Does Regret Explain Why People Search too Little? A Model of
Sequential Search with Anticipated Regret and Rejoicing
Zhiquan(Darren) Weng
March 04, 2009
Abstract
We re-examine the sequential search problem where sellers search for the best price
from a known distribution. Existing theory predicts the optimal strategy to be a unique constant reservation price. However, experimental evidence to date has found that people generally
"search too little" compared to the theoretical benchmarks. We argue the behavioral anomaly
is an artifact of the truncated information structure of the sequential search problem that gives
rise to asymmetric regret and rejoicing. Embedding the problem within a behavioral model
incorporating both anticipated regret and rejoicing based on the regret theory of Loomes &
Sugden (1982) and Bell (1982), we …nd: First, people search too little if and only if they are
more sensitive to regret than to rejoicing. Second, if we revise the feedback structure of the
search problem so that people expect to see what the price would have been had they continued to search (after they stop), search behaviors become observationally indistinguishable from
the benchmarks. Moreover, these results are found to hold even when the price distribution
becomes unknown. By building a dynamic structure between anticipated regret/rejoicing and
experienced regret/rejoicing, our model explains why people sometimes exercise recall. An empirical investigation of 673 separate searches from an experimental dataset con…rms that people’s
(latent) reservation prices do shift with regret in the way predicted by the model. Estimation
results show that regret about the last search being unsuccessful increases the probability of
stopping from 18% to 31% in the current round. Competing explanations for "search too little",
such as risk aversion and satis…cing behaviors, are evaluated in a three-way horserace and are
rejected in favor of regret/rejoicing. One policy implication of the model is that promoting
post-purchase price transparency may induce consumers to search more e¢ ciently.
Key Words: Sequential search, Regret theory, Search too little, Recall, Experiment, Latent
reservation price estimation.
JEL Classi…cation: C91, D03, D11, D12, D83
PhD Candidate. 410 Arps Hall, Department of Economics, The Ohio State University, Columbus, Ohio, USA
43210. Email: weng.27@osu.edu. Phone: (614) 218-2045. I am grateful for Daniel Schunk at the University of Zurich
for sharing his experimental data. All errors in the paper are my own. Preliminary Draft: Please do not cite or
redistribute without the author’s knowledge.
1
1
Introduction
We analyze search behaviors in the simplest possible setting: A seller (for example, a worker)
sequentially searches for buyers’ bid prices (for example, o¤ered wages), which are known to be
drawn independently from a pre-speci…ed distribution. The cost of each search is …xed at a positive
constant c, and moreover, perfect recall for past prices is allowed. The seller can sample in…nitely
many prices and there is no discounting. Established theory predicts that the optimal search rule
in this environment is for the seller to follow an optimal stopping rule, i.e., to set a reservation
price H and to accept the …rst price exceeding H : However, previous experimental studies have
consistently found that (1) people search too little. In other words, sellers accept prices that are
too low relative to the optimal reservation price derived under risk neutrality. 2 . (2) A sizable
portion (usually 10 to 20 percent) of the searches result in the exercises of recall (whereas optimal
stopping rule predicts none whatsoever), and (3) Searcher’s eventual accepted prices seem to be
path dependent (as if they were uncertain of the price distribution after all.)
Two possible explanations, namely, risk aversion and satis…cing behavior, have been advanced
in the literature to account for the "search too little" results. However, neither of the two theories
has been able to satisfactorily resolve the issue of recall, unless people’s risk attitudes or earning
aspirations are assumed to evolve in a systematic way during the search process. The last assertion
has not received any supporting evidence. In this paper, we o¤er a third, and more plausible,
explanation that is able to explain both too little search and recall at the same time. And it is
based on the information/feedback structure unique to the sequential search problem: Namely,
regret and rejoicing are only possible when one continues the search; and the only source of regret
is about "stopping too late". A …ne distinction should be made between the regret/rejoicing
model and a simple learning model: In the regret/rejoicing model, the "search too little" result is
derived from anticipation of future regret and rejoicing whereas in a simple learning model, people
only look at the success or failure of past searches. In an experimental dataset that contains 673
separate searches, we investigate and compare the abilities of risk aversion, satis…cing behaviors
and regret/rejoicing in predicting actual searches in a head-to-head horserace. Risk aversion and
satis…cing behaviors are soundly rejected in favor of the regret/rejoicing explanation.
2
See, for instance, Schotter and Brounstein (1981), Hey (1982,1987), Kogut (1990), Cox and Oaxaca (1996),
Sonnemans (1998), Einav (2005).
2
1.1
Motivation
Bell (1982) and Loomes & Sugden (1982) pioneer the work of regret theory as an alternative
paradigm to the expected utility theory in explaining choice under uncertainty. The premise of the
regret theory is that once the uncertainty of the world is resolved, decision maker will conscientiously
compare the outcome from the chosen alternative and the (counterfactual) outcome had the choice
been made di¤erently, and will experience a cognitively based emotion as a result of this comparison.
She feels pleased if what she gets is better than what she would have gotten, i.e. she rejoices. she
feels displeased if what she gets is worse than what she would have gotten, i. e. she regrets. Thus
even if a choice is ex ante optimal, it may not look as good after the fact. The regret theory futher
hypothesizes that people have the ability to anticipate these positive or negative emotions so they
adjust their choice accordingly prior to the resolution of the uncertainty. In order for anticipated
regret and rejoicing to a¤ect choices, people must also hold the expectation that the resolution of
the unchosen alternatives will be known.
The sequential search problem presents an especially fertile ground to apply the regret theory.
First, the information ‡ows in this problem dictates that regret and rejoicing are asymmetric. While
no feedback (hence regret or rejoicing) is possible once she stops the search, she cannot escape from
feeling either a sense of regret or a sense of rejoicing if she continues to search, because she can
always compare the outcome of the latest search to her earning from the last round, which is the
counterfactual payo¤ had she chosen not to search. In addition, as the distribution of prices one
actually observes in the search process is truncated by her …nal accepted price, her only possible
source of regret is based on prices from the unfavorable part of the distribution which translates into
reactions such as "Price is bad again; I wish I had not wasted my time and energy in running around
and searching!". In other words, one will only regret "stopping too late" but never "stopping too
early". Conversely, one’s only source of rejoicing comes from drawing a good price which translates
into reactions such as "I am glad that I stuck around for one more round because the price turns
out to be really good", but the likelihood and extent of this happening is very limited due to
the truncation from top by the accepted price. While largely ignored by the previous economic
literature on search, the tension between regret and rejoicing turns out to be a very interesting
dynamics that characterizes the actual search that goes on.
A separate strand of literature has developed in the …eld of psychology arguing that regret is a
more intense emotion than rejoicing. If this argument is true, it implies in sequential search that
one might search more conservatively if she is more concerned with minimizing regret than with
maximizing rejoicing. Secondly, given the sequential nature of the search task, every price that
one encounters during the search has the e¤ect of reminding her whether or not the last search has
paid o¤, making the concerns for regret and rejoicing more salient as the search continues and thus
amplifying the impact of anticipated regret/rejoicing in subsequent search decisions. The sequential
3
nature of the task combined with the immediacy of the feedback makes it an ideal setting to test
if anticipations of regret and rejoicing are driving the behaviors. Moreover, by making people’s
sensitivities towards regret and rejoicing endogenous to the experienced regret and rejoicing during
the search process through psychological reinforcing, the model can also shed light on why one
would rationally exercise recall in certain situations.
1.2
Related Literature
Sonnemans (1998) studies what information individuals use to form their search strategies by
utilizing the electronic information board technique as well as by asking subjects to explicitly
record their strategies. He discovers that subjects sought a combination of information including
the last price, the highest price, and the total earning in the search. When explicitly formulating
their strategies, 78% of the subjects admitted total earning or number of bids in their stopping
criteria. However, these explicitly developed stopping rules, when executed by computer on the
same price sequences, resulted in substantially fewer number of recalls than that of actual searches
performed by the same subjects. This discrepancy suggests that either some additional stopping
criteria adopted in the search are not incorporated into the formulated strategies, or there are
dynamic inconsistency in the search behavior. Furthermore, Sonnemans points out the possibility
of one-sided learning to be directing subjects to stopping too early, because late stoppers can learn
much better than early stoppers. In essence, learning in his construct is similar to the concept of
"experienced regret" in the present model. However, unlike in Sonnemans, the "search too little"
result here is derived from the interplay of forward-looking anticipatory regret versus rejoicing,
rather than the backward-looking experienced regret alone (although we do believe experiencing
regret tends to heighten the sensitivity towards future feelings of regret). Finally, using a risk-return
pro…le, Sonnemans concludes that 52% of the searches his subjects make cannot be explained by
risk aversion alone.
Closely related to Sonnemans, Einav (2005) invokes the idea that "observational regret" is
driving the search too little result because sellers, if motivated to minimize this backward-looking
type of regret, can only learn to adjust their reservation downwards. In addition, number of searches
made and search e¢ ciency improve in an experiment treatment in which sellers can observe three
more random prices after they complete the search. This is the …rst piece of evidence to our
knowledge that shows modifying the information structure in the sequential search environment
does mitigate the "too little search" behavior. Yet his idea of observational regret is again backwardlooking and is not embedded in any formal model. Moreover, subjects in Einav’s experiment are
not allowed to revise their reservation prices during the search, preventing them from any chance
of recall. One theoretical …nding in the present paper is knowing that just one more price, rather
than three more, will be revealed following the end of the search is su¢ cient to lead subjects to
search in the most e¢ cient manner.
4
An experimental study by Schunk and Winter (forthcoming)3 investigate the relationship between individuals’heterogeneity in risk attitudes and loss aversions and their behaviors in sequential
search task. They …nd no connection at all between the risk measures developed independently
through the lottery tasks and through the attributed reservation prices in the search task. On the
other hand, they …nd that more loss averse (rather than risk averse) individuals tend to search less.
They also consider the possibility that individuals in the experiment falsely believe that the search
costs are not sunk. Even for these individuals (who try to buy a unit of item at the best price),
optimal reservation prices are declining over time. Therefore, none of the search strategies that
they consider is able to accommodate the possibility of recalls.
In a follow-up paper, Schunk (2008) conjectures that subjects in a sequential search task would
constantly update their reference point to equate their current earning in the search and act according to the prospect theory rather than base their decisions on the absolute payo¤ from the
entire search, which might explain why individual’s measure of loss aversion is related to the
extent of search. It is worth stressing that Schunk’s reference-updating utility speci…cation is
very similar to our regret/rejoicing speci…cation in the sense that both utilities are relative to the
current earning and that a measure of loss aversion is also very similar to sensitivity towards regret. However, Schunk does not derive any theoretical result that directly compares search under
reference-updating preference and search under risk-neutral optimal search while we do under a
regret/rejoicing framework. Due to a lack of dynamic structure, his model is also silent on why
recall takes place.
