.

4.4“Best mixture” of behavioral types and consumer
education

As in the two benchmark cases, the expected surplus is zero if ψ = 1, and it
Rα
is 0 µ(α − 1)vdF if ψ = 0. Recall that, with a binary distribution, the social
surplus is maximized with an interior ψ. This has an important implication
for consumer education: If the “current” ψ is less than that achieves the13

Note that any cross-subsidy from a seller to a buyer is not possible in the optimalmechanism, as the seller’s payoff for each v is uniquely pinned down up to a constant, andthe seller with v = 1 earns zero payoff under the above optimal mechanism.

14

peak, then consumer education can increase the social surplus.14
Not all distributions induce an interior peak of ψ. For example, if v ∼U (0, 1) and α ∈ ( 34 , 2), then the social surplus in the optimal mechanism
is decreasing in ψ. Therefore, any consumer education reduces the social
surplus in this case.
A sufficient condition for an interior peak is that the expected social
surplus is increasing as a function of ψ, locally around ψ = 0:
Proposition 3. If there exists p > αµ such that:Z αµZ pvf (v)dv,(vf (v) − αµf (αµ))dv >0αµthen the expected social surplus is maximized at some interior ψ.
To prove the result with some intuition, consider the following menu
mechanism: {(qm , pm )}M
m=0 = {(0, 0), (1, p), (1, αµ)} with p > αµ. In theassociated equilibrium, the seller with v < v(ψ) chooses (1, αµ), which isaccepted only by the naive buyer; while the seller with v ∈ (v(ψ), p) chooses(1, p), which is accepted only by the rational buyer, where v(ψ) represents
the indifferent type:ψ(p − v(ψ)) = (1 − ψ)(αµ − v(ψ)) ⇔ v(ψ) = αµ −p − αµψ,
1 − 2ψ

which is positive if ψ is sufficiently close to 0.
The expected social surplus is increasing locally around ψ = 0 if:"#
Z v(ψ)
Z 1∂(1 − ψ)(α − 1)vdF + ψ(α − 1)vdF |ψ=0
∂ψ
0
v(ψ)
 Z αµZ p=(α − 1) −vdF − αµf (αµ)(p − αµ) +vdF > 0.
0

(2)

αµ

14

However, a careful subsidy scheme may be necessary in its execution. Obviously, those
who get educated are better off, while the other kinds of buyers, that is, those who are noteducated (either naive before and after the policy, or rational before and after the policy),
can be worse off.

15

The first term in the square brackets in 2 is a direct effect of decreasing
a fraction of the naive buyer: The surplus created by the naive type tradingwith the seller with v < αµ is lost, if a fraction of the naive buyer in thesociety is replaced by the rational buyer. Note that this negative effect does
not exist in the binary case, because no surplus is directly created by thenaive buyer (because he trades with the seller with v = 0 only). The third
term in the square brackets in 2 is a direct effect of increasing the rationalbuyer, because this type of the buyer can enjoy a positive externality madeby the naive type. The second term in the square brackets in 2 is negative
externality made by the rational type on the naive type, based on the above“skimming” logic: With a higher fraction of the rational type in the society,the threshold type of the seller is lower, which implies a larger negative
externality.
Example 3. Let F be a power distribution: F (v) = v γ with γ > 0. If γ < 15and α < 54 , then the expected social surplus is maximized at ψ > 0.

4.5

Exploitation of naivete

As in the binary- or power-distribution cases, in the optimal menu mechanism, the naive buyer earns a negative expected payoff, while the rationalbuyer can earn a strictly positive expected payoff. This suggests a possibilityof cross-subsidization from the rational buyer to the naive buyer, so thatboth types can enjoy non-negative expected payoffs.
The next result shows that this cannot be true in a strong sense. In
any mechanism (not only the one that maximizes expected social surplus,
but any other one, such as the one that maximizes only the buyer surplus)
with non-trivial trading, the naive buyer’s expected payoff must be strictly
negative.
Proposition 4. If there exists a menu mechanism and its associated equilibrium where the buyer’s ex-ante expected payoff is non-negative, then the16expected social surplus must be 0, and in this sense, it is equivalent to theno-trade outcome.
Remark 1. The proof of the above result does not formally consider crosssubsidization, but it is immediate that the result implies impossibility of making both types earn non-negative expected payoffs with cross-subsidization.
Indeed, if both types earn non-negative expected payoffs with cross-subsidization,then it must be the case that the buyer’s ex ante expected payoff is nonnegative, contradicting the proposition. In a companion paper (Murooka andYamashita, 2020), we extend the above result: For any type of behavioral
errors naive buyers make and for any incentive-feasible mechanism, either the
expected payoff of some naive buyer must be strictly negative or the no-tradeoutcome occurs.
Proposition 4 implies that, to achieve a positive social surplus, we cannot“fully” protect the naive buyer by giving him a non-negative payoff. Still,
it may be possible to “partially” protect the naive buyer by preventing himfrom a large negative payoff. We investigate this issue in Section 5.

