Prospect Theory Explains Newsvendor Behavior: The Role of Reference Points

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1 Submitted to Management Science manuscrit (Please, rovide the mansucrit number! Authors are encouraged to submit new aers to INFORMS journals by means of a style file temlate, which includes the journal title. However, use of a temlate does not certify that the aer has been acceted for ublication in the named journal. INFORMS journal temlates are for the exclusive urose of submitting to an INFORMS journal and should not be used to distribute the aers in rint or online or to submit the aers to another ublication. Prosect Theory Exlains Newsvendor Behavior: The Role of Reference Points Xiaoyang Long, Javad Nasiry School of Business and Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong {xlongaa@ust.hk, nasiry@ust.hk} Current understanding in oerations management is that Prosect Theory, as a theory of decision making under uncertainty, cannot systematically exlain the ordering behavior observed in exeriments on the newsvendor roblem. We suggest this is because the newsvendor s reference oint is assumed to be the status quo, i.e., zero ayoff. We roose an alternative based on newsvendor s salient ayoffs and show that Prosect Theory can, in fact, account for exerimental results. Key words : Prosect Theory, reference oints, newsvendor roblem. Introduction In the classic newsvendor roblem, a newsvendor faces stochastic demand yet must make a single inventory ordering decision. The wholesale and retail rices are fixed at w and, resectively. If the demand x [x, x] follows a distribution with cumulative distribution function F (, then the newsvendor s otimal order quantity q solves F (q = w. The fraction w is called the critical fractile, and we refer to q as the otimal rational decision or normative order quantity. However, exeriments have reeatedly shown that decision makers systematically deviate from the normative decision. More secifically: newsvendors consistently underorder in the high-rofit regime (i.e., w <, and overorder in the low-rofit regime (i.e., w > (Schweitzer and Cachon 000, Bolton and Katok 008, Kremer et al. 00. This henomenon is known as the ull-to-center effect because the newsvendor s decision q seems to lie between the normative decision q and the mean demand µ. Schweitzer and Cachon (000, who originally documented the over/underordering attern, rule out Prosect Theory (PT as an exlanation. They study the high and low-rofit regimes under two demand scenarios. The design of their exeriment in the high-demand scenario is such that

2 Long and Nasiry: Reference Points in Prosect Theory and the Newsvendor Problem Article submitted to Management Science; manuscrit no. (Please, rovide the mansucrit number! the newsvendor never realizes a loss and so all rosects are assumed to be evaluated as gains. Because PT dictates that the newsvendor be risk-averse in the domain of gains, he should therefore always underorder (Eeckhoudt et al In the low-demand scenario, the newsvendor may realize losses as well as rofits deending on the order quantity and the realized demand. In this case, Schweitzer and Cachon (000 exlain the over/underordering attern by relying on the risk attitudes toward gains and losses in PT. In articular, in the high-rofit regime, the newsvendor is more likely to exerience gains and because the newsvendor is risk averse over gains, he is more likely to underorder. In the low-rofit regime, the newsvendor is more likely to exerience losses and because the newsvendor is risk seeking over losses, he is more likely to overorder (Eeckhoudt et al More recently, Nagarajan and Shechter (03 reconfirm that the newsvendor always underorders in the low-rofit case by considering a more elaborate descrition of PT that incororates robability weighting when the rosects are evaluated. These authors also show numerically that PT redicts overordering in the high-rofit case. Because these redictions directly contradict existing exerimental results, Nagarajan and Shechter (03 conclude that PT does not exlain the newsvendor s behavior and ose the question of why PT, a well-known and widely acceted framework for decision making under uncertainty, may not aly to a fundamental oerations management roblem. We shall elaborate on the reference oint construct to show that PT can, in fact, exlain newsvendor behavior. Schweitzer and Cachon (000 and also Nagarajan and Shechter (03 imlicitly assume that the decision maker comares the outcomes with the current level of wealth, which is assumed to be zero. Yet this need not be the case because there are situations in which gains and losses are coded to an exectation or asiration level that differs from the status quo (Kahneman and Tversky 979,. 86. In this aer we show that, if a more lausible reference oint is chosen, then PT can indeed exlain observed ordering behavior without relying on risk attitudes or the distortion of robabilities. We suggest in articular that, given the order quantity q, the reference oint is the weighted average of two salient ayoffs: the lowest and highest ossible rofits. We interret the weight assigned to the highest ayoff as the degree of newsvendor otimism. Then we show, consistently with exerimental results, that the newsvendor overorders if w underorders otherwise.. Model The newsvendor s material ayoff for order quantity q and realized demand x is { x wq if x < q, π(q, x = ( wq if x q. is sufficiently large and

