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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

We consider a market for lemons in which the seller is a monopolistic price setter and the buyer receives a private noisy signal of the product's quality. We model this as a game and analyze perfect Bayesian equilibrium prices, trading probabilities and gains of trade. In particular, we vary the buyer's signal precision, from being completely uninformative, as in standard models of lemons markets, to being perfectly informative. We show that high quality units are sold with positive probability even in the limit of uninformative signals, and we identify some discontinuities in the equilibrium predictions at the boundaries of completely uninformative and completely informative signals, respectively.

In many markets, prospective buyers are imperfectly informed about the quality of the items for sale. Sellers, on the other hand, often know much about the quality of their products. In his classical “market for lemons” paper, George Akerlof showed that such informational asymmetry, when taken to the extreme, can lead to adverse selection. Although there is room for gains of trade in all qualities if only buyers were able to distinguish these from each other, the complete lack of such discriminatory power—the usual assumption—may imply crowding-out of high quality items, leaving only the “lemons” in the market; see Akerlof [^{1}

In this paper, we introduce a fairly rich form of asymmetric information in such situations of bilateral trade. In addition to observing the price posted by the seller, the buyer receives a noisy private signal about the product's quality. The noisy signal has a given precision, known also to the seller, and a continuum of different values. The actual value is known only to the buyer. In the metaphorical language of our title: we endow buyers with some capability to detect the scent of lemon. We believe this is a realistic and relevant extension. Consumers typically have the opportunity to study the object for sale, thus getting an impression of its quality, and employers usually interview job candidates, thus obtaining an impression of the candidates' abilities,

By varying the precision of the signal, the present model spans the whole continuum of consumer capabilities to recognize quality, from one extreme to the other. At the one end, when the signal precision is maximal, consumers recognize the true quality without mistake. At the other end, when the signal precision is minimal, it is as if all consumers have no sense of smell and are completely unable to recognize quality. Instead, they have to base their purchasing decision on their prior belief and the price chosen by the seller. This is the case treated in Akerlof's classical model. Central questions in the present study are what happens to the set of equilibria when one varies the signal precision, and, in particular, how robust the standard predictions are for the two extreme cases of “symmetric” information (maximal signal precision) and “one-sided” asymmetric information (minimal signal precision—as in Akerlof's approach).

Our starting point is a standard lemons model: an indivisible unit is available in two qualities, low and high. Buyers are interested in buying at most one unit. The seller knows the quality of the unit for sale. For each quality, the seller's valuation is lower than buyers' willingness to pay, so there are potential gains of trade in both qualities. When the seller and buyer meet, the seller sets a take-it-or-leave-it price for his unit. The buyer observes this price and receives a noisy quality signal. The seller knows the conditional probability distribution for signals, conditional upon the true quality, but does not know the buyer's signal. For the sake of clarity and transparency we focus on an “elementary” situation—only one seller facing one buyer. As the reader will see, even an analysis of such simple situations requires substantial work. We analyze the corresponding non-cooperative game, using pure-strategy (weak) perfect Bayesian equilibrium as solution concept. We characterize equilibrium prices and trading probabilities for all degrees of signal precision.

Our main results are the following. The equilibria are either separating or pooling. Separating equilibria come in two varieties. In partial adverse-selection equilibria, high-quality items are traded with positive probability below one and low quality units are sold with probability one. There is a continuum of such equilibria, even in the limit as the buyer's signal precision is taken down to zero. Hence, high-quality units are sold with positive probability also when buyers have no information at all about the quality of the unit at hand. By contrast, in total-adverse equilibria, high quality units are not sold at all. However, those equilibria are not robust to the elimination of weakly dominated strategies. In pooling equilibria, buyers purchase at the going price only if their quality signal exceeds a certain threshold (determined in equilibrium). Sellers with low-quality units thus have a lower probability of selling than sellers with high-quality goods. In our metaphorical language: a scent of lemon is more likely to come from a low-quality item than from a high-quality item. The equilibrium selling probability for low quality is sufficiently high to deter such sellers from lowering their price. In general, there is a continuum of pooling equilibrium prices. Also here, some discontinuities arise. Firstly, price-pooling at buyers' average valuation—an equilibrium in the classical adverse-selection model (no sense of smell)—need not be an equilibrium outcome if the buyer has even the slightest sense of smell. Secondly, even when this is an equilibrium outcome, there is a discontinuity in trading probabilities. In the classical adverse-selection model, both low and high quality are traded with probability one, while here even a very noisy negative quality signal will deter a buyer from purchase.

