Many theoretical models adopt a normative approach and assume that decision makers are perfect optimiz-ers. In contrast, this paper takes a descriptive approach and considers bounded rationality, in the sense
that decision makers are prone to errors and biases. Our decision model builds on the quantal choice model:While the best decision need not always be made, better decisions are made more often. We apply this frame-work to the classic newsvendor model and characterize the ordering decisions made by a boundedly rationaldecision maker. We identify systematic biases and offer insight into when overordering and underordering mayoccur. We also investigate the impact of these biases on several other inventory settings that have traditionallybeen studied using the newsvendor model as a building block, such as supply chain contracting, the bullwhipeffect, and inventory pooling. We find that incorporating decision noise and optimization error yields resultsthat are consistent with some anomalies highlighted by recent experimental findings.
Key words : bounded rationality; newsvendor; logit choice; random utility; quantal response; supply chain;bullwhip effect; inventory; pooling
History : Received: May 12, 2006; accepted: August 13, 2007. Published online in Articles in Advance.
1. IntroductionThe newsvendor model is one of the main build-ing blocks of inventory theory. The model derives itsname from the canonical setting of a newsvendor fac-ing random demand, who has to decide how manycopies of newspapers to order: Excess quantities thatremain unsold have no value, but ordering too fewcopies means that customers have to be turned awayand potential profits are lost. The newsvendor modelhas a simple and elegant solution that offers insightsinto the optimal balance between the costs of supply-side investment and the costs of potential foregoneprofits. It has extensive applications, including inven-tory management, capacity planning, and pricing andrevenue management.
The current theoretical literature is based primar-ily on the paradigm of perfect rationality. In existingmodels, the newsvendor is a perfect optimizer: With-out fail, he always chooses stocking levels that attainthe maximum possible level of expected profits. Thisinfallible newsvendor is an important workhorse ofinventory theory and many results are based on it.In contrast, recent experimental studies suggest thathuman decision makers do not solve these inventoryproblems as theory predicts. One of the most robust
experimental findings is that subjects in newsvendorexperiments systematically deviate from the optimalcritical fractile solution. Other anomalies have alsobeen identified in more general supply chain settings.These empirical deviations challenge the validity oftheoretical results in the literature. This is a significantissue because in practice, many of these newsvendor-type decisions are made by human decision makerswho suffer from similar psychological biases. We feelthat theoretical results based on perfect rationalityneed to be reconciled with actual human behavior.
This paper investigates the effect of boundedrationality in traditional newsvendor models. Whathappens when newsvendors make mistakes? Are theconsequences of these errors consistent with exper-imental findings? To address these questions, weadopt a descriptive decision framework that allowsfor decision errors. As these errors become negligiblysmall, we obtain the standard normative setting ofperfect rationality as a special case. We hope to gen-eralize existing theoretical results while capturing awider spectrum of empirically observed behavior.
Our decision model of bounded rationality isderived from classical quantal choice theory (Luce1959). When faced with alternatives i ∈ � generating
utility ui, decision makers do not always choosethe utility-maximizing alternative i∗ ∈ argmaxi ui. Allpossible alternatives are candidates for selection, butmore attractive alternatives (yielding higher utility)are chosen with larger probabilities. For analyticalconvenience, we focus on the logit choice model inwhich the probability of choosing alternative i isproportional to eui (McFadden 1981 and Andersonet al. 1992). When applied to newsvendor models,we interpret each possible order quantity x as a can-didate alternative and the corresponding expectedprofit ��x� as the utility. The probabilistic choicesetup implies that the newsvendor is subject to deci-sion noise and may make suboptimal ordering deci-sions. Unlike conventional models, newsvendor orderquantities are no longer deterministic (at the optimalquantity x∗ = argmaxx ��x�); they are now randomvariables. Nevertheless, order quantities that leadto higher expected payoffs are chosen more often.Regarding on the terminology, we stress that boundedrationality refers to a wide range of behavioral phe-nomena (e.g., psychological biases, heuristics or rulesof thumb, cognitive constraints). We focus on noisydecision making as one possible way of incorporatingbounded rationality.
Next, we summarize our main results. Using thelogit decision model, we offer a complete characteri-zation of the boundedly rational newsvendor’s orderquantity (i.e., its distribution). For the special caseof uniform demand, we find that the order quan-tity follows a normal distribution (truncated at theappropriate cutoff points). For the general case, weoffer a method to calculate the choice distribution andother quantities of interest, such as expected ordersand expected profits. Then, we apply this decisionframework to several inventory settings. Table 1 orga-nizes our findings and compares them with conven-tional predictions under perfect rationality. For basicnewsvendor decisions, we show that random decisionnoise yields systematic biases from the optimal criticalfractile solution, and we identify conditions that leadthe boundedly rational newsvendor to overorder orunderorder. In supply contracting, while much of theliterature proposes to achieve coordination by allo-cating a fixed fraction of total profits to the deci-sion maker, we show that this is not feasible in ourmodel of bounded rationality because the reduced
stakes also diminishes the decision maker’s incentiveto make good decisions. Next, our model shows thatthe bullwhip effect may arise when decision makersdo not trust their supply chain partners to order opti-mally and thus take actions to guard against and cor-rect for others’ biases. Finally, in inventory pooling,apart from the benefits that have been associated withvariance reductions resulting from summing randomvariables, we also identify the (additional) behavioralbenefits. As summarized in Table 1, although a largenumber of experimental observations seem to be atodds with conventional theory, our analysis will showthat many of them are consistent with bounded ratio-nality explanations.
There are two main contributions in this paper.First, we apply the quantal choice framework to a fun-damental operations model. The basic premise thatpeople need not optimize, but better decisions aremade more often is intuitively appealing. This quantalchoice approach is sufficiently general to be applica-ble to a wide variety of settings that extend beyondnewsvendor models, while remaining extremely par-simonious in that there is only a single parameter tobe estimated. We stress that this represents an alterna-tive theoretical paradigm in operations management.Instead of solving optimization problems (normativeapproach), we incorporate bounded rationality andcharacterize outcomes based on probabilistic choicemodels (descriptive approach). Second, our results(accounting for bounded rationality) are consistentwith a wide range of experimental observations thatseem to be at odds with conventional theory (assum-ing perfect rationality). Hence, we feel that decisionnoise and optimization error deserve consideration aspossible explanations for these empirical newsvendoranomalies. We hope that our results complement thebehavioral and theoretical rationales that have beenoffered in the literature.
The remainder of this paper is organized as fol-lows. The literature is reviewed in §2. We describe ourmodel of bounded rationality in §3 and apply it to thenewsvendor problem in §4. In §5, we fit our modelto experimental data and show that the data providesempirical support for our specification of boundedrationality. Next, we analyze the effect of boundedrationality in several different contexts, as shown in
Table 1 Summary of Newsvendor-Type Results Under Perfect Rationality and Bounded Rationality
Conventional results Some contradicting experimental/ Can these inconsistencies be fullyContext under perfect rationality empirical/theoretical results explained by bounded rationality?
Basic newsvendor • Critical fractile solution • Systematic overordering and underordering(Schweitzer and Cachon 2000)
• Yes
• Bias toward the mean (Bostian et al. 2007) • Yes• Bias toward low probability demand realizations • Yes• Learning effects (Bolton and Katok 2008, Lurie and
Swaminathan 2005)• No
• Previous-period effects (Ben-Zion et al. 2005) • No
Supply chaincontracting
• Double marginalizationproblem
• Double marginalization may be beneficial • Yes• Coordination is not achieved (Katok and Wu 2006) • Yes
• Different types ofcoordinating contracts
• Stake sizes and the value of committing to standingorders (Bolton and Katok 2008)
• Yes
• Differences between contractual forms (Katok andWu 2006)
• No
• Fairness concerns (Keser and Paleologo 2004, Wuand Loch 2007)
• No
Bullwhip effect • Bullwhip effect(variance of ordersincreases upstream)
• Behavioral causes of bullwhip effect (Sterman1989, Croson and Donohue 2006)
• Yes
• Underweighting of supply line (Sterman 1989,Croson and Donohue 2006)
• Yes
• Physical causes • Need to account for mistakes of supply chainpartners (Croson et al. 2005)
• Yes
• Demand uncertainty and supply uncertainty mayboth propagate upstream
• Yes
Inventory pooling • Pooling benefits • Behavioral benefits of pooling • Yes• No benefit when
demand is perfectlycorrelated
• Cost savings when demand is perfectly correlated • Yes• Cost savings when demand is deterministic • Yes• Pooling of demand uncertainty and supply
uncertainty• Yes
Table 1. Section 6 discusses overordering and under-ordering; §7 explains why coordinating contracts mayfail to coordinate the system; §8 shows how boundedrationality generates the bullwhip effect; and §9 iden-tifies the behavioral benefits of inventory pooling. Weoffer concluding remarks in §10. All proofs are pro-vided in the appendix.
2. Literature ReviewTraditional theory associates rationality with the abil-ity to optimize perfectly: Rational agents will settlefor nothing less than the best. In contrast, the con-cept of bounded rationality recognizes the inherentimperfections in human decision making. The sem-inal work of Simon (1955) proposes satisficing as amore accurate way to model decision-making behav-ior: Rather than optimizing perfectly, agents searchover the choice domain until they find something sat-isfactory. Another broad approach toward bounded
rationality is to study heuristics or rules of thumb;Geigerenzer and Selten (2001). When the bounds onrationality render optimization infeasible, agents mayinstead adopt simple heuristics to make complexdecisions. Several well-studied examples include therepresentativeness heuristic, the availability heuristic,and the anchoring and adjustment heuristic, whichare described in Tversky and Kahnemann (1974). Yetanother approach is to explicitly model agents’ cog-nitive limitations and the computational complexityof decision tasks. Rubinstein (1998) surveys this workand discusses it in the context of economic models ofdecisions and games. For a review of the evolutionand development of bounded rationality, readers arereferred to Simon (1982) and Conlisk (1996).
