Content uploaded by Roland Kirstein
Author content
All content in this area was uploaded by Roland Kirstein
Content may be subject to copyright.
Homo Oeconomicus 26(2): 179–213
•
(2009)
www.accedoverlag.de
Iterative Reasoning in an Experimental
Lemons Market
Annette Kirstein
Faculty of Economics and Management, Otto-von-Guericke-University, Magdeburg, Ger-
many
(eMail: Annette.Kirstein@ovgu.de)
Roland Kirstein
Economics of Business and Law, Faculty of Economics and Management, Otto-von-
Guericke-University, Magdeburg,
Germany
(eMail: md@rolandkirstein.de)
Abstract Inthispaper weexperimentallytest atheoryofboundedly rational behav-
ior in a ‘lemons’ market. We analyze two dierent market designs, for which perfect
rationality implies complete and partial market collapse, respectively. Our empiri-
cal observations deviate substantially from the predictions of rational choice theory:
Even aer repetitions, the actual outcome is closer to eciency than expected. We
examine to which extent the theory of iterated reasoning contributes to the explana-
tion of these observations. Perfectly rational behavior requires a player to perform
an innite number of iterative reasoning steps. Boundedly rational players, however,
carry out only a limited number of such iterations. We have determined the iteration
type of the players independently from their market behavior. A signicant correla-
tion exists between the iteration types and the observed price oers.
Keywords bounded rationality, market failure, adverse selection, regulatory failure, paternal-
istic regulation
. Introduction
Akerlof () has identied asymmetric information as a source of ine-
cient market outcomes and even market collapse. In experimental as well as
© 2009 Accedo Verlagsgesellschaft, München.
ISBN 978-3-89265-071-3 ISSN 0943-0180
Homo Oeconomicus 26(2)
in real world ‘lemons markets,’ however, the empirical extent of market fail-
ure is smaller than predicted by rational choice theory. We have run two
experiments in which the participants had to trade under asymmetric infor-
mation. e prices oered by the uninformed buyers, as well as the amount
of goods traded, were much higher than those predicted by rational choice
theory. A theoretical explanation for this deviation from perfectly rational
behavior can be drawn from the theory of iterative reasoning.
Perfectly rational behavior in a lemons market can be described as the
result of a straightforward maximization problem. It also can be described
as the outcome of an iterative reasoning process with an innite number of
iteration steps. In this context, bounded rationality can be characterized by a
limited ability to perform iteration steps. is theoretical approach leads us
topredict that boundedlyrational buyerswill bidhigher prices thanperfectly
rational buyers. e outcome of a lemons market with bounded rationality
is, therefore, less inecient than the market result if only perfectly rational
buyers are present.
In our experiment, we have examined to which extent the price oers of
an uninformed buyer can be explained by his ‘iteration type,’ i.e., the num-
ber of iteration steps he performs when eliminating dominated strategies.
Our experimental data show a negative correlation between the buyers’ iter-
ation types and theirprice oers. However, this negative correlationcan only
be conrmed for those subjects who perform a positive number of iteration
steps. In the course of the experiments, many decisions appear to have been
made without any elimination of dominated strategies. ese subjects seem
to have picked their prices randomly. Just as the rational choice theory, the
theory of iterativereasoning has little predictive power with regard to players
who act randomly. However, anexplorative analysis of our experimentaldata
indicates that this buyer type, just as the boundedly rational type, has chosen
higher prices than those subjects who were identied as perfectly rational.
e main contribution of our paper lies in the fact that we have deter-
mined the buyers’ iteration types independently of the observable behavior
which the types are supposed to explain. Existing studies on iterative rea-
soning have instead inferred the iteration types from the observed behavior.
A famous example is the ‘guessing game’ experiment in Nagel ().4 For
two reasons, we have chosen a dierent approach: First, if the iteration type
An early example is the ‘acquire-a-company’ experiment by Bazerman/Samuelson ().
Section . of Camerer () explains the ‘levels of reasoning’ concept. Iterative reasoning
has been explored in many experiments; see, e.g., Schotter/Weigelt/Wilson ().
See Costa-Gomes/Broseta/Crawford ().
4See also aler (), Nagel et al. (), Selten/Nagel (), and Ho/Camerer/Weigelt
(). Other examples are Beard/Beil (), Eyster/Rabin (), and Kübler/Weizsäcker
().
A. Kirstein and R. Kirstein: Experimental Lemons Market
Fig. — Iteration Types of Buyers and Observed Prices
Uninformed buyer
@
@R
eory of iterative reasoning
Questionnaire ⇒ iteration type i
@
@R
Type–consistent price intervals
Buyer’s price oer p
@
@R
Research question: relation between p and i?
is directly derived from the observed prices, the former cannot be used as
an explanation for the latter. Secondly, the direct derivation method would
categorize any buyer as rational who oers a very low price. However, this
behavior could as well be caused by the failure to perform any iteration steps
at all. Our method allows to distinguish between perfectly rational buyers
and players who just act randomly.
Another dierence between our experiment and Nagel’s is the focus of it-
erative reasoning. Deviation from the behavior that is predicted under com-
mon knowledge of rationality can be explained by her theory even without
discarding the assumption that all players are fully rational. It is sucient to
assume that a player falsely believes that some of the peers perform only a
limited number of iteration steps, and then reacts optimally to this belief by
staying exactly one iteration step ahead.5 Hence, it is not a cognitive limita-
tion of the player under scrutiny that makes him perform only a nite num-
ber of iteration steps.6 Our model does not focus on the beliefs of the player
under scrutiny. If he deviates from perfectly rational behavior, this is ex-
plained by his performing a nite number of iteration steps, caused by his
limited cognitive ability.7
Our research program is depicted in Figure . We have evaluated two dis-
tinct data sets which were generated independently from each other. e
5A generalization has been presented by Camerer/Ho/Chong () and (): In their the-
ory, a type chooses randomly, while a type k > assumes that other players are of type
through k − , and responds optimally to this belief.
6Arelated conceptisthe‘cursed equilibrium’ in Eyster/Rabin(): A ‘cursed’playerassumes
his opponents to choose their type-contingent optimal behavior with a probability smaller than
one. With the counter-probability, he expects them to choose average behavior, and reacts opti-
mally to this belief.
7Stahl/Wilson (, ) discuss the dierent types of bounded rationality models.
Homo Oeconomicus 26(2)
one variable consists of the observed prices oered by the uninformed buy-
ers in the lemons market, denoted by p. e source of the other variable is a
questionnairelled out in each round by the buyersdirectly aerhavingsub-
mitted their price oers. We are fully aware that relying on verbal statements
given by participants following their decisions bears a risk – the statements
may retrospectively serve as a rationalization of the own behavior. In our
case, however, this problem can safely be neglected for two reasons: First, if a
subjecthas the abilityto performjustoneiterationstep,he is unabletoimitate
a higher type. Secondly, the subjects had to ll out the questionnaire before
they learned the actual outcome resulting from their decisions. Hence, there
was noinformation givenbetween the decision and the verbal statementthat
might have been used for an update.
Weaskedthebuyerstobrieydescribe theirlineofreasoning, andwehave
used these written statements to categorize8 the participants into iteration
types (denoted by i).9 eir self-descriptions indicate that some buyers have
randomly chosen theirprice oers(type-),while othershaveperformed just
one iteration step (type-) or decided in a rather elaborate fashion (type-
and higher). We therefore distinguish only these three categories of iteration
types. We haveappliedthe theory of iterativereasoningtoourlemonsmarket
model and derived price intervals from which we predict a buyer of type i to
choose his price oer.
In the nal step, we compared the type-consistent price intervals with the
observed prices p to answer our research question: Does a negative relation
exist between iteration types and observed prices? We have found two main
results:
• e verbal statementsof most ofthe subjects do not allowforthe inter-
pretation that they have performed iterative elimination of dominated
strategies. ese participants seem to have acted rather randomly.
• A signicant negative correlation between type and price oer exists
forthose types whohaveperformed iterationsteps. Moreover, we have
observedthat most ofthese types’ priceoerswereactually taken from
the type-consistent price intervals.
