GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.176

An Experimental Test of a Search Model under Knightian Uncertainty

Takao Asano, Hiroko Okudaira, and Masaru Sasaki

February 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

An Experimental Test of a Search Model under Knightian
Uncertainty
Takao Asano

Hiroko Okudaira

Masaru Sasakiy

Okayama University

Okayama University

Osaka University

February 15, 2011

Abstract
This paper’s objective is to design a laboratory experiment to explore the e¤ect of Knightian
uncertainty on a subject’s search behavior in a …nite sequential search model. Our …nding isthat the average search duration is shorter when there is Knightian uncertainty in the sense thatthe true point distribution is unknown to subjects, compared to when the point distribution is
known. We also …nd direct evidence that subjects reduce their own reservation point when there
is ambiguity about the point distribution. These results support the implication of Nishimura
and Ozaki (2004). Moreover, ambiguity notably a¤ects the search behavior of risk averse
subjects, but not of either risk neutral or risk prone subjects.
JEL classi…cation: C91, D81
Keywords: experiment, search model, ambiguity, risk attitude, optimal stopping rule
The authors thank Takanori Adachi, Keiko Aoki, Yasuyo Hamaguchi, Youichiro Higashi, Yoichi Hizen, Chiang
Hui-Yu, Keisuke Kawata and the participants in the 2010 JEA Meeting (Fall), the 2010 International Workshop on
Experimental Economics and Finance, and the 2011 APESA Meeting for their help with experimental design andimplementation and comments. Financial support from MEXT (Grant-in-Aid for Scienti…c Research, No. 20530196)
and GCOE and ISER research grants (Osaka University) are greatly acknowledged. All errors are our responsibility.
yCorrespondence: ISER, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka, 567-0047, E-mail: sasaki@econ.osaka-u.ac.jp1

1Introduction

Let us consider the situation in which the prospect of labor market conditions in the future is“uncertain.” In this situation, does an increase in “uncertainty” lengthen or shorten the search
duration of an unemployed worker? In the literature on job search, it is well known that an increasein risk in the sense of Rothschild and Stiglitz (1971) such as a mean-preserving spread lengthens
an individual’s search duration.1 However, from a di¤erent point of view, it can be also consideredthat an increase in “uncertainty”about the prospect of labor market conditions in the future makes
the individual more cautious about the prospect and shortens her or his search duration because
she or he is concerned that more appealing job o¤ers will not be forthcoming. Considering thenotion of Knightian uncertainty or ambiguity that is di¤erentiated from that of risk, Nishimura andOzaki (2004) show theoretically that an increase in Knightian uncertainty shortens an individual’s
search duration. It is noted in this paper that Knightian uncertainty and ambiguity are usedinterchangeably. The purposes of this paper are …rst to design a laboratory experiment of a …nite
sequential search model with ambiguity in the sense that a point distribution from which a subject
draws is unknown and second, to ascertain whether the result of Nishimura and Ozaki (2004) is
supported experimentally. Similarly to Nishimura and Ozaki (2004), Knightian uncertainty is tobe understood in the sense that the true point distribution is unknown to subjects throughout
this paper. This paper …nds that the presence of Knightian uncertainty shortens subjects’search
duration. Our result is consistent with the prediction of Nishimura and Ozaki (2004).
Knight (1921) points out the importance of the distinction between risk and uncertainty. Knight(1921) claims that risk is measured by randomness that can be characterized by a unique probability
measure while uncertainty cannot be captured by a unique probability measure. Based on Knight(1921), Ellsberg (1961) provides some evidence that decision makers prefer to act on known rather
than unknown or vague probabilities.
Ambiguity can be analyzed in the framework of multiple-prior expected utility (MEU) theory.
Gilboa and Schmeidler (1989) axiomatize the MEU theory, showing that a decision maker’s be1

For a survey of job search models, see Lippman and McCall (1976).

2liefs are captured by a set of probability measures and that her preferences are represented by
the minimum of expected utilities over the set of probability measures. MEU has deepened ourunderstanding of a decision maker’s behavior under ambiguity.2 Chen, Katuscak, and Ozdenoren(2007) investigate experimentally the e¤ect of Knightian uncertainty on bidding behavior and revenue on the …rst and second price-sealed bid auctions. They …nd that in the …rst price auction,
bids are lower in the presence of Knightian uncertainty and that the …rst price auction generates
signi…cantly higher revenue than the second price auction, regardless of the presence of Knightian
uncertainty.
This paper focuses on the e¤ect of Knightian uncertainty on individual search activity. The
experimental task of testing sequential search models is tractable, so many experimental studies
have been conducted over the past years (Cox and Oaxaca, 1989; Harrison and Morgan, 1990). The
purpose of these past studies was to test the reservation price property in the sequential searchmodel. Some experimental tasks then deal with the policy e¤ects on search behavior: Boone,
Sadrieh, and van Ours (2009) design a laboratory experiment to test the e¤ect of unemployment
bene…t sanctions on an individual’s search behavior. There is another direction that addresses therelationship between search behavior and heterogeneous preferences. Schunk (2009) …nds that the
search model with loss aversion, rather than with risk aversion, is more suitable for the searchbehavior of a subject in the laboratory. One of the recent topics in this …eld is to explore why
subjects stop searching earlier than the theoretically optimal level (Schunk and Winter, 2009). Our
experimental task focuses on the e¤ect of information available for the search activity on a subject’s
search behavior. Our experiment can contribute to understanding the role of information on thewage distribution from which an individual worker draws randomly during her search activity.
It is notable that our experiment designs two search methods where recall is not allowed. In
the …rst method, a subject decides either to accept or to reject every time a point is drawn by a2

For example, see Epstein and Wang (1994). In a closely related paper, Schmeidler (1989) axiomatizes Choquet

expected utility (CEU) theory and shows that decision maker’s beliefs are captured by a nonadditive measure andher preferences are represented by the Choquet integral. For applications of CEU, see Dow and Werlang (1992),
Nishimura and Ozaki (2004), or Asano (2006).

3computer, while in the second method, the subject ex ante commits to her reservation points for
all rounds subjectively prior to starting a game and then begins to search, based on a list of thesereservation points. The main bene…t of the second method is that we can explicitly observe a long
trend in the reservation point over all the rounds that is usually unobservable in empirical studies.
Because recall in search activity is not allowed in this experiment, in order to adhere closely tothe framework of Nishimura and Ozaki (2004), it is expected from Schunk and Winter (2009) that
subjects are discouraged from searching. If a subject is required to declare her reservation point foreach round before drawing a point and then decides either to accept the point or to reject it andmove on to the next round, we would not obtain enough data to observe a trend in the reservation
point.3 In addition, this second method is designed to induce subjects to behave according to the
optimal stopping rule, so that the subjects would calculate their reservation point for each roundbackward from the last round. If there is no di¤erence in search behavior between the two methods,
we can say that the subjects engage in search activity according to the optimal stopping rule. We…nd that this claim is overall supported, although at the individual level, there are some subjects
who did not behave in accordance with the optimal stopping rule.
Our experiment yields two …ndings. The …rst is that the search duration is on average shorter
when there is Knightian uncertainty, compared to when the point distribution is known. It implies
that subjects are more likely to accept the o¤ered point when the point distribution is unknown
than when the point distribution is known in advance. These results support the implicationof Nishimura and Ozaki (2004). Secondly, we …nd that using the subjective data on a seriesof reservation points obtained from the second method, subjects reduce their own reservation
point over all rounds under Knightian uncertainty, although marginally. This is subjective but
direct evidence supporting the implication of Nishimura and Ozaki (2004). These two …ndings
reinforce one another; that is, these …ndings suggest that ambiguity about the distribution lowers
3

We previously conducted a sequential search model experiment in which a subject was required to declare heror his reservation point for each round before drawing a point. However, because subjects did not search long in theenvironment where recall was not allowed, we could not obtain enough data to estimate a trend in the reservation
point.