This paper models bounded rationality by incorpo-rating stochastic elements into the decision process.Instead of choosing the utility-maximizing alternativeall the time, decision makers adopt a probabilisticchoice rule such that more attractive alternatives are
chosen more often. We focus on the logit choice rule.Our approach is related to three separate streams ofliterature. First, there is a rich academic tradition onstochastic choice rules with the consistency propertythat better options are chosen more often. This approachcan be traced back to Thurstone (1927) and Luce(1959), who set up the mathematical framework anddevelop invariance properties. Blume (1993) motivatesthis stochastic approach by showing that the choicedistributions are analogous to Gibbs states, whichhave proven to be a useful tool in studying Ising mod-els in statistical mechanics even though their station-ary distributions are not completely known. McKelveyand Palfrey (1995) develop a framework that admitsgeneralizations of the better options are chosen moreoften structure and applies it to game-theoretic set-tings. Chen et al. (1997) consider individuals withlatent utility functions and study the stochastic choiceprobabilities that emerge. Our model of boundedrationality is mathematically equivalent to the randomutility approach in discrete choice models (Andersonet al. 1992). In these models, individuals’ utilities overdifferent alternatives have idiosyncratic taste shocksreflecting unobserved heterogeneity. The characteri-zation of stochastic choice models as random utilitymodels was first established by Block and Marschak(1960). The logit choice framework was originallydeveloped by Luce (1959) and McFadden (1981). Thethird stream of related models involves evolutionaryadjustment in decision processes. See, for example,Young (1993), Binmore and Samuelson (1997), Hof-bauer and Sandholm (2002), and Anderson et al.(2004). In these papers, a main interest is characteriz-ing the steady-state distribution of decisions over thelong run. In particular, Anderson et al. (2004) developa model in which agents adjust their decisions towardhigher payoffs, subject to normal error, and show thatthe long run steady-state distribution of this Gaussianprocess agrees with the logit choice rule. In our view,the three streams of work reviewed above providedifferent justifications to our model of bounded ratio-nality. For concreteness, we focus on the first inter-pretation; in other words, when we refer to boundedrationality, we mean that individuals need not alwayspick the best option, but they choose better optionsmore often.
There is a recent stream of work on behavioral oper-ations management; see reviews by Bendoly et al.(2006), Gino and Pisano (2008), and Loch and Wu(2007). This emerging literature points out fundamen-tal inconsistencies between empirical observationsand theoretical predictions in a variety of operationssettings, and underscores the need to reconcile thesefindings. In a similar spirit, our current work appliesquantal choice models to newsvendor-type scenar-ios, and shows that bounded rationality provides apotential explanation for some of these inconsisten-cies. Our goal is theory-building: We seek to extendthe decision-making foundations of existing theory tobetter match empirical observations in the laboratory.
We organize this review into the four areas listedin Table 1. First, for the basic newsvendor model, itis well known that the optimal solution is character-ized by the critical fractile; Porteus (2002). However,Schweitzer and Cachon (2000) present experimentalevidence of decision biases in this basic model. Witha uniform demand distribution, they find that sub-jects tend to order too many low-profit products andtoo few high-profit products. These results are consis-tent with two behavioral explanations: Subjects mayhave a preference to reduce ex-post inventory error, orthey may suffer from the anchoring and insufficientadjustment heuristic (i.e., orders are biased towardthe mean). There are subsequent studies that con-duct experiments to investigate the effect of feed-back and learning in the newsvendor problem. Boltonand Katok (2008) show that requiring newsvendorsto commit to standing orders focuses their attentionon long-term profits and results in better decisionmaking. Ben-Zion et al. (2005) identify a significantprevious-period effect, which weakens over time assubjects learn. Lurie and Swaminathan (2005) showthat more frequent feedback may sometimes degradeperformance. Bostian et al. (2007) find that the bias oforder quantities toward the mean can be explained byan adaptive learning model. In this paper, we demon-strate that bounded rationality, modeled via stochasticchoice rules (i.e., decision noise), can generate someof these laboratory observations.
Second, we position our work with respect to thetheoretical literature on supply chain coordination,which is well developed. When decisions in a sup-ply chain are made by individual parties with mis-aligned interests, the double marginalization problem
arises and system optimal profits cannot be attained.To rectify this situation, various supply contracts havebeen proposed. These contracts coordinate the sys-tem by aligning individual incentives with systemobjectives. Examples include buy-back contracts (e.g.,Pasternack 1985), quantity-flexibility contracts (e.g.,Tsay 1999), markdown money (e.g., Tsay 2001), salesrebates (e.g., Taylor 2002), and revenue-sharing con-tracts (e.g., Cachon and Lariviere 2005). For a reviewof the supply chain contracting literature, readers arereferred to Cachon (2003). Recently, Katok and Wu(2006) investigated the performance of these coordi-nating contracts in the laboratory. They test the per-formance of two mechanisms: buy-back contracts andrevenue-sharing contracts. They observe that, in con-trast to theoretical predictions, coordination is notachieved. In a similar spirit, we find that coordina-tion may not be feasible when the decision maker isboundedly rational.
Third, we discuss the theoretical and experimen-tal literature related to the bullwhip effect. The bull-whip effect is the tendency for the variance of ordersto increase upstream along the supply chain. This isdemonstrated in the important paper by Lee et al.(1997), who also identify four separate causes of thebullwhip effect: demand signal processing, inventoryrationing, order batching, and price fluctuations. The-oretical studies suggest that in the absence of thephysical causes listed above, the bullwhip effect willnot arise. On the experimental side, the first studydemonstrating the bullwhip effect is by Sterman(1989). There are two important contributions in thisseminal paper: First, the study introduces an empir-ical framework for predicting subjects’ choices byassuming that they follow a decision rule based onthe anchor-and-adjust heuristic. Second, it identifiesunderweighting of the supply line as another causeof the bullwhip effect. Because subjects do not fullyaccount for quantities in the supply line, they mayoverorder and generate instability that triggers thebullwhip effect. In a subsequent experiment with acommonly known demand distribution, Croson andDonohue (2006) control for all four physical causesbut find that the bullwhip effect persists. Finally, evenin an experiment in which the demand is constantand publicly known, Croson et al. (2005) find evi-dence of the bullwhip effect. They suggest that it
arises because of coordination risk, when subjects mayplace excessive orders to address the perceived riskthat others will not behave optimally. These experi-mental studies show that, beyond its physical proper-ties, the bullwhip effect is also very much a behavioralphenomenon. In this paper, we apply our frameworkof boundedly rational decision making in such set-tings. To a large extent, we find that the theoreticalpredictions of our model agree with the experimentalfindings reviewed above.
Fourth, we discuss the theoretical literature relatedto inventory pooling. The classic study by Eppen(1979) shows that in a multilocation inventory set-ting, consolidating stocks at a centralized location(instead of holding separate inventories at individ-ual locations) leads to a reduction in total costs; themagnitude of these pooling benefits depends on thecorrelation of demands. Although inventory poolingleads to lower costs, Gerchak and Mossman (1992)show that it does not necessarily lead to lower inven-tory levels. We extend this work by showing thatbeyond the physical benefits, there are also behavioralbenefits to inventory pooling.
From a meta-modeling perspective, the existingbody of work on random supply processes pro-vides a physical analogue to our behavioral notionof bounded rationality. This literature, which tracesback to Karlin (1958), studies the impact of physi-cal phenomena such as yield uncertainty in the pro-duction process, supply unreliability and disruptions(e.g., breakdowns, natural disasters, or labor strikes),and inventory record inaccuracy. For a sample of dif-ferent modeling approaches, readers are referred tothe review by Lee and Yano (1995), as well as morerecent papers by Chen et al. (2001), Tomlin and Wang(2005), Kok and Shang (2006), and Dada et al. (2006).The modeling root in most of this literature, whichalso forms the core of our notion of bounded ratio-nality, is that the supply X is a random variable.The causes may be behavioral or physical in nature,but the consequence is the same: Supply is uncertain.Under physical constraints, the agent is capable ofmaking an optimal decision but may still experiencea suboptimal outcome because the agent is not per-fectly capable of implementing the decision. In con-trast, under behavioral constraints, the agent faces asuboptimal outcome because he/she is not able to
make the optimal decision in the first place (eventhough the decision, once made, can be implementedaccordingly). Although the motivation and practicalcontexts behind these two perspectives are completelydifferent, they can be examined from a similar mod-eling angle.
There are two noteworthy differences between sup-ply uncertainty and bounded rationality as modeledin this paper. First, we capture boundedly rationaldecision making by postulating that better choicesare made more often. This implies that the resultingdecision noise is intricately related to the underlyingdecision problem. In contrast, most practical interpre-tations of supply uncertainty models do not require aclose relationship between supply disruptions and theunderlying economic context; for example, less severedisruptions (analogous to better decisions) do notnecessarily occur more frequently. Second, in manymodels of supply uncertainty, the actual quantitybeing supplied is usually less than the intended quan-tity. On the other hand, in our model of boundedrationality, the chosen quantity may either be largeror smaller than the optimal quantity. The only con-sistency condition we impose is that better decisionsare made more often. Nevertheless, from a man-agerial perspective, the insights that can be gleanedfrom studying random supply processes may haveanalogous interpretations for behavioral settings ofbounded rationality. In our analysis, we draw suchparallels wherever possible.
3. A Model of Bounded RationalityThe standard approach in most normative analysisassumes perfect rationality on the part of the deci-sion maker. Specifically, when faced with a choiceamong different alternatives i ∈� , the perfectly ratio-nal decision maker always chooses the most pre-ferred option(s) i∗ ∈ argmaxi ui. In contrast, to capturebounded rationality, we apply the multinomial logitchoice model and assume that the decision makerchooses alternative i ∈� with probability
i =eui/�∑i∈� eui/�
� (1)
Similarly, the logit choice probabilities over a contin-uous domain � are given by the density
�y�= eu�y�/�∫y∈� eu�y�/�
(2)
with distribution ��y�≡ ∫ y
−��v�dv. In other words,the agent’s choice is a random variable Y ∈ �. Asnoted by Anderson et al. (1992, p. 4), this probabilisticapproach provides a way to model bounded rational-ity. With this logit structure, better alternatives are cho-sen more often. Although the best option is no longerchosen with probability one, it nonetheless will bethe mode of the choice distribution. The logit modelis sometimes called the log-linear model because thelog odds of choosing one alternative over another isproportional to the payoff difference between the twoalternatives.