Methodologically, one recent paper by Filiz and Ozbay (2007) applies the regret theory and similar utility representation of regret to ours to a di¤erent setting and obtains some very interesting
results. They investigate both theoretically and experimentally whether the over/under-bidding in
…rst-price auctions are related to anticipated regrets stemming from varying information feedback
structures: 1) bidders know that they will learn the winning bid if they lose (loser regret condition); 2) bidders know the second highest bid if they win (winner regret condition); 3) or they
will receive no feedback regarding the other bids. They …nd that the loser regret condition does
generate overbidding while the winner regret condition does not lead to underbidding, relative to
the control condition. However, because rejoicing is always absent in the auction setting, they do
not deliberately address the issue of the tension between anticipated regret and rejoicing in decision
making.
The rest of the paper will proceed as follows: Section 2 presents a formal model of the optimal
search strategy under anticipatory regret and rejoicing, constructs a counterfactual search where
feedback becomes possible even after search is stopped, and extends the results to situation in
3
The empirical analyses contained in the second half of the present paper utilize the same experimental dataset
as in the Schunk and Winter study.
5
which price distribution is unknown; Section 3 examines some of the model’s predictions through
an experimental dataset consisting 673 separate searches and conducts a number of robust checks;
Section 4 discusses the distinction between the regret/rejoicing model and two other competing
explanations for the "search too little" result, which are risk aversion and satis…cing behaviors,
and administers a three-way horserace comparing the accuracy of the three in predicting the data;
Section 5 discusses future directions and policy implications of this research.
2
2.1
The Model
The Benchmark Model: Risk Neutral, Optimal Search
The benchmark case throughout our analysis is the simplest variant of the sequential search environment for which formal theory predicts a constant reservation price property. We will …rst
use the example of a seller searching for the best price: In the benchmark case, a seller (e.g. an
unemployed worker) who attempts to sell a unit of good (e.g. her labor) with cost normalized to
0 has to sample bid prices (e.g. wage o¤ers) sequentially. The bid prices are drawn independently
from a known distribution with CDF F (x) and P DF f (x) over interval (a; b). Each price sampled costs the seller a constant amount, c: The search process could last in…nitely and there is no
discounting. Moreover, seller has the ability to recall bid prices she encountered previously. The
capacity to recall is consistent with, for example, the ease with which people can return to past
prices in an online shopping session. The benchmark case is simple enough yet retains the essence
of the relationship between search and information structure that we are interested in.
In the benchmark case, the seller is risk-neutral therefore she aims to maximize the monetary
payo¤ from the search. The preceding search setting presents a dynamic programming problem
where the only state variables are the number of searches that have been conducted, t, and the
highest of those t encountered prices, which we call the candidate price and denote as Ht : Let
V (Ht ; t) represent the value function, with the choice variable being a binary decision between
stopping or continuing the search. (Ht ; t) represents the payo¤ from choosing to stop now.
V (Ht ; t) = M ax { (Ht ; t); E[V (Ht+1 ; t + 1)jHt ]g
(Ht ; t) = Ht
c t
(1)
(2)
Lippman & McCall (1976) show that the solution to the above in…nite horizon problem is
identical to solving a one-period problem myopically, because the option value of searching in the
next round will not be higher than in the current round. Disregarding the sunk search costs that
have been incurred prior to the current decision results in the stationarity of the problem.
As a result, from this point on we suppress the time dimension of the problem. To a seller, the
payo¤ from choosing to stop now is:
6
Stop (H)
=H
(3)
If she continues the search, her payo¤ when the new bid price equals x is:
Search (x; H)
= M axfH; xg
c
(4)
Therefore, the expected payo¤ from continuing the search is:
Search (H)
=
Z
b
Search (x; H)dF (x)
a
= [F (H)H +
Z b
= H + [ (x
Z
b
xdF (x)]
c
H)dF (x)
c]
H
(5)
H
Equation (5) shows that the decision of when to stop the search can be determined by comparing
Rb
the marginal bene…t ( H (x H)dF (x)) and marginal cost (c) from one more search given the
candidate price H. Thus, de…ne a di¤erence function
G(H)
Search (H)
Stop (H)
Z
b
(x
H)dF (x)
c
(6)
H
It is easy to verify that dG(H)=dH < 0: Assuming all the end conditions (i.e. G(a) > 0; G(b) < 0)
are met, the optimal search strategy for a risk neutral seller is therefore of reservation price type.
Speci…cally, she should set a reservation price equal to HRN ; and stop search and accept the …rst
price exceeding HRN ; where HRN is de…ned implicitly by G(HRN ) = 0: In other words,
Z
b
HRN
(x
HRN )dF (x) = c
(7)
Except for a few well-speci…ed density functions, the reservation price HRN will not have a closedform expression. HRN is also a function of the search cost c and the price distribution F (:): Two
interesting comparative statics can be readily derived from taking the total derivatives of Equation
(7).
@HRN
< 0
@c
HRN (F1 ) > HRN (F2 ); if F1 (:) …rst-order dominates F2 (:)
7
(8)
(9)
If we do not restrict the seller’s risk attitudes, the preceding problem can still be solved in a very
similar manner: The only di¤erence is that a properly de…ned utility function U (:) will replace all
of the monetary reward/cost terms in Equation (3) and (5). The solution as in (10) still possesses
the constant reservation price property (subscript RA is short for risk averse subjects).
U (HRA ) = F (HRA )U (HRA
c) +
Z
b
HRA
U (x
HRA
c)dF (x)
(10)
One important implication of the optimal stopping rules outlined above is that a seller conducting a sequential search should NEVER exercise the recall option. This assertion holds so long
as the seller has a constant reservation price throughout the search, regardless of the value of the
reservation price itself. But the previous experiments have revealed that subjects in the lab did
frequently recall past prices, ranging anywhere from 10 to 20 percent of the total searches. This
issue remains puzzling and unsatisfactory even if we take into account the full spectrum of subject’s
risk attitudes in the search process such as in equation (10)
2.2
The Behavioral Model: Sequential Search with Anticipated Regret and
rejoicing
In the ensuing sections, we model the search process of a subject that while staying risk neutral,
feels either a sense of regret or a sense of rejoicing after a search is conducted and a price is revealed.
Note that due to the asymmetrical information structure of the sequential search problem, one only
regrets about wasting the search cost when an unfavorable price is revealed, and one only rejoices
at the discovery of a favorable price. There is neither regret about missing out a good price nor
rejoicing at not wasting the search cost. First, we will make explicit our regret/rejoicing utility
representation.
2.2.1
Regret/Rejoicing Utility Speci…cation
Bell (1982) and Loomes & Sugden (1982) pioneered "Regret/Rejoicing Theory" as an alternative
paradigm to the expected utility theory in explaining choice under uncertainty. Using an axiomatization approach, they proved that a utility function incorporating regret and rejoicing should be
expressed in the following general form:
v(x; y) = u(x) + f (u(x)
u(y)); for some function f
(11)
where x is the choice being made and y is the (only) alternative, u(:) is comprehended as the
standard utility function without concerns for regret/rejoicing. Therefore, f (u(x) u(y)) expresses
the additional utility arising from regret or rejoicing.
8
Following Filiz and Ozbay (2007) in which they adopt the same utility representation to study
the relationship between anticipated regret and over/under-bidding in …rst-price auctions, we decide
to modify a subject’s utility function by incorporating a more explicit functional form for the
regret/rejoicing component of (11):
VSearch (x; H) =
Search (x; H)
where R(y) =
y if y > 0,
+ R(
Search (x; H)
Stop (x; H))
(12)
called the intensity of rejoicing
= 0 if y = 0
=
y if y < 0;
Both ;
called the intensity of regret
are positive.
Our regret/rejoicing utility formulation (labelled V to distinguish it from a standard utility) is
comprised of two additive elements: The …rst element, a "choiceless" utility, is equal to the monetary
payo¤ from the choice outcome, Search (x; H)4 . "Choiceless" is in the sense that the outcome is
evaluated as if it were obtained without an internally processed decision, such as getting $100 as a
result of an income tax credit. The second element, R( Search (x; H)
Stop (x; H)), represents the
subjective utility from the feelings of either regret or rejoicing (R(:) stands for the regret/rejoicing
function). This second element of the utility function depends on the decision maker’s available
choice set and the resolved state of nature. We further simplify the functional form by assuming it
to be linear in the di¤erence in payo¤s between the chosen and forgone alternatives in every state
of the world. But the slopes, which we call the intensities for regret and rejoicing, are set di¤erently
for gains (i.e. rejoicing) and losses (i.e. regret). The di¤erentiation in slopes for regret and for
rejoicing not only allows more ‡exibility but also is more realistic in light of the sizable psychology
literature arguing that regret generates more cognitive and emotional arousal than rejoicing does.
It also makes good sense to make R(0) = 0 when the chosen and forgone alternatives fair equally.
Additionally, in our model the regret and rejoicing intensities are not stationary. In fact, they
are reinforced by the seller’s experienced regret and rejoicing at each search round as a result of
the past search outcomes. Hence they are functions of the price history in the current search,
i.e. t = (p1; p2 ; :::; pt ); t = (p1; p2 ; ::::; pt ) (The exact structure of the relationship will be made
more clear in later sections). This dynamic relationship expresses the notion that the more one
experiences regret or rejoicing in the past, the more sensitive (or mindful) she is to future regret
and rejoicing. The history dependence of (:); (:) o¤ers plausible insights into why sellers might
exercise recall and why their behaviors appear to be path-dependent, even though in fact the sellers
in our model still adhere to a reservation price strategy.
4
Risk neutrality is maintained throughout the discussion surrounding the regret/rejoicing utilities in order to
facilitate comparison with the benchmark case.
9
Before the seller decides to make the investment to search, she is assumed to be able to anticipate
her feelings of regret and rejoicing based on the realized outcomes in every possible state of nature.
So her ex ante utility from continuing the search given the current candidate price H is:
VSearch (H) =
Z
b
VSearch (x; H)f (x)dx
a
=
Z
H
[H
c + ( c)]f (x)dx +
a
+
Z
Z
H+c
[x
c + (x
c
H)]f (x)dx
H
b
[x
c + (x
c
H)]f (x)dx
(13)
H+c
Z
b
H)dF (x) c] [expected monetary payo¤]
Z H+c
[cF (H) +
(c + H x)f (x)dx] [anticipated regret]
H
Z b
+
(x c H)f (x)dx
[anticipated rejoicing]
= [H +
(x
H
(14)
H+c
Equation (14) shows that, VSearch (H) consists of three separate terms: the term in the …rst square
bracket measures the expected monetary payo¤ from the search and thus is identical to Search (H)
in the benchmark model. The second term captures the e¤ect of anticipated regret, The third term
captures the e¤ect of anticipated rejoicing. Generally speaking, anticipated regret makes search
less attractive while anticipated rejoicing makes the search more attractive; therefore, a priori it is
not obvious how the interplay between regret and rejoicing would a¤ect the seller’s search strategy.