4the reservation point. This implies that subjects are encouraged to accept an o¤ered point in an
earlier round, thereby leading to shorter search duration.
We also test whether or not the e¤ect of ambiguity on search behavior di¤ers by the extent of
attitude toward risk. Our …nding is that risk averse subjects respond by lowering their reservation
point, thus shortening their search duration when there is ambiguity about the point distribution,compared to when the point distribution is well recognized, but that search behaviors of those whoare either risk neutral or risk prone are not signi…cantly a¤ected by ambiguity about the point
distribution. This result di¤ered from our theoretical prediction claiming that ambiguity shifts
down a trend in the reservation point, regardless of the extent of attitude toward risk.4
The organization of this paper is as follows. Section 2 provides a brief summary of job searchunder risk and ambiguity. Section 3 provides an experimental design and hypotheses. Section
4 provides the results of our experimental data in this paper. The …nal section provides ourconcluding remarks

2

Search Models under Ambiguity

We consider a simple discrete-time sequential job search model based on Lippman and McCall(1976) and Nishimura and Ozaki (2004). Throughout this section, let (S; BS ) be a measurable
space, where S is a Borel subset of R+ and BS is the Borel -algebra on S. A generic element s 2 S
is regarded as a wage o¤er in each period. In each period, an unemployed worker decides whether
she accepts a wage o¤er and stops searching for a job or obtains unemployment compensationc > 0 and continues searching. She is assumed to know the true wage distribution F0 with thelower bound a and the upper bound b. Let T be the time at which she accepts the o¤er and stops
4

In the literature, there are studies on the correlation between attitudes toward risk and ambiguity but not in thecontext of a search model. Ellsberg’s experiment shows that a risk averse subject is also ambiguity averse (Halevy
and Feltkamp 2005, Halevy 2007). However, Borghans et al. (2009) and Cohen et al. (2010) show that there isessentially no correlation between them.

5searching for a job. Her objective is to maximize her expected life-time income:
"1
#
X
t
E0
yt ;
t=0

where
yt =
and

8
>
< c

>
: sT

for

t<T

for t

T;

2 (0; 1) represents the discount factor. The unemployed worker’s search behavior is de-termined according to the reservation property such that she accepts a wage o¤er s if sR,

but rejects it, receives compensation c, and continues searching if s < R, where R represents thereservation wage. The reservation wage is determined uniquely by the following:
R=c+

1

Z b

[1

F0 (x)]dx:

R

It can be noted that an increase in risk in the sense of a mean-preserving spread increases thereservation wage because stretching the right tail of the wage distribution encourages her to increase
her reservation wage on the one hand, but on the other hand, stretching the left tail does not a¤ect
her reservation wage. If she is assumed to be risk averse, that is, her utility function is characterized
by some concave function, the e¤ect of an increase in mean-preserving spreads on the reservation
wage then is not determined. The e¤ect depends on the curvature of her utility function, that is,the degree of her risk aversion.5
We next consider a job search model within the framework of ambiguity based on Nishimura
and Ozaki (2004). We assume that an individual does not know the true wage distribution F0 fromwhich a wage o¤er is drawn, and that her beliefs are characterized by a set of probability measures
P0 , which is de…ned below.
In order to provide as simple as possible a job-search model under ambiguity, let P0 be theprobability measure on S and F0 be the corresponding probability distribution on R+ . In thissubsection, let S = [a; b] and P0 be the uniform distribution on [a; b], where a < b. Let M be theset of all probability measures on (S; BS ). In this subsection, we also assume that M is the set of5

For further arguments, see Kohn and Shavell (1974) and Nishimura and Ozaki (2004).

6

all wage distributions on [a; b] corresponding to probability measures on BS . The "-contaminationof P0 is the set of the probability measures on S de…ned by the following:
P0
where "

f(1") + "Q j Q 2 M g ;0. If " = 0, then the set of probability measures P0 is a singleton set, that is, P0 = fP0 g.

On the other hand, if " > 0, then P0 is a set of probability measures. The larger " is, the larger theset of probability measures is, which implies that if an unemployed worker’s beliefs are characterizedby P0 , then she becomes less certain about the true probability measure P0 . Therefore, the positive
real number " represents the degree of ambiguity.6
Within the framework of ambiguity, an unemployed worker’s beliefs are assumed to be captured
by P0 ; and her preferences are represented by the minimum of her expected life-time income overthe set of probability measures P0 . Therefore, her objective is to maximize her expected life-time
income:min

Z

S

I(s)P (ds) j P 2 P0 ;where I(s) is her life-time income. Note that I(s) is some bounded measurable function of a wageo¤er s.7 Under the above setting, it can be shown that the unemployed worker’s search behavior
is determined according to the reservation wage property. Moreover, there exists the reservation
wage R by the following:R = c+
= c+
6

1
1

Z b
R

(1

(1

")P0 (fsjs xg)dx
Z 1")[1 F0 (x)]dx:
R

For an axiomatization of "-contaminations, see Nishimura and Ozaki (2006). For applications of "-contaminations

to economics, see Nishimura and Ozaki (2004) or Asano (2010).
7
For any adapted income process 0 y = (y0 ; y1 ; y2 ; : : :) and an initial wage o¤er s0 2 S, the expected life-time
income is de…ned by:
Is0 (0 y)
where=

lim y0 +

t!1

Z

y1 +
S

Z

y2 +
S

Z

yt (dst )S(ds2 )s0 (ds1 )

2 (0; 1) is the discount factor and all the integrations are the Choquet integral with respect to a nonadditive

measure . See Nishimura and Ozaki (2004) for details.

7Within a more general setting, Nishimura and Ozaki (2004) show that an increase in ambiguity
decreases the reservation wage. They also argue that the same implication is obtained even if theutility function is assumed to be risk averse. The next section designs a laboratory experiment totest the implications

3

Experimental Design

3.1

Search Model in Experiment

An individual subject faces games of 20 rounds by way of a …nite sequential search model in which
recall is not allowed. A point is drawn randomly from a given point distribution by a computer
faced by a subject in the …rst round, and then the subject decides either to accept the point or to
reject it. If the subject accepts the point, her or his search activity is concluded, and the accepted
point is converted to her or his payment. If the subject rejects the point, she or he moves on to thesecond round where a point is drawn again from the given point distribution. We here use “point”
to put a search model to the test to prevent subjects from associating “wage”with job search. The
subject can continue to search until the 20th round, and if the subject rejects a point drawn in thelast round, her or his search activity is automatically terminated, and no point is obtained.
We begin to develop the value function of search theoretically. The value of search in eachround is solved backward. The value of search in the 20th round is:

U20 =

Z x

xdF (x);xwhere F (x) represents the point distribution with a lower bound of x and an upper bound of x.
An individual does not receive any point if she or he rejects a point drawn in the last round, so
her or his reservation point must be x in the …nal round (R20 =x). We assume here that utility
is linear in the accepted point, implying that the individual is risk neutral. Backward to the 19thround, the value of search is given by:

8

U19 =Z R19

U20 dF (x) +

x

where

Z x

xdF (x) = F (R19 )U20 +

R19

Z x

xdF (x);R19

represents a discount factor. When a drawn point is lower than R19 , the individual rejects

this o¤er and moves to the 20th round, while in the reserve case, she or he accepts and receives
the point as a one-shot payment in the 19th round.8 The reservation point of the 19th round isRxcalculated by R19 = U20 = x xdF (x). In a similar manner, the value of search in the N th roundis:

UN = F (RN )UN +1 +Z x

xdF (x);(1)

RN

and the reservation point for the N th round is obtained by RN = UN +1 .
As in the previous section, we consider the …nite sequential search model into which ambiguity
is incorporated, based on Nishimura and Ozaki (2004). Let P0 be the uniform distribution on[x; x]; and then the "-contamination of P0 is the set of the probability measures de…ned by P0 =f(1") + "Q j Q 2 M g, where the positive real number " can represent the degree of ambiguity. In

this setting with ambiguity, the value of search in the N th round and the corresponding reservation
point are given by:
UN

= (1

")

F (RN )UN +1 +Z x

xdF (x) ;

(2)

RN

RN

= UN +1 :These equations imply that an increase in ambiguity decreases the value of search in the N thround and therefore its reservation point. A trend in the reservation point is shifted down when
there is ambiguity in the sense that the point distribution is unknown, compared to when the point
distribution is known with certainty.
The model is solved numerically to observe a trend in the reservation point and the e¤ect of
ambiguity on the reservation point. We assume that the point distribution is uniform between a
8

In the standard model, an individual who accepts x in the 19th round can receive x in the 20th round as wellas in the 19th round. That is, the individual’s value of payment is x + x. In this experiment, however, we design
for simplicity that the accepted point is given once as a payment so that subjects can easily understand how to pay
throughout the experiment.