5(Partial) Protection of Naive Buyer

Proposition 4 in the previous section shows that a positive social surplus
and a non-negative payoff of the naive buyer cannot go together, so we cannot “fully” protect the naive buyer under trade. However, it is possible to“partially” protect the naive type in the sense of avoiding large losses, bygiving up (but not entirely) some trade surplus. In practice, a policymaker’s
goal may be to find a right balance between the social surplus and consumer
protection.
In this section, we study the optimal mechanism under partial protectionof the naive buyer. Specifically, consider the same surplus maximization
problem as before, but with the additional constraint that the naive buyer’s

17

(actual) expected payoff is bounded from below by some fixed −L ≤ 0:Z
(αvqσ(v) − pσ(v) )dc (σ(v)) ≥ −L.
v

Call it Problem (1; L). The previous sections consider the case with sufficiently large L, and Proposition 4 implies that L = 0 is possible only under
the no-trade mechanism. We now investigate the case with intermediate
L > 0.
An alternative, equivalent problem is that the principal maximizes aweighted sum of the social surplus and the naive buyer’s (actual) expected
payoff, where the weight is one for the social surplus and φ ≥ 0 for the naive
buyer’s expected payoff. Call it Problem (2; φ). A standard Lagrangian argument shows that each Problem (1; L) is equivalent to Problem (2; φ) forsome φ, and vice versa. This Problem (2; φ) can be interpreted as a situation
where the principal has a higher “Pareto weight” for the naive buyer.
We show that the optimal menu mechanism is again doubly-separating,where the naive buyer trades with relatively low-value sellers (only) and therational buyer trades with relatively high-value sellers (only). In this sense,
the qualitative nature of the optimal mechanism is the same as in Theorem
1. However, in order to protect the naive buyer, the terms of trade may be
modified.
Proposition 5. An optimal menu mechanism {(qm , pm )}M
m=0 has M ≥ 1,∗
and there exists m ∈ {1, . . . , M } such that (qm , pm ) ≤ (1, αµ) for m ≥ m∗and pm > αµ for m < m∗ . It is associated with the following equilibrium:
There exists v ∗ such that the seller with v < v ∗ chooses some m ≥ m∗ , andthe seller with v > v ∗ chooses some m < m∗ .
Example 4. Recall the binary example with v ∈ {0, 1} with Pr(v = 1) = µand αµ < 1. The optimal menu mechanism with any L induces the same
trading pattern: The seller with v = 0 trades with the naive buyer, and theseller with v = 1 trades with the rational buyer. However, the allocationthat the low-value seller offers is not necessarily (q, p) = (1, αµ), in order toprotect the naive buyer.
18

Let (q, p) ≤ (1, αµ) be the allocation that the low-value seller offers andthe naive buyer accepts; and let (q 0 , 1) be the allocation that the high-value
seller offers and the rational buyer accepts.15 Then, the surplus-maximization
problem becomes as follows:max
0q,q ,psub. to(α − 1)ψq 0(1 − ψ)qp ≥ q 0 ψ,p ≤ αµ,−qp ≥ −L,

where the first constraint is the low-value seller’s incentive compatibility, thesecond constraint is the naive buyer’s individual rationality, and the third
constraint corresponds to the protection of the naive buyer. The other constraints (i.e., the high-value seller’s incentive compatibility and the rationalbuyer’s individual rationality constraints) are omitted because they do notbind.
When L is sufficiently large (i.e., L ≥ αµ), the constraint of protecting
the naive buyer does not bind. Indeed, with L ≥ αµ, the same optimalmechanism obtains: q = 1, p = αµ, and q 0 = min{1, 1−ψαµ}.
ψWhen L < αµ, the constraint of protecting the naive buyer is binding.
L}. First,
An optimal mechanism is q = 1, p = L, and q 0 = min{1, 1−ψψthis new mechanism protects the naive buyer by limiting the price to be L
instead of αµ. Second, as a consequence, the trade probability between thehigh-value seller and the rational buyer can also be limited. Because of thissecond effect, the expected social surplus may become lower, but instead, thenaive buyer is better off.
In Example 4, imposing L close to αµ does not reduce the social sur15

It is without loss to assume that the high-value seller’s price in the optimal menu
mechanism is 1. First, it cannot be strictly less than 1 because of the seller’s individual
rationality. Second, for any allocation for the high-value seller (and the rational buyer)
with price strictly greater than 1, we can achieve a weakly higher expected surplus bylowering the price (and keeping the other parts of the allocations the same).

