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Validating Models for Bounded Rationality and DecisionBiases in a Newsvendor Experiment
Li Chen • Andrew M. Davis
The Fuqua School of Business, Duke University, Durham, NC 27708, USAJohnson Graduate School of Management, Cornell University, Ithaca, NY 14853, USA
[email protected] • [email protected]
[Preliminary Draft]
February 3, 2014
A number of experimental newsvendor studies have shown that human subjects make decisionswhich coincide with a “pull-to-center” effect. In general, three behavioral theories have been pro-posed in the literature to explain this behavior: bounded rationality, mean anchoring, and mini-mizing ex post inventory regret. Unfortunately, these models generate similar predictions undersymmetric demand distributions commonly used in newsvendor experiments, such as uniform ornormal, and have thus remained as three equally plausible models for newsvendor decisions. Inthis study, we employ a simple experimental design with asymmetric demand distribution, whichallows us to directly tease out these three explanations. Our results suggest that bounded ratio-nality is useful in explaining variability in decisions, but it is inconsistent with average choices.We follow this by incorporating the mean anchoring and ex post inventory regret biases into thebounded rationality model, and find that the mean anchoring bias model cannot account for de-cisions. However, we do observe that the ex post inventory regret bias, when incorporated withbounded rationality, can consistently explain newsvendor decisions.
Key words: Behavioral Operations, Newsvendor Model, Bounded Rationality, Mean Anchoring, ExPost Inventory Regret
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1. Introduction
Managers often face operational decisions involving uncertain demand scenarios. For example, a
fashion buyer must forecast whether a particular product will be a hit or not in the upcoming
season, and determine how much to order from their supplier (Fisher and Raman 1996, Eppen and
Iyer 1997). Similarly, to meet future market growth, a plant manager must project different demand
scenarios, and decide how much to invest in production capacity (Van Mieghem 1998). A common
modeling framework for addressing these problems is the newsvendor model. The optimal solution
to the newsvendor problem, which maximizes the expected profit payoff, is frequently referred to as
the “critical fractile” point of the demand distribution. However, experimental studies show that
human decision makers tend to deviate from the optimal solution in the newsvendor problem from
period to period, and that they exhibit a “pull-to-center” effect, stocking quantities somewhere
between the critical fractile prediction and mean of demand. These deviations appear to be driven
not only by random errors, but decision biases as well (Schweitzer and Cachon 2000).
There exist three equally plausible theories to account for the newsvendor decision behavior
in the literature: bounded rationality, mean anchoring, and minimizing ex post inventory regret
(Su 2008, Schweitzer and Cachon 2000). The first model captures random errors in decisions,
whereas the last two model newsvendor context-specific biases. Despite the plethora of studies
focusing on newsvendor experiments, no work has successfully allowed one to directly compare
these three competing theories. Instead, all three models have remained as plausible models that
can predict newsvendor decisions. Understanding which biases are at play is important so that
one can begin to determine how to mitigate any sub-optimal behavior. In this paper, we present
experimental studies that allows one to separate and compare the three most common newsvendor
explanations.
Under the bounded rationality model framework, human decision makers are assumed to be
boundedly-rational, such that their decisions are subject to random errors, caused by cognitive
and computational limitations (the “trembling hand” effect). However, their decision choices still
follow a logit model with a theoretically appealing property—“better options are chosen more often”
(Su 2008, p. 569). In the first study of this paper, we conduct a simple newsvendor experiment to
understand how well bounded rationality captures decisions, and find a paradoxical observation—
people reveal their preferences for one option based on the expected profit, but actually choose the
other in the newsvendor problem. In other words, better options are not chosen more often in our
newsvendor experiment. This inconsistency suggests that random errors alone cannot account for
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newsvendor decisions; one must include newsvendor context-specific biases into the model.
Among the models for newsvendor context-specific biases, mean anchoring and minimizing ex
post inventory regret are the two most compelling ones, both receiving considerable support in past
studies (Schweitzer and Cachon 2000, Bolton and Katok 2008).1 However, in our second study we
find that, incorporating the mean anchoring bias into the bounded rationality model cannot account
for newsvendor decisions. On the other hand, incorporating the ex post inventory regret bias (with
an underweighting of leftover inventory) can consistently explain all newsvendor decisions. The
underweighting of leftover inventory effect in our experiment is also consistent with the data in the
previous studies by Schweitzer and Cachon (2000) and Bolton and Katok (2008).
Unlike past newsvendor studies, our experiment is unique in featuring a simple two-point asym-
metric demand distribution (as opposed to uniform or normal demand). Demand is either high or
low, with an uneven probability distribution. In our first study, we allow for two inventory decision
options, which are the same as the two demand points. We set the newsvendor cost parameters
in such a way that the optimal decision is to stock at the low-probability demand point, where
the bounded rationality model predicts that the low-probability demand quantity be chosen more
often than the high-probability demand quantity. However, in our experiment we find the opposite,
suggesting that random errors alone cannot account for the decision behavior.
In our second study, we use the same cost and demand parameters, but include an additional
decision option at the mean of demand. In this design, the mean anchoring model predicts that
subjects should choose the mean more often than the (suboptimal) high-probability demand point.
However, again, we find the opposite in our experiment, suggesting that the mean anchoring bias
cannot account for the decision behavior and only the ex post inventory regret bias can consistently
explain the experimental observations.
In summary, while all three models can explain newsvendor decisions in past studies with
symmetric demand distributions, we find that only the bounded rationality model that incorporates
the ex post inventory regret bias is robust and flexible enough to account for newsvendor decisions
under asymmetric demand distributions. In addition, we find that the decision behaviors predicted
by the three models all exist at the individual level, where a dominant majority can be explained
by the ex post inventory regret bias and the bounded rationality model. Thus, the aggregate level
model fit is essentially driven by the relative proportion of decision behavior at the individual level.
The rest of the paper is organized as follows. We present a brief summary of the relevant
1Schweitzer and Cachon (2000) also suggest an alternative anchoring model based on the prior order quantity,termed the “demand chasing heuristic.” This model only receives weak support from their experimental data. Simi-larly, we find little evidence of demand chasing in our data.
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literature in §2. In §3 we provide the details of our first study, which determines whether bounded
rationality can account for newsvendor decisions, or whether other biases are at play in the newsven-
dor setting. This section will provide details on the bounded rationality model, our experimental
design, results, and a discussion. In §4, we will present the second study, which extends the first
by investigating the most common theories associated with newsvendor decisions. Specifically, in
this section, we will combine bounded rationality with the mean anchoring bias and the ex post
inventory regret bias, present the results of an experiment that can tease out these competing
explanations, and provide a discussion. §5 contains our final summary.