9lower bound of 1 and an upper bound of 3000 and that to be consistent with our experimental
design, there is no discount over the rounds;= 1. Because subjects are discouraged fromsearching longer in an environment where recall is not allowed,= 1 mitigates the disincentive tosearch.9 The utility function for risk neutrality is de…ned as linear in the accept point, whereas theutility function for risk aversion is assumed to show constant relative risk aversion (CRRA) withthe measure of risk aversion being 0.5. Table 1 displays several trends in the reservation point,varying with the degree of risk attitude and ambiguity about the point distribution.
The trends for a risk neutral individual are shown graphically in Figure 1. As we would expect,
the declining trend in the reservation point can be observed in the …nite sequential search model,
regardless of the extent of ambiguity. The reservation point decreases as the end of the search is
closer. Moreover, an increase in the degree of ambiguity shifts down the trend in the reservation
point monotonically, which implies that an individual is encouraged to accept an o¤er in an earlier
round when the point distribution is ambiguous, compared to when there is no ambiguity about
the point distribution. A comparison of eq(1) and eq(2) implies that the declining rate of thereservation point is " at any round according to Table 2.
The values of search in the N th round for a risk averse subject without and with ambiguity
about the point distribution are correspondingly given by:

UN = F (RN )UN +1 +Z xu(x)dF (x);

(3)

Z x

(4)

RN

and

UN = (1")

F (RN )UN +1 +u(x)dF (x) ;RN

where u0 ( ) > 0 and u00 ( ) < 0. In a similar manner, the reservation point of the N th round is
obtained by RN = UN +1 . We see that eq(3) and eq(4) share the same reservation point property
9

Because there are many papers showing that subjects do not search long in laboratory experiments (Schunk andWinter: 2009), we dispense with any discount factor to encourage subjects to search longer. We previously conducted
our experiment with a discount factor taking below a value of one. The results obtained in that experiment werethat subjects terminated their search activity in a very early round.

10

as obtained in eq(1) and eq(2). First, an increase in the extent of ambiguity shifts down thetrend in the reservation point according to Table 1. Figure 2 provides a graphical view. Moreambiguity shifts the trend in the reservation point down even more largely, thus leading to adecrease in the average search duration. Secondly, Table 2 shows that a negative e¤ect of ambiguityon the reservation point does not di¤er in magnitude between a risk neutral individual and arisk averse individual. The risk averse individual reacts as negatively to ambiguity as the risk
neutral individual does. This result is obtained because attitude toward risk and ambiguity areincorporated independently in the model.

3.2

Treatments and Hypotheses

There are four treatments in our experiment. In the …rst treatment (T1), subjects are provided
with common information on a point distribution. We employ a uniform distribution with a lower
bound of 1 and an upper bound of 3000. In the second treatment (T2), the subjects are informedthat a distribution is unknown except for the lower bound of 1 and the upper bound of 3000
and that a di¤erent distribution may be selected in every round by the computer. This prevents
the subjects from updating their information about the true distribution of a point in a Bayesianmanner and rules out the learning e¤ect on search behavior.10 We do not provide them with thetrue distribution of a point. However, to facilitate a comparison of (T1) and (T2), the distributionis actually set the same as the uniform distribution with a lower bound of 1 and an upper boundof 3000. The comparison of these two treatments allows us to identify the di¤erence in searchbehavior caused uniquely by ambiguity by controlling the distribution shape. If the distribution
actually changes every round, we cannot identify whether the di¤erence in the search durations
between the two treatments is attributable to ambiguity or to the variants in distributions.
The third treatment (T3) is a search activity in which a subject ex ante commits to a seriesof her or his reservation points over all the 20 rounds under the uniform distribution with a lower
10

There are some papers that would rather focus on uncertainty and learning of the shape of the wage distribution

through Bayesian updating in a search model (Morgan, 1985; Burdett and Vishwanath, 1988; Dubra, 2004).

11

bound of 1 and an upper bound of 3000. Subjects are requested to …ll in their own reservation
points in all the 20 rounds before starting a game. First, they …ll in the reservation point in the…rst round and then …ll in the reservation point in the second round as if they had moved on to thesecond round. The same procedure is undertaken until the 20th round. After listing 20 numbers of
the reservation point on the sheet, each subject clicks the button to start to search. The computer
randomly picks a …rst-round point and compares it with the …rst-round reservation point. If thepoint drawn is lower than the …rst-round reservation point, her or his search activity continues
with the computer drawing a second-round point. Otherwise, the computer stops searching andgives the subject the drawn point. We suppose that subjects automatically calculate a series ofreservation points backward from the last round and then actually …ll in forward from the …rst
round. Therefore, (T3) is designed in such a way that subjects are induced to behave in accordancewith the optimal stopping rule. The main bene…t of this treatment is that we can directly observea 20 round-long trend in the reservation point. This is usually di¢ cult to observe in an experimentof sequential search without recall because subjects …nish searching long before the …nal round.
This allows us to directly measure the e¤ect of ambiguity on the trend in the reservation point.
A comparison of (T1) and (T3) also allows us to verify whether subjects engage in search activity
following the optimal stopping rule. If there is no di¤erence in search attitude between the twotreatments, it is concluded that subjects actually follow the optimal stopping rule when engaging
in search activity. The fourth treatment (T4) is similar to (T3) except for the realization of thepoint distribution; that is, a subject ex ante commits to a series of her or his reservation points
over all 20 rounds under various unknown distributions with a lower bound of 1 and an upperbound of 3000.
Our experiment consists of two sessions. Each experimental session consists of 11 games. The
di¤erence between the two experimental sessions is the order of the 11 games. The game orders
are as follows in the sessions.
Session 1: (T1-practice), (T2), (T2), (T2), (T2), (T4), (T1), (T1), (T1), (T1), (T1), (T3).
Session 2: (T1-practice), (T1), (T2), (T1), (T2), (T3), (T1), (T2), (T1), (T2), (T1), (T4).
12

In Session 1, subjects engage in search activity in which there is Knightian uncertainty forthe point distribution in the earlier games ((T2) and (T4)), and then they play search games inwhich the point distribution is recognized in advance ((T1) and (T3)). This order rules out thepossibility that subjects infer that the unknown distribution is uniform in (T2) and (T4) games.
In Session 2, subjects alternately engage in search activity under the given and then the unknown
point distributions.
We next display experimental hypotheses to test the theoretical implications of Nishimura andOzaki (2004). We test three hypotheses regarding Knightian uncertainty, the optimal stopping rulein search activity, and risk attitude.
H1: The reservation point is lower when the point distribution is unknown, compared towhen the point distribution is well known ((T1) versus (T2), and (T3) versus (T4)). Therefore, the search duration is shorter under an unknown point distribution than under a given
distribution
A subject’s reservation point is lower, and therefore, she or he accepts a drawn point in an
earlier round when the point distribution is unknown to her or him, compared to when the point
distribution is well known. This results in the shorter search duration in the presence of Knightian
uncertainty.
H2: Subjects engage in search activity following the optimal stopping rule. ((T1) versus(T3), and (T2) versus (T4)).
If there is no di¤erence in the average search duration between (T1) and (T3) (or (T2) and(T4)), it can be said that subjects engage in search activity following the optimal stopping rule;that is, they calculate in advance their own reservation points for all 20 rounds, backward fromthe last round and stop search if a drawn point exceeds the corresponding reservation point butotherwise continue to search.
In addition, our experiment explores a di¤erence in search activity by risk attitude. To identify
each subject’s attitude toward risk, we conducted a questionnaire to all participants after the13experiment and asked them about their attitude toward risk. We asked them what price theywould be willing to pay for a lottery with a 25% chance of winning JPY200, but with a 75% chanceof winning nothing. We then calculated the index measuring the extent of absolute risk aversion
using Cramer et al. (2002).11 Let this index denote risk aversion (A). If this index is positive, asubject is considered to be risk averse, but on the other hand, if negative, the subject is treatedas risk prone. If the index is exactly zero, the subject is risk neutral. Similarly, we calculated
a similar index using the willingness-to-pay price for a lottery of winning much more than forour …rst lottery, in fact, JPY2000 with a 25% chance, but 75% chance of winning nothing. This
index denotes risk aversion (B).12 Comparing the declining rates of the reservation point with andwithout ambiguity in the point distribution (eq(1)-eq(4)), leads us to hypothesize that:
H3: The negative e¤ect of point distribution ambiguity on the search duration does notdi¤er with the extent of risk attitude. There is no correlation between attitude to risk andambiguity.
We investigate a change in the negative e¤ect of ambiguity on the search decision, dependingon the extent of risk aversion.