In Section , we introduce two versions of a lemons market. Under the as-
sumption of perfect rationality the predicted outcomes in the two markets
are complete and partial market collapse, respectively. We then introduce
our notion of iterative reasoning and derive the predicted behavior for dif-
ferent degrees of bounded rationality.
8is categorization was done by an independent research assistant.
9Nagel (, ) mentions that written comments of the subjects in her experiment seem
to support her results, but she has not derived the subjects’ types from these statements.
A. Kirstein and R. Kirstein: Experimental Lemons Market
In Section , we describe our experiments. In the rst experiment, the
subjects play each market setting just once (sections . to .). In Section
., the second experiment is reported, in which the participants repeatedly
played one of the two market designs. Section concludes the article with a
discussion of the possible implications for economic policy, in particular for
the regulation of lemons markets.
. Adverse Selection
. Setup
is section presents two versions of a lemons market model that we have
tested in a series of experiments. In one parameter setting, the market is
expected to collapse completely. In the other setting some trade is predicted
to take place. However, eciency would require all units in both markets to
be traded.
Consider a market in which an unspecied good is traded. We assume
its quality to be uniformly distributed over the interval [, ]and denote the
actual quality of a specic unit as Q. Two groups of agents are active in this
market:
• Sellers, each of whom owns one unit of the good and knows its true
quality. e sellers’ valuation is denoted as a(Q), with a(Q)=βQ.
• Buyers, who cannot observe the true quality of a certain unit of the
good, but know the distribution of quality. eir valuation is denoted
as n(Q)=γ +δQ.
We assume γ ≥ and δ ≥β >. us, foreach qualitylevel Q >, the buyers’
valuation is at least as high as the sellers’.0 We also assume the following
interaction structure: Each buyer makes a price oer. e oer is randomly
assigned to a specic seller, who then decides whether to accept the oer or
not. If the seller accepts, then the unit is traded. If the seller refuses the oer,
then no transaction takes place.
Denote the initial monetary endowment of the players as V
i
≥ and the
(ex post) gain from trade as Π
i
, with i =b, s for buyers and sellers. If a seller
accepts a certain price oer p, then his payo is V
s
+p. If he rejects the oer,
his payo is V
s
+βQ. His gain from trade, therefore, amounts to p −βQ. It
is rational for a seller to accept a price oer only if it exceeds his valuation
of the good (Π
s
> or, equivalently, p > βQ). e simplicity of the sellers’
0Undersymmetric information,the ecient outcomecouldeasilybe achieved. Ifboth market
sides are uninformed, but do know the distribution of quality, then each buyer and seller would
agree to trade a specic unit for a price between their valuations of the average quality.
Homo Oeconomicus 26(2)
decisions later allows us to focus on the buyers’ reasoning process only, and
the buyers’ perception that the sellers make perfectly rational decisions can
be taken for granted.
If a price oer p is accepted by a seller, then the buyer’s payo amounts
to V
b
+γ +δQ −p. If it is rejected, he is le with V
b
. Ex post, his gain from
trade is γ +δQ −p. An uninformed buyer faces a much more complicated
decision problem than a seller. When perfectly rational, he tries to maxi-
mize the expected gain from successfully closing a transaction by choosing
an appropriate price oer p, but he is unaware of the true quality.
. Perfectly Rational Buyers
Any price oer p ≤β divides the interval of possible qualities into three sub-
sets:
• Q <n
−
(p): e oer is accepted, but the buyer suers a loss;
• n
−
(p)<Q <a
−
(p): e oer is accepted with a prot for the buyer;
• Q >a
−
(p): e oer is rejected.
e assumption a(Q) = βQ implies a
−
(p) = pβ. us, the buyer’s ex-
pected gain from trade, conditional on his submitted price oer, is given by
EΠ
b
(p)=
∫
p/β
[n(Q)−p]dQ =
∫
p/β
[γ +δQ]dQ −
p
β
.
A perfectly rational buyer chooses his price oer to maximize EΠ
b
(p).
In our experiment, we will distinguish two dierent market designs which
dier with regard to the parameters in n(Q) = γ +δQ. e rst market
design is characterized by the parameter setting γ = and β < δ < β. e
valuationsofboththe sellersandthebuyersstartinthe origin, andthebuyers’
valuation has greater slope. In the second market design, we have γ = and
δ =β. is market design is is characterized by parallel valuation lines. e
following proposition derives the respective optimal price oer, denoted by
p
∗
, forthe two market designs. e thirdparameter settingmentioned in the
proposition has not been analyzed in the experiments.
Proposition 2.1 Assume a market in which the buyers’ valuation of quality
Q is n(Q)=γ +δQ, and the sellers’ valuation is a(Q)=βQ, with γ ≥ and
δ ≥β >. If
Price oers greater than β are strictly dominated and can, therefore, be neglected: With
p = β, the price oer would attract all possible qualities up to Q = . Hence, a higher price oer
cannot make the buyer better o.
e proof of this proposition is conned to Appendix A.
A. Kirstein and R. Kirstein: Experimental Lemons Market
i) γ = and β <δ <β (rst market design), then the optimal price oer
is p
∗
=, and the average traded quality is ,
ii) γ > and δ = β (second market design), then the optimal price oer
is p
∗
=γ, and the average traded quality equals γβ,
iii) δ ≥β, then the optimal price oer is p
∗
= β, and the average traded
quality is /.
A predicted price p
∗
= impliesthatthe rst market collapses completely.
Even though it would be ecient to trade all units, asymmetric information
makes perfectly rational buyers abstain from positive oers, so no units are
traded. In the second market design, the market collapses only partially:
Units with Q ≤ a
−
(γ) = γβ are traded. e third parameter setting is
added for the sake of completeness; this is a case in which the market does
not collapse, and all units are traded.
. Boundedly Rational Buyers
.. Iterative Reasoning
Now we present a more general model which is based on iterative thinking.
It allows for modeling both boundedly and perfectly rational players. We
start with a buyer who does not analyze the situation at all. He picks his
price oer randomly. We call this type of behavior ‘performing zero iteration
steps.’ Ifanotherbuyeracknowledgesthatthe quality is uniformlydistributed
between and , he would base his decision on the expected quality of /.
Such a buyer would then oer a price ranging between the sellers’ and his
own valuation of the expected Q =. is buyer performs the rst step of
the iterative reasoning process. His maximal willingness to pay is n().
A third buyer may realize in this situation that, even if he oers his maxi-
mal willingness to pay, thesellerswho own the highest qualities would refuse
his oer. If the buyer understands this, then the expected quality of the good
he will actually receive, conditional on his price oer, is smaller than the un-
conditional expected quality his price oer was based on aerthe rst stepof
reasoning. erefore, this buyer will update his oer and bid a lower price. A
buyer who stops here has performed two steps of iterative reasoning. In the
next reasoning steps, a buyer would realize that the lower the price oer, the
smaller the maximum quality the buyer can expect to receive.
Let us denote the expected quality for a buyer who performs k steps of
iterative reasoning as EQ
k
. We assume that such a player represents the dis-
tribution of the quality by this expected value. e buyers’ maximum will-
ingness to pay is denoted as n
k
=n(EQ
k
); k ∈IN.
Homo Oeconomicus 26(2)
.. Complete Market Collapse
In parameter setting (i.e., γ = and δ > β), the maximum willingness
to pay of a buyer who performs only one step of iterative reasoning is n
=
n(EQ
) = δ. We limit our focus to cases where δ < β, which implies
n
< β. If the buyer expects ‘his’ seller to oer the expected (or average)
quality EQ
, the buyer should at least bid a sellers’ valuation of this quality,
i.e., a
=a(EQ
)=β.
At a price oered aer one step of iterative reasoning, all sellers who oer
a quality greater than Q
=a
−
(n
)=δβ will prefer to keep their item for
themselves. It is due to the assumption δ <β that, even if the buyer oers
his maxiumum willingness to pay, the sellers who own units of high quality
can be expected to reject the oer, or: Q
<.
If a buyer performs a second reasoning step, he anticipates Q
to be the
highest possible quality in the market if he oers p = n
. erefore, the ex-
pected quality contingent on the maximal oer during the rst step of itera-
tive reasoning is EQ
=.Q
. erefore, such a buyer has a maximum will-
ingness to pay, contingent on his beliefs, which amounts to n
= n(EQ
)=
δQ
=δ
β. e assumption δ <β implies EQ
<EQ
and n
<n
.