19

plus by much, while it increases the naive buyer’s payoff. Indeed, when1−ψαµ < 1, such a regulation affects neither the rational buyer’s payoff nor
ψthe social surplus. In this sense, this consumer protection with a moderatelevel of L > 0 is in line with asymmetric paternalism: it benefits consumers
who may make errors, while it imposes no (or relatively little) harm on rational consumers (Camerer, Issacharoff, Loewenstein, O’Donoghue, and Rabin,
2003).
The above consumer protection provides a rationale for caps on contract
attributes and other regulations which moderately limit the ex-post lossesof buyers. For example, the European Union’s principle on unfair terms in
consumer contracts prohibits a contractual term that “causes a significant
imbalance [...] to the detriment of the consumer,” so a transaction that is too
disadvantageous to the consumer is disallowed.16 Also, the Consumer Affairs
Agency in Japan stipulates that consumer contract clauses with “a fixedpenalty for contract cancellation in a total amount that exceeds the averageamount of damages that the trader would incur from the cancellation” arevoid, which effectively limits the losses of buyers.17

6

Discussion and Concluding Remarks

In this section, we discuss (i) a model that incorporates a partially-naive
buyer and (ii) analysis under more general mechanisms. While a thoroughinvestigation of these exercises is beyond the scope of this paper, we illustratea framework for the analysis and give a couple of remarks.
16

Article 3 of directive 93/13/EEC of the Council of the European Commissions, available at https://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=
CELEX:31993L0013:en:HTML#d1e245-29-1 (accessed September 1, 2021).
17
Article 9 of the Consumer Contract Act (Japan), available at http://www.
japaneselawtranslation.go.jp/law/detail/?id=3578&vm=&re= (accessed September
1, 2021).

20

6.1

Partial naivety

So far, we have analyzed the model in which a naive buyer is fully naive inthe sense of Eyster and Rabin (2005). We now discuss how our analysis canbe extended to cases in which a fraction of buyer is partially naive.
Motivated by Eyster and Rabin (2005), here we assume that a fraction
1 − ψ of the buyers is partially naive in the following sense (and the otherfraction ψ is rational, as before): Observing that the seller has chosen an
option m ∈ {0, . . . , M }, the partially-naive buyer’s anticipated value of thegood is a weighted average of µ with weight χ ∈ [0, 1] and µ̃m with weight1 − χ, where µ̃ is specified below. Note that our previous analysis covers thecase with χ = 1.
Then, (IR-n) becomes: For each m ∈ {0, . . . , M },dn (m){α[χµ + (1 − χ)µ̃m ]qm − pm }≥ (1 − dn (m)){α[χµ + (1 − χ)µ̃m ]qm − pm }.
The following specifications of µ̃m may be possible depending on the contexts of interest. The first possibility is that µ̃m is exogenously fixed and constant in m. In particular, when µ̃m = E[v|v < ṽ], it is as if the partially-naive
buyer believes that the seller with v > ṽ does not trade at any price. The
second, and perhaps more plausible, specification is that µ̃m = E[v|v < pm ].
In this case, with weight 1 − χ, the naive buyer believes that only a seller
whose value is lower than its associated price wishes to trade. In that sense,(with weight 1 − χ) the naive buyer takes into account the fact that rational
sellers never wish to trade at a price higher than own value. The third caseis, in line with the definition of the partial cursedness in Eyster and Rabin(2005), that µ̃m is based on the expected value. That is, µ̃m = E[v|v ∈ Vm ]where Vm is the set of the seller’s types who choose option m in equilibrium,
and in this sense, the partially-naive buyer’s expectation is a weighted average of the fully-naive case (with weight χ) and rational case (with weight1 − χ).
In all of these three applications, our main messages and insights remain
21

to hold: Naive buyers underestimate the selection and wish to trade at a
relatively low price (which is lower than αµ in these cases), while it createsa possibility of trades between (relatively) high-value sellers and rationalbuyers.

6.2

General mechanisms

In the main part of the paper, we only consider the class of menus. One
may wonder if more general mechanisms, for example, by asking the buyer
to reveal his behavioral type. In this section, we argue that the main qualitative results of the paper remain similar even if we consider more generalmechanisms as long as they satisfy certain features that we think are relevantin the current trading context.
Formally, define a general trading mechanism by (M1 , M2 ; A1 , A2 ; q, p)with finite M1 , M2 . Here, M1 is interpreted as the seller’s message set, M2 isinterpreted as the buyer’s message set, q : M1 × M2 → [0, 1] is the trading
probability, and p : M1 × M2 → [0, 1] is the transfer. For each i ∈ {1, 2},
the binary set Ai = {0, 1} represents i’s acceptance decision, and the tradewith (q, p) happens if and only if both i choose ai = 1. The timing of themechanism is as follows: First, each agent i simultaneously chooses mi ∈ Mi ,and after observing m = (m1 , m2 ), each agent i simultaneously chooses ai ∈Ai = {0, 1}. Consider a (weak) perfect-Bayesian equilibrium as the solution
concept.
Although this formulation might look non-standard, this is to capture thekey feature of the lemon’s problem that the buyer makes an inference about
the seller’s type observing the seller’s choice, and a trade is only upon mutual
agreement.18 See the seminal work by Green and Laffont (1987) which intro18

Indeed, if a mechanism can ask both the seller and the buyer report both mi and ai
simultaneously (which essentially means that the distinction between mi and ai is nonessential), then higher expected social surplus might be possible. As extensively discussedin Green and Laffont (1987), however, we think that the improvement is not well-justified
within the lemon’s trading context.