The parameter � can be interpreted as the extent ofcognitive and computational limitations suffered bythe decision maker. To understand this, observe thatas � → �, the choice distribution in (1) approachesthe uniform distribution over � in the limit. In thisextreme case, the decision maker lacks the abilityto make any informed choices and instead random-izes over the alternatives with equal probabilities.On the other hand, as �→ 0, the choice distributionin (1) becomes entirely concentrated on the utility-maximizing alternative (assuming it is unique). Thiscoincides with the choice of a perfectly rational deci-sion maker. When there are multiple alternatives forattaining the maximum utility, the choice distribu-tion approaches the uniform distribution over theseutility-maximizing alternatives, which is also con-sistent with perfect rationality. Therefore, we inter-pret the magnitude of � as the extent of boundedrationality.
The multinomial logit model described above hasbeen frequently used in different contexts. Under oneinterpretation, the noise terms �i reflect heterogeneitythat is unobserved by the modeler. Despite the pres-ence of noise, the decision maker is perfectly ratio-nal, but is just taking some unobserved factors intoaccount. Alternatively, as in our model, the noiseterms are the explicit result of bounded rationality.Both interpretations are reasonable, and lead to thesame probabilistic choice outcomes. In this paper, weadopt the bounded rationality interpretation becauseit facilitates comparing our results with recent exper-imental findings in which all other external attributeshave been controlled for.
Before proceeding, we put forth two invarianceproperties. First, consider affine transformations of
the decision domain, i.e., instead of choosing y ∈ �,suppose that the decision maker chooses y ∈ ��, wherey ≡ ay + b and �� ≡ a� + b for some constants a
and b. Because such transformations of the decisiondomain do not affect utility, the utility function over�� is given by u�y� = u�y�. This can be interpretedas purely a change in the way choices are namedor labeled. In particular, the multiplicative factor a
changes the units of the choices (e.g., from kilogramsto tonnes), and the additive term b changes the loca-tion of “zero.”
Lemma 1. The choice distribution is invariant to affinetransformations of the decision domain. Specifically, let��y� be the choice distribution over � and let ���y� be thechoice distribution over ��. Then, ��y�= ���y�.
There is a similar result for additive transforma-tions in the utility function, i.e., the decision makerfaces utility function u�y�= u�y�+ a instead of u�y�.
Lemma 2. The choice distribution is invariant to addi-tive transformations in the utility function. Specifically,let ��y� be the choice distribution with respect to u�y� andlet ���y� be the choice distribution with respect to u�y�.Then, ��y�= ���y�.
However, the choice distribution is affected by mul-tiplicative transformations in the utility function. Aswe see in the subsequent analysis, this effect hasimportant implications. Most significant, it suggeststhat stake sizes have an impact on decision outcomes.
4. Newsvendor Model UnderBounded Rationality
We now apply the logit choice framework to thenewsvendor problem. Recall that the canonical set-ting involves a newsvendor who has to determinehow many copies of newspapers to order. Each copycosts c but can be sold at price p, where p > c.The random demand D has density f and distribu-tion F ; we write �F ≡ 1− F . Demand that is not ful-filled is lost, and leftover copies have zero value.(Although it is straightforward to incorporate a sal-vage value, we prefer to suppress it for notationalclarity. By virtue of Lemma 2, the analysis can easilybe modified to include a salvage value s by replac-ing p and c with p − s and c − s.) Given this setup,
the newsvendor’s expected profit when ordering x
copies is
��x�= pEmin�D�x�− cx� (3)
which is uniquely maximized at x∗ = F −1���; here,� ≡ 1− �c/p� is the critical fractile and 1− � ≡ c/p isthe optimal stockout probability.
Let us introduce some terminology. We refer to theprofit-maximizing ordering quantity x∗ as the optimalsolution, which is chosen whenever the newsvendoris perfectly rational. However, under bounded ratio-nality, the newsvendor’s ordering quantity is subjectto noise and becomes a random variable. We refer tothis as the behavioral solution and denote it using x�
(for the realization) and X� (for the random variable).Given the problem data, it is straightforward to
use (2) to write down the behavioral solution of thenewsvendor problem. We assume that the decisiondomain S ⊆ R is the smallest interval containing thesupport of f . In other words, the boundedly ratio-nal newsvendor may order any quantity between thesmallest possible and largest possible demand real-izations. Then, the probability density function of thebehavioral solution is
�x�= e��x�/�∫Se��v�/� dv
= e�pEmin�D�x�−cx�/�∫Se�pEmin�D�v�−cv�/� dv
� (4)
We stress that the decision domain S plays an impor-tant role in this logit choice setup. This will becomeevident in subsequent analysis, where we study thebehavioral solution X� through its density .
4.1. Uniform DemandSuppose that the demand D is uniformly distributedbetween a and b, with b > a≥ 0. Then, the newsven-dor’s profit function in (3) can be simplified into aquadratic function
��x�=Ax2 +Bx+C� (5)
with coefficients A=−p/2�b− a��B= �pb/�b− a�− c�,and C = −pa2/2�b − a�. This quadratic structure isessential and gives us the following result.
Proposition 1. Let D ∼U&a� b'. Then, the behavioralsolution to the newsvendor problem follows a truncated
normal distribution over &a� b', with mean ( and variance)2 given by
(= b− c
p�b− a�� (6)
)2 = �b− a
p� (7)
Corollary 1. Let D ∼ U&a� b'. Then, the expectedbehavioral solution is
EX� =(−) · *��b−(�/)�−*��a−(�/)�
+��b−(�/)�−+��a−(�/)�� (8)
where *�·� and +�·� denote the standard normal densityand distribution functions.
Observe that the parameter ( of the behavioralsolution coincides with the optimal solution x∗. This isnatural because the optimal solution uniquely maxi-mizes expected payoffs and thus should be the uniquemode of the behavioral solution, which occurs at ( fora truncated normal distribution. Next, observe thatthe variance of the behavioral solution )2 is propor-tional to �, as expected because bounded rationality(larger �) increases noise in the decision-making pro-cess. It is also intuitive that the variance increaseswith the range b − a of the demand distribution (awider range generates a more complex decision task)but decreases with p (higher prices increase the stakesand result in better decisions).
The main message from this result is: Underbounded rationality, uniform demand yields normallydistributed choices (with appropriate truncations).From an experimental standpoint, there is no dearthof laboratory data for the uniform demand case.Many experiments are run using uniform demandbecause this is easier to understand for subjects.Therefore, our theory provides implications that canimmediately be put to the test. In particular, givenexperimental data on subjects’ ordering decisions, wemay fit the truncated normal distribution to the datato obtain estimates of the parameters. Because themodel of perfect rationality (with � = 0) is a specialcase of our model, we may test this hypothesis todetect the presence of bounded rationality. Significantevidence for �> 0 would support that presence. Thisprocedure is reported in detail in §5.
4.2. Triangular DemandNext, we consider the special case of triangulardemand distributions. Experimentally, apart from
uniformly distributed demand, the triangular dis-tribution is another special case that can be easilyunderstood by the subjects. Compared to the uni-form distribution, the triangular distribution offers amore accurate representation of demand in practicalsettings.
Here, we consider triangular demand distributionswith range &0�100'. This is without loss of generalityvia a straightforward translation. We use h ∈ &0�100'to denote the peak of the triangular demand density.The probability density function for demand is
f �x�=
x
50h� x≤ h�
100− x
50�100−h�� x > h�
(9)
Under this demand density, it is straightforward toderive the newsvendor profit function ��x� using (3).Observe that while the profit function is quadratic inthe case of uniform demand, it is cubic in the case oftriangular demand. As such, the behavioral solutionX� cannot be characterized using standard probabil-ity distributions. Therefore, we study its propertiesnumerically.
To generate Figure 1, we consider several differenttriangular demand distributions, with peak densitiesat h= 0�20�40�60�80�100; the range is maintained as&0�100'. Each of these demand distributions is rep-resented by one of the six charts in Figure 1. Weset price p = 1. For each demand density, we plotthe expected behavioral orders EX� against the profitmargin (defined as a percentage of price); we do thisfor � = 1�5�10�20. To compute the expected behav-ioral solution EX�, we calculate the behavioral densityusing (4).
Now, we make some observations using Figure 1.Note that in all six plots, along the x-axis, thereis some profit-margin level (call it PML) where thecurves corresponding to different values of � approx-imately intersect. For profit margins below PML, theexpected order quantities EX� tend to increase as thebounded rationality parameter � increases; in contrast,the reverse is true for profit margins above PML. Thissuggests that with triangular demand distributions,bounded rationality leads to an increase in order quan-tities under low-margin conditions, but it leads to adecrease in order quantities under high-margin con-ditions. This is reminiscent of results with the flavor
Triangular distribution withrange [0, 100] and peak = 0
Triangular distribution withrange [0, 100] and peak = 20
Triangular distribution withrange [0, 100] and peak = 40
Triangular distribution withrange [0, 100] and peak = 100
Triangular distribution withrange [0, 100] and peak = 80
Triangular distribution withrange [0, 100] and peak = 60
β = 1β = 5β = 10β = 20
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Exp
ecte
d or
der,
EX�
Exp
ecte
d or
der,
EX�
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Notes. The price is fixed at $1. In each case, we plot expected orders EX� against profit margins (as a fraction of price) for �= 1�5�10�20.
of regression to the mean except that here, boundedrationality is pushing order quantities toward the mid-point m of the range of possible demand realizations(namely, m= 50) instead of the mean, which is �100+h�/3. Another observation is that the threshold profit-margin level PML, which distinguishes low-marginconditions from high-margin conditions, decreases asthe peak density h increases. In our numerical exam-ple, as h increases from 0 to 100, we see that PMLdecreases from 0.75 to 0.25 (approximately).
4.3. General DemandWhen the newsvendor faces a general demand dis-tribution, the behavioral solution cannot be expressed
in terms of explicit distributions such as the (trun-cated) normal. Nevertheless, we show that it is possi-ble to characterize the expected order quantities andexpected profits in the general case.
The key observation is that the behavioral solu-tion X� belongs to an exponential family of proba-bility distributions, parameterized by the price p andcost c.