Although there certainly is a related literature, to the best of our knowledge the present model is new. In Milgrom and Roberts [

It should also be mentioned that the model is different from Spence's [

In our model, the seller is a monopolist who offers an indivisible item for sale at a take-it-or-leave-it (TOL) price. TOL pricing is known to be an expected-profit maximizing mechanism in situations of quotation marks “one-sided” incomplete information when there is only one buyer, granted the item is of a given, and to the seller known, quality; see Riley and Samuelson [

The paper is organized as follows. Section 2 formalizes the model and provides certain necessary conditions on prices to be consistent with pure-strategy perfect Bayesian equilibrium. Section 3 contains the equilibrium analysis including a study of the boundary cases of completely uninformative and fully informative signals, respectively. Section 4 illustrates some points, and provides some additional results, for the special case of normally distributed error terms. Section 5 concludes. All proofs are in the

An indivisible good is available in two qualities, low and high, denoted _{L}_{H}_{L}_{H}_{L}_{L}_{H}_{H}_{L}_{L}_{H}_{H}_{H}_{L}_{L}_{H}_{L}_{H}

The seller sets a take-it-or-leave-it price _{+}.^{2}

Given ^{3}^{4}

The seller is an expected-profit maximizer, where _{L}_{H}_{L}_{H}

Under the assumptions made above, there are positive potential gains of trade in both cases: _{L}_{L}_{H}_{H}

One of the main concerns from a welfare viewpoint is to what extent these potential gains are realized in equilibrium, and how the realized gains of trade depend on the parameters in the model, in particular on the precision

Formally, we model the interaction outlined above as an extensive-form game with incomplete information between a seller, player 1, and a buyer, player 2, as follows:

Nature chooses

The seller observes _{θ}_{L}_{H}^{2} of prices, where _{L}_{H}

The buyer observes the price-signal pair (_{θ}_{θ}^{2} → {0, 1}, where ^{5}

The game ends and the players receive their payoffs,

A decision node for the seller in the infinite game tree is thus a pair (^{2}, specifying the seller's price _{L}_{H}

By contrast, for _{L}_{H}_{θ}

As noted above, this game differs from the standard “lemons model” (as, for example, presented in chapter 13 in Mas-Colell _{θ}

We solve this game by adapting the usual notion of (weak) perfect Bayesian equilibrium to the present setting (see, ^{6}

In the case of an imprecise signal, _{L}_{H}^{2}, and ^{2} → {0, 1}. We will call such a triplet (

At each seller information set {(

If _{L}_{H}_{θ}_{θ}_{θ}_{θ}

If _{L}_{H}_{L}_{L}_{H}_{L}_{H}^{7}

At all information sets: the concerned player's strategy is a best response to the other player's strategy, under the belief induced by

Consistency condition [B1] is met if the seller knows the quality of his item and knows the c.d.f. _{L}_{H}_{L}_{H}^{8}_{H}

In any equilibrium (_{L}_{H}_{L}_{H}

An equilibrium (_{L}_{H}_{L}_{H}_{L}_{H}

The following lemma provides necessary conditions for equilibrium that follow from first principles. For instance, it is sequentially rational for the buyer to buy if the seller asks a price below the buyer's willingness to pay for a low-quality unit, and it is sequentially rational not to buy if the seller asks a price above the buyer's willingness to pay for a high-quality unit:

These simple observations rule out a wide range of price pairs from equilibrium:

For _{L}_{H}

min {_{L}_{H}_{L}

_{L}_{H}_{L}_{L}

_{L}_{L}_{H}_{L}_{H}

_{L}_{H}_{H}

It follows from this lemma that, in separating equilibria, the price of low-quality units equals buyers' willingness to pay for such units (by (a) and (b)), and the price of high-quality units is never below the buyers' willingness to pay for such units (by (c)). Hence, when trade occurs in separating equilibria, the seller reaps all the gains of trade. This is not surprising, since by assumption the seller is a monopolist who can commit to take-it or leave-it offers. Moreover, in pooling equilibria the price is never below buyers' willingness to pay for low-quality units but always lower than buyers' willingness to pay for high-quality units (by (a) and (d)). In sum:

The only remaining candidates for pure-strategy equilibria, when _{L}_{L}_{H}_{H}_{L}_{H}_{L}_{H}

In this section we analyze the game's equilibria and come to our main purpose: an investigation of the robustness of the equilibria—in particular the equilibrium prices and trading probabilities—in the classical boundary cases of uninformative signals (

Under symmetric information, _{θ}_{θ}^{9}_{L}_{H}_{L}_{H}_{L}_{H}

If _{L}_{H}_{H}_{L}

Suppose _{L}_{L} and p_{H}_{H}_{L}_{H}_{L}_{H}

This type of outpricing can occur for any degree of buyer signal precision _{H}_{L}_{L}_{H}

We should add that we do not find these separating equilibria plausible. For a seller with a high-quality couldn't lose, but might gain, by deviating to a price _{H}_{H}_{H}_{L}_{H}_{H}_{H}_{L}_{H}_{H}

We here consider pure-strategy separating equilibria in which the seller's strategy is _{L}_{H}

It follows from first principles that in equilibrium buyers always buy at the low price, _{L}_{H}^{10}_{H}

What can be said about the upper bound on the realization probability _{H}_{H}_{H}_{H}

Hence, the upper bound on the equilibrium realization probability _{H}

If _{H}_{L}

For had the realization probability _{H}_{L}_{H}_{H}_{H}_{H}

It remains to establish existence of equilibria and to find an operational representation of _{H}_{H}_{L}

We note that _{1} > _{0}, since, by hypothesis, _{L}_{H}_{L}_{H}

If

We use this lemma to establish the existence of partially separating equilibria. In these equilibria, the buyer's strategy is to buy from a seller who posts the low price, _{L}_{L}_{H}_{H}

Suppose _{L}_{H}_{L}_{H}

We would like to make some comments to this proposition. First, the continuum of parameterized equilibria is Pareto ranked: the lower the threshold _{H}_{0} Pareto dominates the others.

Secondly, step-functions of the kind used by the buyers in the above equilibria are optimal decision rules in a variety of decision problems related to the one faced by the buyer in the present model. Suppose, for instance, that a decision-maker in a non-strategic environment observes a signal of the form (^{11}

Thirdly, we examine some comparative statics properties of these separating equilibria with respect to the precision _{H}

The upper bound _{H}

What may appear as a surprise is that the present model allows for some trade in high-quality units even when buyers have no information at all about the quality of the unit at hand:

Although buyers receive only a non-informative signal, low-quality sellers do not have an incentive to deviate to the high price, since the trading probability at that price is too low. Until relatively recently, it was commonly presumed that, in classical adverse-selection models, high-quality units are not traded when _{H}^{12}

In sum, the set of equilibrium trading probabilities for high-quality units contains the interval

Fourthly, recall from our second comment that the threshold functions in Proposition 2—where the item is purchased if the quality signal is sufficiently high—are optimal in a wide range of non-strategic circumstances. If the set _{H}_{H}

Suppose _{H}_{H}

We saw in Section 3.1 that pooling equilibria do not exist in the boundary case _{L}_{H}

Consider the decision problem faced by a buyer who receives a quality signal _{L}^{13}

If _{L}_{H}^{14}

The following proposition characterizes the set of prices that can be supported in pooling equilibria.