22

duces and studies some general properties of such posterior implementation.
This class of general trading mechanisms is richer than the class of menu
mechanisms considered in the paper. Indeed, a menu mechanism is a special
case of a general trading mechanism with the additional requirement of trivial
M2 . In this sense, it is potentially possible that some non-menu mechanism
strictly improves the optimal menu mechanism. Although further investigations of general mechanisms are left for future research, such mechanisms
should possess similar qualitative features as in the menu mechanism. In
particular, there must exist v ∗ > 0 such that the seller with v < v ∗ trades
only with the naive buyer. On the other hand, because the naive buyer does
not accept any trade with price greater than αµ, the seller with v > αµ can
trade only with the rational buyer. In this sense, the outcome must exhibit
some form of “double-separation” as in Theorem 1.
Proposition 6. In a (non-trivial) general trading mechanism (M1 , M2 ; A1 , A2 ; q, p),there exists v ∗ > 0 such that the seller with v < v ∗ trades only with the naive
buyer, and the seller with v > αµ trades only with the rational buyer.
For example, when a distribution is binary (i.e., v ∈ {0, 1} with αµ < 1),
Proposition 6 immediately implies that the menu mechanism considered in
the previous sections is optimal among all general trading mechanisms.

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26

A

Proofs

Proof of Theorem 1.
The following lemma collects some key properties of the equilibria. Fix any
menu (qm , pm )M
m=0 and any associated equilibrium (σ, d).
Let Vm ⊆ [0, 1] denote the set of the seller types who choose m, i.e.,
Vm = {v ∈ [0, 1]|σ(v) = m}. By the single-crossing property, each Vm must
be an interval.
Without loss of generality, assume that each Vm for m = 1, . . . , M is
non-trivial in that inf Vm < sup Vm ; and that m > m0 implies v ≤ v 0 for allv ∈ Vm , v 0 ∈ Vm0 .
Let v0 = 1, vm = sup Vm for each m 6= 0, and vM +1 = 0. Because each
Vm is non-trivial for m 6= 0, we have 1 = v0 ≥ v1 > · · · > vM +1 = 0.
For each m ≥ 1, define:
xm = (ψdr (m) + (1 − ψ)dn (m))qm .
ym = (ψdr (m) + (1 − ψ)dn (m))pm .
Without loss of generality, assume xm > 0 for all m = 1, . . . , M . Let m∗ =max{m|dn (m) = 0}, which is well-defined by assuming dn (q0 , p0 ) = 0 without
loss of generality.
Lemma 1.
1. ym − xm vm = ym−1 − xm−1 vm for m = 2, . . . , M ; and y1 −x1 v1 ≥ 0;
< xym
for m = 1, . . . , M − 1;
2. xm+1 > xm and xym+1
m
m+1
3. dr (M ) = 0, and hence, xM ≤ 1 − ψ;
4. ym ≤ αµxm if m > m∗ ;5. ym ≥ αµxm if 1 ≤ m ≤ m∗ ;6. xm∗ ≤ ψ;27

if m ≤ m∗ .
7. αE[v|v ∈ Vm ] ≥ xymm
Proof. The first property is implied by the seller’s incentive compatibility(for m ≥ 2) and individual rationality (for m = 1).
The second property is implied by the seller’s incentive compatibility andthe assumption that each Vm is non-trivial for m 6= 0.
For the third property, if dr (M ) = 1, then the rational buyer’s individual
rationality constraint implies:
0 ≤ αE[v|v ∈ VM ] − pM ≤ αE[v|v ∈ (0, vM )] − vM ,contradicting our assumption of severe adverse selection given by Inequality (1).
For the fourth property, because the naive buyer cannot accept any price
above αµ, we have ym∗ +1 ≤ αµxm∗ +1 . Then, the second property impliesym ≤ αµxm for all m > m∗ .
Conversely, for the fifth property, dn (m∗ ) = 0 implies ym∗ ≥ αµxm∗ , andby the first property, ym > αµxm if 1 ≤ m ≤ m∗ . This also implies dn (m) = 0
if 1 ≤ m ≤ m∗ .
The sixth property is immediate from dn (m∗ ) = 0.
For the last property, because m < m∗ , we have dn (m) = 0. However,
because xm > 0, it must be that dr (m) > 0. Thus, the rational buyer’s
individual rationality constraint implies αE[v|v ∈ Vm ]qm ≥ pm .
Now consider the problem of maximizing the expected social surplus∗
among all possible ((vm , xm , ym )M
m=1 , m ) for some M ≤ M that satisfy theseven properties above:
X Z vmxm (α − 1)vdF
max∗
M ≤M ; ((vm ,xm ,ym )M
m=1 ,m )sub. tom∗vm+1m ∈ {0, . . . , M }, xm ∈ [0, 1], 0 = vM +1 < . . . < v1 ≤ v0 = 1(Properties 1-7 above).