Definition. A family .H01 of probability distribu-tions is said to form an s-dimensional exponentialfamily if these distributions have densities of the form
The parameters 0 are referred to as the naturalparameters.
Observation. The family of choice distributions.�p�c1, where �p�c has density
p�c�x�=e�pEmin�D�x�−cx�/�∫
Se�pEmin�D�v�−cv�/� dv
� (11)
forms a two-dimensional exponential family,with natural parameters 01 = p/��02 = c/��T1�x� =Emin�D� x��T2�x�=−x� l�x�≡ 1, and
A�01�02�= ln(∫
Se�01Emin�D�v�−02v� dv
)� (12)
We proceed to state a well-known fact about expo-nential families.
Fact. Let X be from an exponential family with den-sity (10). Then,
E�Ti�X��= 5
50i
A�0�� (13)
Based on this, it is then straightforward to writedown the following result.
Proposition 2. Let X� be the behavioral solution to thenewsvendor problem with price p and cost c, so the naturalparameters for the choice distribution are 01 = p/� and02 = c/�. Then, we have
EX� =− 5A
502�01�02�� (14)
E��X��= p5A
501�01�02�+ c
5A
502�01�02�� (15)
This result highlights the important role played bythe function A�01�02� in our model of bounded ratio-nality based on logit probabilities. Through this func-tion, we can calculate moments of interest, such asexpected order quantities and expected profits. Thisapproach is useful for empirical and numerical stud-ies in two ways. First, it allows us to replace integra-tion (expectation) with differentiation, which is easierto compute. Instead of taking an expectation by inte-grating the density function (11), we can simply dif-ferentiate A�01�02� and approximate its value usingtwo data points. Second, it offers a way to estimatethe value of � using aggregate data via the method ofmoments. For example, given laboratory data on aver-age order quantities, an estimate of � would be thevalue at which the partial derivative −5A/502, evalu-ated at 01 = p/��02 = c/�, matches the observation.
5. Empirical Evidence forBounded Rationality
The goal provides empirical evidence for our modelof bounded rationality using a data set of news-vendor-type decisions made by individual subjects. Inparticular, we specify a statistical model for newsven-dor decisions. We fit our model to the data to obtainmaximum-likelihood estimates of the bounded ratio-nality parameter �. Finally, we show that our fittedmodel explains the data significantly better than thealternative that does not take bounded rationality intoaccount.
First, we describe the data set. This data set con-sists of a series of newsvendor ordering decisionsmade by human subjects. Each subject participatedeither in the low-profit or high-profit condition andmade a sequence of 100 ordering decisions for thesame parameter values. For the high-profit condi-tion, demand is uniform between 1 and 100, priceis 12, and cost is 3, so the optimal ordering quantityis 75. For the low-profit condition, demand is uni-form between 51 and 150, price is 12, and cost is 9, sothe optimal ordering quantity is again 75. There are20 subjects participating in the low-profit conditionand 18 subjects for the high-profit condition, so thedata consists of 3,800 quantity decisions altogether.Readers are referred to Bolton and Katok (2008) formore details on the experimental procedures used incollecting this data.
Our statistical model is
Yk =X�k + �k� (16)
where Yk is the observed order quantity for deci-sion k, X�
k follows the same distribution as the behav-ioral solution X� obtained in §4, and �k are i.i.d. errorterms. Because the demand is uniformly distributed,we know from §4.1 that the behavioral solution istruncated normal with mean at the optimal quantityx∗ = 75 and standard deviation 7 ≡√
���b− a�/p�. Weassume that
X�k ∼N†�x∗� 72�� (17)
�k ∼N�0�)2�� (18)
where N† denotes the truncated normal distribution.There are two parameters 7 and ) to be estimated.
where L and U denote the lowest and highest pos-sible demand realizations, *�·� denotes the standardnormal probability density function, and ��·� denotesthe probability distribution function of the behavioralsolution X� as given in (17). We stress that the per-fect rationality model, under which 7 = 0 and thusX� ≡ x∗, is a special case of our model. In particular,to investigate whether the data suggests the presenceof bounded rationality, we may test whether 7 = 0.
Our estimation strategy follows a data subsamplingapproach analogous to the bootstrap. We generatebootstrap samples from the data set as follows. Let yijdenote the jth quantity decision made by subject i,where i ∈ .1� � � � � I1 and j ∈ .1� � � � � J 1. Here J = 100is the total number of decisions made by each sub-ject and I is the number of subjects (I = 18 in thehigh-profit condition and I = 20 in the low-profitcondition). Then, to generate each bootstrap sample.z1� � � � � zI 1, we randomly sample each zi uniformlyfrom .yij @ 1≤ j ≤ J 1. Each bootstrap sample is indica-tive of the quantity decisions made by our subjectpopulation. For each bootstrap sample (of size I), weobtain the parameter estimates �7� �) that maximizethe likelihood function (19) above. We used a totalof B= 10�000 bootstrap replicates. In other words, weobtain B estimates (one from each bootstrap replicate)of 7 and ) in the model (16)–(18). Using the 2.5thand 97.5th percentile of these estimates, we obtainbootstrap confidence intervals for our parameters esti-mates �7� �) . Our results are summarized in Table 2.Given these estimates, the log-likelihoods of our fittedmodel are −78.81 and −86.15 for the high-profit andlow-profit conditions, respectively.
Next, as a benchmark for comparison, we fit thedata to the reduced model with 7 = 0, which corre-sponds to perfect rationality. Using the same boot-strap samples generated above, we can obtain themaximum-likelihood estimate �) and then use it tocompute the likelihood of our fitted model. In Table 3,we report the log-likelihood values of our full model(above) and the reduced model here, for both low-profit and high-profit conditions.
Table 2 Maximum-Likelihood Estimates of and UnderBoth Low-Profit and High-Profit Conditions
In terms of fit, we are interested in how well ourmodel (16) performs compared to the reduced modelwith 7 = 0. Because our full model has one additionalparameter, it naturally performs better, so we needto penalize the additional degree of freedom in someway. One common criterion for model selection is theBayes information criterion (BIC),
BIC= l�A�− d log�n�2
� (20)
where l�A� is the log-likelihood of the fitted model, Aare the fitted parameters, d is the number of parame-ters, and n is the data set size. Compared to alterna-tives such as the Akaike information criterion (AIC)and Mallow’s Cp criterion, the BIC is a relatively con-servative model selection criterion that favors simplermodels. In the present context, the BIC of our fullmodel is
BICfull = l��7� �)�− d log�n�2
� (21)
while the BIC of our reduced model is
BICreduced = l� �)�− d log�n�2
� (22)
These values are reported in Table 4. BecauseBICfull > BICreduced, this criterion, despite its conserva-tive nature, chooses the full model over the reducedmodel. This suggests that our model with bounded
Table 3 Log-Likelihoods of Full and Reduced Models Under BothLow-Profit and High-Profit Conditions
High-profit condition Low-profit condition
Log-likelihood of full −78�81 −86�15model, lfull
Log-likelihood of reduced −81�92 −88�97model, lreduced
Table 4 Bayesian Information Criterion of Full and Reduced ModelsUnder Both Low-Profit and High-Profit Conditions
High-profit condition Low-profit condition
BIC of full model −81�70 −89�15BIC of reduced model −83�37 −90�47
rationality is preferred over the alternative model withperfect rationality.
As a final test, we consider the likelihood ratiotest. Here we wish to test the hypothesis that 7 = 0.Because the log-likelihoods of the full and reducedmodels, lfull and lreduced, are given in Table 3, we candirectly compute the test statistic B2 = 2�lfull − lreduced�.This yields B2 = 6�22 and B2 = 5�64 for the high-profit and low-profit conditions, respectively. Underthe assumptions of our full and reduced models, thetest statistic follows a B2-distribution with one degreeof freedom, which has a critical value B2
1 �0�95�= 3�84.Because our test statistics exceed the critical value, wereject the hypothesis that 7 = 0. In other words, thissuggests that there is significant evidence for boundedrationality (with �> 0) in our model.
6. Distortion in Order QuantitiesIn this section, we investigate the effect of boundedrationality on expected order quantities. How doesthe behavioral mean order quantity EX� differ fromthe optimal solution x∗? We would like to distin-guish the situations in which the boundedly ratio-nal newsvendor overorders (i.e., when EX� < x∗) fromthe situations in which he underorders (i.e., whenEX� > x∗). We identify two underlying effects thatmay cause such distortions.
First, when the newsvendor is boundedly rational,order quantities tend to be biased toward the mid-point of the range of possible demand realizations.We call this the midpoint bias. The anchoring heuristic(Tversky and Kahnemann 1974) provides a behavioralrationale for this effect. When the demand density f
has support &a� b', so that a and b are the smallest andlargest possible demand realizations, the midpoint ism≡ �a+b�/2. The following result establishes this biasfor the special case of uniform demand.
Proposition 3. Suppose that the demand density f isconstant over &a� b'. Then, there is underordering whenx∗ >m and overordering when x∗ <m.
There is an equivalent way to describe this result.Let us say that the newsvendor has a high-profit prod-uct if the critical fractile � ≡ c/p < 0�5 and a low-profit product if the critical fractile � ≡ c/p > 0�5.Equivalently, p > 2c for a high-profit product andp < 2c for a low-profit product. Then, for uniformlydistributed demand, there is underordering for high-profit products and overordering for low-profit prod-ucts. This terminology was introduced by Schweitzerand Cachon (2000), who also provide experimen-tal evidence for this result. Specifically, in theirstudy, demand was uniformly distributed between 0and 300, and p = 12. For the low-profit condition,c= 9 (i.e., the critical fractile � = 75%), and for thehigh-profit condition, c = 3 (i.e., the critical fractile� = 25%). Using data from 33 subjects, each making 15newsvendor decisions for each condition, they foundthat in the high-profit condition, the average orderwas significantly lower than the optimal order, and inthe low-profit condition, the average order was signif-icantly higher than the optimal order. Schweitzer andCachon (2000) considered many explanations for theirobservations (including risk aversion, loss aversion,waste aversion, and stockout aversion), and identifiedtwo consistent explanations: preferences to reduceex-post inventory error, and the anchoring and insuf-ficient adjustment bias. Here, we show that underbounded rationality, the midpoint bias provides apossible alternative explanation.