2. Literature Review
Research in behavioral operations management has uncovered various effects that influence human
decisions involving operational problems. These include, but are not limited to, the bullwhip
effect in supply chains (Croson and Donohue 2006), capacity scheduling (Olivares, Terwiesch and
Cassorla 2008), social preferences (Loch and Wu 2008), forecasting (Ozer, Zheng and Chen 2011),
contracting (Lim and Ho 2007, Kalkanci, Chen and Erhun 2011), mental accounting (Chen, Kok
and Tong 2013), and procurement (Davis, Katok and Kwasnica 2013).
One operational topic that has received considerable interest from a behavioral standpoint is
the newsvendor problem. The seminal experimental paper on this topic is Schweitzer and Cachon
(2000). In their work, they identify that decision makers set quantities in a way that systemati-
cally deviates from the normative theory. In particular, they notice a “pull-to-center” effect, where
subjects set a quantity somewhere between the mean of the demand distribution and normative
prediction. Since the work of Schweitzer and Cachon (2000), a number of studies have illustrated
the robustness of the pull-to-center result. In particular, these studies have shown the newsven-
dor pull-to-center effect persists in settings with additional decisions, more feedback relating to
foregone options, higher payoffs, lower decision frequency, multilocation correlation (Bolton and
Katok 2008, Bostian, Holt and Smith 2008, Lurie and Swaminathan 2009, Ho, Lim and Cui 2010),
and experienced managers (Bolton, Ockenfels and Thonemann 2012).
To account for this pull-to-center behavior, Schweitzer and Cachon (2000) posit two explana-
tions: mean anchoring and minimizing ex post inventory regret. However, their experiment does
not allow them to determine which is the primary cause of the pull-to-center effect, leaving both
as plausible explanations. Another accepted explanation for newsvendor decisions is the bounded
rationality model (Su 2008). It states that decision makers prefer better options, but that they
do not make the correct choice 100% of the time. This model differs from the explanations of
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Schweitzer and Cachon (2000) in that it focuses on subjects making errors, rather than incorpo-
rating non-pecuniary payoffs into a decision maker’s expected-utility function.
Bounded rationality is largely based on a model developed in the experimental economics lit-
erature. McKelvey and Palfrey (1995) propose such a model for normal form games (termed the
“quantal response equilibrium”) and apply it to various experimental game data. Successful appli-
cations of the model have been reported in coordination games (Anderson, Goeree and Holt 2001),
auction games (Goeree, Holt and Palfrey 2002), pricing contracts (Lim and Ho 2007, Ho and
Zhang 2008), newsvendor experiments (Su 2008), and capacity allocation games (Chen, Su and
Zhao 2012).
In regards to more recent work on the newsvendor, Lee and Siemsen (2013) find that sepa-
rating ordering decisions into forecasts, uncertainty estimates, and service level choices results in
improved newsvendor decisions. Moritz, Hill and Donohue (2013) identify a strong correlation
between newsvendor behavior and cognitive reflection. Ockenfels and Selten (2013) propose an
“impulse balancing” model for newsvendor decisions. Ren and Croson (2013) conduct a newsven-
dor experiment with normally distributed demand, and find that underestimating the variance of
demand causes orders to deviate in certain ways.
The most closely related work to ours is Kremer, Minner and Wassenhove (2010). They too
include lottery choice questions in a newsvendor experiment, and show that newsvendor context-
sensitive biases exist. However, our first study differs from theirs in 1) presenting subjects with
binary choice lotteries versus seven-choice lotteries, thereby allowing us to easily measure subjects’
utility preferences, 2) examining lotteries within-subject versus between-subject, which permits
one to see if subjects make different decisions between the two contexts, and 3) employing an
asymmetric demand distribution. Furthermore, our second study is unique to theirs in focusing on
the most common newsvendor explanations, mean anchoring, minimizing ex post inventory regret,
and bounded rationality. To the best of our knowledge, our study is the first in the literature to
allow a comparison of these three plausible theories.
3. Study 1: Validating the Bounded Rationality Model
In this section, we present the results of our Study 1. In this study, we investigate whether random
errors, or bounded rationality, can fully explain the newsvendor decision behavior. We begin by
presenting the theory for bounded rationality, followed by the details of the experimental design
and results.
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3.1 The Bounded Rationality Model
Consider a newsvendor problem for a single product. The demand in a period, denoted by D,
takes one of two values: d and d′, with 0 < d < d′. Let δ = d′ − d be the difference between
these two demand values. Let us assume that the probability of demand d occurring is given by
P (D = d) = π. The probability of the alternative demand d′ occurring is 1− π.
The unit purchase cost for the product is c and the unit selling price is p, with 0 < c < p. The
unit salvage value for any leftover inventory is assumed to be zero. With the two-point demand,
the expected profit-maximizing order quantity q∗ is either d or d′, depending on which option gives
a higher expected profit payoff. Specifically, the expected profit for each order quantity is given by
f(d) = (p− c)d
f(d′) = (p− c)d+ (p− c− pπ)δ. (1)
Thus, q∗ = d if π > (p− c)/p, and q∗ = d′ otherwise. The ratio (p− c)/p is commonly referred as
the critical fractile in the newsvendor problem.
Su (2008) proposes a bounded rationality model to account for the random errors in human
decisions observed in newsvendor experiments. Specifically, based on the expected profit function
given in (1), the “noisy” utility function can be written as
u(d) = (p− c)d+ βε
u(d′) = (p− c)d+ (p− c− pπ)δ + βε′, (2)
where β (β > 0) measures the extent of bounded rationality of the decision maker, and ε and ε′
are independent standard Gumbel random variables. If β → 0, the decision maker has perfectly
rational decisions, whereas if β → ∞, the decision maker randomizes between the two choices.
Under this model, it turns out that the choice probability for q = d is given by
P (q = d) = P (u(d) ≥ u(d′)) =e(p−c)d/β
e(p−c)d/β + e[(p−c)d+(p−c−pπ)δ]/β =1
1 + e(p−c−pπ)δ/β. (3)
The choice probability for q = d′ is simply 1−P (q = d). From the above expression, it follows that
q = d is chosen more frequently than q = d′ if and only if π > (p − c)/p. Below we shall refer to
this model as the “bounded rationality model.”