Observe that the above problem admits a solution, even though someconstraints are based on strict inequalities. First, consider a set of restricted28

problems where M is fixed (to any number between 1 and M ), and let Sol(M )denote the solution of this restricted problem with fixed M , in case it exists.
Let M ⊆ {1, . . . , M } denote the set of M where Sol(M ) exists. If this set
is nonempty, then clearly the solution of the unrestricted problem (i.e., withflexible M ≤ M ) is in {Sol(M )|M ∈ M}; and indeed, it is nonempty, because
Sol(1) exists: By Eyster and Rabin (2005), a one-price menu mechanism{(0, 0), (1, αµ)} is known to be optimal.
It is straightforward to see that a solution to the above problem implies
a menu mechanism that indeed achieves the maximized value.
Now we show that the solutions in the above (unrestricted) problem is adoubly-separating menu mechanism (qm , pm )M
m=0 (for some M ) with (q0 , p0 ) =(0, 0) and (qM , pM ) = (1, αµ).
Lemma 2. For the solution in the above problem, m∗ = M −1, yM = αµxM ,and xM = 1 − ψ.
∗Proof. Let ((vm , xm , ym )M
m=1 , m ) denote a feasible policy in the above problem such that one of the stated conditions is violated. If m∗ < M − 1, then−1
> xyM
, and thus, in what follows, we focus on the two cases:
αµ ≥ xyM
M −1M
either yM < αµxM , or xM < 1 − ψ.

Case 1: yM < αµxM∗
In this case, consider the alternative policy ((v̂m , x̂m , ŷm )M
m=1 , m̂ ) where,−1
for sufficiently small ε > 0: xŷM
= xyM
+ ε < min{ xyM, αµ}; v̂M (∈ (vM , 1))MM
M −1
is such that ŷM − xM v̂M = yM −1 − xM −1 v̂M ; and all the other components∗
are the same as ((vm , xm , ym )M
m=1 , m ). Clearly, this new policy is feasible in
the above problem, but achieves a strictly higher expected surplus, as theprobability of trade increases from xM −1 to xM for v ∈ (vM , v̂M ).
Case 2: xM < 1 − ψ.
∗In this case, consider the alternative policy ((v̂m , x̂m , ŷm )M
m=1 , m̂ ) where,
for sufficiently small ε > 0: x̂M = xM + ε(< 1 − ψ); v̂M (∈ (vM , 1)) issuch that xyMx̂M − x̂M v̂M = yM −1 − pM −1 v̂M ; and all the other components
M29∗
are the same as ((vm , xm , ym )M
m=1 , m ). Clearly, this new policy is feasible
in the above problem, but achieves strictly higher expected surplus, as theprobability of trade increases from xM to x̂M for v < v̂M .

Proof of Proposition 1.
Suppose that there is an optimal menu mechanism with M ≥ 3, which wedenote by {(qm , pm )}M
m=0 . It suffices to construct another optimal menu mechanism with M = 2.
m
M
is decreasing in m, pqM= αµ < . . . <It is without loss to assume that pqm
p1≤ 1, qM = 1, and that there exists a sequence of threshold seller types
q1
0 = vM +1 < . . . < v1 ≤ v0 = 1 such that any seller type v ∈ (vm+1 , vm )chooses option (qm , pm ) in the associated equilibrium. More specifically, we
have v1 ≤ pq11 , v2 is the seller type who is indifferent between (q1 , p1 ) and(q2 , p2 ), and so on. Without loss, we can assume that the rational buyer’s
individual rationality given (q2 , p2 ) is binding: αE[v|v ∈ (v3 , v2 )] = pq22 .19
Consider an alternative mechanism, which is exactly the same as the original one, except that, instead of (q1 , p1 ) and (q2 , p2 ), it offers (q20 , p02 ) (together
00with {(qm , pm )}Mm=3 ) where the same v3 is indifferent between (q2 , p2 ) and(q3 , p3 ), and the rational buyer’s individual rationality holds with equality,p0
that is, αE[v|v ∈ (v3 , k20 )] = k20 , where k20 = q20 . Let k2 = pq22 . In what follows,2
we focus on the case where k20 ≤ 1; the other case is similar.
The expected social surplus in the new mechanism is higher than in the
If αE[v|v ∈ (v3 , v2 )] > pq22 , then we can be strictly better off by modifying the mechanism as follows. Fix ε > 0 small, and consider an alternative mechanism {(q̂m , p̂m )}M
m=0 ,with p̂2 = p2 − ε, q̂2 = q2 − vε3 , and the rest is the same as {(qm , pm )}M
.
The
same
m=0
−p1 −εthresholds obtain for the seller types, except that threshold v2 becomes v̂2 = qp22−q
ε .
1− v
3
Note that all the constraints on the buyer side continue to be satisfied. The welfare change
R v2
with small ε is approximately (v2 − v3 )v2 f (v2 ) − v3 vdF > 0 as vf (v) is increasing for a
power distribution.
19