3.3

Administration and Payo¤sThis two-session experiment was conducted on December 17th, 2009 in the experimental laboratoryof the Institute of Social and Economic Research at Osaka University. Subjects consisted of 44
undergraduate and graduate students of Osaka University excluding junior and above economics
11

According to Cramer et al. (2002), the extent of absolute risk aversion is calculated as,

0:5(0:25

0:25 200 price;2002 2 0:25 200 + price2 )where the price is the one that a subject would be willing to pay for a lottery in which she or he has a 25% chanceof winning JPY200, but a 75% chance of winning nothing.
12
The questionnaire also included questions allowing us to produce indices measuring the extent of relative risk
aversion using Kimball, Sahm, and Shapiro (2008). However, we did not obtain robust results and therefore wediscarded these indices.

14majors, with 21 participating in the …rst session and 23 in the second. Each participant was in an
individual booth, from which she or he could not observe other subjects. The experiments wererun entirely on computers using Z-tree (Fischbacher, 2007) laboratory experiment software. Table
3 presents the summary statistics.
Instructors handed an instruction sheet to each subject and presented each with full information
about the search task. The instructors emphasized that (i) subjects’payo¤ was truncated at JPY0
(i.e., they could not incur losses from the search task); and (ii) they would earn an attendance feeof JPY1000.13 Performance pay was determined by one of the results from the 11 games randomly
chosen by each subject, with one point equal to JPY1. The participants were paid on completionof the experiment. The expected total payo¤ was JPY2500–3000, and because the time taken for
the experiment was approximately one and a half hours, the e¤ective hourly payo¤ to participants
was about JPY1600–2000, which is approximately twice as large as the average hourly wage forcollege students. Payments were made to each subject, one by one, while they were completing thequestionnaire.

4

Results

4.1

Search Duration

Table 4 displays average search durations by various treatments and their di¤erences between
treatments. In the …rst column, the average search duration is 6.81 for the treatment in which thepoint distribution is well known to subjects as a uniform distribution with a lower bound of 1 andan upper bound of 3000 (T1) and 5.63 in the treatment with ambiguity in the sense that the point
distribution is unknown except for a lower bound of 1 and an upper bound of 3000 (T2). The
former is longer than the latter, and the null hypothesis (H1) that the di¤erence of (T1)–(T2) isequal to or shorter than zero is rejected at the 1% level of signi…cance. This implies that subjects
reduce their reservation point when facing ambiguity about the point distribution, thereby leading
13

One US dollar was equivalent to JPY82.92 as of January 15, 2011.

15

to shorter search duration. This result is consistent with the prediction of Nishimura and Ozaki(2004). The second column shows the e¤ect of ambiguity about the point distribution on average
search duration, given that a subject ex ante commits to reservation points prior to her or his gameand then decides either to reject or to stop searching in light of precommitted reservation points.
The average search duration is shorter with ambiguity about the point distribution, but the null
hypothesis is not signi…cantly rejected. We now compare average search durations in the …rst row,
capturing the e¤ect of a di¤erence in search activity method on search duration under the uniform
distribution with a lower bound of 1 and an upper bound of 3000 ((T1) versus (T3)). Hypothesis(H2), that there is no di¤erence in average search duration, regardless of search activity methods,
is not signi…cantly rejected. It implies that subjects engage in search activity following the optimal
stopping rule because (T3) is designed such that subjects are induced to follow the optimal stopping
rule. The same result is obtained if there is ambiguity about the point distribution ((T2) versus(T4)).
Tables 5 and 6 show average search durations, by the extent of absolute risk aversion. The
sampled data are analyzed by risk aversion index (A) in Tables 5-1, 5-2 and 5-3 and risk aversion
index (B) in Tables 6-1, 6-2 and 6-3. According to the …rst columns of Tables 5-1 and 6-1 con…ned
to risk averse subjects, the search duration is shorter when the point distribution is unknown (T2),relative to when it is well known to subjects as a uniform distribution with a lower bound of 1 andan upper bound of 3000 (T1). Therefore, the null hypothesis (H1) is signi…cantly supported. The
di¤erences are larger in magnitude in Tables 5-1 and 6-1 than in Table 4, which implies that the
search duration is further shortened for risk averse subjects when there is Knightian uncertainty. On
the other hand, the …rst columns of Tables 5-2, 5-3, 6-2, and 6-3 show that for risk neutral and riskprone subjects, the di¤erences in search duration between (T1) and (T2) are statistically minimal.
These results suggest that ambiguity about the point distribution does not a¤ect search behaviorof risk neutral and risk prone subjects. These results di¤er from our theoretical prediction and thus
do not support (H3); that is, our experimental outcomes imply that when the point distribution is
ambiguous in terms of Knightian uncertainty, the trend for the reservation point is shifted down

16

only for risk averse subjects. There is essentially a correlation between attitude toward risk andambiguity.
Table 7 estimates linear models for determinants of search duration. The …rst two columns
use data from (T1) and (T2) while the last three columns include data from all treatments ((T1)–(T4)). The dependent variable is the search duration of each game that each subject experienced.
In columns (1) and (2), Knight is a dummy variable representing 1 if subjects draw a point fromunknown distributions with a lower bound of 1 and an upper bound of 3000 (T2) and 0 if subjects
draw a point from a uniform distribution with a lower bound of 1 and an upper bound of 3000
(T1). Its coe¢ cient is negative at the 1% level of signi…cance in column (1). The result remains thesame after controlling for individual subject and session e¤ects (see column (2)) capturing not only
an individual subject’s risk attitude (risk averse or risk prone), but loss attitude (loss averse or loss
prone) and other unobservables. Because the subject dummies indicating individual characteristics
and the session dummy are not correlated with the Knight term at all, its coe¢ cient remains thesame even after these dummies are included. Our …nding is that the search duration is shorter
when the point distribution is unknown because of Knightian uncertainty, relative to when the
point distribution is known. This result implies that subjects reduce their own reservation point
if there is ambiguity about the point distribution. We therefore support (H1), and this result is
consistent with the main prediction of Nishimura and Ozaki (2004). The coe¢ cient on Knight
remains signi…cant using all observations, inclusive of precommitted data ((T3) and (T4)) (column(3)).
Columns (4) and (5) add two dummy variables, one each for ambiguity and search method andtheir cross term. Similarly to the estimates of columns (1)–(3), Knight is a dummy indicating 1 if
subjects draw a point from unknown distributions with a lower bound of 1 and an upper boundof 3000 (either (T2) or (T4)) and 0 if subjects draw a point from a uniform distribution with alower bound of 1 and an upper bound of 3000 (either (T1) or (T3)). Precommitted is anotherdummy representing 1 if subjects ex ante commit to their reservation points over all the 20 rounds
in advance and then decide either to reject or to stop searching, based on precommitted reservation