It is worth noting that the Gumbel noise assumption is not essential in the above model predic-
tion. The same prediction can be obtained from the normal noise assumption. To see this, suppose
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that ε and ε′ are independent standard normal random variables. Then ε − ε′ follows a normal
distribution with N(0,√
2), and we have
P (q = d) = P (u(d) ≥ u(d′)) = P (β(ε− ε′) ≥ (p− c− pπ)δ) = 1− Φ
((p− c− pπ)δ√
2β
),
where Φ(·) is the standard normal cumulative distribution function. From the above expression,
we have P (q = d) > 1/2 if and only if π > (p − c)/p, which is identical to what is implied in (3).
Thus, the above prediction is robust to different noise assumptions under the two-point demand.
3.2 Experimental Design
In our experiment each participant played the role of a manager facing uncertain demand, whose
task was to decide how many units of inventory to stock for a single demand realization. In each
of 30 rounds, demand was either 100 or 400, independent and identically distributed each round,
and participants were given the option of ordering either 100 or 400 units. Note that the demand
mean was not a feasible decision choice in the design, differentiating our experiment from past
newsvendor experimental studies (Schweitzer and Cachon 2000, Kremer et al. 2010).
In our experiment the selling price was fixed at p = 4 per unit, the cost per unit was fixed at c =
1, and the P (D = 100) = 0.70. We chose a sample size of 30 to match the number of rounds played
by each participant, thus allowing the statistical power to be comparable between any analysis at
the aggregate level, where a single subject constitutes an independent observation (n = 30), and
individual level, where each round is an observation (n = 30). Also, with a sample size of 30, the
underlying binomial distribution can be well approximated by a normal distribution. Given these
parameters, the bounded rationality model yields a probabilistic prediction, P (q = 100) < 1/2 (see
§3.1). In Table 1, we display the actual profits for each demand realization, the expected profits,
and the bounded rationality prediction.
Table 1: Experimental profits and predictions for Study 1.
Decision options
Demand scenario q = 100 q = 400
D = 100 300 0 (×0.70)
D = 400 300 1200 (×0.30)
Expected profit 300 360
Bounded rationality prediction (probability) < 1/2 > 1/2
After participants completed the newsvendor experiment, all 30 rounds, they were presented
with two sequential lottery choice questions that explicitly displayed the same random profit payoffs
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as in the newsvendor problem (outlined in Table 1). For this task subjects chose between (A) a
100% chance of 300 profit, and (B) a 70% chance of 0 profit and a 30% chance of 1200 profit. In
the first of the two lottery questions, the decision was played as a single-play lottery, emulating a
one-round newsvendor problem. For the second question, the decision was played as a repeated
30-play lottery, i.e., after the participant made their choice, the lottery was automatically played 30
independent times and the earnings were summed up, emulating a 30-round newsvendor problem.2
Participants were paid based on the results of both lottery choice questions, to ensure that the
monetary incentive was the same as in the newsvendor task. The purpose of these lottery choice
questions was to elicit subjects’ utility preferences for the relevant payoffs, and see whether any
potential biases are specific to the newsvendor context.3
We conducted Study 1 in the spring of 2013 at a northeast U.S. university. Participants were
mostly undergraduate students. At the start of the session, each participant was given about five
minutes to read the instructions, which included a summary of the game and how their profits
would be calculated. Following this, we read the instructions aloud and answered any questions
(instructions available upon request). Participants were recruited through an online system where
cash was the only incentive offered. Average compensation was around $22, with a minimum of
$5 and a maximum of $40. Each session lasted approximately 40 minutes and utilized the zTree
system for software implementation (Fischbacher 2007).
3.3 Results
We begin by presenting summary statistics, using the frequency that a subject orders a quantity
of 100, P (q = 100), as our primary metric of interest, and its equivalent decision in the lottery
questions. Figure 1 plots the round by round decisions for how often a quantity of 100 was selected
as well as the bounded rationality prediction (grey area), and normative benchmark. We do not find
evidence of learning effects, as a Pearson’s chi-squared test to check the independence of decisions
across 30 rounds provides no significant effect of rounds (χ2(29) = 28.52, p = 0.490).
Table 2 shows the probability that q = 100 was selected in the newsvendor rounds and lottery
questions, where standard errors are calculated across subjects and presented in the brackets. When
we compare the newsvendor and lottery data to each other, we observe a paradox. Comparing the
first-round newsvendor data and the single-play lottery data, a one-sided proportions test reveals a
2We emphasize that subjects made a single decision for each lottery, single and repeated, not 30 separate decisions.3To avoid any order effects, we did not provide any information about the lottery exercise until subjects completed
the newsvendor task. This is similar to other experimental studies that elicit participant preferences, such as Davis,Katok and Kwasnica (2011), who conduct the risk aversion lottery exercise of Holt and Laury (2002) after subjectscompleted all decisions of an auction experiment.
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0.000.100.200.300.400.500.600.700.800.901.00
0 5 10 15 20 25 30
P(q=
100)
Round Bounded Rationality Observed Normative
Figure 1: Probability of quantity of 100 across time for Study 1.
significant difference (p = 0.035). Between the 30-round average newsvendor data and the repeated-
play lottery data, a one-sided proportions test also reveals relative significance (p = 0.067). This
discrepancy suggests that there exists a newsvendor specific bias compared to the the profit-based
utility preferences.
Table 2: Summary of experimental results for Study 1.
Newsvendor Newsvendor Single RepeatedRound 1 Average Lottery Lottery
P (q = 100) 0.67 0.56 0.43 0.37[0.09] [0.05] [0.09] [0.09]
Standard errors are in brackets. Average = 30 round average.
From the choice probability of P (q = 100) given in (3), we can derive a maximum-likelihood
estimate (MLE) of the bounded rationality measure β (see the Appendix for the detailed derivation).
Recall that β > 0 is required under the bounded rationality model assumption. Thus, based on
the sign of the β estimate, we can determine the goodness of fit of the bounded rationality model.
Table 3 shows the estimation results based on the aggregate-level data, where standard errors are
presented in the brackets. We observe that the sign of the bounded rationality parameter β from
the newsvendor data is incorrect and significant, but the signs of β from the lottery question data
are both correct.
We also explore how the bounded rationality model fits our data when allowing for heterogene-
ity. Based on the expected-profit utility, we calculate that 67% of subjects’ choice probabilities
do not conform to the bounded rationality model prediction, i.e. 67% set P (q = 100) < 1/2.