Next, we describe the second decision bias. Con-sider the case where the demand density is monotone.An increasing density suggests that demand is morelikely to be high, whereas a decreasing density sug-gests that demand is more likely to be low. Our nextresult shows that the behavioral solution is distortedin the direction of low-probability demand realiza-tions. We call this the rare-event bias. One potentialbehavioral explanation is that decision makers tend toplace excessive weight on rare occurrences (as in theprobability weighing function of cumulative prospecttheory in Tversky and Kahneman 1992). To control formidpoint bias described earlier, we assume that theoptimal solution occurs precisely at the midpoint m.
Proposition 4. Suppose that x∗ = m. Then, there isoverordering when f is decreasing over &a� b' and there isunderordering when f is increasing over &a� b'.
Figure 2 Newsvendor Profit Functions (Solid Lines) for Linear Probability Densities (Dashed Lines) over �0�100�
0 20 40 60 80 1000
50
100
150
200
Exp
ecte
d pr
ofit,
π(x
)
Exp
ecte
d pr
ofit,
π(x
)
0
0.005
0.010
0.015
0.020
Pro
babi
lity
dens
ity
Pro
babi
lity
dens
ity
0 20 40 60 80 1000
50
100
150
200
Order quantity, xOrder quantity, x
0
0.005
0.010
0.015
0.020
Notes. On the left, p= 10� c= 2�5, so the optimal quantity is 50. On the right, p= 10� c= 7�5, so the optimal quantity is again 50.
This result becomes quite intuitive when one visu-alizes the shape of the profit function ��x�. Let usconsider the two examples in Figure 2, with linearlyincreasing or decreasing demand densities. Given thatthe optimal solution x∗ occurs at the midpoint m,when f is decreasing (as in the left panel), profitsfall from the optimal level faster when x is decreasedbelow x∗ compared to when x is increased above x∗.This implies that overordering is less costly comparedto underordering, and hence occurs more frequently.The combined effect is that the expected behavioralorder EX� exceeds the optimal quantity x∗, so there isoverordering on average. Similarly, the reverse is truewhen f is increasing (as in the right panel). Note thatthe rare-event bias may distort order quantities awayfrom both the mean and the median. For example,in Figure 2, when the demand density is decreasing(on the left panel), the mean and median demandsare both smaller than m, but the expected order quan-tity EX� is greater than m. This suggests that resultsof regression toward the mean (typically for uniformdemand settings) do not capture the complete picture.
In the analysis above, we have made assumptionsto isolate the two decision biases from each other.In Proposition 3, to control for the rare-event bias,we focus on uniform demand with all realizationsequally likely. In Proposition 4, to suppress the mid-point bias, we assume that the optimal solution islocated at the midpoint. However, in most instancesof the newsvendor problem, both effects are presentand may run in opposite directions.
Next, we provide a numerical example to illustratethe combined effect of both decision biases. In thisexample, we assume that the demand follows a trun-cated normal distribution with ( = 100, ) = 20 andsupport &0�200'. We assume that p = 10 and sepa-rately consider cases with c = 1 and c = 9. The opti-mal solution x∗ is obtained using the critical fractile,and the mean behavioral solution EX� is computedusing the procedure outlined in Proposition 2. Theresults are summarized in Figure 3, with each panelcorresponding to each value of c. In each panel, thesolid lines represent the behavioral expected ordersEX� and the dashed lines represent the optimal orderquantities x∗. When c = 1, the optimal order quantityx∗ exceeds (, so the midpoint bias distorts decisionsdownwards; however, because the normal density isdecreasing at x∗, the rare-event bias distorts decisionsupwards. When these two effects are put together,we observe overordering for small values of � andunderordering for larger values of �. This suggeststhat the rare-event bias is dominant for small valuesof �, while the midpoint bias is dominant for largevalues of �. When c = 9, the directions of both deci-sion biases are reversed. Nevertheless, a similar argu-ment leads to the same conclusion: As � increases,decisions are first governed by the rare-event bias andthen the midpoint bias takes over at larger valuesof �. The reason becomes clear when one recalls thatas � → �, the choice distribution becomes uniformover the choice domain, and the expected behavioralorder EX� coincides with the midpoint m in the limit.
Figure 3 Behavioral (Solid) and Optimal (Dashed) Solutions for Truncated Normal Demand with �= 100� = 20 and Support �0�200�, p = 10, andc= 1 (Left), c= 9 (Right)
0 200 400 600 800 1,000110
115
120
125
130
135
140
β
Ord
er q
uant
ity
Ord
er q
uant
ity
0 200 400 600 800 1,00060
65
70
75
80
85
90
β
It is therefore not surprising to find that the midpointbias dominates for large values of �.
7. Supply Chain CoordinationThe newsvendor model is an indispensable buildingblock in the operations literature on supply chaincoordination and contracting. The literature recog-nizes the problems caused by decentralized decisionmaking in a supply chain: When different parties actaccording to their own interests, systemwide opti-mal performance cannot be attained. The generalparadigm to solve these double marginalization prob-lems is to design coordinating contracts that alignthe incentives of individual parties with the objec-tives of the entire supply chain. In this section, wesee that in the presence of bounded rationality, sup-ply chain coordination can no longer be achieved inthis manner.
The basic model of a decentralized supply chainconsists of a single manufacturer and a single retailer.The manufacturer produces the good at unit cost c
and sells it to the retailer at the wholesale pricew ≥ c. The retailer decides how many units x toprocure before selling to the market at price p. Hisprofit function is �R�x� = pEmin�D�x� − wx, whiletotal profits for the supply chain is given by �S�x�=pEmin�D�x�− cx. Although it is well known that theorder quantity x∗
S that maximizes total supply chainprofits satisfies �F �x∗
S�= c/p, the retailer’s profit-maxi-mizing order quantity x∗
R satisfies �F �x∗R�= w/p ≥ c/p.
Therefore, the retailer orders too little. Standard the-ory concludes that system profits are lost, but under
bounded rationality, it is possible to construct exam-ples in which double marginalization is beneficial.(Intuitively, this may occur when the centralized sellersystematically overorders and double marginalizationhelps to correct for these errors.)
How can the supply chain optimum be achieved?The literature has proposed many alternative solu-tions. First, consider the buy-back contract. Under thiscontract, the manufacturer agrees to buy back unsoldunits at the buy-back price b. In this case, the retailerfaces the profit function �R�x�= �p− b�Emin�D�x�−�w − b�x. When the contractual parameters w�b arechosen such that for some E ∈ &0�1', b= �1−E�p, andw= �1−E�p+Ec, the retailer’s profit function becomes�R�x�= E&pEmin�D�x�− c'= E�S�x�, which is a con-stant fraction of total supply chain profits. Therefore,by maximizing his own profits, the retailer is alsomaximizing supply chain profits because the payoffis always a constant fraction E of total profits. In thisway, the social optimum can be attained even thoughthe ordering decision is made by the retailer consider-ing only his own profits. Another possible alternativeis the revenue-sharing contract. Under this contract,the retailer agrees to share sales revenue withthe manufacturer, specifically, to pay the manufac-turer r for every unit sold. The retailer’s profit func-tion becomes �R�x�= �p− r�Emin�D�x�−wx, whichreduces to �R�x�= E&pEmin�D�x�− c'= E�S�x� whenwe choose r = �1 − E�p and w = Ec. As before,the retailer now enjoys a fixed E share of totalsystem profits. The supply chain is thus similarlycoordinated.
As we have seen, one general approach in supplychain coordination is to align individual incentiveswith social objectives by devising contractual trans-fers so that the decision maker’s payoff function�R�x� is a constant proportion E of the social wel-fare function �S�x�. While this approach is validunder perfect rationality, the next result shows that itdoes not always achieve coordination under boundedrationality.
Under boundedly rational decision making, we saythat the system is coordinated when decentralizedcontrol yields centralized profit levels. In other words,verifying system coordination involves comparing:(i) a decentralized system in which inventory deci-sions are made by a boundedly rational retailer, and(ii) a centralized system controlled by a boundedlyrational newsvendor. The decision makers in bothcases share the same bounded rationality parame-ter �. When decentralized expected profits reach thesame level as centralized profits, we say that systemcoordination is achieved.
Proposition 5. Let X�1 and X�
2 denote the behavioraldecisions of two decision makers with the same boundedrationality parameter �, but facing different utility func-tions u1�x�= E1��x� and u2�x�= E2��x�, where E1 >E2.Then, E��X�
1� > E��X�2�.
Corollary 2. Consider a supply chain facing news-vendor profit function ��x�= pEmin�D�x�− cx. Let X�
i
denote the behavioral solution when the decision makerenjoys Ei share of the total profits. Then, when E1 >E2, wehave E��X�
1� > E��X�2�.
In a centralized system, the newsvendor’s share oftotal profits is E = 1. In a decentralized system, thedecision maker (retailer) enjoys only a reduced shareof E < 1. Assuming the same bounded rationalityparameter � for both decision makers, our result indi-cates that total expected profits are lower in the decen-tralized system. This shows that even when incentivesare perfectly aligned, coordination cannot be attainedwhen decision makers are boundedly rational.
This theoretical result is consistent with recent ex-perimental findings in the literature. Katok and Wu(2006) designed experiments on supply chain con-tracting and demonstrate that under contracts that aretheoretically proven to coordinate the system, humansubjects make ordering decisions that do not lead to
perfect coordination. Their findings are based on buy-back contracts and revenue-sharing contracts. Theynot only discover that both contracts do not achievecoordination, but also identify systematic differencesbetween these two contractual forms. Our currentmodel of bounded rationality explains why coordina-tion does not occur, but does not distinguish betweendifferent contractual forms that align incentives in thesame way.