Figure displays EQ
, a
, n
, Q
, and EQ
. Quality is shown on the hori-
zontal axis, the valuations of both sellers and buyers on the vertical axis. e
upperdiagonalline represents thebuyers’valuation, n(Q), and the lowerone
represents the sellers’ valuation, a(Q). Clearly, Q
k
as well as n
k
decrease as
the number of iteration steps k increases. Iterative reasoning leads to lower
priceoers,thegreaterthenumberofreasoningstepscarried out. Foranin-
nite number of steps, the buyer reaches the price oer predicted for perfectly
rational buyers: He oers zero, and no unit is traded. Boundedly rational
players, however, carry out only a limited number of steps. For any num-
ber of reasoning steps k a player performs, we can derive an interval [a
k
, n
k
]
from which this theory predicts the player to choose his price oer.
.. Partial Market Collapse
Forthesecondparametersetting(γ >andδ =β), Figuredemonstratesthe
situation of a decision-maker who performs one step of iterative reasoning.
Such a buyer assumes an expected quality EQ
=. us, he should oer a
price between a
=a(EQ
)=β and n
=n(EQ
)=γ +β.
If a buyer carries out a second step, he would realize that, even if he bids
n
, the sellers holding a unit of the highest quality would reject his oer. e
highest possible quality which a buyer actually expects to achieve during the
rst step of reasoning is Q
= a
−
(n
) = (γ +β)β. us, this buyer ex-
pects a quality that equals Q
= (γ +β)β. Aer an innite number of
A. Kirstein and R. Kirstein: Experimental Lemons Market
Fig. — Complete market collapse: rst step of iterative reasoning
-
6
n(Q) = δQ
a(Q) = βQ
n, a
δ
β
δ/2 = n
1
β/2 = a
1
Q
1
0
EQ
1
= 1/2
a
−1
(n
1
) = Q
1
EQ
2
Fig. — Partial market collapse
-
6
n(Q) = γ + δQ
a(Q) = βQ
n, a
γ + β
β
γ + β/2 = n
1
β/2 = a
1
γ
Q
1
0
EQ
1
= 1/2
Q
1
EQ
2
1/3
Homo Oeconomicus 26(2)
iteration steps, a perfectly rational buyer oers p =γ, and qualities below
are traded.
. e Experiment
. Experimental Design
e experimental parameter settings with complete and partial market col-
lapse are labeled as (comp), and (part), respectively. In the (part) market, we
chose δ =, and γ =. Hence, the buyers’ valuation was n(Q)= +Q. In
the (comp) market, we chose δ = and γ = , leading to n(Q) = Q. In
both designs, the sellers’ valuation was xed as a(Q)=Q (thus β =). We
conducted two experiments with two treatments each.
Experiment
• treatment A: rst (part), then (comp);
• treatment B: rst (comp), then (part).
Experiment
• treatment C: rounds (comp);
• treatment D: rounds (part).
In treatments A and B, each subject played (part) and (comp) once. We
added treatments C and D in order to examine whether the observations of
the rst two treatments had merely been rst-round eects. e experi-
ments were conducted with students of Karlsruhe University (Germany)
who participated in experimental sessions (ve sessions each for treat-
ments A and B, and four sessions each for C and D). e group size ranged
from to participants per session. Each of the subjects participated in
only one session. Most of the participants were studying Business Engineer-
ing at the undergraduate level. At the time of the experiment, none of them
had enjoyed any formal training in contract theory.
In each session, the group was split in half and randomly assigned to two
dierent rooms. e participants were not permitted to communicate with
each other. e written instructions were distributed and read aloud. Ques-
tions were asked and answered only in private.
e rst experiment was not computerized, i.e., paper and pencil were
used. eparticipantsin each oftheroomsrst actedas buyers(they submit-
ted price oers to the other room), and then acted as sellers (they received
price oers from the other room). We let subjects take over both roles be-
cause each seller only had to make the simple decision of whether or not
e instructions for (part) in treatments A and B are included in Appendix B. e highly
similar instructions for (comp) as well as for the second experiment are available on request.
A. Kirstein and R. Kirstein: Experimental Lemons Market
a certain price oer exceeded the valuation of his unit of the good.4 Every
buyer wrotea price oer on a prepared form. An administrator in each room
rstcollectedall the price oers,andthen he endowedtheplayersin his room
with one unit of the good.5 e price oers were randomly allocated to the
participants in the other room, and the sellers’ decisions were made.
Aer having submitted their price oers in each round, and before hav-
ing learned the actual results, the buyers were asked to write down, in their
own words, the line of reasoning that led to the corresponding price oer.6
Finally, the subjects learned their individual outcomes in private. Only those
buyerswhose oerswereacceptedlearnedaboutthequalitytheir anonymous
partner was endowed with. e second round was carried out in the same
way as the rst, but with a dierent market design.
While acting as buyers, participants received an initial endowment of
Euros per round, which ensured that their willingness to pay did not exceed
their ability to pay. As sellers, the subjects received one unit of the good and
an additional show-up fee of Euros which compensated for the possibility
of being endowed with a poor-quality good. Aer the two rounds, the sub-
jects were paid their earnings in cash. e chosen parameters resulted in an
average payment of about Euros, and the experiment lasted approximately
minutes.
e second experiment was computerized. Each subject played repe-
titions of only one of the above market designs, i.e., (comp) or (part). e
subjects were seated and instructed the same way as under treatments A and
B.7e buyers were endowed with ECU (experimental currency units) per
round. e sellers received one unit of the good (the quality of which could
be dierent in each round), and ECU per round to compensate for the pos-
sibility of receiving low qualities of the good. In every round, each buyer was
randomly and anonymously rematched with one of the sellers. Aer each
round, the buyers were asked to write down their reasoning regarding the
prices they oered in a questionnaire (we used the same wording as in treat-
ments A and B). en the subjects were informed about their own outcome
fromthe preceding round. Aer rounds, subjects werepaid their earnings
4In the rst session of both treatments A and B, the subjects played only one role, either that
of buyer or seller. From the second session on, we switched to the above procedure.
5is guaranteed that the quality of participants’ units (as sellers) did not aect their price
oers (as buyers).
6e exact wording of the question was, now translated into English, ‘Please briey describe
– in each round – the reasoning that led to your particular price oer in that round.’
7e procedures diered only slightly from treatments A and B in that the subjects stayed
in the randomly assigned role of either buyer or seller during all rounds. Even though the
sellers’ situation was of the same simplicity as under treatments A and B, it appeared reasonable
notto switch roles. is experiment was computerized, andwe wantedto avoidthe possibility of
subjectsmixingupthetworolesifconfrontedwithdierentcomputerscreensinrapidsequence.
Homo Oeconomicus 26(2)
in cash. ECU amounted to . Euros. e sessions lasted about one hour,
and the participants were paid about Euros on average.
. One-shot Play in Treatments A and B
.. Description of Individual Data
Figures and give an overview of all price oers made in both rounds of
each design. Treatment A, i.e., (part) in the rst round and (comp) in the
second, contains observations. Treatment B (rst (comp), then (part))
consists of observations per round. e bold symbols represent rejected
oers (no trade), and the open ones represent accepted prices (trade). e
dots depict the rst round of play, i.e., (part) in Figure , and (comp) in
Figure , and the triangles represent the second round of play, i.e., (part)
and (comp). e line represents the sellers’ valuation of their quality. For all
seller decisions to be rational, no bold symbol should appear abovethe lineas
the oered price exceeded the seller’s valuation. Moreover, no open symbol
should appear beneath the line since the price is short of the valuation. Only
a negligible number of the sellers’ decisions deviate from perfectly rational
behavior.8
.. Does the Ordering of the Market Designs Matter?
e rst step in evaluating the experimental data relates to the question of
whether the ordering of the two market designs in treatments A and B has
a signicant inuence on the oered prices.9 us, our rst null hypothesis
is:
Hypothesis 1 In both market designs, the oered prices in rst-round play
do not dier from those in second-round play.