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Table 3: Maximum-likelihood estimates of the bounded rationality model for Study 1.
Newsvendor Single RepeatedAverage Lottery Lottery
β -248.8∗∗∗ 223.7 109.8∗
[69.3] [307.2] [76.1]Correct sign No Yes YesStandard errors are in brackets.Average = 30 round average; ∗∗∗p < 0.001, ∗p < 0.1.
Moreover, 50% of subjects’ choice probabilities reject the prediction based on a more stringent,
two-sided proportions test with a 5% level of significance. Figure 2 illustrates a histogram of the
choice probabilities, P (q = 100), at the subject level, where the grey range represents the bounded
rationality prediction.
5
0 0
1
3
1
7
4 4 4
1
0
1
2
3
4
5
6
7
8
P(q=100)
Freq
uenc
y
Bounded Rationality Observed
Figure 2: Histogram of P (q = 100) in the experimental treatment for Study 1.
In Figure 2, one may note that there is somewhat of a mode in the range of 0.55-0.85, which
roughly coincides with the probability of demand equalling 100, P (D = 100) = 0.70. This resembles
probability matching behavior (Vulkan 2000), where a common strategy is “win-stay-lose-shift.”
To evaluate the win-stay-lose-shift effect, we define a “win” as a match between the order decision
and the realized demand, and “lose” as a mismatch between the two. Conditional on an event of
win or lose, we evaluate whether there is a change in a subject’s subsequent order decision. Out of
the 30 subjects, we find only 40% of the subjects employed the “win-stay” strategy and 0% used
the “lose-shift” strategy with p < 0.05. This is not entirely surprising as studies have shown that
probability matching behavior disappears in the presence of financial incentives (Vulkan 2000).
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Another common explanation for experimental newsvendor decisions is the “demand chasing
heuristic” (Schweitzer and Cachon 2000). To evaluate the demand chasing effect in our data, we
applied the technique used in Bolton and Katok (2008), which measures the correlation between the
current period’s quantity and the previous period’s demand, by subject. This method was recently
highlighted as one of the best means for identifying demand chasing (Lau and Bearden 2013). We
find that 0% of our subjects had a correlation coefficient that is positive and significant at the
p < 0.05 level. Thus, there is no support for demand chasing in our data.
3.4 Robustness Treatment
In an effort to see whether our results extend to a different set of parameters, we conducted an
additional treatment that mirrored our first experiment from Study 1. Specifically, we used all of
the same experimental protocols as before, except, in this new treatment, P (D = 100) = 0.30, and
the cost per unit c = 3 such that the critical fractile is 0.25 (and n = 31). Our objective with this
treatment was to confirm whether our earlier results, particularly that bounded rationality does
not fully explain newsvendor decisions, are robust to different experimental parameters.
In Table 4, we display the outcomes and predictions for this robustness treatment. Let us
highlight a few observations. First, as shown in Table 4, stocking q = 100 yields a fixed, positive
payoff with certainty. Hence, stocking q = 400 is a riskier choice than stocking q = 100. Addi-
tionally, stocking q = 400 could potentially result in a loss (-800 when demand is 100). Thus, if
subjects tend to set a quantity of 400 more frequently, contradicting the prediction of the bounded
rationality model, they have to overcome both risk aversion and loss aversion. In the case of the
lottery questions in this treatment, subjects had to choose between (A) a 100% chance of 100 profit,
and (B) a 30% chance of -800 profit and a 70% chance of 400 profit. Again, these emulate the
newsvendor outcomes and present them explicitly.
Table 4: Experimental profits and predictions in the robustness treatment for Study 1.
Decision options
Demand scenario q = 100 q = 400
D = 100 100 -800 (×0.30)
D = 400 100 400 (×0.70)
Expected profit 100 60
Bounded rationality prediction (probability) > 1/2 < 1/2
Figure 3 plots the round by round decisions for how often a quantity of 100 was selected as
well as the bounded rationality range, and normative prediction. As with the primary treatment,
11
a Pearson’s chi-squared test to evaluate the independence of decisions across 30 rounds provides no
significant effect of rounds (χ2(29) = 37.11, p = 0.143).
0.000.100.200.300.400.500.600.700.800.901.00
0 5 10 15 20 25 30
P(q=
100)
Round Bounded Rationality Observed Normative
Figure 3: Probability of quantity of 100 across time in the robustness treatment for Study 1.
Table 5 shows the probability that q = 100 was selected in the newsvendor rounds and lottery
questions, where standard errors are calculated across subjects and presented in the brackets. We
again see a newsvendor paradox. Between the first-round newsvendor data and the single-play
lottery data, a one-sided proportions test reveals a significant difference (p = 0.009). Between the
30-round average newsvendor data and the repeated-play lottery data, a one-sided proportions test
also reveals marginal significance (p = 0.090). This discrepancy further supports the theory that
there exists a newsvendor specific bias compared to the the profit-based utility preferences.
Table 5: Summary of experimental results in the robustness treatment for Study 1.
Newsvendor Newsvendor Single RepeatedRound 1 Average Lottery Lottery
P (q = 100) 0.23 0.44 0.52 0.61[0.08] [0.04] [0.09] [0.09]
Standard errors are in brackets. Average = 30 round average.
When we structurally fit the bounded rationality model to our data using the MLE method,
we find the same results as in the original experimental treatment. In particular, the sign of the
bounded rationality parameter is incorrect and significant (β = −262.1 p < 0.01) but positive for
both lotteries (although not significant).
With respect to subject-level behavior, 52% of subjects’ choice probabilities do not conform
to the bounded rationality model prediction, and 45% of subjects’ choice probabilities reject the
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prediction based on a two-sided proportions test with a p < 0.05 level of significance. Again, we
find little evidence of probability matching behavior as well (19% follow the “win-stay” strategy
and 0% employ the “lose-shift” strategy). In addition, we find little support for demand chasing
behavior in our data; there is only 3% of our subjects have a correlation coefficient that is positive
and significant at with p < 0.05 (Lau and Bearden 2013).
3.5 Discussion
From Study 1, we observe that, while there exist random errors in newsvendor decisions over
time, the bounded rationality model alone cannot fully explain the newsvendor decision behavior.