3 Long and Nasiry: Reference Points in Prosect Theory and the Newsvendor Problem Article submitted to Management Science; manuscrit no. (Please, rovide the mansucrit number! 3 We assume a iecewise-linear value function for the reference-deendent newsvendor. That is, { η y if y 0, v(y = λ η y if y < 0; here η catures the strength of reference effects and λ is the coefficient of loss aversion. A higher value of η imlies more sensitivity to deviations from the reference oint; a higher λ indicates more sensitivity to losses in comarison to gains. This oerationalization of PT abstracts from the newsvendor s risk attitudes and is sufficient to exlain the ordering behavior. In Aendix B we demonstrate that the results are robust to more general forms of the value function. The newsvendor s ex ost utility is U(q, x = π(q, x + v(y, which consists of the realized rofit and a sychological comonent that catures how the realized ayoff comares with the reference ayoff (see Kőszegi and Rabin 006 and Herweg 03 for similar model setus. We hyothesize that the reference ayoff for an order quantity q is a convex combination of the maximum ossible ayoff ( wq and the minimum ossible ayoff x wq. That is, we assume r(q = β( wq + ( β(x wq. ( The arameter β [0, ] can be interreted as the decision maker s level of otimism. A high value of β means that the newsvendor holds high exectations for the final outcome whereas a low β imlies that he anchors more on the worst-case scenario. The following roosition summarizes our major findings. Proofs are in Aendix A. Proosition. Assume that demand x is uniformly distributed on [x, x] and that the newsvendor s reference oint is as in equation (. Then the following statements hold. (i The newsvendor s otimal order quantity q satisfies F ( q = w > t(λ, β = (λ β +β (λ β + and underorders otherwise. w+η( β. (ii The newsvendor overorders if +η(λ β +η Hence, the newsvendor overorders if the margin is sufficiently low, w > t(λ, β, and underorders if the margin is sufficiently high, w < t(λ, β. This redicted behavior is in line with the results from exerimental research. Figure deicts a scenario where we set λ = and β = so that t(λ, β =. In this case, the newsvendor underorders (F ( q < F (q if the otimal rational order q exceeds mean demand and overorders (F ( q > F (q otherwise. This shows that even without loss aversion (λ =, reference deendence in PT could redict the newsvendor s under/overordering attern. Moreover, because 0 t(λ, β for 0 β, our model accommodates ordering behavior from always overordering (β = 0 to always underordering (β =. Our model then hels exlain the findings of Lau et al. (04, who reort that individual behavior varies widely in the newsvendor exeriment and that the ull-to-center effect is not evident in the decisions of all subjects.

4 Long and Nasiry: Reference Points in Prosect Theory and the Newsvendor Problem 4 Article submitted to Management Science; manuscrit no. (Please, rovide the mansucrit number! Critical fractile 3 4 rational order 4 actual order w Figure The newsvendor s order quantity comared with the rational order quantity. η =, λ =, β =. Asymmetries in Ordering Behavior Our model also exlains the asymmetries observed in ordering behavior between the high-rofit and low-rofit regimes. Schweitzer and Cachon (000 find that subjects exhibit greater bias in the low-rofit than in the high-rofit case. That is, the actual order is farther away from the normative order in the low-rofit case. Bostian et al. (008 observe a like asymmetry, albeit to a lesser extent. In contrast, Ho et al. (00 find a greater bias in the high-rofit case. Still other works on the newsvendor roblem do not reort any asymmetry (Bolton and Katok 008, Kremer et al. 00. We next show that our model can redict the asymmetry in either direction. Define B : (0, [0, ] as a function that mas the fraction w to the degree of bias in the ordering decision, i.e., B( w = F ( q F (q. Proosition. Assume that demand x is uniformly distributed on [x, x] and that the newsvendor s reference oint is as in equation (. Let a (0, ]. (i If t(λ, β >, then B( a > B( + a, i.e., there is a greater bias in the high-rofit regime. (ii If t(λ, β <, then B( a < B( + a, i.e., there is a greater bias in the low-rofit regime. In other words, if t(λ, β >, the order quantity is farther away from the normative order quantity in the high-rofit regime; on the other hand, if t(λ, β <, the order quantity is farther away from the normative order quantity in the low-rofit regime. Most exerimental literature in suort of the ull-to-center effect follows Schweitzer and Cachon (000 in testing only two cases (e.g., w = 4 for the high-rofit case and w = 3 4 for the low-rofit case. Yet by anchoring on µ when interreting the results, researchers imlicitly assume that the newsvendor makes the rational decision (i.e., q = q = µ at w =. To the best of our knowledge, However, they use a Normal distribution for demand.