A Wilcoxon test shows for each market design that the prices oered in the
rst round did not dier signicantly from the observed prices in the second
round.0 us, the null hypothesis cannot be rejected for both of the market
8In Figure , we observe rejected oers that should have been accepted, i.e., no bold symbol
appearsabove the line, and accepted oers that should have been rejected, i.e., open symbols
appearbelow the line. InFigure,onlyrejectedoer shouldhavebeenaccepted,andaccepted
oers were better rejected.
9We have used SPSS version .., a statistical soware packages from SPSS Inc., to evaluate
the data. All tests were conducted at a percent signicance level.
0For each market design, we compared the results of the rst and second round play by using
a Mann-Whitney U-test. e mean ranks of . in (part) and of . in (part) do not dier
signicantly at a %-level of signicance (Z = −., two-sided asymptotic probability of p =
A. Kirstein and R. Kirstein: Experimental Lemons Market
Fig. — Price Oers in (part)
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Price
Seller Valuation
rejected prices (part1)
accepted prices (part1)
Indierenz
rejected prices (part2)
accepted prices (part2)
a(Q) = 3Q
Fig. — Price Oers in (comp)
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Price
Seller Valuation
rejected prices (comp1)
accepted prices (comp1)
Indierenz
rejected prices (comp2)
accepted prices (comp2)
a(Q) = 3Q
.). We obtained a similar result for the (comp) markets, in which the mean ranks amount
to . and . in (comp) and (comp), respectively (Z = −., two-sided asymptotic
probability of p = .).
Homo Oeconomicus 26(2)
designs, and we have derived our rst result.
Result 1 e observed price oers are independent of the order in which
the market designs were played.
is result encouraged us to evaluate the data generated for each market
design without regard to whether it was generated in the rst or the second
round.
.. Do Buyers Oer Rational Prices?
e proposition in Section . and the theoretical analysis in . show that
fully rational buyers in each of the two market designs need to perform an
innite number of iterative reasoning steps. Many recent experimental stud-
ies, however, reveal thatiterative reasoning seems to stopaer very fewsteps,
if it starts at all. us, we conjecture a considerable number of subjects to be
boundedly rational when formulating the following null hypothesis:
Hypothesis 2 In the(comp)market, only p = is oered, while inthe (part)
market, only p = is oered
According to the Proposition in Section ., the average traded quality in
(comp) should be zero, whereas in the (part) market it is expected to be ,
if the above null hypothesis is true. e descriptive aggregate data of both
(comp) and (part) are provided in Table . It shows the minimum, maxi-
mum, and average values of the price oers, qualities, and traded qualities,
as well as the buyers’ and sellers’ gains from trade in each market design.
In(part), % ofthe price oers are accepted, andthe averageprice of .
is signicantly greater than the predicted p =.4 e average traded quality
of . is nearly twice as high as the theoretical prediction of ..
In (comp), % of all prices oered are accepted. e average price oer
amounts to . Euros, and the average traded quality is ., both of which
are obviously far greater than zero. Clearly, the market does not collapse
Since the level of signicance for dierences in the (part) markets is rather low, we have also
evaluated the data of the two rounds separately, which leads to conclusions that are identical to
those subsequently derived.
As mentioned above, the subjects acted either as buyers or sellers in the rst session. ere-
fore, the number of observations is not exactly the half of the number of participants.
e table only shows the gains and losses from trade (the sellers’ show-up fee, their endow-
ments with the good, and the buyers’ monetary endowment are excluded).
4etwo-sided one-sample t-testshowsthattheempiricalaverageis signicantlygreaterthan
the theoretical average of . e test results are as follows: average = ., t = ., and
p < ..
A. Kirstein and R. Kirstein: Experimental Lemons Market
Table — Basic Data per Round (in Euros, endowments excluded)
p Q traded Q Π
b
Π
s
min 0.00 0.00 0.00 -1.20 -1.16
(part) average 1.66 0.51 0.34 0.12 0.47
101 observations max 3.00 1.00 0.94 2.16 2.20
min 0.00 0.00 0.00 -1.71 -1.26
(comp) average 1.31 0.51 0.29 -0.21 0.34
101 observations max 3.40 1.00 0.94 2.18 2.08
completely under the (comp) design, so we can reject the null hypothesis
also for this market design.
Result 2 In both market designs, observed prices are higher than predicted
for perfectly rational players.
Since some goods are traded, buyers in the (part) design earn an average
payo of . Euros but make an average loss of . in the (comp) market.
Sellers in (part) earn ., whereas in (comp) they only earn . Euros per
round on average.
.. Does Limited Iterative Reasoning Explain the Price Oers?
In this section, we examine the data with regard to our claim that itera-
tive thinking may provide an explanation for the observation that prices and
traded qualities are higher than predicted by rational choice theory. e ar-
gument proceeds in four steps:
. Aer each round, the subjects gave descriptions of their own reason-
ing. Only fromthese statements,and independently of theirsubmitted
priceoers,the participants’iteration types havebeen determined. We
denotethe number ofiterative reasoning steps a subject apparently has
carried out according to his self-description, as ‘i’ and call the subject
‘type-i.’
. According to the theory of iterative reasoning and the valuations
a
i
, n
i
presented in Section ., we derive the predicted, i.e., the type-
consistent price interval for each type-i.
. We then observe the actual price oer p.
Homo Oeconomicus 26(2)
. Finally, we are interested to see whether a negative correlation exists
between the type-i of a participantandhisactual price oer. Moreover,
we explore whether the observed price oer has been chosen from the
type-consistent price interval. If not, then the theory of iterative rea-
soning would have no explanatory power with regard to the observed
behavior.
e self-descriptions have been sorted into three type-i categories.5 If a
self-description did not contain an expected quality of / nor any further
systematic evaluation of the market situation, this subject was categorized
into type-.6 Participants who expressly mentioned they were calculating
with an expected quality of / were encoded as type-.7 All individuals who
performed moreiterativereasoningstepsweregrouped into the lastcategory,
called type-+.8.’ Most of the written statements indicate that players either
perform , , , or an innite number of iteration steps.9
Table — Type-i-consistent Price Oer Intervals
buyer’s type-i min oer max oer
0 0.00 4.00
comp 1 1.50 2.00
2+ 0.00 1.33
0 0.00 4.00
part 1 1.50 2.50
2+ 0.00 2.25
5In Appendix C, we present an overview of some typical verbal statements of each type. e
encoding of the verbal statements was done without any knowledge of the oered prices. e
lled in questionnaires are available on request.
6For instance, typical lines of reasoning that were categorized as type- subjects were ‘I chose
p such that quality gets better’, or ‘I had no idea, I just gambled’, or ‘I analyzed what the seller’s
quality must be, compared to my price oer.’ e third statement could as well be made by a
subject who understood the market mechanism well, but was unable or unwilling to describe
this in more detail. However, this statement is too ambiguous to be anything else than type-.
Overall, we instructedtheassistantwhodid the encoding to be ratherhesitantwhencategorizing
a statement into type- or type-+.
7Subjectsoftype- could easily beidentied. Typical examplesfora type-statement are‘E(Q)
= / and a(Q) = ., thus my oer is .’, or ‘I calculated E(Q) = / and wanted to make some
prots.’
8e subjects’ self-descriptions rendered it impossible to distinguish, e.g., type- from type-.
A typical type-+ statement was, e.g., ‘e possible loss is always higher than the possible gain,
thus on average there is always a loss
9In this, our observations are in accordance with studies such as Nagel (), or
Kübler/Weizsäcker ().