Specifically, fitting our newsvendor decision data with the bounded rationality model, we obtain an
incorrect sign for the bounded rationality measure β. On the other hand, the bounded rationality
model can still explain people’s decisions when the cue of newsvendor context is weakened (i.e.,
when the newsvendor problem is presented as an equivalent lottery choice problem). We note that
this discrepancy extends the result reported by Kremer et al. (2010). In their experiment, they
find discrepancies between the lottery choice and newsvendor decisions under a uniform demand
distribution. However, their newsvendor decision observations can still be fit by the bounded
rationality model with a correct sign.
To better explain the newsvendor behavior in our experiment, we need to incorporate certain
newsvendor context-specific biases into the original bounded rationality model. In our analysis, we
tested both the win-stay-lose-shift strategy and the demand chasing heuristic; neither can explain
the observed behavior. However, if we incorporate into the model a decision bias based on either
mean anchoring or minimizing ex post inventory regret (Schweitzer and Cachon 2000), we can
explain the observed decisions. To see this, in our first experiment in Study 1 the mean of demand
was 190. Therefore, if a subject set a quantity of 100, it agrees with a mean anchoring bias (100
is closer to 190 than 400 is).4 However, at the same time, a quantity of 100 will also minimize
ex post inventory regret. Thus, if a subject has a strong decision bias for either mean anchoring
or minimizing ex post inventory regret, then she will choose the quantity of 100 more often than
400, which agrees with our experimental observation. In fact, we can show that these two decision
biases are mathematically equivalent under our two-choice model (see the Appendix).
4In this treatment, the average order across all subjects is 232, falling between the mean 190 and the normativeoptimal decision 400. The average order appears to confirm the hypothesis that subjects tend to anchor on the meanand adjust (insufficiently) toward the optimal decision (Schweitzer and Cachon 2000). In our experiment, however,there are only two decision choices: 100 and 400. If subjects were indeed adjusting toward the optimal decision, thenthe optimal quantity 400 should be chosen more often. But the contrary is observed in our experiment. Therefore,we believe that a more plausible mean anchoring explanation is to assume that subjects anchor on the mean andadjust toward the option with the smaller deviation from the mean (i.e., quantity 100 in this case).
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To further tease out which decision bias model can better explain the newsvendor behavior, we
introduce a third quantity decision option, representing the mean of demand, into our experiment in
Study 2. With this additional choice option, the mean anchoring model and the ex post inventory
model will yield different model predictions. It is important to note that these two competing
theories yield the same pull-to-center prediction under a symmetric demand distribution, such as
the uniform or normal distribution. As a result, in past newsvendor studies researchers have not
been able to compare these two theories directly.
4. Study 2: Validating Models for Newsvendor Decision Biases
In this section, we present the results of our Study 2. In this study, we investigate whether
including the mean anchoring or the ex post inventory regret bias into the bounded rationality model
can explain the newsvendor decision behavior. We begin by presenting the theory for bounded
rationality model with decision biases, followed by the details of the experimental design and
results.
4.1 The Bounded Rationality Model with Decision Biases
Consider again a newsvendor problem for a single product with a two-point demand. In this case,
we introduce an additional decision option at the mean of demand m = πd+ (1− π)d′. Moreover,
we extend the bounded rationality utility function (2) by including a newsvendor-specific decision
bias for each option:
u(d) = (p− c)d− αg(d) + βε1
u(m) = (p− c)d+ (p− c− pπ)(1− π)δ − αg(m) + βε2
u(d′) = (p− c)d+ (p− c− pπ)δ − αg(d′) + βε3, (4)
where g(·) is the psychological cost induced by the decision bias for each option, α (α > 0) is the
relative weight of such a cost with respect to the expected profit and random errors. We note that
εi (i = 1, 2, 3) are independent standard Gumbel random variables as in (2).
Let a1 = (p − c − pπ)(1 − π)δ and a2 = (p − c − pπ)δ. Also, define b1 = g(d) − g(m) and
b2 = g(d) − g(d′). Given the above utility function, the choice probability for each option can be
14
written as
P (q = d) = P (u(d) ≥ u(m), u(d) ≥ u(d′)) =1
1 + e(a1+αb1)/β + e(a2+αb2)/β
P (q = m) = P (u(m) ≥ u(d), u(m) ≥ u(d′)) =e(a1+αb1)/β
1 + e(a1+αb1)/β + e(a2+αb2)/β
P (q = d′) = P (u(d′) ≥ u(d), u(d′) ≥ u(m)) =e(a2+αb2)/β
1 + e(a1+αb1)/β + e(a2+αb2)/β. (5)
4.1.1 Mean Anchoring Bias
In the mean anchoring bias model, we use the following psychological cost function for each option:
g(d) = |d−m| = (1− π)δ
g(m) = 0
g(d′) = |d′ −m|γ = (πδ)γ ,
where γ captures the potential overweighting (γ > 1) or underweighting (0 < γ < 1) of the
upward deviation from the mean. When γ = 1, we have equal weight of the downward and upward
deviations from the mean. From the above expression, it is clear that these cost functions capture
people’s tendency to anchor on the mean. We will test the model with γ values in the neighborhood
of γ = 1, i.e., γ = 0.9, 1 and 1.1, representing underweighting, equal weighting, and overweighting
of the upward deviation from the mean, respectively.
4.1.2 Ex Post Inventory Regret Bias
In the ex post inventory regret model, we assume the following psychological cost for each option:
g(d) = (1− π) · |d′ − d| = (1− π) · δ
g(m) = π · |d−m|γ + (1− π) · |d′ −m| = π · ((1− π)δ)γ + (1− π)πδ
g(d′) = π · |d− d′|γ = π · δγ ,
where γ captures the potential overweighting (γ > 1) or underweighting (0 < γ < 1) of the regret of
having leftover inventory. When γ = 1, we have equal weight of the regret of having either inventory
stockout or leftover. In this particular case, we have b1 = (1 − 2π)(1 − π)δ and b2 = (1 − 2π)δ.
It follows that a1/b1 = (p − c − pπ)/(1 − 2π) = a2/b2. Hence, when γ = 1, we cannot estimate
α and β separately based on the choice probabilities specified in (5). Therefore, we will instead
estimate the model with γ values in the neighborhood of γ = 1, i.e., γ = 0.9 and 1.1, representing
underweighting and overweighting of leftover inventory, respectively.