On the other hand, the expected utility of stopping the search remains at H. This is because
stopping the search will preclude any information feedback as to what the price would have been,
implying R(:) will not be operative here. Hence,
VStop (H) = H
(15)
De…ne L(H) to be equal to the di¤erence between the expected utilities from continuing the
search and from stopping the search given H:
L(H)
VSearch (H) VStop (H)
Z b
Z H+c
= [ (x H)dF (x) c]
[cF (H) +
(c + H
H
H
Z b
+
(x c H)f (x)dx
H+c
10
x)f (x)dx]
(16)
Therefore, the seller’s decision rule is to continue to search () L(H)
2.2.2
0
Optimal Search Rule under Anticipated Regret/Rejoicing
It is easy to see that in Equation (16) L0 (H) < 0:This is intuitive because not only does the monetary
return to search decreases with H; but also the likelihood and the extent of regret increase with
H while those of rejoicing decrease with H: It implies that a seller, in anticipation of regret and
rejoicing, should once again adhere to a unique reservation price strategy with the reservation price
HRR (c; F; ; ) which solves L(HRR (c; F; ; ) ) = 0: We can further simplify the three components
in L(H):
1) Anticipated Regret:
cF (H)
Z
Z
H+c
(c + H
x)f (x)dx =
H
2) Anticipated Rejoicing:
Z b
(x
H+c
F (x)dx
(17)
H
c
H)f (x)dx = (b
c
Z
H)
H+c
b
F (x)dx
(18)
H+c
In aggregate, we now have
Z b
L(H) = [ (x
H)dF (x)
H
(c + H) + b
c]
Z b
Z
H+c
F (x)dx
H
F (x)dx
(19)
H+c
Ultimately our goal is to diagnose whether and how concerns for regret and rejoicing will drive
the seller to revise her search strategy relative to the risk-neutral benchmark case. Alternatively,
the benchmark case can be viewed as a special case under the regret/rejoicing model where =
= 0: Because it is impossible to derive the precise explicit solutions for HRN and HRR ; a direct
comparison between the two cuto¤ points is infeasible. To circumvent this problem, an alternative
strategy is devised in which we examine the value of L(H) evaluated at HRN : To that end, we …rst
utilize the property that HRN is the implicit solution to the Equation (7). Hence,
Z H +c
Z b
RN
L(HRN ) = G(HRN )
F (x)dx
(c + HRN ) + b
F (x)dx
=
Z
HRN
HRN +c
HRN
F (x)dx
(c + HRN ) + b
11
Z
HRN +c
b
HRN +c
F (x)dx
(20)
Further simplifying Equation (20) (detailed derivation contained in the Appendix), we obtain the
following crucial equation:
L(HRN ) =
(
)
Z
HRN +c
F (x)dx
(21)
HRN
Equation (21) shows that if > > 0, then L(HRN ) < 0: Then it must be true that a <
HRR < HRN because L(a) > 0; L0 < 0: On the other hand, if 0 < < ; then it must be true that
HRN < HRR < b: If
= ; then HRR = HRN : A graphic summary of the results is in Figure 1 in
which H1 =.HRR and H0 = HRN :Therefore the following proposition holds.
Figure 1
A Comparison Between Reservation Prices
Proposition 1:
When both anticipated regret and anticipated rejoicing are present in the search
process, the seller continues to adopt a reservation price strategy. Nevertheless, the
value of the reservation price will be lower than that in the benchmark model if and
only if the intensity of regret is stronger than the intensity of rejoicing, i.e.
> ;
12
resulting in a higher hazard rate of stopping the search as well as lower expected number
of searches than the benchmark case. Conversely, the seller will search more than the
benchmark case if and only if the intensity of rejoicing is stronger than the intensity of
regret, i.e. < . When the two intensities are equal; a seller will search in the exact
same way as a risk-neutral seller without any concerns for regret and rejoicing.
The preceding proposition makes it clear that anticipatory concerns for regret would indeed
lead to "too little search", provided that it is stronger than the counteracting force of rejoicing.
Next, we derive some comparative statics of how the reservation prices under the regret/rejoicing
framework would respond to the changes in the underlying parameters c; ; : Setting L(HRR (c; F; ; )) =
0 as in Equation (19) and take the total derivatives with respect to the parameters of interest will
give:
@HRR (c; F; ; )
@c
@HRR (c; F; ; )
@
@HRR (c; F; ; )
@
< 0;
(22)
< 0;
(23)
> 0
(24)
Note if a purchasing framework is considered wherein a buyer is trying to shop at the lowest price,
all the comparative statics are exactly reversed.
Therefore, under a regret/rejoicing model, a seller’s reservation price will decrease if her regret
coe¢ cient gets larger, and vice versa for rejoicing. Put di¤erently, a seller will search for prices
less aggressively en ante if she becomes more sensitive to the feeling of regret ex post, other things
being equal.
A further rami…cation of the statement is that, because as we have hypothesized, the values of
the regret and rejoicing intensities (:); (:) are subject to the reinforcement from the experienced
regret and rejoicing based on the history of realized prices in the search, the reservation prices
HRR will also be dependent on past prices. For example, a consecutive string of bad prices might
cause the seller to be more sensitive to regret and therefore lower her reservation prices over time,
possibly causing her to recall a past price. This o¤ers one plausible explanation why actual search
behavior in the lab will appear history-dependent and why recall is often opted for, even though
the distribution is known. Thus a complete behavioral model encompassing anticipatory regret and
rejoicing can coherently explain the three empirical anomalies that the benchmark model cannot
not explain.
13
2.2.3
Counterfactual Search with Availability of Feedback even when Search is Stopped
In most of the real life search settings including the one we have just modelled, stopping the search
will preclude any information feedback. In other words, a seller never gets to or expects to observe
what the next price would have been. Consequently, regret or rejoicing only occurs when one
continues the search. Hence, if regret is the more prominent force here than rejoicing (e.g. let
= ! 1), stopping the search becomes the preferred, regret-minimizing choice.
Nevertheless, there are certain situations in life where you may …nd out what price you have
missed out after you stop the search. Suppose that you are shopping for a new Sony Playstation at
a couple of local electronic stores for your son’s birthday this coming Wednesday. You happen to
also have a plan to go to the local Walmart to run some errands this Saturday so there is a chance
that you may see Walmart carry the same game console at a lower price. How does this piece
of information a¤ect your search for the console in the …rst place? In fact, you might decide to
check out the price at Walmart …rst before you buy it at any other store such as Bestbuy. Or you
may decide to purposefully avoid looking up the price of the console at Walmart on your weekend
shopping trip. Can this kind of behavior be predicted by our regret/rejoicing search model?
In this section, we simulate the preceding situation by allowing information feedback even if
the search is stopped. More concretely, in the new "counterfactual" (not all that "counterfactual"
considering the real-life scenario that we just presented) search environment, the seller expects to
observe one more price quote even after she stops the search and sells the item. Given this new
information structure, the seller will have the opportunity to experience another kind of regret:
This happens following the end of the search when the next price exceeds the current price by more
than the search cost. Correspondingly, there is also the opportunity of experiencing another kind
of rejoicing when the price revealed after the search turns out to be unfavorable.
As opposed to the original search problem, the expected information feedback on the Stop the
Search option will augment its expected utility to account for the additional elements of regret and
rejoicing:
VStop (H) = H
Z
b
(x
H
c)f (x)dx + cF (H) +
H+c
Z
H+c
(c + H
x)f (x)dx
(25)
H
The expected utility of the Continue the Search option, however, is unchanged.
Clearly, VStop (H) H increases with the value of H because of diminished regret and heightened
rejoicing. Similarly, VSearch (H) H decreases with the value of H: De…ne a di¤erence function
M (H)
VSearch (H) VStop (H): We have M 0 (H) < 0: Simplifying the expression for M (H), we
have:
14
M (H)
VSearch (H) VStop (H)
Z b
Z H+c
(x H)dF (x) c ( + )[cF (H) +
(c + H
H
H
Z b
(x c H)f (x)dx
+( + )
x)f (x)dx]
(26)
H+c
Note that the functional form of M (H) in Equation (26) is identical to that of L(H) in Equation
(16), except that in M (H) the intensity of regret and the intensity of rejoicing are identically equal
to the sum of and : According to Proposition 1, whenever the intensities of regret and rejoicing
are equal, reservation price under regret/rejoicing is identical to that of a risk-neutral optimal
search. Therefore, the search outcomes in the search setting with two additional elements: 1)
anticipation of regret and rejoicing 2) counterfactual information feedback after search is stopped
become behaviorally indistinguishable from those of the benchmark model where both are absent.
This forms the basis of our second proposition:
PROPOSITION 2:
If, after stopping the search and selling an item, a seller with anticipated regret/rejoicing
freely observes one more price quote, then her behaviors in the search will be observationally equivalent to those in the benchmark case. In other words, HRN is the chosen
reservation price in this hypothetical search problem, no matter what the intensities
of regret and rejoicing are. The implication is, behaviors in this counterfactual search
environment should be path independent, and recall should never happen.
2.2.4
Extension to Search under Unknown Distribution
We …rst examine search behaviors in which the underlying distribution is completely known because
theorists and to a larger extent experimentalists wish to avoid the additional complexity based
upon people’s heterogeneity in their ability in forming a prior regarding the price distribution and
in updating the distribution by processing new price information. However, this simpli…cation also
makes the model possess much less external validity. The seminar paper in this area is Rothschild
(1974) where he used a simple Bayesian updating process on a …nite set of possible prices to model
search from unknown distribution. In this section, we would try to make a very limited attempt
15
in shedding some lights on search from an unknown distribution under the anticipation of regret
and rejoicing. Our direct concern, once again, is how the search behaviors with regret and rejoicing
compare with the behaviors in absence of these elements. Will people also search too little? The
answer seems to be yes.
Our overarching strategy here is to approximate a Bayesian updating process from an unknown
distribution by a Bayesian updating process between the probabilities of making draws from n
known distributions. In other words, the problem is turned into one in which the seller is sure that
the price comes from one of the n known distributions, but in the meantime she is unsure as to
which one it is. Therefore with each new price revelation she updates her belief about which one
of the n distributions she is choosing from. Our sense is that this approximation is always possible
given the appropriate choices of the number of distributions n, the cumulative probability functions
of each of the distributions F1 (x); F2 (x); :::; Fn (x) (with Fi (x) de…ned on [ai; bi ]; respectively) and
the seller’s initial belief 0 but we still need to locate the precise statistical theory.