The novel finding herein is that it is not sufficientto align the ratio of marginal costs and benefits; theactual margins must be aligned to achieve coordina-tion. In other words, for the newsvendor problem, itis important to align the actual overage and underagecosts (two numbers) instead of simply aligning theretailer’s critical fractile (one number). In particular,when the retailer receives a fixed share of total sys-tem profits (this aligns the ratio of margins but scalesdown both underage and overage costs), there is anefficiency loss. One possible explanation is that whenpayoffs are scaled down, the decreased stakes held bythe decision maker make choices more prone to errorsand biases. This suggests that for an agent’s decentral-ized decisions to coincide with the system’s central-ized decisions, the agent must be the sole stakeholderof the system. This points to the strategy of selling thefirm to the agent, which can be implemented usinga two-part tariff. Here, the manufacturer charges theretailer a fixed fee T and then sells to him at cost.In this case, the manufacturer’s transfer payment T
does not affect the retailer’s decisions, which are thusmade from the perspective of a sole owner. Therefore,two-part tariffs can coordinate the supply chain underour model of bounded rationality.
We may even go one step further and argue thatin our model, there is potential for supercoordination.That is, expected profits may be higher in a decentral-ized system (under bounded rationality) relative to acentralized system (also under bounded rationality).By Proposition 5, this may occur if each ordering deci-sion made by the retailer accounts for E > 1 timesof total system profits. This can be implemented inseveral ways. First, suppose that a manufacturer sellsto a retailer who is constrained to place the sameorders over n periods. Under some contract that givesthe retailer E share, the payoff from each decision isessentially nE times of the system’s per period profits.
Decentralization is thus beneficial as long as nE > 1.Bolton and Katok (2008) experimentally demonstratethe benefits of having retailers commit to their stand-ing orders over multiple periods. However, we stressthat these benefits should be attributed to commit-ment (i.e., to placing the same orders several times)rather than decentralization per se, although the lattermay facilitate such commitment.
Another way to achieve supercoordination is to usesales rebates. Suppose that the manufacturer chargesa wholesale price w but offers a sales rebate of r perunit (so the retailer makes p+ r from each unit sold).For some E > 1, let r = �E − 1�p and w = Ec. Let usalso use a fixed transfer T to allocate some surplusto the manufacturer (so this is a two-part tariff cou-pled with a sales rebate). Then, the retailer’s payofffunction is E�S�x�−T . Because the transfer T does notaffect retailer decisions, and because E > 1, expectedprofits are higher here than in the centralized systemwith profit function �S�x�.
In general, increasing the monetary stakes associ-ated with each decision epoch generates better deci-sions. From a behavioral standpoint, this is intuitivebecause people tend to allocate more cognitive andcomputational effort into more important tasks. In ourmodel, note that increasing E achieves the same effectas decreasing �; essentially, a decision maker facingincreased stakes is akin to one who is more rational.Because E cannot increase indefinitely (i.e., E → �),this suggests that all the scenarios discussed in thissection cannot match the ideal centralized benchmarkwith a perfectly rational decision maker (with �= 0).Therefore, the reader should be cautious when inter-preting our notion of supercoordination.
In summary, we have seen how bounded rationalitycan enrich the supply chain coordination framework.While the conventional normative approach predictsthat perfect coordination is attained as long as thedecision maker’s payoff function is a fixed E share oftotal profits, our descriptive model of bounded ratio-nality shows that the magnitude of E also plays an im-portant role. Specifically, when E< 1, as in most casesstudied in the literature, perfect coordination is notachieved. In contrast, contractual arrangements maygive rise to individual decisions with E > 1, underwhich there is supercoordination.
8. Bullwhip EffectIn this section, we discuss the relationship betweenbounded rationality and the bullwhip effect. Thebullwhip effect refers to a commonly observed phe-nomenon in supply chains: The variance of orderstends to increase dramatically as we move upstreamalong the supply chain. In a seminal analysis, Leeet al. (1997) identify four different sources of the bull-whip effect: demand forecasting, inventory rationingdue to supply constraints, order batching, and pricefluctuations. They show that each of these factors canindependently generate the bullwhip effect.
We analyze a model that eliminates the physicalcauses of the bullwhip effect that have been identifiedin the literature. In other words, in our setup, a stan-dard normative analysis following the conventionalparadigm of perfect rationality would not yield thebullwhip effect. However, we find that once boundedrationality is introduced, the bullwhip effect emerges.This suggests that apart from the physical causes ofthe bullwhip effect, which have been well studied,there are also behavioral causes such as boundedrationality.
The model consists of a serial supply chain indexedby i = 1� � � � �n. Market demand is fulfilled at stagei= 1, each member at stage i procures supply fromthe adjacent upstream member at stage i+1, and pro-duction is initiated at stage i = n. We assume thatmarket demand D, with distribution F , is indepen-dent across periods. We consider an infinite-horizon,discrete-time model, with the following sequence ofevents in each time period. First, units previouslyshipped from the upstream neighbor are received (atstage i = n, units entered into production are com-pleted). Let xi denote the inventory position at stage i
at this point. Second, demand Di is realized at eachstage i; this refers to market demand D at stagei= 1 and orders from downstream neighbors at stagei= 2� � � � �n. Third, units are shipped to fulfill demand,lowering the inventory position at stage i to xi −Di.If this is positive, leftover inventory is carried overto the next period, incurring a holding cost of hi perunit. If this is negative, there is a backordering cost ofbi per unit. (Alternatively, we may assume that thereis an alternative supply source from which units canbe borrowed at a cost of bi per unit.) Finally, unitsare ordered from the upstream neighbor. We use Oi to
denote the order submitted by member i to memberi+ 1 at the end of the period.
Let us see how this model eliminates all four phys-ical causes of the bullwhip effect. First, there are nodemand forecasting effects because market demand isi.i.d. across periods with a commonly known distri-bution. Second, there is no inventory rationing, whicheliminates strategic gaming effects. Third, becausethere is no fixed cost, order batching is irrelevant.Finally, there are no price fluctuations in this model.
Under perfect rationality, this model can be solvedusing a standard newsvendor analysis. Consider thedecision problem at stage i= 1 of choosing the inven-tory level xi. The trade-off is between ordering toomuch (incurring excessive holding costs) and order-ing too little (incurring excessive borrowing costs).The problem can be formulated as
minx∈S
hE&x−D'+ + bE&x−D'− (23)
⇔ minx∈S
�h+ b�E&x−D'+ − bE�x−D� (24)
⇔ maxx∈S
−�h+ b�E&x−D'+ + bx (25)
⇔ maxx∈S
�h+ b�Emin�D�x�−hx� (26)
where the subscript i has been omitted for brevity.This is a newsvendor problem with p = h + b andc= h, so the optimal solution x∗ satisfies F �x∗� =b/�h+ b�. The optimal order-up-to level x∗ representsa target inventory position that the decision makerwould like to maintain. At stage i = 1, for a givenorder-up-to level x∗
1 , orders submitted at the end ofeach period are always equal to the demand real-ization in that time period, so that inventory wouldbe brought back up to the target level x∗
1 in thenext period. This implies that demand faced by theadjacent upstream member i = 2 is equal to mar-ket demand, but lagged by one time period. Thisupstream member thus faces the same demand dis-tribution, solves a similar newsvendor problem, andsubmits orders that are equal to market demand (butnow lagged by two periods). Using an inductive argu-ment, it then follows that the orders placed by eachsupply chain member are all lagged versions of thesame market demand, so these orders follow the samedistribution F . In other words, the variance of ordersremains constant throughout the supply chain, andthere is no bullwhip effect.
Now let us see what happens under bounded ratio-nality. For simplicity, we begin with a two-echelonmodel (n = 2), although the analysis carries forwardto the general case. We also assume that the decisiondomain S = � is stationary. This yields the followingresult.
Proposition 6. Suppose that the decision makers atstage i = 1�2 are boundedly rational with parameter �i.Then, there exist independent random variables �1� �2such that
O1 =d D+ �1� (27)
O2 =d D+ �1 + �2� (28)
and Var��i�= 0 if and only if �i = 0.
Corollary 3. (i) Var�Oi� = Var�D�+∑ij=1 Hj , where
Hi = 0 if and only if �i = 0.(ii) Var�Oi� ≥ Var�Oi−1�, with equality if and only if
�i = 0.
This result demonstrates that the bullwhip effectcan persist even when its physical causes have beenremoved. As long as some agents in the supply chainare boundedly rational (with �i > 0), the variance oforders will strictly increase upstream along the supplychain.
There is ample experimental evidence in the liter-ature showing that the bullwhip effect persists evenwhen its physical causes have been controlled for.In a seminal study that predates the taxonomy ofthe causes of the bullwhip effect in Lee et al. (1997),Sterman (1989) controls for three causes (inventoryrationing, order batching, and price fluctuations), andleaves demand signal processing as a potential causebecause subjects were not informed of the demanddistribution. Subsequently, the experiment in Crosonand Donohue (2006) also controls for the fourthcause by using a publicly known (uniform) demanddistribution. In both studies, the bullwhip effect isobserved. Furthermore, both studies identified thefailure to account adequately for the supply line as animportant cause of the bullwhip effect; that is, subjectsdo not fully account for orders that have been placedbut have not yet arrived. The framework of analysis isdue to Sterman (1989). Subjects are assumed to makeordering decisions using the anchor-and-adjustmentheuristic (Tversky and Kahnemann 1974): They first
anchor on the expected demand rate, and then adjusttheir order quantities to correct discrepancies betweendesired and actual stock, both on hand and in the sup-ply line. In both studies, fitting experimental data tothis heuristic decision rule yields parameter estimatesthat demonstrate underweighting of the supply line,which leads to overordering and system instability. Incontrast to these studies, there is no supply line in oursetup. Orders placed at the end of each period willalways arrive at the beginning of the next period. Yet,our analysis suggests that the bullwhip effect remains.What, then, is causing the bullwhip effect?
Our results provide an alternative behavioral expla-nation, which we demonstrate analytically. We pointout that the bullwhip effect is potentially a phe-nomenon that arises whenever individuals attempt toguard against and correct for the mistakes that oth-ers may make. The underlying mechanism is madetransparent in Proposition 6. In the normative model,orders at every stage should follow the same distri-bution as the market demand, so Var�Oi� = Var�D�
for each i. Now, suppose that the decision maker ati= 1 is boundedly rational but the one at i= 2 is per-fectly rational, i.e., �1 > 0 and �2 = 0. Then, while itis clear that Var�O1� > Var�D�, we also end up withVar�O2� >Var�D�. The order variance at i= 2 has beenincreased from the normative benchmark of Var�D�
even though the decision maker there is perfectlyrational. This increase arises solely because of theneed to recognize the decision biases of downstreammembers. Of course, if the decision maker at i = 2is also boundedly rational, the order variance will beincreased further. This in turn increases the burden onupstream members to account for his decision biases,and the variance increase is propagated upstream.The bullwhip effect thus arises.