30

original mechanism by at least ∆, where:
∆ =

q20

Z k20

xdF − q1

xdF − q2

∝
v3

xdF
v2

v3

v3

Z k20

Z v1

Z v2

k2 − v3
xdF 0−
k2 − v3

Z v1

Z v2
xdF −

xdF
v2

v3

k2 − v2.
v1 − v2

Note that the new mechanism has exactly one less option than in the original
mechanism. Therefore, if ∆ ≥ 0, then we complete the proof by induction.
From here on, the notation is modified as follows: We use v instead of v3 ,βv instead of v2 , and δv instead of k20 . Note that β ≥ 1 is a free parameter,while δ is determined as a function of F and α:R δv
α v xdF.
δv =F (δv) − F (v)By construction, we have δ ≥ β ≥ 1.
We now show that ∆ ≥ 0. First, observe that v1 ≤ δβv. Thus:
Z δv
Z βv
Z v1
k2 − vk2 − βv
∆∝
xdF
−
xdF −
xdF
.
δv − v
v1 − βv
v
v
βv
Because xf (x) is increasing, this is at least:
Z δv

Z δβv

Z βv

k2 − βv
δβv − βv
βv
v
v
Z δβv
Z βv
Z δv
xdF (k2 − βv)∝ β
xdF (k2 − v) − βv
xdF (δ − 1) −k2 − vxdF−
δv − v

xdF −

v

v

xdF

βv

0

≡ ∆.
Note thatk2

R βv
α v xdF=
F (βv) − F (v)R βv(F (δv) − F (v)) v xdF
=
δv,
R δv
(F (βv) − F (v)) v xdF
31and thus,
R βv
R δv
δ(F (δv) − F (v)) v xdF − (F (βv) − F (v)) v xdF
v,
k2 − v =R δv(F (βv) − F (v)) v xdF
R βv
R δv
δ(F (δv) − F (v)) v xdF − β(F (βv) − F (v)) v xdF
k2 − βv =v.
R δv(F (βv) − F (v)) v xdF
Hence:


Z δv
Z βv
Z δv
0
xdF
xdF − (F (βv) − F (v))
xdF δ(F (δv) − F (v))
∆ ∝ β
v
v
v
Z βv
Z δv
−β
xdF (δ − 1)(F (βv) − F (v))
xdF
v
v


Z δv
Z βv
Z δβv
xdF
xdF − β(F (βv) − F (v))
xdF δ(F (δv) − F (v))
−
= β



Z δv
Z βv
xdF
xdF − (F (βv) − F (v))
xdF δ(F (δv) − F (βv))
βv

v

v

Z δβv
−

v

v

βv

Z δv



Z βv
Z δv
xdF δ(F (δv) − F (v))
xdF − β(F (βv) − F (v))
xdF .

βv

v

v

Rb

Because F (δv)F (βv) = F (δβv)F (v) and a xdF = c(bF (b) − aF (a)) for
any a < b for some constant c > 0, we finally have:


Z βv
Z δv
∆0 ∝ F (v) δ(F (δv) − F (βv))
xdF − (F (βv) − F (v))
xdF
v

βv



Z δv
Z βv
xdF
xdF − β(F (βv) − F (v))−F (βv) δ(F (δv) − F (v))
vv

= δF (v)(F (δv) − F (βv))(βvF (βv) − vF (v)) − F (v)(F (βv) − F (v))(δvF (δv) − βvF (βv))−δF (βv)(F (δv) − F (v))(βvF (βv) − vF (v)) − βF (βv)(F (βv) − F (v))(δvF (δv) − vF (v))= 0.