17

points ((T3) or (T4)) and 0 if subjects decide whether to reject or accept every time a point isdrawn in each round ((T1) or (T2)). Knight*Precommitted is the cross term of the two dummy
variables.
Knight is negative at the 1–5% level of signi…cance in columns (4) and (5), so we con…rm thesame result as columns (1)–(3); that is, the search duration is shorter when there is Knightian
uncertainty, compared to when the point distribution is well known. Ambiguity about the point
distribution decreases a subject’s reservation point, thus encouraging her or him to accept in an
earlier round.
Precommitted is statistically insigni…cant, implying that there is no di¤erence in search behavior
between the decision rule in which a subject ex ante commits to reservation points for all 20 rounds
prior to a game and that in which a subject decides every time a point is drawn in each round.
(T3) and (T4) are designed so that subjects would follow the optimal stopping rule in the sensethat they ex ante are subject to calculate their reservation point in each round backward from thelast round. Subjects therefore engage in search activity following the optimal stopping rule, which
supports (H2).
Tables 8 and 9 estimate determinants of search duration by the extent of absolute risk aversion,index (A) in Table 8, and index (B) in Table 9. Columns (1)–(3) of both tables use the subsample
from nonprecommitted games ((T1) and (T2)), while columns (4)–(6) use all observations including
data from precommitted games ((T1), (T2), (T3), and (T4)). We begin with columns (1)-(3) ofboth tables. Con…ned to risk averse subjects, the coe¢ cient on Knight remains negative at the
1–5% level of signi…cance in column (3), and its coe¢ cient is larger in absolute magnitude than that
displayed in Table 7. On the other hand, columns (1) and (2) show that the signi…cance of Knight
is lower when using the subsamples of risk neutral and risk prone subjects. We con…rm that the
search duration is furthermore shorter for risk averse subjects when there is Knightian uncertainty,
relative to when the point distribution is well known among subjects, but that ambiguity about
the point distribution does not a¤ect the search behavior of risk neutral and risk prone subjects.
These results are consistent with those from Tables 5 and 6 showing a correlation between attitude18

toward risk and ambiguity.

4.2

Determinants of the Reservation Point

This subsection reports the direct e¤ect of ambiguity about the point distribution on the reservation
point. Table 10 shows the estimates of determinants of the reservation point using the round-based
data from (T3) and (T4) in which subjects commit in advance to their own reservation points forall 20 rounds and then start to search. The bene…t of this method is to allow us to detect a change
in the reservation point from direct but subjective viewpoints.
In column (1), Knight is statistically insigni…cant, contrary to our expectation. One reason
for this result is that subjects who do not follow the optimal stopping rule are included in theestimation. We then extract the data from those who precisely did follow the optimal stopping
rule (referred to as OSR-consistent) and then re-estimate the determinants of the reservation point.
A subject is among the OSR-consistent subsample if the following conditions are satis…ed;for any given round, a subject accepts a point drawn in (T1) if the point is no less than thecorresponding reservation point in (T3);
for any given round, a subject accepts a point drawn in (T2) if the point is no less than thecorresponding reservation point in (T4);
for any given round, a subject rejects a point drawn in (T1) if the point is lower than thecorresponding reservation point in (T3);
for any given round, a subject rejects a point drawn in (T2) if the point is lower than thecorresponding reservation point in (T4),and zero otherwise. At the individual level, not all subjects follow the optimal stopping rule.
We here impose the strict assumption to distinguish between subjects who precisely follow the
optimal stopping rule and those who do not. According to Table 11, 96% of round-based decisions
are consistent with the optimal stopping rule, but when grouped by game, only 21% of games are19

consistent with the optimal stopping rule. In addition, only 20% of subjects follow the optimal
stopping rule while searching. It is also interesting to note that, on average, OSR-consistent subjects earn more than OSR-inconsistent subjects. Among those who follow the optimal stopping
rule, the average payment is larger when the point distribution is well known as a uniform distribution, relative to when the point distribution is unknown. This is consistent with our prediction.
However, among our other subjects the opposite is the case.
As seen in column (2) of Table 10 using the OSR-consistent subsample, Knight turns out
to be negative at the 10% level of signi…cance. It implies that subjects marginally reduce theirown reservation point when the point distribution is unknown because of Knightian uncertainty,
compared to when the point distribution is known to subjects. We con…rm the same results from
Table 7, and therefore (H1) is again supported. It should be emphasized that although we reachthis conclusion indirectly from the estimates of individual search duration from Table 7, Table 10
provides direct evidence to support the implication of Nishimura and Ozaki (2004); that is, thereservation point is lower in the presence of Knightian uncertainty about the point distribution.
Next we explore how the e¤ect of ambiguity on the reservation point di¤ers by the degreeof attitude toward risk. Tables 12 and 13 estimate the determinants of the reservation point,
separating the sample by the extent of absolute risk aversion. Table 12 uses risk aversion index(A) while Table 13 uses risk aversion index (B). It is noted that estimates shown in both tables
are obtained, using the OSR-consistent subsample. According to column (3) of Table 12, thecoe¢ cient on Knight is negative and larger in absolute magnitude than that displayed in column(2) of Table 10, but its signi…cance is reduced. Similarly, Knight is statistically insigni…cant forrisk averse subjects in column (3) of Table 13. When limited to those who are either risk neutralor risk prone, columns (1) and (2) of Tables 12 and 13 show that Knight remains insigni…cant.14
However, from di¤erences in magnitude of the coe¢ cients, we can say that risk averse subjects
lower their reservation point slightly and thus complete searching in an earlier round when the14

It is noted that there is no estimate for risk prone subjects in Table 13 because there are no observations of riskprone subjects who correctly follow the optimal stopping rule when the sample is analyzed by risk aversion index
(B).

20

point distribution is unknown.

5

Concluding Remarks

This paper designed a laboratory experiment to explore the e¤ect of Knightian uncertainty on a subject’s search behavior in a …nite sequential search model and tested the implications of Nishimura
and Ozaki (2004). We utilized various approaches to test the e¤ect of ambiguity about the point
distribution on sequential search behavior.
It is particularly worth noting that we designed two search methods. In the …rst a subject
decides either to accept or to reject every time a point is drawn, while in the second a subject
commits to their reservation points for all 20 rounds prior to a game and only then begins tosearch. The second method is designed to embody the optimal stopping rule in which a subject
must calculate their reservation points for all 20 rounds backward from the last round. If there isno di¤erence in search behavior between the two methods, we can say that subjects always follow
the optimal stopping rule. This was supported by our estimates although there are some subjects
who do not follow the optimal stopping rule.
One …nding is that the average search duration is shorter with ambiguity rather than certainty
about the point distribution, implying that subjects are more likely to accept the o¤ered point
when the point distribution is unknown, compared to when the point distribution is well known
in advance. This result supports the prediction of Nishimura and Ozaki (2004). We also showthat subjects explicitly reduce their own reservation point when there is Knightian uncertainty,
using the data from (T3) and (T4) in which subjects commit in advance to their own reservation
point for all 20 rounds and then start to search. Although the data are obtained subjectively,
this is direct evidence supporting the implication of Nishimura and Ozaki (2004). The estimatedresults from these di¤erent approaches reinforce each other, supporting the view that introducing
ambiguity about the point distribution lowers the reservation points, thus encouraging subjects
to accept an o¤ered point in an earlier round and therefore leading to shorter search duration.
Ambiguity about the point distribution has a signi…cantly negative impact on search duration for21risk averse subjects, but not for those who are either risk neutral or risk prone.
This experiment used a simple and speci…c environment of individual sequential search, controlling information about the point distribution and search methods. However, the essence of theresults can be applied to the labor market. The presence of Knightian uncertainty about the wage
distribution induces individuals to be more cautious about search activity, which lowers the reservation wage and thereby the accepted wage. It implies that the presence of Knightian uncertainty
reduces the welfare level of individuals. These experimental outcomes are useful to understand therole of information in determining the level of individual welfare.

