OSIPP Discussion Paper: DP-2020-E-002

A Note on Adverse Selection and Bounded Rationality
March 9, 2020
Takeshi Murooka*
Osaka School of International Public Policy (OSIPP), Osaka University

Takuro Yamashita†
Toulouse School of Economics, University of Toulouse

Abstract: There is accumulating evidence that some consumers are behavioral in the sense that they
may make suboptimal decisions. This paper investigates adverse selection with general types of such
behavioral biases. In our model, some buyers (i.e., consumers) may take actions that do not necessary
optimize own payoffs, which encompass virtually any type of biases including subjective probability,framing, model misspecification, random errors, and inferential naivety. We focus on a situation inwhich there exists severe adverse selection where only no-trade outcome is possible under rational
agents. We show that the no-trade theorem remains to hold without imposing any additional assumption
on buyers' behavior. That is, if there is any trade under a mechanism which is incentive compatible forsellers, then the expected payoff from the trade is negative (i.e., ex ante individual rationality constraintis violated) for some type of buyers. Our result sheds light on a new trade-off between social surplus andpayoff losses of boundedly-rational buyers.

JEL Codes: D82, D83, D86, D90, D91
Keywords: adverse selection, bounded rationality, mechanism design, no-trade theorem*Email: murooka@osipp.osaka-u.ac.jp†Email: takuro.yamashita@tse-fr.eu

Adverse selection with general behavioral types
Takeshi Murooka∗

Takuro Yamashita†

March 9, 2020

1Introduction

There is accumulating evidence that some consumers are behavioral in thesense that they may make suboptimal decisions.1 This paper investigates
adverse selection with general types of such behavioral biases. In our model,
some buyers (i.e., consumers) may take actions that do not necessary optimize own payoffs, which encompass virtually any type of biases including
subjective probability, framing, model misspecification, random errors, andinferential naivety. We focus on a situation in which there exists severe adverse selection where only no-trade outcome is possible under rational agents.
We show that the no-trade theorem remains to hold without imposing any
additional assumption on buyers’ behavior. That is, if there is any tradeunder a mechanism which is incentive compatible for sellers, then the expected payoff from the trade is negative (i.e., ex ante individual rationality
constraint is violated) for some type of buyers.
∗Osaka School of International Public Policy, Osaka University. murooka@osipp.osakau.ac.jp†Toulouse
SchoolofEconomics,UniversityofToulouse,Capitole.
takuro.yamashita@tse-fr.eu
1
See Heidhues and Kőszegi (2018) for evidence on consumer behavior and Beshears,Choi, Laibson, and Madrian (2018) on household decision makings.

1Because rational buyers never wish to trade whenever their expected payoff is negative, our theorem implies that generating trade is possible onlywith permitting losses of behavioral buyers. Our theorem does not pose alimit on the social surplus, however. For example, Murooka and Yamashita(2020) extensively investigate adverse-selection problems when some buyers
have inferential naivety (Eyster and Rabin, 2005; Jehiel, 2005). Murooka andYamashita (2020) derive an incentive-feasible mechanism, in which rationalbuyers (as well as naive buyers) trade even under severe adverse selection,and the condition for the social surplus to be strictly higher under the mechanism. In this sense, our theorem sheds light on a new trade-off between thesocial surplus and payoff losses of behavioral buyers.

2Severe adverse selection with rational types

A seller has private information, denoted by θ ∈ Θ, about the goods tobe traded. The distribution of θ, denoted by F ∈ ∆(Θ), is assumed to becommon knowledge. A buyer has no private information. A (deterministic)
trading outcome is denoted by y ∈ Y , which may include the information asto which goods are traded, the associated monetary transfers, and so on. The
seller’s ex post payoff is denoted by uS (y, θ), and the buyer’s ex post payoff
is denoted by uB (y, θ). The trading outcome includes a “no-trade outcome”
y = 0 ∈ Y , and we assume uS (0, θ) = uB (0, θ) for normalization. A feasible
allocation is a stochastic trading outcome, denoted by x ∈ ∆(Y ). For eachRi ∈ {S, B}, let ui (x, θ) = y ui (y, θ)dx denote the expected payoff given x.
A fundamental observation in the literature (Akerlof, 1970; Samuelson,1984) is that adverse selection can be so severe that only no-trade outcome
is incentive feasible (i.e., incentive compatible for the seller, and individually
rational for both parties) by rational traders. Our goal is to obtain a different
but related observation in case the buyer is not necessarily rational in thestandard sense.
Specifically, consider an allocation rule (x(θ))θ that satisfy:2• (ICS ) incentive compatibility for the seller: for any θ, θ0 ,uS (x(θ), θ) ≥ uS (x(θ0 ), θ),• (IRS ) individual rationality for the seller: for any θ,uS (x(θ), θ) ≥ 0,
• (IRB ) individual rationality for the buyer:
ZuB (x(θ), θ)dF ≥ 0.
θAssumption 1 (Severe adverse selection). An allocation rule (x(θ))θ satisfies(ICS ), (IRS ), and (IRB ) if and only if x(θ) = 0 for (F -almost) all θ.
Example 1. The seller has an indivisible object. A deterministic tradingoutcome specifies whether the trade of the object occurs and the associatedmonetary transfer from the buyer to the seller. Both parties are risk neutral
in monetary transfer, and hence, an allocation is identified by a pair x =(q, p) ∈ [0, 1] × R, where q represents the probability of trading the object
and p represents the expected monetary transfer from the buyer to the seller.
The seller’s ex post payoff is given by uS (q, p, θ) = p − qθ, where θ ∼U (0, 1) can be interpreted as the seller’s opportunity cost of trading. Thebuyer’s ex post payoff is uB (q, p, θ) = qαθ − p, where αθ can be interpreted
as the buyer’s valuation for the object. Assume α ∈ (1, 2).
By the standard argument based on the envelope theorem, the combination of (ICS ) and (IRS ) implies:
Z 1p(θ) ≥ q(θ)θ +q(z)dz.
θThen, the buyer’s expected payoff is at most:Z 1
Z 1q(θ)αθ − q(θ)θ −q(z)dz dθ0θ
Z 1=q(θ)(α − 2)θdθ,0which is negative unless q(θ) = 0 for almost all θ.
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Severe adverse selection with behavioral types