15
It is worth noting two points here. First, our ex post inventory regret model resembles the
reference dependence model proposed by (Ho et al. 2010) that captures the difference between the
psychological costs of leftovers and stockouts. However, their model assumes the same linear cost
function with two different weights, whereas here we assume two different cost functions with the
same weight.5 Second, if we assume that subjects choose an order quantity based on their demand
forecast, then minimizing ex post inventory regret is the same as minimizing demand forecast
errors. Interpreted in this way, our model above also captures the potential underweighting and
overweighting of over-forecast with different γ values.
4.2 Experimental Design
We modified our experiment from Study 1 by adding a third quantity option representing the mean
of demand, 190. We kept all experimental protocols and parameters the same as in Study 1, p = 4,
c = 1, P (D = 100) = 0.70, 30 rounds, and included 30 new participants from the same subject
pool during the fall of 2013. We also included one additional post treatment question, which asked
for subjects to estimate the mean of the demand distribution, and compensated them $1.00 if they
entered a value between 170 to 210. Table 6 illustrates the payoffs for each demand scenario and
quantity decision, along with the the bias for each model outlined in §4.1.
Table 6: Experimental profits and decision biases for Study 2.
Decision options
Demand scenario q = 100 q = 190 q = 400
D = 100 300 210 (×0.70) 0 (×0.70)
D = 400 300 570 (×0.30) 1200 (×0.30)
Expected profit 300 318 360
Mean anchoring bias cost (γ = 1) 90 0 210
Ex post inventory regret bias cost (γ = 1) 90 126 210
A few comments are in order. First, note that similar to Study 1, stocking 400 results in the
highest expected profit. Therefore, stocking 400 suggests that people are making decisions in line
with the bounded rationality model (or the normative benchmark if they make zero errors and
stock 400 every round). Second, stocking the mean, 190, yields the smallest psychological cost of
deviating from the mean. Lastly, stocking 100 yields the smallest psychological cost of the ex post
inventory regret. As a result, each of the three quantity choices is roughly tied to one of the models
outlined in §4.1 (and depending on the specific values of β and α).
5The primary reason why we chose this model over the model proposed by (Ho et al. 2010) is because α and βcannot be estimated separately based on the linear cost function under the three-choice model.
16
4.3 Results
Figure 4 illustrates the probability of stocking 100, and the cumulative probability of stocking 100
and 190, over time. A Pearson’s chi-squared test of independence reveals no significant effect of
rounds (χ2(58) = 50.12, p = 0.760).
0.000.100.200.300.400.500.600.700.800.901.00
0 5 10 15 20 25 30
Prob
abili
ty
Round P(q=100) P(q=100)+P(q=190)
Figure 4: Probability of stocking quantities across time for Study 2.
Table 7 depicts the summary statistics from the newsvendor decisions and lottery decisions.
As with Study 1, we observe a paradox. Comparing the first-round newsvendor data and the
single-play lottery data for P (q = 100), a one-sided proportions test reveals a significant difference
(p = 0.033). Between the 30-round average newsvendor data and the repeated-play lottery data
for P (q = 100), a one-sided proportions test also reveals significance (p = 0.052). Although not
depicted, the average estimate of the mean of demand among subjects was 180.4, which is not
statistically different from the true mean of 190.6
Table 7: Summary of experimental results for Study 2.
Newsvendor Newsvendor Single RepeatedRound 1 Average Lottery Lottery
P (q = 100) 0.53 0.51 0.30 0.30[0.09] [0.06] [0.08] [0.08]
P (q = 190) 0.37 0.23 0.47 0.30[0.09] [0.05] [0.09] [0.08]
P (q = 400) 0.10 0.26 0.23 0.40[0.05] [0.05] [0.08] [0.09]
Standard errors are in brackets.
6This excludes 5 subjects who entered a number less than 100, suggesting they did not understand this post-gamequestion. When we include them, the average becomes 161.2.
17
In order to directly compare the models outlined in §4.1, we turn to structural estimation.
From the choice probabilities of given in (5), we can derive the MLE estimates for the decision bias
weight α and the bounded rationality measure β (see the Appendix for the detailed derivation).
Table 8 shows the estimates for β for the bounded rationality model (without decision biases), and
estimates of β and α for the mean anchoring model with γ = 0.9, 1, 1.1, and the ex post inventory
regret model with γ = 0.9, 1.1. First, the bounded rationality model results in a worse fit compared
to the other four estimations, evidenced by the lower log-likelihood and the incorrect sign of β,
which suggests that there exist strong decision biases in the newsvendor data. Turning to the other
models, the value of β is negative and significant for all estimations except the ex post inventory
regret bias model with γ = 0.9. Therefore, in a direct comparison among these models, it appears
that the ex post inventory regret bias model (with an underweighting of the leftover inventory)
results in the best fit on aggregate.7
Table 8: Maximum-likelihood estimates of the bounded rationality, mean anchoring, and ex postinventory regret models for newsvendor decisions in Study 2.
Bounded Mean Mean Mean Inventory Inventoryrationality anchoring anchoring anchoring regret regret
(γ = 0.9) (γ = 1) (γ = 1.1) (γ = 0.9) (γ = 1.1)β -93.9∗∗∗ -70.2∗∗∗ -49.8∗∗∗ -38.6∗∗∗ 20.9∗∗∗ -4.54∗∗∗
[12.8] [6.52] [4.63] [3.59] [1.94] [0.42]α - 0.40∗∗∗ 0.22∗∗∗ 0.13∗∗∗ 2.58∗∗∗ 0.20∗∗∗
[0.05] [0.02] [0.01] [0.06] [0.00]LL -959.8 -931.7 -931.7 -931.7 -931.7 -931.7
Correct signs No No No No Yes NoStandard errors are in brackets. ∗∗∗p < 0.001.
We further conducted an individual level analysis. Specifically, we put subjects into four cat-
egories based on their choice decisions: 1) weak or no decision bias if P (q = 400) ≥ 50%; 2)
strong mean anchoring bias if P (q = 190) ≥ 50%; 3) strong ex post inventory regret bias if
P (q = 100) ≥ 50%; and 4) other if all choice probabilities are less than 50%. We found that 50% of
the subjects with strong ex post inventory regret bias, 20% with weak or no decision bias, another
20% with strong mean anchoring bias, and the final 10% belong to other. These individual level
results indicate that, while all three decision behaviors hypothesized by the three respective theories
(i.e., bounded rationality without biases, mean anchoring and minimizing ex post inventory regret)
exist at the individual level, the bounded rationality model with an ex post inventory regret bias
7We have conducted additional structural estimations based on different γ values. It appears that the ex postinventory regret model yields the correct signs for β and α as long as γ < 1. Thus, our conclusion here does notdepend on the particular choice of γ = 0.9.