There is experimental evidence supporting this ex-planation. Croson et al. (2005) conduct an experi-ment in which the demand is constant (four units perperiod) and commonly known, and the system beginsin equilibrium. With this setup, it is optimal to orderfour units every period at each stage. Yet, in theirexperiment, this does not happen and the bullwhipeffect is still observed. Postexperimental question-naires suggest that although subjects realized whatthe optimal policy was, they were uncertain whether
the other players understood it. Lack of trust in oth-ers’ actions may generate suboptimal order quantities.Once initial deviations from optimal orders occur, thesystem is knocked into disequilibrium and the bull-whip effect eventually occurs. The authors call thiscoordination risk. Our model of bounded rational-ity complements this work by offering a quantitativeframework to model coordination risk. For example,applying our framework to their experimental setupwith D ≡ 4, we can use Proposition 2 to iterativelycompute the behavioral solutions X�
1, X�2, X
�3, X
�4, using
the order distribution at each stage as the demanddistribution at the next upstream stage. Experimentalresults can then be used to estimate the values of �i
at each stage.In summary, our model of bounded rationality
complements the recent behavioral operations liter-ature by showing that the bullwhip effect can ariseeven in the absence of its physical causes. Even inthe absence of a supply line (i.e., zero lead time), ourmodel still generates the bullwhip effect. This bringsout another relevant behavioral phenomenon (apartfrom underweighting of the supply line): Decisionbiases may not be errors in themselves, but rather,appropriate safeguards taken when one is not suffi-ciently confident in others’ actions.
Nevertheless, the conclusion that there are behav-ioral causes of the bullwhip effect stops short of amore general treatise. One of the fundamental goalsof any supply chain is to match supply with demand,and in this regard, orders generated within the sys-tem are driven by both demand and supply processes.Demand processes trigger order incidence, and corre-sponding supply processes trigger order fulfillment.Naturally, there is uncertainty in both demand andsupply processes, and this generates variance ampli-fication upstream along the direction of order flow.Although the causes of demand uncertainty, partic-ularly those pertinent to the bullwhip effect, havebeen well studied, the causes of supply uncertaintyhave received less attention. Bounded rationality andits associated behavioral phenomena are one suchcause because they introduce noise into an other-wise deterministic decision-making process. Similarly,other sources of supply uncertainty, such as randomyield, have analogous interpretations. In general, our
results suggest that, after controlling for the physi-cal causes of demand uncertainty, the bullwhip effectmay persist as a result of supply uncertainty. Inter-estingly, recent experimental findings by Rong et al.(2006) suggest that under supply disruptions, theremay be a reverse bullwhip effect that causes varianceamplification downstream rather than upstream.
9. Inventory PoolingThis section investigates the impact of boundedrationality on the benefits of inventory pooling, acommonly used strategy in multilocation inventoryproblems. When demand occurs at different locations,instead of holding separate stocks for each source ofdemand, firms may alternatively pool inventory atsome centralized location. Most of the literature oninventory pooling is based on the newsvendor model.The general conclusion is that centralization generatespooling economies that lead to lower costs (specifi-cally, holding costs and backordering costs) for thesystem. We are interested in whether this result con-tinues to hold when the decision makers are bound-edly rational.
We use the following setup in our analysis. This issimilar to the setting used in Eppen (1979). Considerthe following single-period multilocation newsvendorproblem. There are n different sources of demand,each occurring at a different location. Let Di be thedemand at location i for i= 1� � � � �n, and let Fi be itsdistribution. We assume that the demand vector fol-lows a multivariate normal distribution. Let (i and )2
i
denote the mean and variance of Di, and let )2ij and Iij
denote the covariance and correlation coefficient of Di
and Dj . First, we treat the decentralized case; that is,separate inventories are maintained at each location.In this case, the decision variables xi are the quantitiesto hold on hand at each location i. There is a hold-ing cost of h for each unit left unsold at the end ofthe period, and there is a backlogging penalty of b foreach unit of demand that cannot be fulfilled. In otherwords, the goal at each location i is to minimize thecost function
Ji�xi�= hE&xi −Di'+ + bE&xi −Di'
−� (29)
This is equivalent to maximizing the newsvendorprofit function
�i�xi�= pEmin�Di� xi�− cxi� (30)
where p ≡ h+ b and c ≡ h. Clearly, the optimal solu-tion x∗
i at each location satisfies Fi�x∗i � = 1 − �c/p�.
Next, we treat the centralized case. Here, insteadof solving n separate newsvendor problems for thestocking levels xi at each location, there is only onecentralized stocking decision x to make. Let DT ≡∑n
i=1Di denote the total demand from all sources, andlet FT be its distribution. We use (T and )2
T to denotethe mean and variance of the total demand, so (T =∑n
i=1(i and )2T =∑n
i�j=1 )2ij . This total demand is to be
met from the same pool of inventory xT . The goal isto minimize total cost
JT �xT �= hE&xT −DT '+ + bE&xT −DT '
−� (31)
which is equivalent to maximizing the newsvendorprofit function
�T �xT �= pEmin�DT �xT �− cxT � (32)
As before, the optimal centralized solution satisfiesFT �x
∗T � = 1− �c/p�. The benefits of inventory pooling
can then be studied by comparing the optimal totalcosts in the two cases:
∑ni=1 Ji�x
∗i � in the decentralized
case and JT �x∗T � in the centralized case.
Under perfect rationality, when demands are nor-mally distributed, it is well known that optimaldecentralized costs
∑ni=1 Ji�x
∗i � and optimal central-
ized costs JT �x∗T � are, respectively, proportional to the
sum of standard deviations∑n
i=1 )i and the standarddeviation of total demand )T (with the same pro-portionality constant). Note that )T ≤ ∑n
i=1 )i, withequality, if and only if all the demands are perfectlycorrelated (i.e., Iij = 1 for all i� j). This implies thefollowing two results. First, the total cost in a decen-tralized system is at least as high as that in a central-ized system; in other words, inventory pooling savescosts. Second, these cost savings depend on the cor-relation between individual demands; in particular,there are no cost savings when all the demands areperfectly correlated with one another. These resultsare due to Eppen (1979).
Next we check whether these predictions are robustagainst bounded rationality. Let X�
i denote the behav-ioral solutions at each location i, and let X�
T denotethe behavioral solution at the centralized location.The following result provides sufficient conditions forpooling benefits to persist.
Proposition 7. Suppose that the following inequalitieshold:
(a) )T ≤∑ni=1 )i,
(b) )T ≥ )i for every i= 1� � � � �n.Then, we have EJT �X
�T � <
∑ni=1 EJi�X
�i �. In other words,
inventory pooling leads to a strict reduction in total costs.
Let us examine the two conditions of Proposition 7in greater detail. We already know that (a) alwaysholds, with equality, if and only if the demands at alllocations are perfectly correlated with Iij = 1. Condi-tion (b) says that the standard deviation of aggregatedemand is at least as large as the standard deviationin any one location, which is likely to hold in practi-cal situations (unless substantial negative correlationsexist between demand sources). Hence, under broadconditions, inventory pooling continues to generatecost savings.
Proposition 7 also shows that the reduction in totalcosts is strictly positive. For the extreme case with per-fectly correlated demands, there is no cost reductionunder perfect rationality. However, under boundedrationality, we observe strictly positive gains. Thissuggests that the benefits of inventory pooling extendbeyond the physical benefit that is related to thereduction in variance resulting from summing sepa-rate random demands.
There may also be behavioral benefits from inven-tory pooling. To see the intuition behind these ben-efits, let us consider the following example. Supposethat demand at each separate location is determin-istic (i.e., )i = 0). In this case, it is trivially optimalto stock the quantity that matches demand exactly.Therefore, under perfect rationality, there is no differ-ence between decentralization (with individual stocksx∗i = (∗
i ) and centralization (with total stock x∗T = (T )
because zero cost is attained in both cases. In con-trast, under our model of bounded rationality, deci-sion errors create a disparity between these two cases.Under decentralization, decision errors �X�
i − (i� ateach location accumulate and separately contributeto aggregate costs. However, under centralization,these decision errors may cancel out, thus decreasingexpected total costs. This example illustrates that in anenvironment with no demand uncertainty, inventorycentralization helps by pooling decision errors acrosslocations.
More generally, inventory pooling achieves twoeffects: It pools demand uncertainty as well as supplyuncertainty. The former is well understood and arisesbecause of variance reduction. The latter, however,deserves elaboration. Bounded rationality (in partic-ular, our model of decision noise) injects uncertaintyinto the supply process, and we have seen that pool-ing reduces the aggregate impact of such decisionerrors. Similarly, inventory pooling serves to attenuateother sources of supply uncertainty, for example, ran-dom yield, record inaccuracy, and processing errors.Small wonder that in Proposition 7, the benefits ofpooling persist even when demands are perfectly cor-related: These gains are the result of pooling supply(rather than demand) uncertainty.
10. ConclusionThe classic quantal choice paradigm posits that peopledo not make the best decision all the time, but theymake good choices more often than worse ones. Inthis paper, we use this framework to capture boundedrationality and apply it to several newsvendor-typeinventory settings. Our analysis generalizes existingresults and reconciles them with empirical obser-vations. This suggests that accounting for decisionnoise and optimization error is one possible way toenhance the predictive accuracy of theoretical models.We hope that our modeling approach serves to con-nect the theoretical and experimental literatures, andin so doing, stimulate future research on behavioraltheory in operations management.