A. Kirstein and R. Kirstein: Experimental Lemons Market
Table — (comp) played by Type-i
(101 possible observations, 4 descriptions missing)
Type-
Price oer interval 0 1 2+ Sum
p > 2 5 0 0 5
1.5 ≤ p ≤ 2 30 22 1 53
1.33 < p < 1.5 2 2 0 4
p ≤ 1.33 20 5 10 35
Sum 57 29 11 97
Average price by type 1.45 1.47 0.29 –
Median price by type 1.50 1.50 0.00 –
Table — (part) played by Type-i
(101 possible observations, 4 descriptions missing)
Type-
Price oer interval 0 1 2+ Sum
p > 2.5 2 0 0 2
2.25 < p ≤ 2.5 4 4 0 8
1.5 ≤ p ≤ 2.25 36 20 4 60
p < 1.5 20 1 6 27
Sum 62 25 10 97
Average price by type 1.66 1.91 1.17 –
Median price by type 1.63 2.00 1.00 –
Table displays the price interval from which a certain iteration type
would consistently choose his price oer, as we have demonstrated with our
theoretical analysis in Section .. We have encountered three specics:
• Asubjectoftype-is expectedtooerpricesfromtoinbothmarket
designs. Hence, any price oer would be type-consistent. us, our
theory does not provide falsiable hypotheses with regard to type-.
• Inthe (comp)marketdesign, prices between.and . canneitherbe
related to type-, nor to type-+. Such prices were oered only twice.
• e predicted price intervals in (part) overlap. Prices between . and
. would be consistent with type- and type-. Nevertheless, any
price below . would be consistent only with type-+.
Homo Oeconomicus 26(2)
Tables and show the frequencies of chosen prices0, where the rst
column lists the price oer intervals as presented in Table and discussed
above. In the bottom two rows, the types’ average and median prices are
depicted.
e self-descriptions of % of the subjects in the (comp) and of % in
the (part) market design are consistent with our denition of type-. Ex-
tremely high prices, i.e., prices located in the rst interval, have seldom but
solely been chosen by types-. Since type- chooses a price randomly, any
price oer is consistent with type- (type-consistent choices are printed bold
in Tables and ). In (comp), % of types- and % of types- oer type-
consistent prices. In (part), the percentages amount to % and %, re-
spectively. us, regarding the descriptives, our observations are to a large
extent in line with the theory. As our theory generates restricted price oer
intervals only for type- and type-+, we initially conjecture a negative rela-
tion between oered prices and type-i for i = ,+. We test this against the
(converse) null hypothesis for type- and +:
Hypothesis 3 e higher the type, the higher the price in both market de-
signs.
e highly signicant rank order correlations amount to -. in (comp),
andto-. in (part). Tables, and haveshownthatthe negativerelationsof
types andprices are based on alargenumber oftype-consistentprice choices.
erefore, we draw the following conclusion:
Result 3 e iteration types- and + derived from the subjects’ self-
descriptions are signicantly negatively correlated with the observed price
oers.
ough restricted price intervals are theoretically derived only for type-
and type-+, we can explore dierences in median price oers among all
three types in each market (see Tables and for the median prices). e
Kruskal-Wallis ANOVA on ranks in (comp) reveals that the three groups dif-
0Ineachmarket, fourdescriptionsaremissing,asfoursubjects didnotllinthequestionnaire.
A χ
-test would clearly support this result, but its application faces the problem that too
many entries in Tables and equal zero.
e tests each reveal a (one-sided) p < .. We used the prices and self-descriptions gen-
erated by types- and + in each market to conduct the tests.
We do not test group dierences in mean prices by using a one-way ANOVA, as the data in
Treatments C and D did neither pass the normality tests nor the equal variance tests. us, we
applied the non-parametric alternative, i.e., the Kruskal-Wallis ANOVA on ranks to all Treat-
ments A through D. We used the prices and self-descriptions generated by types-, and + in
each market to conduct the tests.
A. Kirstein and R. Kirstein: Experimental Lemons Market
fer signicantly in median prices (H = ., d f = , p < .). Also in
(part), the dierences in median prices are signicant among the three types
(H =., d f =, p =.).
e pairwise comparisons (Dunn’s method)4 show that, in both market
designs, types-+ chose signicantly lower prices than types-, which con-
rms the above stated result . What is more, types-+ have submitted lower
price oers than types-, which, however, is signicant only in (comp). e
comparison between types- and types- exhibits no signicant dierence in
both markets. Overall, there is no evidence that lower iteration types have
chosen lower prices than higher types. Hence, this explorative analysis sus-
tains the idea that iterative thinking may contribute to explaining the ob-
served deviations from perfect rationality.
.. Is Limited Iterative Reasoning Eciency-enhancing?
In the previous sections, we derived the conclusion that bounded rationality
on the buyers’ side prevents one-shot lemons markets from a complete or
partial collapse. Figure shows which market sideproted orlost fromtrade
in treatments A and B.
e point labeled ‘no trade’ or ‘rational(comp)’ represents the situation
without trade, as well as the outcome which rational choice theory predicts
for the (comp) market. e lower diagonal indicates the iso-welfare line (for
a utilitarian welfare function, which denes welfare as the sum of the parties’
outcomes) for the resulting zero welfare level. Point ‘data(comp)’ is the ob-
served outcomeunder the (comp) design: e total gains fromtrade amount
to . Euros for the sellers, and to -. Euros for the buyers. Trade has
earned thegroup of sellersa remarkablegain whicheven exceedsthe loss suf-
feredby thegroup of buyers. Trade has increased totalwelfare, butonlyin the
Kaldor-Hicks sense. Voluntary trade does not lead to a Pareto-improvement.
Boundedly rational buyers would prefer prohibition over voluntary trade if
this were the only way to protect them from their losses.
e analysis comes to dierent results for the (part) design. e theo-
retical prediction, assuming perfect rationality, is represented by the point
‘rational(part)’: If the buyers oer a price p =, then only units with quality
Q < are traded. Trading one unit generatesa welfaregain of . With a uni-
formdistributionof quality and buyers, the expected welfare gain is ..
e price p =, which is predicted by rational choice theory, distributes this
welfare gain evenly among the two market sides, so both sides receive ..
eupperdiagonalrepresentsthe welfarelevelachieved inthisoutcome. e
actual result, however, is shown at the point labeled ‘data(part)’: e average
4Dunn’s method is used as a post hoc test and is conducted to a %-level of signicance.
Homo Oeconomicus 26(2)
Fig. — Average Gains from Trade
-
6
Π
b
Π
s
t
rational(comp)
no trade
0
t
rational(part)
16.83
16.83
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
t
data(part)
48.5
12.4
t
data(comp)
34.5
−21.2
earnings of the sellers accrue to a total of ., while the buyers receive a to-
tal of . Euros. Welfare is higher than under perfect rationality, but – as
in the (comp) market – at the buyers’ expense. e sellers prot from the
existence of bounded rationality among the buyers, while the boundedly ra-
tional buyers are (on average) worse o than perfectly rational buyers would
be. However, in the (comp) market, both sides gain from voluntary trade, as
it induces a Pareto-improvement. Hence, for this market design our study
provides no justication for prohibition.5
. Repeated Play in Treatments C and D
According to section .., many subjects seem to have performed only a
limitednumberofiterativereasoningsteps. is explainssignicantlyhigher
price oers than predicted by rational choice theory. It is possible that these
results are due to the fact that only one round per market design was played.
e subjects may learn to perform more iterative steps when playing several
repetitions of the game. erefore, we let subjects who did not take part in
treatments A or B play rounds of either the (comp) design – subsequently
5e welfare analysis for treatments C and D comes to the same pattern and, hence, does not
yield any other insight.
A. Kirstein and R. Kirstein: Experimental Lemons Market
denoted as treatment C – or the (part) design – treatment D. We explore the
following questions:
. In Section ..: Do prices and traded qualities decline to the level pre-
dicted by rational choice theory?
. InSection..: Are thesubjects’ types-i stable,or do they changeover
time?
. In Section ..: Does a negative relation exist between types-i and
observed prices over rounds?
.. Data Description
In the repeated (comp) market, % of price oers during all rounds are
accepted, while the acceptance rate in treatment D is %. As in the one-
shot play, we observe higher acceptance rates in the (part) than the (comp)
market, and sellers behaved very rationally.6
Table displays the prices and qualities, as well as the gains and losses
from trade to the buyers and the sellers. e data aggregate rounds with
observations per round under (comp) and rounds with observations
per round under (part). Prices and payos show a tendency to be higher in
the repeated (part) than in the repeated (comp) market. As in treatments A
and B, some buyers face severe losses, especially in the (comp) design.