Proof of Proposition 2.
We consider a two-option mechanism of the form: {(0, 0), (q, p), (1, αµ)},where q ' 0, p = kq with k > αµ. We show that, if vf (v) is (locally) strictly
32increasing at v = αµ, then this two-option mechanism achieves strictly higher
expected welfare than the (optimal) one-option mechanism {(0, 0), (1, αµ)}.
Given the two-option mechanism, let v ∗ denote the seller type who isindifferent between offering (q, p) and (1, αµ):(1 − ψ)(αµ − v ∗ ) = ψ(kq − qv ∗ );.
equivalently, v ∗ = (1−ψ)αµ−ψkq
1−ψ−ψqTherefore, the social welfare increases by:
Z αµ
Z k
vdF
vdF − (1 − ψ)
ψq
v∗
v∗
Z k
dv ∗
' q[ψ
vdF − v ∗ f (v ∗ )(1 − ψ)
]q=0
dq
αµ
Z k
vdF − αµf (αµ)ψ(αµ − k)]
= q[ψ
αµ

Z k
(vf (v) − αµf (αµ))dv],∝
αµwhich is positive if vf (v) is (locally) strictly increasing at v = αµ.

Proof of Proposition 3.
Consider the following menu mechanism: {(qm , pm )}M
m=0 = {(0, 0), (1, p), (1, αµ)}with p > αµ. In the associated equilibrium, the seller with v < v(ψ) chooses(1, αµ), which is accepted only by the naive buyer; while the seller withv ∈ (v(ψ), p) chooses (1, p), which is accepted only by the rational buyer,
where v(ψ) represents the indifferent type:ψ(p − v(ψ)) = (1 − ψ)(αµ − v(ψ)) ⇔ v(ψ) = αµ −
which is positive if ψ is sufficiently close to 0.

33

p − αµψ,
1 − 2ψ

The expected social surplus is increasing locally around ψ = 0 if:#"
Z 1
Z v(ψ)∂(α − 1)vdF + ψ(α − 1)vdF |ψ=0
(1 − ψ)∂ψ0v(ψ)
 Z αµZ p
vdF − αµf (αµ)(p − αµ) +vdF > 0.
=(α − 1) −
0

(3)

αµ

Proof of Proposition 4.
Fix any menu mechanism and its associated equilibrium. Let x(v) denote theprobability of trading conditional on the seller type v, and let y(v) denotethe expected transfer conditional on that the seller type is v.
Suppose that the buyer’s ex ante expected payoff (i.e., the buyer’s expected payoff before realizing his own type) is non-negative in this equilibrium:Z(αvx(v) − y(v))dF ≥ 0.
v

It suffices to show that we must have x(v) = 0 for all v.
Recall that our assumption of severe adverse selection, Inequality (1),means that only the no-trade outcome is incentive-feasible if there is no
naive buyer (Samuelson, 1984). Precisely, if {(x(v), y(v))}v∈[0,1] satisfies:(IC-r) :y(v) − vx(v) ≥ y(v 0 ) − vx(v 0 ), ∀v, v 0(IR-r) :y(v) − vx(v) ≥ 0, ∀v
Z(αvx(v) − y(v))dF ≥ 0,

(EAIR-B) :v

then we must have x(v) = 0 for all v.

Proof of Proposition 5.
Consider Problem (2; φ) for any given φ ≥ 0. Lemma 1 holds in this problem,
because it is purely the implication of the incentive constraints of the seller
34and buyer, and does not depend on the principal’s objective. Existence ofm∗ and v ∗ that satisfy the above conditions is then immediate.

Proof of Proposition 6.
Fix any mechanism in the class of general trading mechanisms and its associated equilibrium. Given the buyer’s behavior in the equilibrium, theseller is facing the problem of choosing m1 ∈ M1 optimally (assuming that
m2 , a1 , a2 are optimally chosen), where each choice is associated with the
expected continuation trade outcome denoted by (q(m1 ), p(m1 )). Without
loss of generality, assume that different choices of the seller induce different
expected trade outcomes.
Then, the seller’s choice of m1 ∈ M1 must induce the interval-partition
structure because of the single-crossing payoff function. That is, there exist{vm }M
m=0 for some M ∈ N such that 0 = vM +1 > vM > . . . > v0 = 1 andthose in the same interval [vm+1 , vm ] choose the same message in M1 in theequilibrium.
In particular, take the interval [0, vM ]. Given the seller’s choice of m1 andthe rational buyer’s choice of m2 , the severe adverse selection assumption
implies that either the seller chooses a1 = 0 or the rational buyer chooses
a2 = 0. Thus, trade is not possible between them. However, the mechanism
being non-trivial implies that the seller must trade with the naive buyer.
Therefore, we complete the proof by setting v ∗ = vM .