18
is flexible enough to explain at least 70% of subjects’ behavior (50% with strong ex post inventory
regret bias plus 20% with weak or no bias). As a result, at the aggregate level, this model yields
the best fit under structural estimation.
4.4 Robustness Treatment
As with Study 1, we conducted a separate robustness treatment with c = 3 and P (D = 100) = 0.30,
and added a third decision option for the mean of demand, 310. Table 9 reflects the payoffs for each
demand outcome and quantity decision, along with the bias for each behavioral model. As with
the robustness treatment from Study 1, note that subjects could incur losses if picking a quantity
of 310 or 400, and that they coincide with riskier choices, compared to stocking 100.
Table 9: Experimental profits and decision biases in the robustness treatment for Study 2.
Decision options
Demand scenario q = 100 q = 310 q = 400
D = 100 100 -530 (×0.30) -800 (×0.30)
D = 400 100 310 (×0.70) 400 (×0.70)
Expected profit 100 58 40
Mean anchoring bias cost (γ = 1) 210 0 90
Ex post inventory regret bias cost (γ = 1) 210 126 90
Figure 5 shows the probability of stocking 100, and the cumulative probability of stocking 100
and 310, over time. A Pearson’s chi-squared test of independence reveals no significant effect of
rounds (χ2(58) = 50.12, p = 0.760).
0.000.100.200.300.400.500.600.700.800.901.00
0 5 10 15 20 25 30
Prob
abili
ty
Round P(q=100) P(q=100)+P(q=310)
Figure 5: Probability of stocking quantities across time in the robustness treatment for Study 2.
19
Table 10 displays the average stocking quantities and lottery decisions for the robustness treat-
ment of Study 2. Focusing on the average newsvendor quantities, overall, in the robustness treat-
ment in Study 2, there is a somewhat equitable split between all three quantities, compared to the
primary treatment. Comparing the first-round newsvendor data and the single-play lottery data
for P (q = 100), a one-sided proportions test does not reveal a significant difference (p = 0.296),
although this difference is marginally significant between 30-round average newsvendor data and
the repeated-play lottery data (p = 0.081). Lastly, the average estimate of the mean of demand
among subjects, after excluding four subjects who entered values less than 100, was 301, which is
not statistically different from the true mean of 310.
Table 10: Summary of experimental results in the robustness treatment for Study 2.
Newsvendor Newsvendor Single RepeatedRound 1 Average Lottery Lottery
P (q = 100) 0.33 0.39 0.40 0.57[0.09] [0.06] [0.09] [0.09]
P (q = 190) 0.33 0.28 0.37 0.30[0.09] [0.05] [0.09] [0.08]
P (q = 400) 0.33 0.34 0.23 0.13[0.09] [0.06] [0.08] [0.06]
Standard errors are in brackets.
Table 11 shows the MLE results for the robustness treatment in Study 2. First, the sign of the
bounded rationality model is actually correct for the newsvendor decisions, suggesting a possible
weak effect of decision biases in this treatment. When looking at the remaining four estimations,
as with the primary treatment in Study 2, only the ex post inventory regret model with γ = 0.9
has the correct signs for both β and α. Therefore, after conducting a likelihood-ratio test between
the bounded rationality model (without decision bias) and the ex post inventory regret model, we
find that the ex post inventory regret model with γ = 0.9 is significantly better after adjusting for
the additional parameter (χ2(1) = 9.84, p < 0.01). This result is consistent with our finding in the
primary treatment that the ex post inventory regret model (with an underweighting of the leftover
inventory) results in the best fit on aggregate. Also, this result is quite strong when one considers
that this robustness treatment provides a slight advantage to the bounded rationality model, as
it is generally tied to a stocking quantity of 100, which is the only option that does not have any
variability or potential for losses (i.e. behavior which matches the ex post inventory regret model
in this robustness treatment requires one to overcome both risk and loss aversion).
As in the primary treatment, we further conducted an individual level analysis. We found
20
Table 11: Maximum-likelihood estimates of the bounded rationality, mean anchoring, and ex postinventory regret models for newsvendor decisions in the robustness treatment for Study 2.
Bounded Mean Mean Mean Inventory Inventoryrationality anchoring anchoring anchoring regret regret
(γ = 0.9) (γ = 1) (γ = 1.1) (γ = 0.9) (γ = 1.1)β 294.9∗∗∗ -281.3∗∗∗ -681.6∗∗∗ 1610.1∗∗∗ 2.04∗∗∗ -20.3∗∗∗
[113.5] [69.7] [169.0] [399.2] [0.51] [5.03]α - 0.65∗∗∗ 1.29∗∗∗ -2.37∗∗∗ 0.38∗∗∗ 1.24∗∗∗
[0.10] [0.27] [0.63] [0.00] [0.03]LL -985.4 -980.5 -980.5 -980.5 -980.5 -980.5
Correct signs Yes No No No Yes NoStandard errors are in brackets. ∗∗∗p < 0.001.
in this treatment that 33% of the subjects with weak or no decision bias, 30% with strong ex
post inventory regret bias, 20% with strong mean anchoring bias, and the final 17% belong to
other. These individual level results again confirm that all three decision behaviors hypothesized
by the three respective theories exist at the individual level. In this case, the bounded rationality
model with an ex post inventory regret bias is again flexible enough to explain at least 63% of
subjects’ behavior. As a result, at the aggregate level, this model yields the best fit under structural
estimation.
4.5 Discussion
From Study 2, we observe that, the bounded rationality model with a mean anchoring bias cannot
account for the newsvendor decisions at the aggregate level, yielding incorrect signs for the MLE
estimates. On the other hand, the bounded rationality model with an ex post inventory regret bias
that underweights the leftover inventory can consistently account for the newsvendor decisions at
the aggregate level for both the primary and robustness treatments. Our observation of under-
weighting of leftover in the ex post inventory regret model is also consistent with what has been
observed in the previous studies by Schweitzer and Cachon (2000) and Bolton and Katok (2008).
Therefore, based on the results of our Studies 1 and 2, we are able to tease out the three plausible
theories (i.e., bounded rationality without decision biases, mean anchoring, and minimizing ex post
inventory regret) for the newsvendor decision behavior: While all three can explain newsvendor
decisions under symmetric demand distributions, it appears that only the bounded rationality
model with an ex post inventory regret bias is robust and flexible enough to account for newsvendor
decisions under asymmetric demand distributions.