We conclude with some suggestions for futureresearch. The first and most impending directionis experimental. While some of our findings havebeen experimentally validated by previous studies,many others remain untested. For example, stud-ies of newsvendor decision biases under nonuniformdemand would be a good starting point. The seconddirection is applications oriented. There is a wide vari-ety of situations in which decision makers (whetherthey are firms, workers, or customers) may displaysome extent of bounded rationality. In these cases,how are the current theoretical results affected? This isa broad question that equally applies across differentareas, such as operations strategy with boundedlyrational firms, revenue management with bound-edly rational customers, and staffing and human
resource management with boundedly rational work-ers. Another possible research focus is understand-ing the fundamental behavioral mechanisms respon-sible for biases and errors. Specifically, how does thebounded rationality parameter � depend on the natureof the operational task (e.g., complexity and context)?What are the effects of learning on �? How does indi-vidual heterogeneity in � affect the aggregate? Froman experimental viewpoint, how can � be manipu-lated? Finally, it would also be worthwhile to studythe combined effects of bounded rationality (decisionnoise) and other behavioral regularities. We believethat the quantal choice framework, being general yetparsimonious, is well suited to complement otherbehavioral theories.
AcknowledgmentsThe author thanks Elena Katok and Gary Bolton for shar-ing the data from their paper, Bolton and Katok (2008). Thecomments from the editors and reviewers have improvedthis paper significantly. The author has also received use-ful feedback from participants at the Behavioral OperationsConference and INFORMS Annual Meeting, as well as sem-inar participants from Columbia University, New York Uni-versity, Northwestern University, University of California,San Diego, and University of Washington at Seattle. Finally,the author would like to thank the production team for theirgracious assistance.
AppendixProof of Lemma 1. Because y ≡ ay+b, we have d y/dy =
a. Therefore, for any y0� y0 satisfying y0 ≡ ay0 + b, we have∫ y0
−�eu�y�/� d y = a
∫ y0
−�eu�y�/� dy� (33)
which implies that
���y0� =∫ y0−� eu�y�/� d y∫ �−� eu�y�/� d y = a
∫ y0−� eu�y�/� dy
a∫ �−� eu�y�/� dy
=∫ y0−� eu�y�/� dy∫ �−� eu�y�/� dy
=��y�� (34)
as desired. �
Proof of Lemma 2.
���y� =∫ y
−� eu�v�/� dv∫ �−� eu�v�/� dv
=∫ y
−� e�u�v�+a�/� dv∫ �−� e�u�v�+a�/� dv
= ea/�∫ y
−� eu�v�/� dv
ea/�∫ �−� eu�v�/� dv
=∫ y
−� eu�v�/� dv∫ �−� eu�v�/� dv
=��y�� � (35)
Proof of Proposition 1. The density of the behavioralsolution, from (4) and (5), is given by
�x�= e�Ax2+Bx+C�/�
∫ b
a e�Av2+Bv+C�/� dv� (36)
The density K�x� of a truncated normal random variableover &a� b' with mean ( and variance )2 is
K�x�= e−�x−(�2/2)2
∫ b
a e−�v−(�2/2)2 dv� (37)
Therefore, over domain &a� b', we have
�x�∝ e�Ax2+Bx�/�� (38)
K�x�∝ e−�1/2)2�x2+�(/)2�x� (39)
which implies that the behavioral solution has the truncatednormal distribution with parameters ( and )2 satisfying
A≡− p
2�b− a�=− 1
2)2�� (40)
B≡ pb
b− a− c= (
)2�� (41)
Solving these two equations yields the desired values of (and )2. �
Proof of Proposition 3. When the demand density isconstant, we have uniformly distributed demand. Recallfrom Corollary 1 that we have
EX� =(−) · *��b−(�/)�−*��a−(�/)�
+��b−(�/)�−+��a−(�/)�� (42)
where ( = b − �c/p��b − a�, )2 = ���b − a�/p�, and *�·� and+�·� denote the standard normal density and distributionfunctions. When x∗ = ( > m, we have �b − (� < �a − (�,so *��b − (�/)� > *��a − (�/)�, implying that EX� < ( =x∗, so there is underordering. The same argument showsoverordering when x∗ <m. �
Proof of Proposition 4. We first consider the casewhere f is decreasing over &a� b'. Denote r ≡ �b − a�/2, sob=m+ r and a = m− r . Thus, for any y ∈ �0� r', we havef �m+ y� < f �m− y�. Now, the assumption x∗ = m impliesthat c= p �F �m�, because �F �m�= �F �x∗�= c/p is satisfied at theoptimal solution x∗. It then follows that for any y ∈ �0� r',
Because the density of the behavioral solution X� satisfies�x�∝ e��x�/�, which is increasing in ��x�, we have �x∗+y�> �x∗ − y� for every y ∈ &0� r'. Together with the fact that
EX� =∫ b
ax�x�dx=
∫ x∗+r
x∗−rv�v�dv
=∫ r
−r�x∗ + v��x∗ + v�dv (51)
= x∗ +∫ r
−rv�x∗ + v�dv (52)
= x∗ +∫ r
0v&�x∗ + v�−�x∗ − v�' dv� (53)
we conclude that EX� > x∗, so there is overordering whenf is decreasing over &a� b'. The case with increasing f istreated similarly. �
Proof of Proposition 5. Let the density of X�i be
i�x� = eui�x�/�/∫S e
ui�v�/� dv for i = 1�2. Let K ≡∫S e
u1�v�/� dv/∫S e
u2�v�/� dv. Then, we have 1�x� > 2�x� ifand only if
e�u1�x�−u2�x��/� > K (54)
⇔ ��x� >� lnKE1 −E2
� (55)
Therefore, for any k≥ � lnK/�E1 −E2�, we have
P���X�1�≥ k� =
∫.v@��v�≥k1
1�v�dv >∫.v@��v�≥k1
2�v�dv
= P���X�2�≥ k�� (56)
Similarly, for any k≤ � lnK/�E1 −E2�, we have
P���X�1�≥ k� = 1−
∫.v@��v�≤k1
1�v�dv>1−∫.v@��v�≤k1
2�v�dv
= P���X�2�≥ k�� (57)
This shows that ��X�1� stochastically dominates ��X�
2�, sowe have our result. �
Proof of Proposition 6. Let X�i and x∗i denote the
behavioral solutions and optimal solutions at stage i= 1�2.Then, using time subscripts t to avoid ambiguity, we have
O1� t =X�1� t+1 − �X�
1� t −Dt�� (58)
This is because after period t, the inventory position hasbeen lowered from the previous target of X�
1� t by an amountequal to current demand Dt , and the order O1� t is placedto replenish inventory to the new target X�
1� t+1. Note that
although the targets X�1� t and X�
1� t+1 follow the same choicedistribution, the actual realizations may differ across timeperiods. Now, let us define �1 as the difference between twoindependent realizations of the random variable X�
1. Then,we have the first relation (27). Next, for stage i= 2, we sim-ilarly have
O2� t = X�2� t+1 − �X�
2� t −D2� t� (59)
= X�2� t+1 − �X�
2� t −O1� t−1� (60)
= �X�2� t+1 −X�
2� t�+ �X�1� t −X�
1� t−1�+Dt−1� (61)
Now, defining �2 as the difference between two independentrealizations of the random variable X�
2, we have (28). Finally,it is straightforward to show that
�i = 0 ⇔ P�X�i = E�X�
i ��= 1 ⇔ Var�X�i �= 0 (62)
using the Chebyshev inequality. Therefore, it follows thatVar��i�= 0 if and only if �i = 0. �
Proof of Proposition 7. In the proof, we maximize overprofit functions �i�xi� rather than minimize over cost func-tions Ji�xi�. For the stocking problem at each location i,consider the following transformation. Instead of choosingthe stocking quantities xi, we choose the standardized stock-ing quantities zi ≡ �xi − (i�/)i. The profit function over zishould satisfy �i�zi�=�i�xi�, so we have
�i�zi�= )ipEmin�Z�zi�−)iczi + �p− c�(i� (63)
where Z is a standard normal random variable. Let Z�i
denote the behavioral solution of maximizing �i�zi�. ByLemma 1, we know that behavioral solutions are invariantagainst affine transformations of the decision domain, soX�
i =(i +)iZ�i and E �i�Z
�i �= E�i�X
�i �. This justifies working
in the standardized decision domain. Next, by Lemma 2, weknow that behavioral solutions are not affected by transla-tions in the utility function, so Z�
i is also the behavioral solu-tion when maximizing zi over the centered profit function
��i �zi�= )ipEmin�Z�zi�−)iczi� (64)
Therefore, at each location i, we use the standardized andcentered profit function ��
i �zi� to characterize the behavioralsolution Z�
i . This argument also applies to the centralizedcase: we can use the problem of maximizing
��T �zT �= )T pEmin�Z�zT �−)T czT (65)
over zT to characterize the behavioral solution Z�T .
Let us define the following canonical newsvendor prob-lem: Choose z to maximize
P�z�= pE�Z�z�− cz� (66)
which we refer to as the canonical profit function. Let *and + denote the standard normal density and distribu-tion functions. Then, it is easy to see that the solution to
the canonical newsvendor problem z∗ satisfies +�z∗�= 1−c/p, and the optimal objective function satisfies P�z∗� =p∫ z∗−� v*�v�dv≤ 0. In other words, P�z�≤ 0 for every z ∈�.Now, observe that ��
i �zi�= )iP�zi� and ��T �zT �= )TP�zT �.
Therefore, for each i, applying Proposition 5 and condi-tion (b) yields the result that EP�Z�
T �≥ EP�Z�i �, with equal-
ity holding if and only if (b) holds with equality.Finally, we can put our conclusions together to write
n∑i=1
E�i�X�i � =
n∑i=1
E �i�Z�i �=
n∑i=1
E ��i �Z
�i �+ �p− c�
n∑i=1
(i (67)
=n∑
i=1
)iEP�Z�i �+ �p− c�(T (68)
≤n∑
i=1
)iEP�Z�T �+ �p− c�(T (69)
≤ )T EP�Z�T �+ �p− c�(T (70)
= E ��T �Z
�T �+ �p− c�(T
= E �T �Z�T �= E�T �X
�T �� (71)
Note that inequality (70) holds because of condition (a) andP�z� ≤ 0. Now, observe that (70) binds if and only if (a)holds with equality, and (69) binds if and only if (b) holdswith equality. However, because )i <
∑ni=1 )i, (a) and (b) can
not both bind at the same time. Therefore, strict inequalitymust hold in our result. �
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