Figure displays the development of average prices over rounds. Even
in round , both in the (comp) and the (part) design, the markets did not
collapse to the extent predicted by rational choice theory. In the repeated
(comp) market, the average price oscillates around . during the last seven
rounds, which is far more than the theoretically predicted price of zero. e
overall average traded quality is . (see Table ), which also substantially
deviates from the prediction of zero. Under the (part) design, the average
price ranges from . to . during the second half of the experiment. Even
aermany repetitions, theoeredpricesexceed the perfectly rationalpredic-
tion of p =. In each round, the observed prices dier signicantly from the
theoretically predicted price.7 e overall averagetraded quality of. (see
Table ) is almost twice the . which was predicted by rational choice the-
ory. Moreover, prices decline both more rapidly and to a larger extent under
the (comp) than under the (part) design. is implies our next result.
6As to the sellers’ behavior, in treatment C, we observed only unprotably accepted oers,
and disadvantageously rejected oers in rounds of play. In treatment D, they amounted to
, and , respectively.
7We exemplarily give the two-sided one-sample t-test results for the last two rounds, testing
for a mean of . Round : mean = .; t = .; p < .; round : mean = .; t = .;
p < ..
Homo Oeconomicus 26(2)
Table — Basic Data per Round (in ECU, endowments excluded)
p Q traded Q Π
b
Π
s
min 0.00 0.00 0.00 -3.00 -0.56
20 times (comp) average 0.93 0.49 0.23 -0.19 0.57
max 3.30 1.00 0.95 1.33 3.00
min 0.00 0.00 0.00 -1.68 -1.94
20 times (part) average 1.58 0.50 0.29 0.09 0.44
max 3.00 1.00 0.98 2.94 2.68
Fig. — Price Oers in repeated (comp) and (part)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Price
Round
Average p 20(comp)
Average p 20(part)
Result 4 Even aer rounds of repeated play, prices and traded qualities
do not decline to the level predicted by rational choice theory.
A. Kirstein and R. Kirstein: Experimental Lemons Market
Fig. — Percentages of Types in rounds (comp)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Round
Percent Type 2
Percent Type 1
Percent Type 0
Fig. — Percentages of Types in rounds (part)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Round
Percent Type 2
Percent Type 1
Percent Type 0
.. e Development of the Types
e average prices show a tendency to decrease over time under both treat-
ments. Inlightofourtheoryofboundedrationality, thisshouldcoincidewith
an increase in the level of reasoning, the more rounds are played. Figures
and reveal the percentage of types- to + in the two markets.8
8Note that types are not necessary stable over time. A certain subject’s type-i may be adjusted
upwards or downwards if the participant describes his reasoning accordingly. Moreover, an
individual’s development is not necessarily monotonic.
Homo Oeconomicus 26(2)
Table — Overview of Self-descriptions
through periods
Subjects’ Self-descriptions 20(comp) 20(part)
20 rounds type-0 11 8
20 rounds type-1 1 5
20 rounds type-2+ 1 –
From type-0 to 1 – 2
From type-0 to 2+ 3 3
From type-1 to 2+ 1 3
From type-0 to 1 to 2+ – 1
From type-1 to 0 6 3
Forth and back 6 7
Missing 2 –
Sum 31 32
During the whole rounds of (comp) (see Figure ), a stable percentage
of about % to % of participants are type-. Types- very quickly almost
vanish from the market and, aer round , constitute only a small share of
%. e percentage of types-+ varies between % and %. Figure shows
that only one half of the subjects are of type- in the repeated (part) market.
e share of types-+ is almost of the same size as in the repeated (comp)
market. From round on, the percentage of types- amounts to about %,
which is much higher than under the (comp) design. Overall, the data allow
us to draw the conclusion:
Result 5 Inboth marketdesigns, thepercentage of type-+growsovertime.
Type- subjects almost vanish during the rounds of (comp). In (part), the
percentageoftype-is almoststable. e numberoftypes-slightlyincreases
in the repeated (comp) market, and slightly decreases in the repeated (part)
market.
Table providesan overview of thesubjects’ developments. We trackeach
buyer individually with regard to his self-described type-i through the
rounds. e rst column indicates the observed developments. ‘Forth and
back’ at the bottom of Table labels subjects who – according to their self-
description – changed from low to high type, and back.9 e label ‘Missing’
9A development forth and back may happen if a subject starts with ‘trying’, then calculates
E(Q) and, in the following, explains in detail that high qualities vanish from the market, and
A. Kirstein and R. Kirstein: Experimental Lemons Market
indicates subjects who did not (completely) ll out their questionnaires. e
remaining nominations are self-explanatory. e entries display numbers of
subjects, which add up to in rounds of (part), and to in (comp),
respectively.
About one third of subjects remain the same type throughout the
rounds. Another third shows a development from type- to type-, or forth
and back. e last third of the subjects moves from lower to higher types.
Puretypes-can be observed in the (part) market, but are almostnonexistent
in the repeated (comp) market.
.. Correspondence of Types-i and Price Oers
e percentage of type-+ grows from a very small percentage in the be-
ginning to about % during the last third in both treatments. is would
explain the observation that average prices decrease (see Figure ). In this
section, we investigate whether all types-i choose their price oers from the
type-consistent intervals throughout the rounds.40
We test our conjecture that price oers are type-consistent and, therefore,
types-+ should bid lower prices than types-. If this were true also in the
repeated game, the observed growing number of high types can be made re-
sponsible for the decreasing price. Analogously to the examinations of treat-
ments A and B, Tables and display the frequencies of price oers in treat-
ments C and D (type-consistent choices are printed bold). e bottom two
rows depict the types’ average and median price oers.
Similar to the one-shot treatments A and B, the highly signicant rank
order correlations that relate types- and + to their price oers reveal that,
the higher the type, the lower the price. Spearman’s rho amounts to -. in
treatment C, and to -. in treatment D.4 Tables , and show that these
relations of types and prices are based on a large number of type-consistent
price choices (the bold gures in the two tables). erefore, we conclude:
Result 6 During rounds of repeated play, the types-i contribute to ex-
plaining the observed prices as the self-described iteration types- and +
are negatively correlated with the observed price oers.
nally turns to ‘gambling’. Such behavior would have been coded as a sequence ‘type-, , , and
back to ’.
40Because buyers and sellers were newly matched aer each round, each of the ⋅ =
price oers under (comp), and of the ⋅ = under (part) are treated as independent
observations. In (comp), however, subjects lled in the questionnaire only until round ve,
and round seven, respectively, hence self-descriptions are missing.
4e tests each reveal a (one-sided) p < .. We used the prices and self-descriptions gen-
erated by types- and + in each market to conduct the tests.
Homo Oeconomicus 26(2)
Table — (comp) by Type-i
( possible observations, descriptions missing)
type-
price oer interval 0 1 2+ Sum
p > 2 57 1 0 58
1.5 ≤ p ≤ 2 103 39 18 160
1.33 < p < 1.5 6 5 9 20
p ≤ 1.33 218 27 108 353
Sum 384 72 135 591
Average price by type 1.09 1.14 0.50 –
Median price by type 1.03 1.50 0.04 –
Table — (part) by Type-i
( observations)
type-
price oer interval 0 1 2+ Sum
p > 2.5 17 6 0 23
2.25 < p ≤ 2.5 17 14 0 31
1.5 ≤ p ≤ 2.25 223 122 12 357
p < 1.5 65 28 136 229
Sum 322 170 148 640
Average price by type 1.75 1.73 1.04 –
Median price by type 1.75 1.60 1.00 –
Even though the theory predicts restricted price intervals only for type-
and type-+, we explore dierences in median price oers among all three
types in treatments C and D (see Tables , and for the median prices).4
e Kruskal-Wallis ANOVA on ranks in the repeated markets reveals that
the dierences in median prices among all three types are signicant, H =
., d f = , p < . in (comp), and H = ., d f = , p <
. in (part). Pairwise comparisons (Dunn’s method)4 show that, in
both market settings, types-+ oer signicantly lower prices than types- or
types-, which conrms our Result .
4As the data did neither pass the normality tests nor the equal variance tests, we use the non-
parametric alternative, i.e., the Kruskal-Wallis ANOVA on ranks. We used the prices and self-
descriptions generated by types-, and + in each market to conduct the tests.