It is also interesting to note that the average order across all subjects in our primary treatment
21
in Study 2 is 198, falling between the demand mean 190 and the optimal order quantity 400. Based
on the average order, one may be tempted to hypothesize that subjects anchor on the mean 190 and
make insufficient adjustment to the optimal decision 400. In our experiment, however, there are
only three decision choices: 100, 190 and 400; and we observe subjects actually chose the quantity
100 more often than the combination of 190 and 400. Thus, by simply looking at the average order
quantity, one may reach the incorrect conclusion that mean anchoring works well here.
We also find that all three decision behaviors exist at the individual level. Thus, the aggregate
level model fit is essentially driven by the relative proportion of each decision behaviors at the
individual level. These individual level results agree with informal conversations we had with
participants after the treatments in Study 2. In general, a small portion of participants said that
they preferred to focus on average demand (mean anchoring), while others said they felt more
comfortable setting the quantity equal to the most likely demand outcome (minimizing ex post
inventory regret).
5. Conclusion
In this paper we design a simple newsvendor experiment with an asymmetric demand distribution
to validate three plausible theories for the newsvendor decision behavior. Our experimental results
suggest that, besides random errors, there also exist newsvendor context-specific biases in people’s
decision behavior. More specifically, we find that the ex post inventory regret model can account for
such decision biases quite well. It is also interesting to note that, if we assume subjects choose an
order quantity based on their demand forecast, then minimizing ex post inventory regret is the same
as minimizing demand forecast errors. From our post-experiment conversations with participants,
it appears that subjects did try to minimize their demand forecast errors by setting the quantity
equal to the most likely demand outcome.
Based on our experimental findings, one can design strategies to mitigate the decision bias in the
newsvendor problem. For example, one could try to reframe the newsvendor problem to emphasize
on the expected profit payoff structure, such as presenting the problem as a lottery choice problem,
so as to eliminate the newsvendor context. Alternatively, one could try to weaken the salient cue for
either inventory stockout or leftover (or demand forecast errors), such as giving delayed feedback
of demand realizations (Lurie and Swaminathan 2009).
In summary, our experimental findings demonstrate that human decision makers are susceptible
to the tendency of weighing too much on minimizing ex post inventory regret (or demand forecast
errors), rather than choosing the rational decision to maximize the expected profit. From the supply
22
chain point of view, our results suggest that random errors and decision biases in a downstream
newsvendor’s decisions may cause excessive order variability to the upstream party. In this sense,
our experimental study provides an interesting behavioral cause for the widely-spread supply chain
bullwhip effect, complementing the normative causes identified by Lee, Padmanabhan and Whang
(1997) and other behavioral causes identified by Croson and Donohue (2006).
Acknowledgements
The authors would like to thank Elena Katok, Vishal Gaur, Joel Huber, Rick Larrick, Greg Fischer,
Jack Soll, Adrian Camilleri, and seminar participants at Duke University and University of Science
and Technology in China for their valuable comments and suggestions.
Appendix
Maximum-Likelihood Estimation for the Two-Choice Model
Suppose that there are a total of n observations, with k observations of quantity d and n − k
observations of quantity d′. Then, according to the choice probability (3), the log likelihood function
is given by
L(β|n, k) = k ln
(1
1 + e(p−c−pπ)δ/β
)+ (n− k) ln
(e(p−c−pπ)δ/β
1 + e(p−c−pπ)δ/β
)= (n− k)(p− c− pπ)δ/β − n ln(1 + e(p−c−pπ)δ/β).
Note that the log likelihood function is concave. Thus, by the first-order condition, we can derive
the maximum likelihood estimator as
β =(p− c− pπ)δ
ln(n/k − 1).
The standard error estimate of β is given by
σβ =(p− c− pπ)δ
(ln(n/k − 1))2
√n
k(n− k).
Maximum-Likelihood Estimation for the Three-Choice Model
Let k1 be the number of observations of d, k2 be the number of observations of m, and n be the total
number of observations. Thus, based on the choice probabilities specified in (5), the log likelihood
23
function is given by
L(α, β|k1, k2, n) = k1 ln
(1
1 + e(a1+αb1)/β + e(a2+αb2)/β
)+ k2 ln
(e(a1+αb1)/β
1 + e(a1+αb1)/β + e(a2+αb2)/β
)
+(n− k1 − k2) ln
(e(a2+αb2)/β
1 + e(a1+αb1)/β + e(a2+αb2)/β
)
=k2(a1 + αb1)
β+
(n− k1 − k2)(a2 + αb2)
β− n ln
(1 + e(a1+αb1)/β + e(a2+αb2)/β
).
By the first-order condition, we can obtain the MLE estimates of α and β by solving the
following two equations simultaneously:
a1e(a1+αb1)/β + a2e
(a2+αb2)/β
1 + e(a1+αb1)/β + e(a2+αb2)/β=
a1k2 + a2(n− k1 − k2)n
,
b1e(a1+αb1)/β + b2e
(a2+αb2)/β
1 + e(a1+αb1)/β + e(a2+αb2)/β=
b1k2 + b2(n− k1 − k2)n
.
From the MLE estimates α and β obtained from the above equations, let c1 = (a1 + αb1)/β and
c2 = (a2 + αb2)/β. According to the second-order derivatives, we can obtain the standard error
estimates for α and β as
σα =|β|(1 + ec1 + ec2)√
n(b21 · ec1 + b22 · ec2 + (b1 − b2)2 · ec1+c2),
σβ =|β|(1 + ec1 + ec2)√
n(c21 · ec1 + c22 · ec2 + (c1 − c2)2 · ec1+c2).
Equivalence between Mean Anchoring and Minimizing Ex Post Inventory Regretunder the Two-Choice Model
Under the two-choice model, the utility function with the mean anchoring bias can be expressed as
u(d) = (p− c)d− α(1− π)δ + βε
u(d′) = (p− c)d+ (p− c− pπ)δ − απδ + βε′,
where α ≥ 0 is the weight on the mean anchoring bias. Similarly, we can show that the utility
function with the ex post inventory regret bias can be expressed as
u(d) = (p− c)d− α(1− π)δ + βε
u(d′) = (p− c)d+ (p− c− pπ)δ − απδ + βε′,
where α ≥ 0 is the weight on the ex post inventory regret bias. It is clear that these two models
are identical. Thus, one cannot distinguish these two decision biases under a two-choice model.
24
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