The process of a seller sequentially searching for the highest prices from an unknown distribution
is equivalent to the process of her searching from n known distributions with the probabilities of
each distribution being represented
by her belief t ; a n-element simplex, at each decision period
P
t; t = (p1t; p2t; p3t;::::; pnt ):& i pit 1: t is updated according to the Bayes rule after each new
price observation xt 1 :We’ll …rst derive the results for the benchmark case where the concerns for
regret and rejoicing are not present.
2.2.5
Search from an Unknown Distribution with No Anticipated Regret/Rejoicing
First, we would customarily use subscript t to denote the decision and information variables at
decision period t, because now the problem is no longer time invariant, due to the fact that each
realized price on the path would change the perceived price distribution from which the search is
made. Nonetheless, it is clear that the return to no search remains is still the highest standing
price whether or not regret/rejoicing is part of the utility function.
t
VStop
=
t
Stop
= Ht
Here Ht denotes the current candidate price after t price draws. The expected payo¤ from search
will be a probabilistically weighted sum of the expected payo¤ from conducting one more search
from each of the possible distribution as in the following expression:
16
t
Search
n
X
=
i=1
n
X
=
pit [(Ht
c)Fi (Ht ) +
Z
bi
(x
c)dFi (x)]
Ht
pit [(Ht
c) +
Z
bi
(x
Ht )dFi (x)]
Ht
i=1
= Ht +
n
X
pit
i=1
Z
bi
(x
Ht )dFi (x)
c
(27)
Ht
As before, the value of conducting one more search consists of the option value of
R bt ; and
Precall H
the cost of search c, and the expected net return to search in probabilistic form, ni=1 pit Hit (x
Ht )dFi (x):
R bi
Pn
Ht )dFi (x) c: Apparently Gt (Ht ) still
Let us set Gt (Ht ) = search
stop =
i=1 pit Ht (x
retains the property dGt (Ht )=dHt < 0: Therefore, the solution to the seller in search of a best price
will still be of the optimal stopping form. The major di¤erence between the known and unknown
distribution cases, notably, is that now the reservation price is path dependent due to the fact that
all the (p1t; p2t; p3t;::::; pnt )0 s are subject to Bayesian updating. Hence, the seller should sell at search
period t whenever the revealed price exceeds HRN;t ; and HRN;t is the solution to the following
equation:
0=
n
X
pit
Z
bi
(x
HRN;t
i=1
HRN;t )dFi (x)
c
(28)
where Pit is subject to the following Bayesian updating process,
Pit =
Pi(t
n
P
1) fi (xt 1 )
Pj(t
; with
0
given.
(29)
1) fj (xt 1 )
j=1
The optimal stopping rule and the beliefs of the seller should be su¢ cient to completely pin
down her search behaviors in this no regret/rejoicing problem. Moreover, since the reservation price
at any search period HRN;t is just the weighted average of the reservation prices the seller would
have adopted if she knew for sure that she was searching from one of the n distributions weighted
by the relative probabilities of the n distributions, then all the …rst-order comparative statics are
also retained as in the search from known distribution case.
17
2.2.6
Search from an Unknown Distribution with Anticipated Regret/Rejoicing
In essence, the value of search from an unknown distribution with the incorporation of anticipatory
regret and rejoicing is just a probabilistic version of the one we had before for the case when the
distribution is known. To see this, …rst, because the feedback on the no search option is still absent,
t
VStop
= Ht: In contrast, the expected value of search should encompass anticipation of regret and
rejoicing experienced on searching from each of the n possible distributions:
t
VSearch
n
X
=
i=1
Z
pit [Ht +
bi
(x
Ht )dFi (x)
c
x)dFi (x) +
Z
Ht
Z
Ht +c
(c + Ht
cFi (Ht )
bi
(x
c
Ht )dFi (x)]
(30)
Ht +c
Ht
Simplifying the preceding expression, we have
t
VSearch
= Ht +
n
X
pit
i=1
n
X
pit [
Z
bi
(x
Ht )dFi (x)
c+
Ht
Z
Z
Ht +c
Fi (x)dx + (bi
c
Ht )
Fi (x)dx]
(31)
Ht +c
Ht
i=1
bi
t
t
VStop
Let HRR;t satis…es Lt (HRR;t ) 0: Again, since HRR;t is a
As before, set Lt (Ht ) = VSearch
weighted average of the reservation wages that a regret/rejoicing driven seller would have chosen
if she was searching from just one of the n possible distributions, all the …rst-order comparative
statics would still hold.
To compare the values of Ht0 and Ht1 ; we again calculate the value of Lt (Ht ) evaluated at HRN;t :
Lt (HtRN )
= 0+
n
X
pit [
i=1
=
(
)
n
X
Z
=
(
)
i=1
Fi (x)dx + (bi
Ht0
pit
i=1
n
X
0 +c
Htt
pit
Z
0 +c
Htt
Fi (x)dx +
Ht0
Z
HtRN
< ( )0 if and only if
n
X
i=1
HtRN +c
Fi (x)dx
>( )
18
c
Z
Ht0 )
pit (bi
bi
Ht0
Ht0
Z
Fi (x)dx +
Z
Ht0 +c
Ht0
Fi (x)dx]
bi
Ht0
Fi (x)dx
c)
(32)
The preceding result says that the reservation prices will be lower and search is less than the
benchmark case if and only if the regret intensity, in anticipatory terms, is stronger than the
intensity of rejoicing. Proposition 3 summarizes the major results in this section.
PROPOSITION 3: When a seller is searching for the best price from an unknown
distribution, at every period she still adopts a reservation price type strategy: stop
search if and only if Ht
HRN;t (or HRR;t if regret/rejoicing are present), but her
reservation price is going to be dependent on the price history Hto = Hto (x1; x2;:::; xt 1 )
[analogously for Ht1 ] due to Bayesian updating. Nevertheless, on the same realized
price path, a seller with anticipated regret & rejoicing will search less than someone
without them if and only if she feels regret more strongly than rejoicing. Therefore, the
expected number of searches will be fewer for a seller with anticipated regret/rejoicing
if and only if > .
2.3
Competing Explanations for Why People Search Too Little
Previously, researchers puzzled by the observed "search too little" results have o¤ered two chief
explanations: 1) human subjects may display risk aversion on monetary payo¤s, 2) human subjects
may not be able to make the precise optimization calculation (i.e. fully rationality is costly), and
therefore they will satis…ce at reaching a vicinity of the optimum.
2.3.1
Risk Aversion:
Risk aversion is a natural …rst instinct in explaining why people accepts reservation prices that are
too low. Essentially, conducting one more search is like taking a gamble, waging the cost of search
in order to win potential higher price than the current price. If people are risk averse, on the
margin the uncertain prospect of higher earning looks less enticing than the certain loss in search
cost. Presumably the larger the search cost, the more people will shave their reservation price. For
example, in the empirical investigation section to follow, we use a sequential search dataset that
asks subjects to search for the lowest possible price to purchase a particular item that they value
at $500. The price distribution is a discrete normal distribution with mean=$500 and standard
deviation=$10 (truncated at end points $460 and $540). If we also assume a CARA utility function
for the subject on the monetary outcomes, the following graph depicts the relationship between a
subject’s risk coe¢ cient and her constant reservation price. 5 = 0 corresponds to risk neutrality.
Apparently, higher does lead to higher reservation price and more conservative search here.
5
CARA utility of the following form is used:
X Xmin
u(x) = 1 (1 exp(
))
Xmax Xmin
19
Figure 2:
2.3.2
Satis…cing Behaviors
The other concern is: People may not be able to calculate the optimal reservation price accurately
due to limited cognitive processing abilities. But is there a reason why they tend to search less
rather than more? For the same distribution as above, we calculate the expected payo¤ from
adopting reservation prices of di¤erent values and translate them into a percentage term of the
maximum expected payo¤ (obtained at R=490 when = 0) in the Figure 3:
20
Figure 3: Expected Payo¤s from Various Constant Reservation Strategies as % of the Maximum
Expected Payoffs (R=480:R=499)
10
49
8
49
6
49
4
49
2
49
0
48
8
48
6
48
4
-5
48
2
0
48
0
Expected Payoff
5
-10
-15
-20
-25
Reservation Price
The expected values are calculated with the following equation:
E (L) = $500
Ep (pjp
L)
c
F (L)
(33)
In this purchasing task, because the e¢ ciency of choosing reservation prices higher than the
optimum is relatively high compared to that of choosing reservation prices lower than the optimum,
a satis…cing seller on average will choose to search too little.
3
Empirical Investigation
The most crucial prediction from the behavioral model in the preceding section is that, once we
model for a search agent’s auxiliary utilities related to her ex ante anticipation of regret and
rejoicing which are in turn based on the ex post evaluation of her decision making abilities, she has
a tendency to search more conservatively than someone who is both risk-neutral and unmotivated
by these behavioral concerns, provided that the intensity of regret
in the behavioral utility
model we de…ned is larger than the intensity of rejoicing . Although the ability of anticipated
regret/rejoicing to explain the "search too little" result due to the asymmetric information structure
of the sequential search problem is a theoretical novelty in its own merit, what is more important
21
is to show these concerns do underpin the actual search behaviors. If anticipation of regret and
rejoicing is in fact recognized to play a role in the actual search process, does it have a more
prominent role than other competing factors, such as risk attitudes and satis…cing behaviors?
In order to answer these questions, we borrowed an experimental dataset that contains 673
individual search sequences with random price o¤ers.6 One key testable proposition we have from
the model is that search agent’s reference price shifts according to the intensities of regret and
rejoicing. Furthermore, we will make one additional behavioral assumption that people’s sensitivities to future regret or rejoicing are reinforced by their past experiences of regret or rejoicing
which could happen within the same search sequence. Therefore, our empirical methodology is
focused on estimating the latent reservation prices at a particular point of the search as a function
of experienced regret and rejoicing up to that point during the same search. Moreover, the estimation results also make a convincing case why anticipation of regret and rejoicing matters more
than individual’s risk attitudes or satis…cing behaviors. Finally, it is worth underscoring that our
theoretical model is based on forward-looking regret and rejoicing which enables us to make the
connection to backward-looking regret and rejoicing. Hence, our explanations are fundamentally
di¤erent from a simple learning model such as a directional learning model.
3.1
Data Description
The dataset we examine is from a search experiment conducted at the University of Mannheim in
the fall 2003.7 A total of 64 subjects participated in the experiment divided into two sessions, with
each of them completing either 10 or 11 separate sequential search tasks.