Suppose _{L}_{H}

The first (second) condition in the proposition guarantees that a seller with a low (high) quality unit does not have an incentive to deviate. We note that condition (_{H}_{L}_{H}_{L}_{H}_{L}_{L}_{L}_{L}^{15}

For the boundary case _{L}

Hence, the set of pooling equilibrium prices is non-empty at _{H}_{L}_{H}

Is the equilibrium correspondence that maps buyers' precision _{H}_{L}_{L}_{H}

The case _{L}_{L}_{H}_{H}

The model in Akerlof [_{L}_{H}_{H}_{H}_{H}_{L}_{L}

In the final lines of subsection 3.4, we mentioned a discontinuity in the set of pooling equilibria: although _{L}_{H}

Clearly _{L}_{H}

If ^{2}), then _{L}_{H}_{H} and

We would like to make some comments to this proposition. Firstly, in our model, pooling equilibrium is possible even when _{H}_{L}_{L}_{H}

Secondly, the set of pooling equilibrium prices, if non-empty, is an interval. This interval has max{ _{L}_{H}

Thirdly, by (_{L}_{H}

Fourthly, no price above _{L}_{L}_{H}

On a somewhat different note, let us compare the equilibrium profit to the seller in pooling equilibria with the maximal expected profit in separating equilibria with partial adverse selection. The expected profit in a pooling equilibrium with price

We compare this with the expected profit in separating equilibria. Since the prices are the same in all such equilibria, _{L}_{H}

These equations are useful when comparing the expected profit in pooling and separating equilibria in the limit case when the signal precision _{0} of pooling equilibrium prizes in this limit case is empty if _{H}_{H}_{L}_{H}

If _{H}

In other words, if the buyer's average willingness to pay exceeds the seller's valuation of high quality, then the seller prefers pooling equilibria over separating equilibria. Moreover, in the limit as the buyer's quality signal becomes uninformative, potential gains of trade are virtually realized. The qualifier “virtually” refers to the fact (established in the proof of the proposition) that while there exist pooling equilibria that result in profits arbitrarily close to

We finally note that when the buyer's quality signal is virtually perfect,

In other words, the welfare loss due to asymmetric information vanishes in the limit as the buyer's quality signal becomes perfect.

The goal of this paper was to study equilibria when noisy inspection is introduced in an otherwise standard “lemons” model; more specifically, when the buyer does not know the quality of the unit for sale but receives a noisy signal about its quality while the seller knows the quality but not the buyer's signal. We believe that this extension is relevant and provides a more realistic description of asymmetric information in markets. A prospective buyer of a used car may walk around it, take a look under the hood, check whether the bumper falls off when it receives a slight kick etc. and thus get at least some impression of the car's quality, rather than, as in Akerlof's seminal analysis, no impression at all. Although the seller knows the quality of his car, he usually can only guess at the exact impression on the prospective buyer. We chose to study an elementary situation—only one seller facing one buyer at a time—devoid of all the bells and whistles that could make the model richer and more realistic, for the sake of clarity and transparency. Moreover, we focussed on pure-strategy equilibria. In the present model, this is a rich set of strategies, offering a continuum of options for the seller and allowing the buyer to condition the purchasing decision on signal values in a continuum. Arguably, this provides a behaviorally plausible form of randomization—in the form of hesitation in face of quality uncertainty.

While the focus of this paper is admittedly narrow, we felt it natural to first establish a clear-cut and transparent analysis of the set of pure strategies, before embarking on studies of richer, and thus more complex, market settings. Evidently, there are numerous relevant directions for further research. We here briefly name two:

The model allows for multiple equilibria with distinct outcomes. This raises the question whether there are relevant refinements or stability conditions that one could impose in order to cut down the equilibrium set. This is a very interesting avenue for future research, although non-trivial, since standard refinements are defined only for finite games, while the present game is infinite.

Highly relevant extensions would be to analyze cases of multiple quality levels, multiple sellers and/or buyers, and a variety of pricing mechanisms.