B

Optimal Mechanism and Its Properties under Power Distribution

This section investigates properties of the optimal menu mechanism underPower distribution (with M = 2, thanks to Proposition 1).
To investigate the naive type’s payoff under the optimal mechanism withincentive-feasible cross-subsidization in Example 2.2, we focus on the case in35

which rational type’s IR is not binding; when rational type’s IR is binding,there is no incentive-feasible cross subsidization from the rational type to thenaive type. In this case, (q1 , p1 ) = (1, 1) and (q2 , p2 ) = (1, αµ). The thresholdtype v ∗ is obtained from the seller’s indifference condition: (1−ψ)(αµ−v ∗ ) =. Note that v ∗ < αµ if and only if αµ < 1 andψ(1 − v ∗ ) ⇔ v ∗ = (1−ψ)αµ−ψ
1−2ψ
∗
1−αµ
ψ < 12 . Note also that dv= − (1−2ψ)
2 < 0.
dψ
R v∗
Under this mechanism, naive type’s actual expected payoff is 0 (αv −R v∗∗γv 1+γ −αµv γ ]v0 = −αµ(1−v ∗ )(v ∗ )γ <αµ)dF = 0 (αγv γ −αµγv γ−1 )dv = [α 1+γR1R1
0. Also, rational type’s expected payoff is v∗ (αv − 1)dF = v∗ (αγv γ −γ
γv γ−1 )dv = [α 1+γ
v 1+γ − v γ ]1v∗ = αµ − 1 + (v ∗ )γ − αµ(v ∗ )1+γ .
The expected buyer surplus is:− (1 − ψ)αµ(1 − v ∗ )(v ∗ )γ + ψ αµ − 1 + (v ∗ )γ − αµ(v ∗ )1+γ
= − (1 − 2ψ)αµ(1 − v ∗ )(v ∗ )γ − ψαµ(v ∗ )γ + ψ [αµ − 1 + (v ∗ )γ ]= − (1 − 2ψ)αµ(1 − v ∗ )(v ∗ )γ − ψ(1 − αµ)[1 − (v ∗ )γ ] < 0.
Although the ex ante expected buyer surplus is always non-positive and maximized at ψ = 1 as in Proposition 4, it can be non-monotonic. For example,at ψ = 0, the ex ante expected buyer surplus is decreasing in ψ if α and γ
are sufficiently small.
As discussed in Example 2.2, naive type’s actual expected payoff with the
maximal cross-subsidization is:− αµ(1 − v ∗ )(v ∗ )γ +ψ 
αµ − 1 + (v ∗ )γ − αµ(v ∗ )1+γ .
1−ψ36Its derivative with respect to ψ is:


1
∗ γ
∗ 1+γ
αµ
−
1
+
(v
)
−
αµ(v
)
(1 − ψ)2ψψ
1 − αµ
∗ γ−1
∗ γ
∗ γ−1
∗ γ
−
−αµγ(v )+ αµ(1 + γ)(v ) +γ(v )−αµ(1 + γ)(v )(1 − 2ψ)2
1−ψ
1−ψ


1
∗ γ
∗ 1+γ
=
αµ
−
1
+
(v
)
−
αµ(v
)
(1 − ψ)2
1 − αµψ
1 − 2ψ
(1 − ψ)αµ − ψ∗ γ−1−−αµγ +
γ−αµ(1 + γ)(v )(1 − 2ψ)2
1−ψ
1−ψ
1 − 2ψ
{z
}
|
1
= 1−ψ
(α−1)γ[(1−ψ)αµ−ψ]=
1 − αµ ∗ γ−1 11
αµ − 1 + (v ∗ )γ − αµ(v ∗ )1+γ −(v )(α − 1)γv ∗2(1 − ψ)(1 − 2ψ)
1−ψ



=



1
αµ − 1 + (v ∗ )γ − αµ(v ∗ )1+γ − 1 − ψ (1 − αµ)(α − 1)γ(v ∗ )γ (1 − ψ)2 
1 − 2ψ
|{z}=1−v ∗=

1{αµ − 1 + [1 − (α − 1)γ − γv ∗ ](v ∗ )γ } .
2(1 − ψ)Because [1 − (α − 1)γ − γv](v)γ subject to v ∈ (0, αµ) is maximized at
v = 1 − αµ, the derivative is at most




1
γ
αµ
−
1
+
[1
−
(α
−
1)γ
−
γ(1
−
αµ)]
(1
−
αµ)
|
{z
}

(1 − ψ)2 
=1−αµ

= −

1
(1 − αµ) [1 − (1 − αµ)γ ] < 0.
(1 − ψ)2

Hence, the naive type’s actual expected payoff with the maximal crosssubsidization is decreasing in ψ, implying that the payoff of the naive typeis strictly lower than the one-price mechanism.

37

We next turn to Example 3. The expected social surplus is:
Z 1
Z v∗(α − 1)vdF + ψ(α − 1)vdF(1 − ψ)v∗
1+γ v ∗= (1 − ψ)(α − 1)[µv ]0 + ψ(α − 1)[µv 1+γ ]1v∗
0

= (1 − ψ)(α − 1)µ(v ∗ )1+γ − ψ(α − 1)µ(v ∗ )1+γ + ψ(α − 1)µ= (α − 1)µ (1 − 2ψ)(v ∗ )1+γ + ψ .
The derivative at ψ = 0 is positive, for example, if γ < 51 and α < 54 .

38