The search setting is identical to our model except that it is phrased in terms of purchasing at
the lowest price rather than selling at the highest price: Each subject can sequentially search, for an
unlimited number of times, prices from a known distribution in order to purchase an item that they
value at $500. There is no discounting, and the ability to recall past prices. Subjects are informed
that the prices will be drawn randomly from a discrete truncated normal distribution with mean
$500 and standard deviation $10, truncated at $460 and $540. Each price that they sample costs
$1. In addition, the subjects are permitted to practice the search task for an unlimited number of
times during a certain time period before going into the actual paid tasks. They are paid for their
earning in one of the 10 or 11 search tasks One distinguishing feature of this dataset is that it is one
of the few in which the price sequences are randomized across tasks and across individuals, which
gives us more estimation power. In total, there are 673 individual searches, and 3,408 stop-and-go
decisions in the dataset. Each stop-and-go decision will serve as one observation in our estimation.
6
We thank Prof. Daniel Schunk from University of Zurich for generously o¤ering to make his experimental data
available to our use.
7
See Schunk & Winter (2005) for more details regarding the dataset.
22
But …rst, we will o¤er some descriptive analysis related to the question: Do people search too little
(yet again!) in our dataset?
3.1.1
Do subjects search too little?
In order to see if subjects in the experiment follow optimal search reasonably well or otherwise,
Figure 4 depicts the distribution of actual number of searches in bars against the density of number
of searches if an optimal rule is followed. In the latter case, the optimal stopping price will be
491, and the number of searches would exhibit a geometrically declining pattern. In 134 of the
673 tasks, buyers accept the …rst price that they obtained and in 95 of them they searched for
just two prices. In contrast, on a few occasions, buyers make as many as 30 or 40 searches, which
are clearly unpro…table. The di¤erence between actual and optimal searches are apparent from
Figure 4: exhibited search behaviors are concentrated more on shorter searches (t < 7) and less on
longer searches (t 7) than what is optimal. Similarly, under optimal risk-neutral assumption, the
expected number of prices searched would be 6.26. The actual number of prices searched is 5.06 on
average. So here buyers search for roughly one fewer price. The expected payo¤ from each search
under optimality is 8.53. The actual average payo¤ is 5.71, an average e¢ ciency loss of 33%.
Figure 4
23
It is also interesting to look for any pattern of convergence or learning among subjects over
time. Figure 5 presents the average and median number of price searches across the 64 subjects
over number of tasks completed. Neither evidence of convergence nor of learning is discernible from
the picture. The number of searches is persistently lower than the optimal level up to the …nal
task. Median number of search is even lower due to a few outliers who go through lengthy and
costly searches. E¢ ciency of actual search does not improve with experiences either. It goes from
58% in Task 1 to 77% in Task 11 with no consistent upward trend (Figure 6).
Figure 5
24
Figure 6
3.2
Estimation
In our behavioral model, agents in a sequential search task optimizes based on a utility function
that incorporates both monetary payo¤ and the anticipation of the psychological a¤ect of regret and
rejoicing. Consequently, a search agent’s reservation price is in‡uenced by her own sensitivities to
regret and rejoicing in a manner predicted by the model. That is, she would search less aggressively
if she tends to worry more about not obtaining a good price and therefore wasting her search costs.
Conversely, anticipated rejoicing would motivate her to search more aggressively.
In the last section we also propose that the intensity coe¢ cients t and t are not time invariant.
Instead, they are functions of the history of prices that the searcher has encountered in the current
search period. t = (p1; p2 ; :::; pt ); t = (p1; p2 ; ::::; pt ) The logic here is simple: past regrets from
observing unfavorable prices would reinforce current regret intensity t ; and past rejoicing from
observing favorable prices would reinforce current rejoicing intensity t . In turn, reservation price
becomes a function of past experiences of regret and rejoicing through t ; t . This statement forms
our main testable implication from the model.
This also separates a regret/rejoicing driven search from a risk-aversion driven search: Although
both are able to predict less search than that in a risk-neutral model, reservation price is supposedly
held constant throughout the search process for a risk-averse searcher (it’s hard to imagine one’s
risk attitude changes as the search unfolds) whereas a subject anticipating regret and rejoicing holds
a shifting reservation price based on the realization of prices. In addition, because the distribution
25
of observed prices is truncated by the …nal accepted price, searchers are more likely to come across
unfavorable prices during the search than favorable prices, resulting in higher frequency of regrets
than rejoicing. In response, reservation prices will often change in a way that makes recall a tenable
option as the search goes on.
3.2.1
De…ning Measures of Experienced Regret/Rejoicing
In order to test whether the reservation prices move with experienced regret and rejoicing, we’ll
operationalize the regret/rejoicing reinforcing functions t = (p1; p2 ; :::; pt ); t = (p1; p2 ; ::::; pt ) in
the following way. First, we de…ne two possible types of backward regret.
In a framework where a buyer is trying to purchase a good value at V at the lowest possible
price by comparison shopping, let LT represents the lowest price of all the ones the buyer has
obtained up to the current decision period T ; i.e. Lt = min fp1; p2 ; :::; pt g: If the buyer stops
search now, her payo¤ would be T = V LT T c: Her counterfactual payo¤, had she chosen to
stop search last period, would have been T 1 = V LT 1 (T 1) c: If T 1 > T ; she would
feel a sense of regret about continuing the search into this round. Note this preceding condition is
equivalent to LT 1 LT < c; indicating whether the last round of search has paid o¤ or not. This
forms our …rst measure of regret, which is called one-step regret. It is either 1 or 0 in our dataset
due to the fact that the search cost c is equal to 1.
One-Step Regret:
reg_1_stepT
= M axf
T ; 0g
T 1
= M ax f(V
= M axfLT
LT
LT
(T
1
1
(34)
1) c)
(V
LT
T
c) , 0g
+ c; 0g
(35)
(36)
Analogously, to evaluate the merit of her decisions, a buyer can also compare her current
payo¤ against the highest (counterfactual) payo¤ she could have earned, by stopping earlier at
any of the previous rounds from t = 1 to T
1. This counterfactual payo¤ can be computed
as max V
Lt t c. So if the current payo¤ is lower than this counterfactual payo¤, the
1 t T 1
buyer also feels a sense of regret about continuing the search into the current round. We will call
this multi-step regret. The distinction between one-step regret and multi-step regret helps us learn
against which reference outcome the subject evaluates her payo¤ in the decision making process.
Multi-Step Regret:
reg_t_stepT = M axf M ax (V
1 t T 1
Lt
26
t c)
(V
LT
T
c) ; 0g
(37)
Two measures of backward rejoicing can similarly be developed. They are positive whenever
current payo¤ is favored over the counterfactual payo¤s which corresponds to a sense of rejoicing
over the decision to search to the current round.
One-Step Rejoicing:
rej_1_stepT
= M ax { (V
= M axfLT
LT
1
T
LT
c)
[V
LT
(T
1
1) c], 0g
c; 0g
(38)
Multi-Step Rejoicing
rej_t_stepT = M axf(V
LT
T
c)
M ax (V
1 t T 1
Lt
t c) ; 0g
(39)
According to how the terms are de…ned, the following properties can easily be veri…ed to hold
amongst there four measures:
reg_i_step > 0 ) rej_i_step = 0, i = 1 or t
rej_i_step > 0 ) reg_i_step = 0; i = 1 or t
(40)
rej_t_step > 0 ) rej_1_step
(41)
reg_1_step > 0 ) reg_t_step
3.2.2
reg_1_step > 0
rej_t_step > 0
Postulated Relationship between Reservation Price and Regret/Rejoicing Measures
With all the backward regret and rejoicing measures, the intensity functions can be expressed more
concretely as functions of these measures. Moreover, it’s natural to assume that all the partial
derivatives here are positive:
t
=
(reg_1_stept ; reg_t_stept ); and @ t =@reg1t > 0; @ t =@regtt > 0
(42)
t
=
(rej_1_stept ; rej_t_stept ); and @ t=@rej1t > 0; @ t =@rejtt > 0
(43)
Moreover, inserting Equation (42) and Equation (43) into the reservation price function LRR;t =
LRR (c; F; t ; t ) (in a purchasing context), we discover the following relationship hold between
reservation price and the regret/rejoicing measures.
27
LRR;t = LRR (c; F; (rej_1_stept ; rej_t_stept ); (reg_1_stept ; reg_t_stept ))
=
(reg_1_stept ; reg_t_stept ; rej_1_stept ; rej_t_stept ; c; F )
(44)
The following …rst-order partial e¤ects from Equation(44) o¤er some very interesting testable hypotheses:
@LRR;t
@reg_i_stept
@LRR;t
@rej_i_stept
3.3
@LRR;t
=
@ t
> 0; for i = 1; t
@reg_i_stept
(+)
@ t
< 0; for i = 1; t
@rej_i_stept
(+)
@ t
(+)
@LRR;t
=
@ t
( )
(45)
(46)
Latent Reservation Price Estimation
Every continue-or-stop decision in each of the 673 price search sequences can be represented by a
vector (Yij1; :::; YijT ) = (0; :::; 0; 1); with Y =0 denoting search and Y =1 denoting stopping. Search
tasks in which the buyers accept the …rst price (134 of them) will be excluded from the estimation,
leaving us with a total of 2735 observations. For simplicity, assume the latent reservation price at
each decision point of the search LRR;t has a simple linear relationship with all the likely covariates
in the decision process, including all the regret and rejoicing measures as in the following equation (We are going to suppress the subscript RR from this point on unless a digression from the
regret/rejoicing model is explicitly pointed out) :
Lijt =
0
Xijt +
0
ijt
+
1
ijt
+
2 Searchijt
+ Ii + "ijt
(47)
in which,
i = 1; ::::; 64 indexes subjects participating in the search tasks;
j = 1; :::::; 11 indexes the search tasks;
t = 1; :::::; T indexes the search rounds;
Xijt = (reg_1_stepijt ; reg_t_stepijt ; rej_1_stepijt ; rej_t_stepijt )0 is the vector of regret/rejoicing
measures;
ijt is a vector of other factors that may a¤ect search such as subject’s total earning in the
experiment at t;
ijt is subject’s payo¤ in the current search if she stops now;
searchijt is the number of prices encountered so far in the current search;
28
Ii ’s are individual-speci…c dummies, re‡ecting subject’s innate di¤erences in a variety of factors
that in‡uence the search outcomes, including but not limited to di¤erences in their risk attitudes
and regret/rejoicing intensities and abilities to process the task at hand;
"ijt are i.i.d error terms, distributed N ormal(0; ! 2 ):
The estimation is carried out via Maximum Likelihood in a way that is akin to a standard Probit
model. There are two kinds of possible outcomes, each implying a certain relationship between the
latent reservation price Lijt and the lowest price observed Lijt :
Pr(Yijt = 0) = Pr(Lijt > Lijt )
Pr(Yijt
= Pr(Lijt > 0 Xijt + 0
1
=
( (Lijt ( 0 Xijt +
!