6

Appendix: Instruction

Note: Session 1 Instructions. Session 2 di¤ers in the order of games.
Welcome to our experiment! In this experiment, you will be asked to play 11 games. In eachgame, within a limited number of 20 rounds, you will be asked to choose either to receive a pointthat is randomly selected from a selected distribution or to refuse this point and move on to thenext round to wait for a higher point. If you are willing to accept an o¤ered point, you click “Y”
displayed on your PC screen, but if not, you click “N”. If you do not accept a point o¤ered in the…nal round, your score will be zero automatically. Your score will be decided on the basis of thepoints that you accept.
We have prepared 11 games and would like you to play them. Before starting the experiment,
we would like you to try the following practice. Please let us know if you have any questions.
Practice: In each round, the computer randomly selects a point from a uniform distribution
with a lower bound of 1 and an upper bound of 3000. You decide whether or not to accept
the point drawn from this distribution. If you accept the point, you then …nish your searchand the point is your score. If you do not accept the point, you move on to the next roundand observe a point newly drawn by the computer. You can continue to search up to 20
rounds.
22

Before starting each game, we explain how to play it in more detail.
Game 1: In each round, the computer randomly selects a point from an unknown distribution
except that it has a lower bound of 1 and an upper bound of 3000, and a di¤erent distribution
may be randomly selected every round by the computer. You decide whether or not to accept
the point drawn from the unknown distribution. If you accept the point, you then …nish yoursearch and the point is your score. If you do not accept the point, you move on to the next
round and observe a point newly drawn by the computer. You can continue to search up to
20 rounds.
Game 2: the same as Game 1.
Game 3: the same as Game 1.
Game 4: the same as Game 1
Game 5: In each round, the computer randomly selects a point from an unknown distribution
except that it has a lower bound of 1 and an upper bound of 3000, and a di¤erent distribution
may be randomly selected every round by the computer. Please type in the minimum pointthat you are willing to accept (hereafter the reservation point) for all 20 rounds before starting
this game. First, please type in your reservation point in the blank for the …rst round on
the PC screen, and then type in your reservation point in the blank for the second round,
as if you had moved on to the second round. Please repeat this until the 20th round. After…nishing your list of 20 reservation point numbers on your PC Display, click “OK”. The
computer then starts to draw a point from an unknown distribution and compares it withyour reservation point for the …rst round. If the drawn point is equal to or higher than yourreservation point for the …rst round, you then …nish your search and the point is your score. If
the drawn point is lower than your reservation point for the …rst round, the computer moves
on to the second round and draws a point. Your search activity is left to the computer.
Game 6: In each round, the computer randomly selects a point from a uniform distribution
with a lower bound of 1 and an upper bound of 3000. You decide whether or not to accept
23

the point drawn from this distribution. If you accept the point, you then …nish your searchand the point is your score. If you do not accept the point, you move on to the next roundand observe a point newly drawn by the computer. You can continue to search up to 20
rounds.
Game 7: the same as Game 6.
Game 8: the same as Game 6.
Game 9: the same as Game 6.
Game 10: In each round, the computer randomly selects a point from a uniform distribution
with a lower bound of 1 and an upper bound of 3000. You decide on whether or not to
accept the point drawn from this distribution. Please type in the minimum point that you
are willing to accept (hereafter your reservation point) for all 20 rounds before starting thisgame. First, please type in your reservation point for the …rst round on the PC screen andthen type in your reservation point for the second round, as if you had moved on to thesecond round. Please repeat this until the 20th round. After …nishing listing 20 reservation
point numbers on your PC Display, click “OK”. The computer then starts to draw a pointfrom the uniform distribution and compares it with the reservation point of the …rst round.
If the drawn point is equal to or higher than your reservation point for the …rst round, youthen …nish your search and the point is your score. If the drawn point is lower than yourreservation point for the …rst round, the computer moves on to the second round and draws
a point. Your search activity is left to the computer.
After the experiment, please respond to our questionnaire. You will be paid an attendance feeof JPY1000. Your performance pay is determined by one of the points from the 11 games randomly
chosen by you. This payment treats one scoring point as JPY1. You will be paid both fees as soon
as the experiment is completed. Please be quiet and do not communicate with other participants
during the experiment. Thank you for your participation.

24

References[1] Asano, T. (2006): “Portfolio Inertia under Ambiguity,”Mathematical Social Sciences 52, 223232.
[2] Asano, T. (2010): “Portfolio Inertia and Epsilon-Contaminations,” Theory and Decision 68,
341-365.
[3] Boone, J., A. Sadrieh, and J.C. van Ours (2009): “Experiments on Unemployment Bene…t
Sanctions and Job Search Behavior,” European Economic Review 53, 937-951.
[4] Borghans, L., B.H. Golsteyn, J.J. Heckman, and H. Meijers (2009) “Gender Di¤erences in Risk
Aversion and Ambiguity Aversion,” Journal of the European Economic Association, 7(2-3),
649-658.
[5] Burdett, K., and T. Vishwanath (1988): “Declining Reservation Wages and Learning,”Review
of Economic Studies, 55(4), 655-665.
[6] Chen, Y, P. Katušµcák, and E. Ozdenoren (2007): “Sealed Bid Auction with Ambiguity: Theory
and Experiments,” Journal of Economic Theory 136, 513-535.
[7] Cohen, M., J-M. Tallon, and J-C. Vergnaud (2010): An Experimental Investigation of Imprecision Attitude and its Relation with Risk Attitude and Impatience, forthcoming in Theory
and Decision.
[8] Cox, J.C., and R.L. Oaxaca (1989): “Laboratory Experiments with a Finite-Horizon JobSearch Model,” Journal of Risk and Uncertainty 2, 301-330.
[9] Cramer, J.S., J. Hartog, N. Jonker, and C.M. Van Praag (2002) “Low Risk Aversion Encourages the Choice for Entrepreneurship: An Empirical Test of a Truism,” Journal of Economic
Behavior & Organization, 48, 29-36.
[10] Dow, J., and S.R.C. Werlang (1992): “Uncertainty Aversion, Risk Aversion, and the Optimal
Choice of Portfolio,” Econometrica 60, 197-204.
25

[11] Dubra, J. (2004): “Optimism and Overcon…dence in Search,”Review of Economic Dynamics,
7, 198-218.
[12] Ellsberg, D. (1961): “Risk, Ambiguity, and the Savage Axioms,” Quarterly Journal of Economics 75, 643-669.
[13] Epstein, L.G., and T. Wang (1994): “Intertemporal Asset Pricing under Knightian Uncertainty,” Econometrica 62, 283-322.
[14] Fischbacher, U. (2007) “Z-Tree Zurich Toolbox for Ready _Made Economic Experiments,”
Experimental Economics, 10, 171-178.
[15] Gilboa, I., and D. Schmeidler (1989): “Maxmin Expected Utility with Non-Unique Priors,”
Journal of Mathematical Economics 18, 141-153.
[16] Harrison, G.W., and P. Morgan (1990): “Search Intensity in Experiments,”Economic Journal
100, 478-486.
[17] Halevy, Y. (2007): Ellsberg Revisited: An Experimental Study, Econometrica, 75(2), 503-536.
[18] Halevy, Y., and V. Feltkamp (2005): A Bayesian Approach to Uncertainty Aversion, Review
of Economic Studies, 72, 449-466.
[19] Kimball, M.S., C.R. Sahm, and M.D. Shapiro (2008): Imputing Risk Tolerance From Survey
Responses, Journal of the American Statistical Association, 103(484), 1028-1038.
[20] Knight, F. (1921): Risk, Uncertainty and Pro…t, Boston: Houghton Mi- in.
[21] Lippman, S.A., and J.J. McCall (1976): “The Economics of Job Search: A Survey, Part I:
Optimal Job Search Policies,” Economic Inquiry 14, 155-189.
[22] Morgan, P.B. (1985): “Distributions of the Duration and the Value of Job Search with Learning,” Econometrica, 53(5), 1199-1232.
[23] Nishimura, K.G., and H. Ozaki (2004): “Search and Knightian Uncertainty,”Journal of Economic Theory 119, 299-333.
26

[24] Nishimura, K.G. and H. Ozaki (2006): “An Axiomatic Approach to Epsilon-Contamination,”
Economic Theory 27, 333-340.
[25] Rothschild, M. (1974): “Searching for the Lowest Price When the Distribution of Price is
Unknown,” Journal of Political Economy 82, 689-711.
[26] Rothschild, M., and J. Stiglitz (1970): “Increasing Risk I: A de…nition,”Journal of Economic
Theory 2, 225-243.
[27] Rothschild, M., and J. Stiglitz (1971): “Increasing Risk II: Its Economic Consequences,”
Journal of Economic Theory 3, 66-84.
[28] Schmeidler, D. (1989): “Subjective Probability and Expected Utility without Additivity,”
Econometrica 57, 571-587.
[29] Schunk, D. (2009): “Behavioral Heterogeneity in Dynamic Search Situations: Theory andExperimental Evidence,” Journal of Economic Dynamics and Control 33, 1719-1738.
[30] Schunk, D., and Winter, J. (2009): “The Relationship Between Risk Attitudes and Heuristics
in Search Tasks: A Laboratory Experiment,”Journal of Economic Behavior and Organization,
71, 347-360