4Dunn’s method is used as a post hoc test and is conducted to a %-level of signicance.
A. Kirstein and R. Kirstein: Experimental Lemons Market
. Conclusion
We have discussed a lemons market in which the buyers do not know the
actual quality of the traded good. In our experiments, we have examined
two dierent market designs: Under one design, labeled (comp), perfectly
rational players are predicted to conclude no transactions at all. us, the
market is expected to collapse completely. Under the other market design,
called (part), perfectly rational players are expected to trade only some units
of low quality. In both market designs it would be ecient that all units be
traded. According to the empirical results for both market designs, the aver-
age prices oered by the buyers and the average traded qualities are higher
than the predictions for perfectly rational players.
Many of the buyers in our experiment do not play their subgame perfect
Nash equilibrium strategy. Similar observations have oen been made in ex-
periments on other games.44 e explanation of such observation proposed
in thispaper drawson thetheory of iterativereasoning. Playerswho perform
only a limited number of iteration steps to eliminate dominated strategies
are boundedly rational. is theory includes perfectly rational behavior as a
limit case: Such a decision-maker is able to carry out an innite number of
iteration steps. For all iteration types, the theory allows us to derive a type-
consistent price interval from which a buyer of a certain type is predicted to
choose his price oer.
During the experiments, we have determined each individual buyers’ it-
eration type from written self-descriptions, independently of the observed
price oers. ree types could be identied: e type- did not start an iter-
ation, but picked his price oer randomly. Type- was able to carry out just
one iterationstep. Type-+ decided rather elaborately, i.e., undertook at least
two iteration steps.
For these types, we have compared the corresponding type-consistent
price interval with the prices which were actually oered. e vast major-
ity of prices were chosen from the respective type consistent price-interval.
Moreover, for type- and type-+, we observed a signicant negative corre-
lation between types and oered prices. is correlation did not vanish in
the repeated play experiment. Additionally, the data indicate that type-+
subjects have chosen signicantly lower prices than type- subjects. is
empirical result supports the hypothesis that the theory of limited iterative
reasoning contributes to explaining the behavior of buyers in lemons mar-
kets. e behavior of type-, however, is not captured by this theory which,
just as the theory of perfect rationality, does not say anything about such
actors. With our method, we were able to identify each player’s iteration type
in each round without referring to their price oers. Furthermore, we could
44For an overview and discussion, see Binmore/Shaked ().
Homo Oeconomicus 26(2)
determine how many subjects actually have performed iterative reasoning at
all.
e dierence between the two market settings, (comp) and (part), can
be interpreted as the existence of a quality insurance(e.g., by a contractual or
a mandatory warranty). With a full insurance, the valuation function of the
buyers would be horizontal. Hence, the (part) market reects partial insur-
ance, while buyers in the (comp) market bear the full quality risk. e results
of our experiments show that a partial warranty may induce the buyers to
oer higher prices and to conclude a higher number of transactions. Note
that this eect of warranty is not caused by signaling, nor does it depend on
risk-aversion on the part of buyers.
e collapse of markets that suer from asymmetric information is an
inspiring theoretical phenomenon. If, however, bounded rationality (in the
form of limited iterative reasoning) of the uninformed market participants is
taken into account, the ineciency derived under the assumption of perfect
rationality might be greatly exaggerated. Institutional means to preventmar-
ket failure, such as mandatory insurance, warranties, building of reputation,
may therefore go too far and be too costly. ey may perhaps do even more
harm than good.
is policy implication of our experiment, however, suers from a seri-
ous drawback: Successfully completed transactions may inict losses upon
the buyers. ey may have submitted their oer based on overly optimistic
expectations. In such a case, having concluded a transaction may not be a
Pareto-improvement. In our (comp) market, boundedly rational buyers are
even worse o than without trade. ese consumers would be interested in
regulation that protects them from participation in voluntary trade. With
regard to (comp) markets, such a regulation would not harm the perfectly
rational buyers.45
Appendix
A Proof of the Proposition
We rstderive thegeneral conditionforan optimumprice oer. Sellersvalue
quality Q ∈ [, ] with a(Q) = βQ, while the buyers value quality with
n(Q) = γ +δQ. We assume γ ≥ and δ ≥ β > . We can disregard price
oers p > β since they are strictly dominated by p = β. For any price oer
p ∈[, β], the respective buyer’s expected payo is
45is can be interpreted as an example of ‘asymmetric paternalism,’ following Camerer et al.
().
A. Kirstein and R. Kirstein: Experimental Lemons Market
V
b
+Eπ
b
(p)=V
b
+
∫
a
−
(p)
[n(Q)−p]dQ
=V
b
+
∫
p/β
n(Q)dQ −
p
β
=V
b
+
γ
β
p +
δ
β
p
−
p
β
=V
b
+
γ
β
p +
δ −β
β
p
e rst derivative with respect to p is
∂Eπ
b
(p)
∂p
=
γ
β
+
δ −β
β
p
and the second derivative is (δ −β)β
. If δ ≥ β, then both the rst and
second derivatives are positive. Hence, the corner solution p =β maximizes
the buyer’s payo, which proves our result iii).
If δ < β, then the second-order condition for an internal maximum is
satised. e rst derivative equals zero if
p
∗
=
βγ
β −δ
.
In our rst market design withγ = and β <δ <β, the maximum payo
is, thus, obtained with p =. is result establishes our result i) according to
which the market collapses completely under this parameter setting.
In our second market design (γ > and β = δ), the second-order con-
dition for a maximum is also satised, and the rst-order condition can be
simplied to
p =
βγ
β −β
=
βγ
β
=γ.
isestablishesourresultii), accordingtowhichthemarketcollapsesonly
partially.
B e Basic Instructions: Treatment A
You are taking part in an economic experiment. Each participant makes his
decisions in isolation from the others and enters them into an answer sheet.
Homo Oeconomicus 26(2)
Communication between participants is not allowed. Male forms like ’he’
will be used to refer to anyone.
In the experiment, there are two types of players, ‘buyers’ and ‘sellers,’ in
the market for good X. You take both the role of a ‘buyer’ and the role of a
‘seller.’ e subjects you interact with are not located in your room but in the
room opposite to yours. ere are as many subjects in your room as in the
opposite one.
e experiment consists of rounds. In each of the two rounds, one seller
interacts with one buyer. In both rounds, buyers and sellers will be matched
randomly anew. ereby, a subject from this room in the role of a seller ran-
domly interacts with a buyer from the opposite room. Likewise, a subject
from the opposite room randomly interacts as seller with a buyer from this
room. erefore, in the role of a seller, you always sell your X to the other
room. ere is only a small chance that you as a buyer interact with a seller
from the other room who simultaneously acts as buyer of your X. In each of
the two rounds, it will be randomly chosen which buyer and seller interact.
Even aer the experiment, you will not be informed about who you traded
with.
In each round, each seller is endowed with one unit of good X, and each
buyer has Euros at his disposal.
In each of the two rounds, the situation is as follows: e sellers oertheir
X. Each unit of good X has a certain quality that is only known to its seller.
e qualities of X are uniformly distributed on the interval [,], that is each
quality between and is equally probable. us, indicates the worst and
the best quality. is probability distribution is known to both buyers and
sellers. e actual quality of a unit of good X is labeled Q.
e buyersvalue good quality more highlythan bad quality. e valuation
of a certain quality in Eurosis described by a function n(Q). e exact shape
of the function n(Q) will be explained later in the instructions. No buyer
can discover the real quality prior to his decision to buy; he only knows the
probability distribution of quality. Not until aer a purchase does each buyer
learn about the real Q of his unit of X.
Aer each round, the buyers are credited a payo following this rule:
• If trade has taken place at price p, the buyer gets −p +n(Q)Euros,
• If no trade has taken place, the buyer gets Euros.
As for the sellers, the function a(Q)=Q denotes their value of good X
in Euros: If X is not sold in one round, the seller receives a(Q)Euros in that
round. If, in contrast, a seller sells his X, he obtains the respective sales price.
e totaled payos of the two rounds are the earnings of buyers and sellers.