= 1) = Pr(Lijt H1ijt )
ijt
+
1
0
ijt
+
ijt
+
1
ijt
2 Searchijt
+
+ Ii + "ijt )
2 Searchijt
+ Ii ))
0
= Pr(Lijt
Xijt + 0 ijt + 1 ijt + 2 Searchijt + Ii + "ijt )
1
=
( ( 0 Xijt + 0 ijt + 1 ijt + 2 Searchesijt + Ii Lijt ))
!
where (:) denotes the CDF of standard normal distribution.
(48)
(49)
0 s.
The objective is to identify (b; b; b) that maximizes the probability of observing all the Yijt
(b; b; b) = arg max
P
log Pr(Yijt = 1) +
Y ijt=1
P
log Pr(Yijt = 0)
(50)
Y ijt=0
Equations (45)-(46) lead us to be particularly interested in testing if the following hypotheses
will hold in the estimation results:
1. Buyers’ reservation prices will be shifting through the search process, rather than staying
constant. This is true even during a single search sequence, as a result of experienced regret
and rejoicing. 8 If it holds, simple risk aversion cannot explain satisfactorily why people tend
to search too little. In addition, we have direct evidence that search is history-dependent.
2. In a purchasing context, buyer’s reservation price increases (i.e. search less aggressively) with
greater regrets and decreases (i.e. search more aggressively) with greater rejoicing.
3. Quantitatively, regret moves the reservation price more than rejoicing does.
8
To my knowledge, previous experimental studies have only examined the changes in reservation prices between
two consecutive searches by having the subjects directly report their reservations before the search begins. See
Sonnemans (1998) and Einav (2005).
29
4. The occurrence of recall is more frequent in search sequences where regrets are more present.
5. To the extent that satis…cing behavior exists, reservation prices should fall at roughly the
rate of c with each additional search.
3.4
Estimation Results
3.4.1
Descriptive Statistics
Table 1 below presents the summary statistics of the main variables of interest to be used in the
MLE estimations.
Table 1: Summary Statistics
Variables
Mean Std. dev.
Endogenous Variables
Last Round
0.20
0.40
Explanatory Variables
Search
6.42
5.53
Prices
499.73
9.81
Lowest Prices
493.10
6.28
Search Pro…t if stops now
0.48
6.76
Balance
41.21
31.70
Regret-not stopping last round
0.63
0.48
Regret-not stopping earlier
2.77
4.18
rejoicing-not stopping last round
2.76
5.44
rejoicing-not stopping earlier
2.34
5.17
Sample Size (N=2,735)
Note: 773 observations are deleted from the sample
in which the search sequence length equals 1.
Min
Max
0
1
2
467
467
-27
-14
0
0
0
0
40
535
519
31
149
1
32
38
38
After eliminating searches with stops after only observing the …rst price, 2,735 stop-or-go decisions from 539 individual searches remain in our sample. The number of searches in a task ranges
from 2 to 40. The random prices appear to follow the distribution very well. Search pro…t if
stopping now tracks buyer’s payo¤ from stopping at each of the decision point. In the event that
all the subjects in the experiment try to reach a certain aspired level of payo¤, this measure should
exert a very large impact on their decisions as to whether to stop or continue the search. The variable, balance, measures the subject’s total earning from previously completed search tasks which
is inserted here as another possible control variable.
30
The …nal four explanatory variables on the list have the preponderance of our interest. First,
notice that one step regret literally acts as a dummy variable due to the fact that if current round
payo¤ is worse than last round payo¤, the di¤erence will be exactly equal to the search cost c
that’s wasted. In 63% of the cases, the realized price is no better than the previous price, resulting
in one-step regrets. On the other hand, in roughly 35% of the cases, the buyer does discover
a better price than the last round, resulting in one step rejoicing. The magnitude of one step
rejoicing is sometimes much larger (the highest being 38). In roughly 28% of the cases, the buyer
discovers a much better price in the sense that her current payo¤ beats any counterfactual payo¤
had she stopped in any of the earlier rounds. This translates into a positive multi-step rejoicing.
Nevertheless, over 50% of times after she has obtained such a good price as de…ned by positive
multi-step rejoicing, the buyer continues to search for at least one more price, demonstrating a
tendency to become more aggressive while experiencing rejoicing. We’d also like to point out that
one step rejoicing and multi-step rejoicing are very close in values in our sample.
Regarding the question of when buyers decide to stop in the search process, our sample shows
that 73% of them ultimately decide to stop when getting a better price than the last round, of
which 85% also have a positive multi-step rejoicing when they stop. In 26% of the stopping rounds,
the buyer actually does not do as well as if she has stopped at one of the earlier rounds, of which
70% have a lower payo¤ than they had in the last round. The overall recall percentage in our
sample is 17.4%, which is similar to what’s found in previous experiments.
Table 2 reports the Pearson correlation matrix of all the major explanatory variables within
our estimation. All of the correlations in the table are found to be signi…cantly di¤erent from zero
at the 5 % level. In particular, the variable Number of Search has very positive correlation with
multi-step regret at 0.88. This may arise from the fact that buyers usually prolong their searches
confronted with a string of unfavorable prices which also results in large mutli-step regrets. Also,
it is noteworthy that the positive correlation between one-step regret and multi-step regret is 0.41
whereas the positive correlation between one-step rejoicing and multi-step rejoicing runs as high as
0.98. The pair of rejoicing variables seem to both arise from the discovery of a very low price in
the sequence. All of the regret and rejoicing variables are negatively correlated, as expected.
31
Table 2: Pearson Correlations between Independent Variables
Variables
Search
t
Balance
Reg_1
Reg_t
Rej_1
Rej_t
Note: All
3.4.2
Search
Balance Reg_1 Reg_t Rej_1
Rej_t
t
1
-0.492
1
-0.118 0.193
1
0.216 -0.388
-0.087
1
0.881 -0.586
-0.131
0.414
1
-0.205 0.524
0.103
-0.658 -0.317
1
-0.246 0.506
0.101
-0.589 -0.300 0.978
1
correlations reported here are signi…cantly di¤erent from zero at p=.05
Main Results
All of the succeeding estimations are run using Maximum Likelihood algorithm based on Equations
(48) through (50). The algorithm is very similar to a standard Probit estimation with the caveat
that throughout the estimations, the coe¢ cient on the variable current (lowest) price is …xed at
1:
Pr(Yijt = 1) =
( 0 Xijt +
0
ijt
+
1
ijt
+
2 Searchijt
+ Ii + ( 1) Pijt )
where Pijt denotes the lowest price including the current period price.
32
(51)
Table 3: Maximum Likelihood Estimation
(1)
Ind. variables
prelim.a
coef.
Std. Err.
Const.
491.10**
0.48
Reg: 1 step
5.05**
0.77
Reg: t step
-0.26**
0.05
Rej: 1 step
-0.42**
0.12
Rej: t step
0.34**
0.12
Search
t
of the Latent Reservation Prices
(2)
(3)
Complete b
w/ Spline Fn.c
coef.
Std. Err.
coef.
Std. Err.
494.24**
1.77
493.72**
1.89
4.02**
0.69
5.07**
0.59
0.08
0.16
0.03
0.19
-0.14**
0.04
-0.12*
0.05
-0.41**
-0.13
0.00
0.06
0.12
0.10
0.13
0.03
0.23
0.09
-0.35**
-0.11
0.00
0.03
0.13
0.13
0.03
0.23
Balance
Task
Search*Reg1step
Reg: t step-10
0.33
0.26
Reg: t step-20
-0.62
0.42
Reg: t step-30
9.42**
1.55
Rej: 1 step-10
0.09
0.12
Rej: 1 step-20
-0.77*
0.32
Sample Size
2,735
2,735
2,735
Notes:
a :Error terms clustered at the subject level.
b : 63 individual dummies inserted. Coe¢ cients on dummies not shown here.
c : 63 individual dummies inserted. Spline functions have knots at 10,20,30.
a;b;c : robust std err applied. **: signi…cant at 1% level. * signi…cant at 5% level.
Table 3 reports the main results from the estimations. The speci…cation in the estimation has
enabled us to interpret the coe¢ cients in the result as unit changes in the latent reservation prices
given marginal changes in the set of independent variables. Column (1) presents a preliminary
estimation by including only the regret and rejoicing variables. The result shows that all of these
four measures have signi…cant e¤ects on deciding the reservation price for the buyer. Moreover,
one step regret and one step rejoicing are found to shift the reservation price in the directions
predicted by our model. Most striking of all is the very strong quantitative e¤ect of one step regret
in making the buyers revise their reservation prices upwards by as many as 5.05, which is a very
sizable adjustment given that the standard deviation of the price distribution is 10. Compared to
one step regret, multi-step regret, one step rejoicing and multi-step rejoicing all seem to be much
less relevant in the decision, although keep in mind that the average values of these measures are
much larger than one step regret so the overall e¤ect needs …ner calculation. Finally, since one
step rejoicing and multi-step rejoicing are so highly correlated and their coe¢ cients in (1) are of
33
opposite signs, it is worthwhile to ask if they are jointly signi…cant. A likelihood ratio test soundly
rejects the null hypothesis that they are jointly insigni…cant (p=0.0012).
Column(2) presents the full-‡edged version of the estimation. It inserts an additional set of
control variables into the estimation, including task number, search rounds, current search payo¤,
total earning balance, and an interaction term between search and one-step regret in order to test
whether regret matters more in longer search sequence. Besides, individual dummies are added
to control for subject level variations in determining reservation price that are unrelated to the
observed prices. Multi-step rejoicing is omitted from the estimation to avoid any collinearity issue
with one-step rejoicing.
One-step regret remains very signi…cant in the complete version of the estimation. Other things
equal, the regret generated by seeing this round’s search to be less pro…table than not searching at
all will trigger the buyer to revise her reservation price upwards by 4.02. At the average values of
other covariates, the marginal e¤ect of one step regret will raise the probability of stopping in the
next round from 0.18 to 0.31, a 72 percent increase. Mutli-step regret raises the reservation price
as well, although not very signi…cantly. Rejoicing is found to signi…cantly reduce one’s reservation
price, yet the marginal e¤ect is much smaller in scale. All these are in line with the behavioral
model’s predictions. In the meantime, a risk-aversion based story of sequential search appear all
but dead, because the constant reservation property does not appear to hold at all.