Eyster and Rabin (2005) show that some trade becomes possible if the buyer
is “cursed” in that she underappreciates the relationship between other players’ actions and these other players’ information. Under such a behavioral
bias, the total surplus in the society can be higher than with no trade. However, the buyer makes a loss in such a case. Murooka and Yamashita (2020)
consider a generalized environment of Eyster and Rabin (2005) where thebuyer is either a cursed type or a rational type (i.e., a standard Bayesiantype). They show that the trades between the cursed buyer and the rational
seller as shown in Eyster and Rabin (2005) generates positive externalities
to others: there exists a mechanism in which the rational buyer trades withthe seller even under severe adverse selection. The rational buyer earns nonnegative (and sometimes strictly positive) expected payoff, while the cursed
buyer still makes a loss.
Although those two papers consider very specific behavioral types, weshow that the same property occurs under any kind of behavioral types.
Namely, some buyer type must make a loss unless it is a no-trade mechanism.
To formally state our main result, let k = 1, . . . , K denote the buyer’s
behavioral type, where gk ∈ (0, 1) denotes the probability of each type k. We
assume θ, k are independent, which means that the buyer’s behavioral type
is not informative about the seller’s value.2 Let x(θ, k) denote the allocation
if the agents’ types are (θ, k). For the seller, we require:• (ICS ) incentive compatibility for the seller: for any θ, θ0 ,XX
gk uS (x(θ, k), θ) ≥gk uS (x(θ0 , k), θ),kk2

We think this is a natural assumption. However, generalization to the correlated caseis also possible, as long as we strengthen the seller’s incentive compatibility and individualrationality to their ex post version. This strengthening is in order to avoid a CremerMcLean type mechanism, which extracts the seller’s information rent by asking the seller
to bet on the buyer’s type realization.

4• (IRS ) individual rationality for the seller: for any θ,Xgk uS (x(θ, k), θ) ≥ 0.
k

RWe say that buyer type k makes a loss if θ uB (x(θ, k), θ)dF < 0. A key
assumption is that the buyer’s payoff of any type k does not directly dependon k — the buyer’s payoff is evaluated based on a single measure, whichcorresponds to the payoff of the rational type.
Theorem 1. Suppose that an allocation rule (x(θ, k))(θ,k) satisfies (ICS ) and(IRS ). Then, unless x(θ, k) = 0 for all k and (F -almost) all θ, some buyertype makes a loss.
Proof. We show a slightly stronger claim: If an allocation rule (q(v, k), p(v, k))(v,k)
satisfies (ICS ), (IRS ), and makes the buyer’s ex ante expected payoff (i.e.,the buyer’s expected payoff before realizing own type) non-negative, thenq(v, k) = 0 for all k and (F -almost) all v.
Specifically, the buyer’s ex ante expected payoff is:X Z
gkuB (x(θ, k), θ)dF .
θkDefine x(θ) =Pk gk x(θ, k) for each θ. Then, (ICS ) becomes:uS (x(θ), θ) ≥ uS (x(θ0 ), θ),(IRS ) becomes:uS (x(θ), θ) ≥ 0,and the buyer’s ex ante expected payoff becomes:
ZuB (x(θ), θ)dF.
v

Therefore, by Assumption 1, if the buyer’s unconditional expected payoff is non-negative, we must have x(θ) = 0 for (F -almost) all θ, implyingx(θ, k) = 0 for all k and for (F -almost) all θ.
5Theorem 1 points out a fundamental problem of an adverse selectionenvironment. Namely, whatever decision rules of the buyer one introduces inorder to avoid the no-trade outcome, such avoidance is possible only at therisk of some buyer types, or more precisely, at the risk of the buyer’s ex antepayoff.
Notice that we have little restriction on which kinds of behavioral types
we consider. It includes any type of misinference (Eyster, 2019), inattention(Gabaix, 2019), random errors such as Goeree, Holt, and Palfrey (2008)
and Gabaix, Laibson, Li, Li, Resnick, and de Vries (2016), or heterogeneous
priors for the value. There may exist multiple kinds of behavioral typescoexisting, with possibly different degrees. Theorem 1 is also agnostic about
the class of mechanisms and the equilibrium concept, as long as the seller
is best-responding. For example, non-dependence on mechanisms allows forframing-based biases and model misspecifications such as Ahn and Ergin(2010), Spiegler (2016), and Esponda and Pouzo (2016).

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