We are currently working on a few such directions, in joint work with Yukio Koriyama. The analysis being quite demanding, it appears that generalization in one dimension typically requires simplification in another. We hope that the present study will inspire future research on market interactions in which some or all parties have noisy signals about product quality.

We thank Carlos Alós-Ferrer, Tore Ellingsen, Yukio Koriyama, Lars-Göran Mattsson, Jianjun Miao, and Zvika Neeman, and two anonymous referees for comments. Voorneveld thanks the Wallander/Hedelius Foundation and the Netherlands Organization for Scientific Research for financial support.

If a seller of type _{L}_{θ}_{θ}_{L}

Suppose _{L}_{H}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{L}

Suppose _{L}_{L}_{H}_{L}_{H}_{H}_{H}_{H}_{H}_{L}_{L}_{L}_{H}_{L}_{H}_{L}

Suppose first _{L}_{H}_{H}_{L}_{L}_{L}_{L}_{H}_{H}_{H}_{L}_{H}_{L}_{H}_{H}_{H}_{L}_{L}_{L}

To prove necessity, suppose (_{L}_{H}_{H}_{L}_{L}_{L}_{H}_{H}_{H}_{L}_{H}_{L}_{H}_{L}_{L}_{H}_{H}_{L}_{H}_{L}_{L}_{H}_{L}_{H}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{H}_{L}_{H}_{L}_{H}

To establish the last claim in the proposition, consider any equilibrium (_{L}_{L}_{H}_{H}_{L}_{H}

Recall that the c.d.f. for

If _{H}_{L}

If _{H}_{L}

In order to prove that _{L}_{L}_{H}_{L}_{H}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{H}_{L}_{H}_{L}_{L}_{L}_{L}_{H}_{H}

To prove the second claim, let (_{L}_{H}_{L}_{H}

This defines a pure-strategy pair, (

Let _{L}_{H}_{L}_{L}_{H}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{L}_{H}_{H}

Hence charging the price _{L}_{L}_{H}_{L}_{L}_{H}_{L}_{H}_{H}

Hence charging price _{H}

Assume the buyer's strategy at price _{H}_{H}_{H}

Since these probabilities are decreasing in

Solving this equation for _{0} and that _{H}_{H}_{0}, which is exactly _{H}

To prove necessity, let (_{L}_{H}_{L}

Moreover, such a seller can guarantee himself a profit arbitrarily close to _{L}_{L}_{L}_{H}

Moreover, such a seller can guarantee himself zero profit by setting _{H}_{L}_{H}_{L}_{L}_{H}_{L}_{H}_{L}_{L}_{L}

Condition (_{H}_{L}_{L}_{H}_{L}

It thus only remains to study prices _{L}_{H}_{H}_{θ}^{2}). Hence,

The exponential function being strictly increasing, the inequality above holds iff

Using (

As expected, _{H}_{L}_{L}_{H}_{H}

Notice that, with

Hence

It follows from (

If

If

If

Let _{L}_{L}

We cannot possibly do justice to this rich literature here. The interested reader is referred to Riley [

To be more precise, we assume

See section 3.5 for a discussion of the subtle difference between pure noise and no signal.

The alternative and more conventional representation

More generally, the buyer may randomize, in which case the range of the function

The present game has infinitely many information sets and thus departs from the usual setting for sequential equilibrium. For exact definitions, see Fudenberg and Tirole [

In the boundary case _{θ}_{L}

Expressed differently, [B3] requires the probability _{L}

The buyer's information sets then are of the form (_{θ}_{θ}

In order to buy from a high-quality seller with probability one the buyer has to buy irrespective of the quality signal, since the error term has full support.

See Mattsson, Voorneveld and Weibull [

Notice, however, that this requires that the parties deliberately randomize. In our model, the required randomization is obtained by pure strategies conditioned on the random signal.

Recall that if _{L}

Indeed, it is the buyer's unique best reply to buy if

This is true even when