In Column (2) current search pro…t is not a signi…cant predictor of one’s reservation price,
contradicting the premise of a satis…cing model in which search agent wants to reach certain aspiration payo¤ level. If higher payo¤ will make one more likely to quit the search, we would expect
a positive coe¢ cient on t: The other possibility is that if we assume heterogeneous aspiration
levels among subjects, the higher one’s aspiration payo¤ is, the more aggressively she would have
to search, which would also create results that appear similar to Column (2). Therefore, it is too
early to completely discount the "satis…cing" explanation of too little search. In the next section,
we’ll run a direct horserace to demonstrate why regret matters more than satis…cing behaviors.
Finally, the coe¢ cient on the number of search turns out to be signi…cantly negative, implying
that buyers become more aggressive searchers the longer the search goes on . This directly contradicts a type of behavioral strategies that some experimenters have previously thought would apply
in this setting, which is "keep searching until the number of searches exceeds XX". In contrast, the
negative relation between number of searches and reservation price tends to lend support to either
an escalating commitment story (i.e. keep searching until one …nds some favorable price that would
justify her lengthy search) or a satis…cing story in which one’s reservation price falls proportionally
with more searches. On the other hand, the variables, task, balance, and the interaction between
one-step regret and search do not appear to be important here.
34
Since people seem to be more concerned with one-step regret than with multi-step regret whose
values come from a much wider range, the reason that the e¤ects of multi-step regret or rejoicing
appear to be weak could possibly be attributed to people’s increasing insensitivities towards larger
values of regret or rejoicing. In other words, the curvatures of the regret/rejoicing intensity functions
may matter also. Consequently, in Column (3) we divide multi-step regret and multi-step rejoicing
into several numerical intervals and institute a spline function on each of them. For example, the
variable, Reg: t step-10, subtracts 10 from the original value of multi-step regret (if >10) and helps
to test whether there is some additional change in slope once the value of multi-step regret exceeds
10. In the results, we discovered: First, adding the spline functions do not change the e¤ects of
remaining variables. Secondly, regtstep_10, regstep_20 as well as rej1step_10 do not appear to
have signi…cant impact on the search process. In other words, intermediate values of regret and
rejoicing do not change the reservation price much. However, once the regret value exceeds 30 and
the rejoicing value exceeds 20, their e¤ects grow much stronger and signi…cant. In particular, one
unit increase in regret once it exceeds 30 will raise the reservation price by 9.42, literally halting
the search right away. The implication is simple: People really hate very extended searches, and
perhaps they are prepared to give up immediately once they have drawn many disappointing prices
(even though the underlying price distribution has not changed a bit!).
3.5
Robustness Checks
There are two main types of arguments that could cast into doubt whether the revisions of reservation prices are truly caused by concerns for regret and rejoicing in the sequential search task, one
behavioral and one statistical.
First, whether or not the subjects in the experiment truly trust that prices are drawn randomly
from the distribution they are told or whether or not they have a basic comprehension of the
underlying (truncated normal) distribution could be an issue. In either of the two scenarios, subjects
will be undertaking some extent of learning of the price distribution while searching for the best
price. In that case, regrets (rejoicing) that are associated with unfavorable (favorable) price draws
would also signal to the buyers of an unfavorable (favorable) distribution. Consequently, it is only
natural for the buyers to decide to search less (more) extensively. Bounded rationality on the
subject’s part causes a possible spurious relation between regret/rejoicing and the reservation price
in the observed manner.
Secondly, if due to the computational demands of the search task, people are unable to correctly
pinpoint the optimal reservation price. Sometimes they set it too low; sometimes they set it too
high. Facing the same (unfavorable) price sequences, those who set the reservation too low will
always undertake longer searches than those who set it too high, resulting in a higher frequency
of regrets based on higher expenditure on search costs. These subjects will also have tendency
35
to adjust their reservation upwards over time when they learn to correct the computation errors.
Again, computational de…ciency together with statistical reasons causes a possible spurious relation
between regret/rejoicing and the reservation price in the observed manner.
In order to clarify these two issues, we run three additional robustness checks in the hope of
corroborating with the basic …nding.
1. Prior to the start of the …rst paid search task, each subject in the experiment is allowed
to practice freely with the software within a certain time limit. They can generate from zero up
to an unlimited number of price searches during this time without any concerns for committing
mistakes. We tabulated the number of prices drawn by each subject from the same distribution as
in the later paid tasks. The numbers di¤er widely, ranging from a low of 7 prices to a high of 756
prices. The median number of prices searched during the trial period is 80.5. So for anyone who
has seen at least 80 price draws during the trial, familiarity with the distribution should not be an
issue. In Robustness Check (1), only subjects who have made above median number of free price
searches during the practice will be included in the re-estimation.
Figure 7:
2. Experimental subjects’concerns about the distribution and their inability to …nd the
optimal reservation prices should be mitigated with more search tasks completed. An average
subject would have observed around 30 price draws by the time she …nished half of the search
tasks. As a result, we are going to repeat the estimation with data from Task 6 to Task 11 to see
if the results hold (Robustness Check 2)
3. We are also going to separate the data into two groups: subjects who have recalled at
least once through the 11 tasks, and subjects who have never recalled. If search sequences in which
recalls happen in the end are more likely to identify those who set reservations too low and who
stumble upon bad price draws (See Figure 8 for an illustration of this point), estimating exclusively
36
on the group who has never recalled would be instructive as to whether the relationship between
reservation price and regret/rejoicing is barely statistical or not (Robustness Check 3).
Figure8
Table 4: Results from the three robustness checks
Ind. variables
Const.
Reg: 1 step
Reg: t step
Rej: 1 step
Search
t
(1)
Practices>80.5 a
coef.
Std. Err.
492.72**
1.96
4.19**
0.98
0.07
0.23
-0.14**
0.05
-0.28*
0.13
-0.04
0.14
0.02
0.04
0.01
0.30
0.13
0.12
1,419
(2)
Task>5 b
coef.
Std. Err.
494.48**
3.51
4.83**
1.09
0.01
0.30
-0.19**
0.07
-0.25
0.19
0.00
0.23
0.08
0.07
-0.45
0.23
0.12
0.09
1,401
Balance
Task
Search*Reg1step
Sample Size
Notes:
a;b;c : 63 individual dummies inserted. robust std err applied.
**: signi…cant at 1% level. * signi…cant at 5% level.
37
(3)
Never Recalled c
coef.
Std. Err.
496.58**
0.50
1.69*
0.87
0.03
0.08
-0.02
0.03
-0.45**
0.09
-0.15
0.11
0.02
0.02
-0.16
0.14
0.12
0.09
1,066
All the three robustness tests clearly con…rm that the strong relationship between buyer’s
reservation price and one-step regret is neither the result of behavioral uncertainty regarding the
price distribution nor the result of sheer statistical coincidence. All of the quantitative e¤ects
resemble very closely those derived in the main estimation. In particular, in the …rst two subsamples
containing experienced subjects, buyers are found to adjust their reservation prices upwards by
roughly $4 once they experience a one-step regret. In the third subsample, buyers who never recall
a past price would adjust reservation price upwards by a slightly smaller amount, nevertheless, the
e¤ect is still robust at 5 percent level.
4
4.1
Discussions and Extensions
Why risk aversion does not explain "too little search"?
One of the most publicized explanations in the present literature to explain "too little search"
is based on risk aversion in the search agent’s utility function. In essence, risk averse searchers
view each search decision as a gamble, trading o¤ potential gains from …nding a better price
against a safer option which is to save on the search cost. The more risk averse one person is, the
more conservative her search pattern will be. However, as the early discussion in this paper has
pointed out, regardless of the degree of risk aversion, there should be no dynamic aspects to the
search problem at all. Basically, a constant reservation price prevail throughout the search process.
However, this is what we have observed in the current dataset. On the contrary, it is found that the
subject’s reservation price during the search ‡uctuates in a predicted manner with her experienced
regret and rejoicing. And this could explain why subjects search too little in connection with our
theory incorporating anticipated regret and rejoicing into the search.
Other authors have also raised questions about the plausibility of risk aversion in explaining
search. Schunk and Winter (2005) in their paper explicitly compared two sets of risk coe¢ cients: the
…rst set is elicited through a pre-search questionnaire directly surveying subject’s risk preferences;
the second set is inferred from their actual search behavior assuming risk aversion is the only driver
of search. Surprisingly the two sets of risk coe¢ cients are found to have very little correlation.
Sonnemans (1998) also provided evidence that rejects risk aversion in explaining actual search
behaviors. His argument is that if risk aversion is interpreted as a willingness to accept lower
average payo¤ in exchange for lower variability of payo¤, then pure risk aversion is only able to
explain about 20 percent of the situations where people search too little.
38
4.2
4.2.1
Three-way Horserace between Regret/Rejoicing, Risk Aversion, and Satis…cing Models
Methodology:
A model’s explanatory power ultimately rests on its ability to predict actual behaviors. Of the three
possible models we have to explain too little search, which one could predict the actual pattern of
search the best? A three-way horserace among them helps reveal this point:
First, our set of candidate strategies contains the range of all plausible satis…cing strategies,
the range of all plausible reservation price strategies re‡ecting pure risk aversions, as well as the
strategy of our principle interest, the moving reservation price strategy based on our estimation
result from the last section. Speci…cally, the predicted reservation prices used in the horserace are
predicted values from the main estimation equation (51).
Let us use the example of one particular experiment subject so as to illustrate how the horserace
is run. For instance, Subject 33 in the experiment performed a total of 11 separate searches. For
any one of those 11 searches, if a search strategy predicts the same set of stop or continue decisions
as what were made by the subject, then we’ll say the strategy predicts the search, or otherwise,
the strategy fails to predict the search. Between a pair of strategies, the one that predicts a higher
proportion of a subject’s total searches is said to win the horserace for that subject. Here is how
one possible satis…cing strategy "Satis…ed7 "(i.e. stop whenever current search payo¤ exceeds $7),
one possible risk-aversion strategy "RP490" (i.e. hold a constant reservation price equal to 490),
and the Regret/Rejoicing predictions "RR" perform against each other in the horserace for Subject
33.
Table 5: Horserace Example (Subject 33)
Candidate Strategy RR Satis…ed7 RP490
Predicts
9/11
5/11
7/11
So RR wins the three-way race for Subject 33.