27

Table 1
Trends in the Reservation Points
No Ambiguity
Ambiguity (Epsilon=0.1)
Ambiguity (Epsilon=0.25) Ambiguity (Epsilon=0.5)
Risk Neutral Risk Averse Risk Neutral Risk Averse Risk Neutral Risk Averse Risk Neutral Risk Averse

R1
r1
R2
r2
R3
r3
R4
r4
1
2749.32738.42474.32464.62061.92053.81374.6
1369.2
22737.82726.1
2464.02453.52053.32044.6
1368.91363.132725.22712.52452.72441.32043.92034.41362.61356.3
42711.32697.52440.22427.82033.52023.21355.71348.852695.92680.92426.32412.82021.92010.61347.91340.462678.72662.22410.82396.02009.01996.61339.31331.172659.32641.1
2393.42377.01994.5
1980.91329.71320.682637.42617.22373.72355.5
1978.1
1962.91318.71308.692612.42589.82351.12330.8
1959.3
1942.31306.21294.9102583.42558.02325.12302.2
1937.6
1918.51291.71279.0112549.62520.72294.72268.6
1912.2
1890.51274.81260.3122509.52476.32258.62228.6
1882.1
1857.21254.81238.1132461.12422.42215.02180.11845.81816.81230.51211.2142401.32355.52161.22120.01801.01766.71200.71177.8152325.52270.22092.92043.21744.11702.71162.71135.1162225.42157.1
2002.91941.4
1669.11617.81112.71078.5172086.2
1998.8
1877.6
1798.91564.71499.11043.1999.4
18
1875.41758.31687.81582.41406.51318.7
937.7
879.1
19
1500.51334.21350.51200.81125.41000.7
750.3
667.1
20
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Note: The utility function for risk neutrality is defined as linear in the accepted point (x) while the utility
function for risk aversion is CRRA with the measure of risk aversion, 0.5.

Round

Table 2
Decrease Rates of the Reservation Point by Risk Attitude and Ambiguity
Risk Neutral Risk Averse Risk N[eutral Risk Averse Risk Neutral Risk Averse(R2-R1)/R1(r2-r1)/r1(R3-R1)/R1(r3-r1)/r1(R4-R1)/R1(r4-r1)/r1

-0.10

-0.10

-0.25

-0.25

-0.50

-0.50

Note: The decrease rate is the same for all rounds. The decrease rate is thesame as the epsilon value indicating the extent of ambiguity.

28

Table 3
Summary Statistics# of subjects# of females
Average age
Average absolute risk aversion (A)# of subjects (risk aversion (A)>0)# of subjects (risk aversion (A)=0)# of subjects (risk aversion (A)<0)
Risk aversion (A) unavailable
Average absolute risk aversion (B)# of subjects (risk aversion (B)>0)# of subjects (risk aversion (B)=0)# of subjects (risk aversion (B)<0)
Risk aversion (B) unavailable# of subject-games
Average outcome (JPY)
Average search duration# of subject-game-rounds
Average reservation point
Round 1Round 2Round 3Round 4
Round 5
Round 6
Round 7
Round 8Round 9Round 10
Round 11
Round 12
Round 13
Round 14
Round 15
Round 16
Round 17
Round 18
Round 19
Round 20

29

Session 1

Session 2

Total

21
11
20.6
0.00121136
10.00038
16
2
2
1
252
2703.5
7.4
4620

23
7
19.4
0.0014115
6
10.00032
15
5
2
1
276
2661.2
5.4
5060

44
18
20
0.001322812
20.00035
31
7
4
2
528
2681.36.4
9680

2623.02609.62594.92592.02587.92589.82577.92572.92571.42556.42492.32450.92404.02352.82320.42229.42157.82022.21778.6569.5

2440.82414.62381.52366.22362.82334.52300.72283.22255.92241.62163.32139.22095.72053.2
1974.81891.01775.71601.71288.5437.52527.72507.72483.42474.02470.32456.32433.02421.42406.52391.92320.32288.02242.82196.22139.72052.5
1958.01802.41522.4500.5

Table 4
Average Search Durations (Total)
Difference
Nonprecommitted games
Precommitted games
6.81 [T1]
7.20 [T3]-0.40(0.90)
5.63 [T2]
6.16 [T4]-0.53(0.77)
1.18
1.05(0.50)(1.23)
Note: Standard errors in parentheses. Averages are calculated on the treatment base.
Uniform
KnightDifference

Table 5-1
Average Search Durations by the Extent of Risk Attitude (A) (Risk Averse Subjects Only)
Difference
Nonprecommitted games
Precommitted games
Uniform
6.66 [T1]
6.27 [T3]0.39(1.22)
Knight5.05 [T2]
6.22 [T4]-1.17(1.05)
Difference
1.61
0.05
(0.67)(1.68)
Note: Standard errors in parentheses. Averages are calculated on the treatment base. The extent of risk
attitude is calculated from a subject's response to the post-experimental questionnaire: how much is asubject willing to pay for a lottery of a 25% chance of winning JPY200, but a 75% chance of winningnothing according to Cramer, Hartog, Jonker, and van Praag (2002).
Table 5-2
Average Search Durations by the Extent of Risk Attitude (A) (Risk Neutral Subjects Only)
Difference
Nonprecommitted games
Precommitted games
7.85 [T1]7.00 [T3]0.85(2.19)
6.91 [T2]4.88 [T4]2.03(2.04)
0.94
2.13(1.30)(2.52)
Note: Standard errors in parentheses. Averages are calculated on the treatment base. The extent of risk
attitude is calculated from a subject's response to the post-experimental questionnaire: how much is asubject willing to pay for a lottery of a 25% chance of winning JPY200, but a 75% chance of winningnothing according to Cramer, Hartog, Jonker, and van Praag (2002).
Uniform
KnightDifference

Table 5-3
Average Search Durations by the Extent of Risk Attitude (A) (Risk Prone Subjects Only)
Difference
Nonprecommitted games
Precommitted gaames
Uniform
6.55 [T1]8.83 [T3]-2.28(1.78)
Knight6.04 [T2]7.08 [T4]-1.04(1.45)
Difference
0.51
1.75
(0.92)(2.75)
Note: Standard errors in parentheses. Averages are calculated on the treatment base. The extent of risk
attitude is calculated from a subject's response to the post-experimental questionnaire: how much is asubject willing to pay for a lottery of a 25% chance of winning JPY200, but a 75% chance of winningnothing according to Cramer, Hartog, Jonker, and van Praag (2002).

30

Table 6-1
Average Search Durations by the Extent of Risk Attitude (B) (Risk Averse Subjects Only)
Difference
Nonprecommitted games
Precommitted games
6.63 [T1]6.90 [T3]-0.27 (1.06)
5.21 [T2]
6.13 [T4]-0.92 (0.36)
1.42
0.77
(0.57)(1.52)
Note: Standard errors in parentheses. Averages are calculated on the treatment base. The extent ofrisk attitude is calculated from a subject's response to the post-experimental questionnaire: howmuch is a subject willing to pay for a lottery of a 25% chance of winning JPY2000, but a 75%
chance of winning nothing according to Cramer, Hartog, Jonker, and van Praag (2002).
Uniform
KnightDifference

Table 6-2
Average Search Durations by the Extent of Risk Attitude (B) (Risk Neutral Subjects Only)
Difference
Nonprecommitted games
Precommitted games
Uniform
6.83 [T1]7.14 [T3]-0.31 (2.00)
Knight6.93 [T2]5.00 [T4]1.93 (2.04)
Difference
-0.10
2.14
(1.24)(2.38)
Note: Standard errors in parentheses. Averages are calculated on the treatment base. The extent ofrisk attitude is calculated from a subject's response to the post-experimental questionnaire: howmuch is a subject willing to pay for a lottery of a 25% chance of winning JPY2000, but a 75%
chance of winning nothing according to Cramer, Hartog, Jonker, and van Praag (2002).
Table 6-3
Average Search Durations by the Extent of Risk Attitude (B) (Risk Prone Subjects Only)
Difference
Nonprecommitted games
Precommitted games
Uniform
8.65 [T1]9.00 [T3]-0.35 (3.58)
Knight7.25 [T2]9.00 [T4]-1.75 (2.82)
Difference
1.40
0.00
(1.91)(4.93)
Note: Standard errors in parentheses. Averages are calculated on the treatment base. The extent ofrisk attitude is calculated from a subject's response to the post-experimental questionnaire: howmuch is a subject willing to pay for a lottery of a 25% chance of winning JPY2000, but a 75%
chance of winning nothing according to Cramer, Hartog, Jonker, and van Praag (2002).