Each round passes as follows:
. First, the buyer makes his decision and enters his proposal for a sales
A. Kirstein and R. Kirstein: Experimental Lemons Market
price in his form (there are separate forms for each of the two rounds).
All forms will then be collected by the experiment supervisor and ran-
domly distributed to the sellers in the other room. Each seller receives
exactly one form from an anonymous buyer.
. Each seller gets assigned a certain quality. en he decides whether or
not he wants to sell his unit X at the price proposed by the buyer. He
enters this decision in the form. If a sale is made, he also enters the
actual quality of the unit sold.
. Again, the forms will be collected by the experiment supervisor and
given back to the respective buyers. If a purchase has taken place, the
buyer is informed about the real quality of the good X that he bought.
. is completes one round.
. Aer the two rounds, each player gets paid his total payos in cash.
Instructions for Buyers: 1st round46
Your subject number is:
Duringthis round,thesituationonthe X-marketis as follows(alsoseeFigure
):
• Each buyer owns exactly Euros, and eachseller ownsexactly oneunit
of X.
• e buyer’s valuation of the quality of good X in the rst round is
n(Q) = +Q. us, for example, one unit of good X with quality
Q = . is worth n(.)=. Euros to each buyer.
• e sellers value X by a(Q) = Q. erefore, the same unit is worth
a(.)=. Euros to the seller.
Example — We assume a buyer to purchase an X at price p = . Euros,
and the real quality of that X to be Q = .. us, p > n(Q). en, the
buyer receives an amount of ( −. +.) = . Euros out of this round.
If, in contrast, he buys this unit (with Q = .) at price p = . Euros, then
p <n(Q). His earnings will then be ( - . + .) Euros = . Euros.
46e instructions for the second round are the same, except for the altered n(Q) which then
is n(Q) = Q.
Homo Oeconomicus 26(2)
Fig.
1
3
0.5 1
n(Q) = 1 + 3Q
a(Q) = 3Q
Q
Euro
Oer Form (Round 1)47
e decision of a buyer
Your subject number is:
My price oer:
I want to buy one unit of X at price p = ...
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
e decision of a seller
Your subject number is (please ll in!): ...
My decision :
( ) I decline the oer.
( ) I accept. My unit of X is of quality Q = ....
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
e Questionnaire
Description of buyers’ reasoning:
Your subject number is:
Please briey describe - in each round - the reasoning that led to your par-
ticular price oer in that round: Round : Round :
47e form for Round is similar.
A. Kirstein and R. Kirstein: Experimental Lemons Market
C Typical Statements
Here we present some typical verbal statements of our participants.
Type- is supposed to not even calculate an expected quality. Some of the
written statements that we coded as types- are, for example:
• ‘I chose p such that quality gets better,’
• ‘I had no idea, I just gambled,’
• ‘Seller only sells if p >Q; my choice wasarbitrary –best choicewould
have been Cent above Q,’
• ‘Defensive behavior - better to be le with the good on my hands,’
• ‘I analyzed what the seller’s quality must be, compared to my price of-
fer,’
• ‘Prots rise with higher risk – no alternative seems to have decisive
advantages, so I chose the middle course.’
Type- is expected to explicitly use an expected quality of / in their cal-
culations. Some examples are:
• ‘E(Q)= and a(Q)=.; thus, my oer is .,’
• ‘SinceQ isuniformlydistributed,I used Q < (risk-averse). Because
a(Q)=Q, I chose p =.,’
• ‘With E(Q)=. a price p =. is accepted with probability /,’
• ‘I calculated E(Q)=. and wanted to make some prots.’
Type-+ performs at least one more step of iterative reasoning than type-
. erefore, type-+ knows that the conditional expected quality clearly is
smaller than / and a loss is to be expected with too high a price. Some
examples (from the (part) market) are:
• ‘I compared possible gains and losses in a table; the chance to gain is
: compared to the chance to lose; this is too risky,’
• ‘e possible loss is always higher than the possible gain; thus, on av-
erage there is always a loss,’
• ‘e expected gains are always smaller than ; an oer is advantageous
only if the slope of n(Q)is at least twice as much as the slope of a(Q),’
• ‘E.g., at p = . the seller sells if Q < .: with Q = . prots are
cents, with Q = . prots are zero, with Q = . losses are cents,
and so on; thus, there is a negative expected prot.’
Homo Oeconomicus 26(2)
Acknow ledgements
We aregrateful to Max Albert, George Akerlof, Ted Bergstrom, Friedel Bolle,
Vincent Crawford, Maher Dwik, Mathias Erlei, Ralph Friedmann, Rod Gar-
rat, Hans Gerhard, Manfred Holler, Wolfgang Kerber, Manfred Königstein,
Clemens Krauß, Rosemarie Lavaty, Göran Skogh, Dieter Schmidtchen, Jean-
Robert Tyran, to other seminar and conference participants in Hamburg,
Karlsruhe, Kassel, Saarbrücken, SantaBarbaraandZürich, andtotwoanony-
mous referees for valuable comments (the usual disclaimer applies). Part of
this research was done while we enjoyed the hospitality of the University of
California in Berkeley (Law School) and Santa Barbara (Economics Depart-
ment). e Deutsche Forschungsgemeinscha provided nancial means to
run the experiments.
References
Akerlof, G. (), e Market for Lemons, Quarterly Journal of Economics : –
.
Bazerman, M.H. and Samuelson, W.F. (), I Won e Auction But I Don’t Want
e Prize, Journal of Conict Resolution : –.
Beard, T.R. andBeil, R.O.(), Do People Rely on theSelf-interested Maximization
of Others? An Experimental Test, Management Science : –.
Binmore, K. and Shaked, A. (), Experimental Economics: Science or What?
Mimeo, London and Bonn; forthcoming in Journal of Economic Behavior and
Organization.
Camerer, C.F. (), Behavioral Game eory. Experiments in Strategic Interaction,
Princeton University Press.
Camerer, C., Issacharo, S., Loewenstein, G., O’Donoghue, T., and Rabin, M. (),
Regulation for Conservatives: Behavioral Economics and the Case for ‘Asymmet-
ric Paternalism’, University of Pennsylvania Law Review : –.
Camerer, C.F., Ho, T.-H., and Chong, J.-K. (), Behavioral Game eory: ink-
ing, Learning, andTeaching; unpublishedmimeo, presented attheNobel Seminar
.
Camerer, C.F., Ho, T., and Chong, J.(), A Cognitive Hierarchy Model of Games,
Quarterly Journal of Economics, : –.
Costa-Gomes, M., Crawford, V.P., and Broseta, B. (), Cognition and Behavior in
Normal-Form Games: An Experimental Study, Econometrica : –.
Eyster, E. and Rabin, M. (), Cursed Equilibrium, Econometrica : –.
Ho, T.-H., Camerer, C., and Weigelt, K. (), Iterated Dominance and Iterated Best
Responses in Experimental ‘p-Beauty Contests’, American Economic Review :
–.
Kübler, D./Weizsäcker, G. (), LimitedDepthofReasoningandFailure ofCascade
Formation in Laboratory, Review of Economic Studies : –.
A. Kirstein and R. Kirstein: Experimental Lemons Market
Nagel, R. (), Unraveling in Guessing Games: An Experimental Study, American
Economic Review : –.
Nagel, R., Bosch-Domènech, A., Satorra, A., and Garcïa-Montalvo, J. (), One,
Two, (ree), Innity: Newspaper and Lab Beauty-Contest Experiments; Discus-
sion Paper, University Pompeu-Fabra, Barcelona.
Schotter, A.W., Weigelt,K., andWilson, C. (), ALaboratory Investigation of Mul-
tiperson Rationality and Presentation Eects, Games and Economic Behavior :
–.
Selten, R. and Nagel, R. (), Das Zahlenwahlspiel - Hintergründe und Ergebnisse,
Spektrum der Wissenscha, February: –.
Stahl, D.O. and Wilson, P.W. (), On Players’ Models of Other Players: eory and
Experimental Evidence, Games and Economic Behavior : –.
aler, R. (), Giving Marketsa Human Dimension, Financial Times,sectionMas-
tering Finance, June th: .