Table 6 showcases the outcomes of the horserace run between the regret/rejoicing model and the
whole range of plausible satis…cing strategies. To reiterate, in the regret/rejoicing model, a subject
stops the search whenever the randomly received price is lower than the predicted reservation
prices estimated from Equation (51). In the satis…cing model, a subject stops the search whenever
the randomly received price results in a total search payo¤ in excess of a level prescribed by the
strategy. The table makes a forceful case that the regret/rejoicing model performs a much better
job in describing the data than the satis…cing model. The former beats every conceivable satis…cing
strategy in terms of predicting correctly a higher proportion of search outcomes for more individuals
in the sample. Even if we allow each individual to have di¤erent aspiration level and correspondingly
pick the best satisfying strategy for every subject, the regret/rejoicing model still wins the horserace
(see the last column in the table).
39
Table 6: Regret/Rejoicing Predictions vs. All Satis…cing Strategies (S2-S15)
Satis…cing
Strategies
S2
S3
S4
S5
S6
S7
Wins
43
43
46
44
43
49
Ties
10
9
9
12
12
7
11
12
9
8
9
8
Losses
Net W32 W31 W37 W36 W34 W41
Note: 1) out of a total of 64 subjects.
2) S-Best assumes individual heterogeneity by selecting
individual respectively.
S8
56
4
4
W52
S9
59
3
2
W57
S10
60
2
2
W58
S11
62
0
2
W60
S12
62
1
1
W61
S-Best
24
20
20
W4
the best satis…cing strategy for each
Similarly, the horseracing results between the regret/rejoicing strategy and the whole range
of risk aversion strategies are listed in Table 7. Various degrees of risk aversion is mapped into
constant reservation price strategies with various cuto¤s. Once again, the regret/rejoicing model
predicts searches better in more individuals in the data than any constant reservation price strategy
when applied to the whole sample. When heterogeneity in risk attitudes is allowed and for each
individual we select the best constant reservation price as her strategy, the regret/rejoicing strategy
does fall behind in its predictive power (see the last column in Table #).
Table 7: Regret/Rejoicing Predictions vs. Risk Aversion Strategies (RP488-RP498)
Constant RP RP
RP
RP
RP
489
490
491
Strategies 488
Wins
45
34
33
34
12
20
18
17
Ties
Losses
7
10
13
13
Net W38 W24 W20 W21
Note: 1) out of a total of 64 subjects.
2) S-Best assumes individual heterogeneity
for each individual respectively.
RP
492
31
17
16
W15
RP
493
28
23
13
W15
RP
494
33
17
14
W19
RP
495
34
15
15
W19
RP
496
46
10
8
W38
RP
497
51
3
10
W41
RP
498
52
6
6
W46
by selecting the best constant reservation strategy
Overall, the horserace results suggest clearly that the data in themselves favor the risk/rejoicing
explanation over the risk aversion and the satis…cing behavior explanations.
for any one of the 673 random price sequences in the dataset, if a particular search strategy is
able to predict the exact same set of stop-or-go decisions as is actually made by the subject, that
search strategy will score one point for that individual on that particular search. An individual will
appear to favor one of any two competing search strategies more frequently if that search strategy
40
RP
Best
2
19
43
L41
score more points than the other one among the 10 or 11 search tasks that the individual conducted
in the data. Consequently a strategy that’s favored by a higher proportion of 64 total subjects will
win the head-to-head horserace with another strategy. In particular, we are interested in examining
if our regret/rejoicing model which possess a reservation price structure as estimated in equation
(51) can defeat both the risk aversion and the satis…cing search strategies.
5
Conclusion
This paper tries to answer the question why people in the lab have a tendency to search too little
compared to the risk-neutral benchmarks by o¤ering an entirely new perspective: anticipatory
regret and rejoicing could be a prominent factor here. In addition to deriving this analytically by
embedding the original regret theory into the search model, we also show that the regret/rejoicing
explanation organizes the experimental data better than risk aversion and satis…cing behavior.
However, one issue that we shun away from is whether people do use a search strategy of the
reservation form, or they use some other bounded rational rules. Nor do we have the con…dence to
claim that anticipated regret/rejoicing is the only factor operating here.
Our theory stresses that it is because of the feedback information structure of the sequential
search problem that makes regret and rejoicing not only asymmetric but also one-sided. If people
can anticipate these future feelings correctly and if regret dominate rejoicing, people will end
up searching too little. Empirically, both our anticipatory regret/rejoicing model and a simple
learning rationale put forward by Sonnemans (1998) and Einav (2005) will deliver similar results;
nevertheless, Proposition 2 predicts that by modifying the information ‡ow in the search task,
the two motives can be disentangled. More speci…cally, if revealing one more price following the
end of search can e¤ectively induce subjects to search more aggressively, then it must be the
anticipation of future regret and rejoicing that a¤ect people’s reservation prices. More strongly,
Proposition 2 predicts that in the counterfactual search environment, people’s behaviors should be
path-independent, and recall should never happen. How much of the gap between observed search
behaviors and e¢ cient search can be closed by a change in the information ‡ow should be subject
to future experimental investigation.
Finally, it is worth highlighting that the regret/rejoicing model has rich empirical implications
in the marketplace. For example, search is going to be closer to optimum in markets in which
consumers are more likely to obtain post-purchase information. The di¤erences between consumer’s
accessibility towards post-purchase price information can occur naturally or by government decree,
such as local markets vs. online markets, states with or without banned advertising on certain goods.
Policies that enhance post-purchase price transparency should induce more e¢ cient consumer search
in the …rst place. Second, the "price-matching" clauses that a lot of retailers use could also mitigate
41
anticipatory regret and therefore lead to few searches and presumably higher market power by …rms
that implement these marketing tactics.
42
REFERENCES
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Cox, J. and R. Oaxaca. 1996. "Testing Job Search Models: The Laboratory Approach".
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Einav, Liran. 2005. "Information Asymmetries and Observational Learning in Search". The
Journal of Risk and Uncertainty, Vol. 30, No.3, pp: 241-250
Filiz, E. & E. Ozbay. 2007. "Auctions with Anticipated Regret: Theory and Experiment".
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Hey, John. 1982. "Search for Rules for Search". Journal of Economic Behavior & Organization. Vol 3, pp: 65-81.
Hey, John. 1987. "Still Searching". Journal of Economic Behavior & Organization. Vol 8,
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Houser, D. and J. Winter. 2004. "How do Behavioral Assumptions A¤ect Structural
Inference? Evidence from a Laboratory Experiment." Journal of Business and Economic Statistics.
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Kogut, C. 1990. "Consumer Search Behavior and Sunk Costs." Journal of Economic Behavior
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Loomes, G. & R. Sugden. 1982. "Regret Theory: An Alternative Theory of Rational Choice
Under Uncertainty" The Economic Journal, Vol. 92, No.368, pp:805-824.
Lippman, S. and J. McCall. 1976. "The Economics of Job Search: A Survey". Economic
Inquiry. Vol 14, pp 155-189.
Rothschild, M. 1974. "Searching for the Lowest Price When the Distribution of Prices is
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43
Schunk, Daniel. 2008. "Sequential Decision Behavior with Reference Point Preferences".
working paper, University of Zurich.
Schunk, D. & J. Winter. 2009. "The Relationship Between Risk Attitudes and Heuristics
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44
A
Proof of Proposition One
Lemma 1 L0 (H) < 0
This is intuitive. Not only does the monetary return to search decreases with H; but also
the likelihood and the extent of regret increase with H while the likelihood and
the extent of
R H+c
0
0
rejoicing decrease with H: More concretely, we have L (H) = G (H) cf (H)
[ H 1f (x)dx
cf (H)]+
Rb
H+c (
1)f (x)dx = [F (H)
1]
[F (H + c)
F (H)]
[1
F (H + c)]: Note that all the
three terms in the preceding expression are negative which coincides with our previous intuition.
Hence L0 (H) < 0 throughout its domain [a; b]: Furthermore, since in order to make the question
interesting it must be the case that L(a) > 0; L(b) < 0; lemma 1 thus implies that a seller, in
anticipation of regret and rejoicing, should once again adhere to a unique reservation price strategy
with the reservation price equal to H1 (c; F; ; ) which solves L(H1 (c; F; ; ) ) = 0:
We can further simplify the three components in the expressions for L(H) :
1) Monetary Payo¤:
Z
b
(x
H)dF (x) c
Z b
=
xdF (x) H dF (x) c
H
H
Z b
= xF (x)jbH
F (x)dx HF (x)jbH
H
Z b
= b H c
F (x)dx
H
Z b
c
(52)
H
2) Anticipated Regret:
cF (H)
Z
H+c
(c + H x)f (x)dx
Z H+c
Z H+c
[(c + H)
dF (x)
xf (x)dx]
H
H
Z H+c
[ cF (H) +
F (x)dx]
H
=
cF (H)
=
cF (H)
Z H+c
=
H
F (x)dx
(53)
H
45
3) Anticipated Rejoicing:
Z
b
(x
c
H+c
=
(c + H)
Z
H)f (x)dx
Z
b
dF (x) +
(c + H)(1
=
(b
c
xf (x)dx
H+c
H+c
=
b
F (H + c)) +
Z
H)
Z
[xF (x)jbH+c
b
F (x)dx]
H+c
b
F (x)dx
(54)
H+c
In aggregate, we now have
L(H) = [b
H
Z
Z
b
F (x)dx
c]
H
Z
H+c
F (x)dx
(c + H) + b
H
b
F (x)dx
(55)
H+c
Ultimately our goal is to diagnose whether and how concerns for regret and rejoicing will drive
the seller to revise her search strategy relative to the case without those concerns. Because it is
di¢ cult to derive the precise explicit solutions for H0 and H1 ; a direct comparison between the
two cut-o¤ points is unavailable. To circumvent this problem, an alternative strategy is devised
in which we examine the value of L(H) evaluated at H0 : Whether L(H0 ) > 0 (< 0) will provide
su¢ cient evidence as to whether H1 lies to the left or right of H0 ; To that end, we …rst utilize the
fact that H0 is the implicit solution to the equation G(H0 ) = 0. That is,
G(H0 ) =
Z
Z
b
(x
H0 )dF (x)
H0
c=b
H0
b
F (x)dx
c=0
H0
Hence,
L(H0 ) = G(H0 )
Z
=
H0 )
=
=
(b
(b
(
H0 +c
F (x)dx
H0
c
c
)
Z
(c + H0 ) + b
H0 +c
F (x)dx
H0
H0 )
(
)
Z
F (x)dx
Z
b
H0 +c
F (x)dx
b
F (x)dx
H0 +c
F (x)dx
H0
H0
H0 +c
Z
[
Z
Z
H0 +c
F (x)dx]
H0
(b
c
H0 )
(56)
H0
46
The last equation shows that if
>
> 0, then L(H0 ) < 0: Then it must be true that
a < H1 < H0 because L(a) > 0; L0 < 0: On the other hand, if 0 < < ; then it must be true
that H0 < H1 < b: If
= ; then H1 = H0 :
47