31

Table 7
Search Duration under Knightian Uncertainty

Knight (=1)

Nonprecommitted games(1)(2)-1.178-1.178(0.488)***(0.451)**(3)-1.139(0.456)Precommitted (=1)
Knight*Precommitted
Subject dummies
Session dummy

All observations(4)-1.178**(0.488)**0.395( 1.010)0.133(1.314)

Yes.
Yes.

(5)
-1.178(0.455)0.395( 0.872)
0.133(1.175)Yes.
Yes.

R-squared
0.014
0.266
0.012
N
396
396
484
Note: Robust standard errors in parentheses. *** 1%, ** 5%, * 10% significance.

32

0.014
484

0.253
484

***

Table 8
Search Duration under Knightian Uncertainty by the Extent of Risk Attitude (A)
(1)
(2)
(3)
Risk prone
Risk neutral
Risk averse-0.5081.215-1.607Knight (=1)(0.862)-(0.780)(0.631) **Precommitted (=1)
Knight*Precommitted
Subject dummies
Session dummy

Yes.
Yes.

Yes.
Yes.

Yes.
Yes.

(4)
Risk prone-0.508(0.877)2.283(1.759)-1.242(2.349)(5)
Risk neutral-0.944(1.222)-0.850(1.818)-1.181(2.747)

Yes.
Yes.

Yes.
Yes.

(6)
Risk averse-1.607(0.657) **-0.391(1.073)1.561(1.534)Yes.
Yes.

R-squared
0.190
0.243
0.298
0.244
0.223
0.274
N
108
72
198
132
88
242
Note: Robust standard errors in parentheses. *** 1%, ** 5%, * 10% significance. The extent of risk attitude is calculated from a subject's
response to the post-experimental questionnaire: how much is a subject willing to pay for a lottery of a 25% chance of winning JPY200, but a
75% chance of winning nothing according to Cramer, Hartog, Jonker, and van Praag (2002).
Table 9Search Duration under Knightian Uncertainty by the Extent of Risk Attitude (B)
(1)
(2)
(3)
Risk prone
Risk neutral
Risk averse-1.4000.100-1.423Knight (=1)(1.808)(1.209)(0.533) ***Precommitted (=1)
Knight*Precommitted
Subject dummies
Session dummy

Yes.
Yes.

Yes.
Yes.

Yes.
Yes.

(4)
Risk prone-1.400(1.866)0.350(2.241)
1.400(3.611)(5)
Risk neutral
0.100(1.223)0.314(2.255)-2.243(2.520)

Yes.
Yes.

Yes.
Yes.

(6)
Risk averse-1.423(0.564) **0.271(0.921)0.648(1.316)Yes.
Yes.

R-squared
0.153
0.177
0.290
0.226
0.168
0.921
N
36
63
279
44
77
341
Note: Robust standard errors in parentheses. *** 1%, ** 5%, * 10% significance. The extent of risk attitude is calculated from a subject's
response to the post-experimental questionnaire: how much is a subject willing to pay for a lottery of a 25% chance of winning JPY2000, but a
75% chance of winning nothing according to Cramer, Hartog, Jonker, and van Praag (2002).
33

Table 10
Knightian Uncertainty on Reservation Point
Reservation point in precommitted games
(1)
(2)
0.965
-52.788
Knight (=1)
(13.127)(28.365)
Subject dummies
Round dummies
Session dummy

Yes.
Yes.
Yes.

*

Yes.
Yes.
Yes.

R-squared
N

0.831
0.859
1760
360
All observation
OSR-consistent subjects only
Note: Robust standard errors in parentheses. *** 1%, ** 5%, * 10%
significance. A subject is among the OSR-consistent subsample if thefollowing conditions are satisfied; (1) for any given round, a subject
accepts a point drawn in (T1) if the point is no less than the corresponding
reservation point in (T3); (2) for any given round, a subject accepts a point
drawn in (T2) if the point is no less than the corresponding reservation
point in (T4); (3) for any given round, a subject rejects a point drawn in(T1) if the point is lower than the corresponding reservation point in (T3);(4) for any given round, a subject rejects a point drawn in (T2) if the point
is lower than the corresponding reservation point in (T4), and zerootherwise.

34

Table 11
Detecting Inconsistency with the Optimal Stopping Rule
Consistent with OSR

Inconsistent with OSR

Number of observations
2395 [96%]round-base
312 [21%]game-base
9 [20%]
subject-base
Average final payoff
2732.04
uniform
2711.73knightNote: Standard deviations in parentheses.

Difference

94 [4%]
84 [79%]35 [80%]2495.402536.5835

236.64 (47.03)
175.15 (60.08)

Table 12
Knightian Uncertainty on Reservation Point by the Extent of Risk Attitude (A)Reservation point in precommitted games
(1)
(2)
(3)
risk pronerisk neutral
risk averse-10.000-29.417-55.000
Knight (=1)
(6.882)(48.261)(41.616)
Subject dummies
Round dummies
Session dummy

Yes.
Yes.
Yes.

Yes.
Yes.
Yes.

Yes.
Yes.
Yes.

R-squaredN

0.99940.75720.751140
16080
OSR-consistent subjects OSR-consistent subjects OSR-consistent subjects
Note: Robust standard errors in parentheses. *** 1%, ** 5%, * 10% significance. The extent ofrisk attitude is calculated from a subject's response to the post-experimental questionnaire: howmuch is a subject willing to pay for a lottery of a 25% chance of winning JPY200, but a 75%
chance of winning nothing according to Cramer, Hartog, Jonker, and van Praag (2002).
Table 13Knightian Uncertainty on Reservation Point by the Extent of Risk Attitude (B)Reservation point in precommitted games
(1)
(2)
(3)
risk pronerisk neutral
risk averse-43.333-19.250Knight (=1)(38.944)(44.507)
Subject dummies
Round dummies
Session dummy

Yes.
Yes.
Yes.

R-squared
N

Yes.
Yes.
Yes.

0.905
0.654
120
160
No observations
OSR-consistent subjects OSR-consistent subjects OSR-consistent subjects
Note: Robust standard errors in parentheses. *** 1%, ** 5%, * 10% significance. The extent ofrisk attitude is calculated from a subject's response to the post-experimental questionnaire: howmuch is a subject willing to pay for a lottery of a 25% chance of winning JPY2000, but a 75%
chance of winning nothing according to Cramer, Hartog, Jonker, and van Praag (2002).

36

Figure 1:Trends in the Reservation Point by the Extent of Ambiguity (Risk Neutral Agent)
3000
2500

point

2000
1500
1000
500
0
1

3

5

7

9

11

13

15

17

19

round
R1

R2

R3

R4

R1: No ambigutiy
R2: Ambiguity (Epsilon = 0.1)
R3: Ambigutiy (Epsilon = 0.25)
R4: Ambigutiy (Epsilon = 0.5)

Figure 2:Trends in the Reservation Point by the Extent of Ambiguity (Risk Averse Agent)
3000.02500.0point

2000.01500.01000.0
500.0
0.0
0

2

4

6

8

10

12

14

16

round
r1

r2

r3

r1: No ambigutiyr2: Ambiguity (Epsilon = 0.1)
r3: Ambigutiy (Epsilon = 0.25)
r4: Ambigutiy (Epsilon = 0.5)

37

r4

18

20