5 Long and Nasiry: Reference Points in Prosect Theory and the Newsvendor Problem Article submitted to Management Science; manuscrit no. (Please, rovide the mansucrit number! 5 Bostian et al. (008 are unique in testing this assumtion. They find that the newsvendor slightly overorders when w =. In our model this would imly that t(λ, β <, and hence there should be a larger bias in the low-rofit regime. Bostian et al. (008 observe such an asymmetry. In our model, underordering for w = imlies that F ( q F 4 (q = ( β 3 4 [(λ β +] < 0 and overordering for w = 3 imlies F ( q F 4 (q = ( β 4 [(λ β +] > 0. If, as claimed by Schweitzer and Cachon (000, the degree of bias is higher in the low-rofit case, then 3 4 [(λ β +] ( β ( β 4 [(λ β + ]; this exression simlifies to ( β [(λ β + ] 0. If we assume that λ = on average, then these inequalities imly that the tyical subject studied by Schweitzer and Cachon (000 is not very otimistic (0. < β < Discussion and Conclusion In this aer, we show that a simle version of Prosect Theory can exlain the behavioral deviations in the newsvendor roblem without relying on risk references or distortion of robabilities. This demonstration is made ossible by a novel interretation of the reference oint, which is a key concet and oen arameter (Rabin 03 in PT. We find that PT exlains the ordering behavior observed in newsvendor exeriments rovided the newsvendor anchors on the most and least favorable outcomes. We also establish that individual characteristics of the newsvendor, such as his degree of otimism, significantly affect the ordering decision and may exlain the heterogeneous behavior of the subjects in newsvendor exeriments. We conclude by highlighting two critical issues. First, our results are not driven by the assumtion of a iecewise-linear value function. We show in Aendix B that more general value functions that also account for risk attitudes over gains and losses yield the same qualitative insights. Second, our formalization of the reference oint dearts from the notion of reference oint as a fixed benchmark, as the reference oint in our model is a function of the order quantity. An interretation of our aroach is that the newsvendor forms an exectation about the unknown ayoff which deends on the order quantity. We are not the first to roose this aroach however. Kőszegi and Rabin (006 suggest that, in an uncertain environment, the reference oint is stochastic and determined in a ersonal equilibrium by the decision maker s rational exectations which deend on his actions. The advantage of the ersonal equilibrium framework is that it does not imose an exogenous structure on the reference oint. Herweg (03 alies this framework to the newsvendor s roblem and finds that the newsvendor always underorders. Overall, our results shed light on the imortance of roerly calibrating behavioral theories to the secific roblems in oerations management. See Ho and Zhang (004.

6 Long and Nasiry: Reference Points in Prosect Theory and the Newsvendor Problem 6 Article submitted to Management Science; manuscrit no. (Please, rovide the mansucrit number! Acknowledgments The authors thank deartment editor Serguei Netessine and an anonymous associate editor for their constructive feedback that heled imrove the aer. References Bolton, G. E., and E. Katok Learning by doing in the newsvendor roblem: A laboratory investigation of the role of exerience and feedback. Manufacturing & Service Oerations Management 0 (3: Bostian, A. A., C. A. Holt, and A. M. Smith Newsvendor ull-to-center effect: Adative learning in a laboratory exeriment. Manufacturing & Service Oerations Management 0 (4: Eeckhoudt, L., C. Gollier, and H. Schlesinger The risk-averse (and rudent newsboy. Management Science 4 (5: Herweg, F. 03. The exectation-based loss-averse newsvendor. Economics Letters 0 (3: Ho, T.-H., N. Lim, and T. H. Cui. 00. Reference deendence in multilocation newsvendor models: A structural analysis. Management Science 56 (: Ho, T. H., and J. Zhang Does format of ricing contract matter? Working aer, Haas School of Business, University of California, Berkeley, CA. Kahneman, D., and A. Tversky Prosect theory: An analysis of decision under risk. Econometrica: Journal of the Econometric Society 47 (: Kőszegi, B., and M. Rabin A model of reference-deendent references. The Quarterly Journal of Economics (4: Kremer, M., S. Minner, and L. N. Van Wassenhove. 00. Do random errors exlain newsvendor behavior? Manufacturing & Service Oerations Management (4: Lau, N., S. Hasija, and J. N. Bearden. 04. Newsvendor ull-to-center reconsidered. Decision Suort Systems 58: Nagarajan, M., and S. Shechter. 03. Prosect theory and the newsvendor roblem. Management Science 60 (4: Rabin, M. 03. Incororating limited rationality into economics. Journal of Economic Literature 5 (: Schweitzer, M. E., and G. P. Cachon Decision bias in the newsvendor roblem with a known demand distribution: Exerimental evidence. Management Science 46 (3: Tversky, A., and D. Kahneman. 99. Advances in rosect theory: Cumulative reresentation of uncertainty. Journal of Risk and Uncertainty 5 (4:

7 Long and Nasiry: Reference Points in Prosect Theory and the Newsvendor Problem Article submitted to Management Science; manuscrit no. (Please, rovide the mansucrit number! 7 Aendix A: Proofs Proof of Proosition. the argument in r(q is EU(q = q x λη We ut F (q = F (q. Then the newsvendor s exected utility (suressing (x wqdf (x + ( wq F (q r+wq x q (r x + wqdf (x + η r+wq Given r(q as in (, EU(q is concave. This is because EU(q = ( w F (q λη(r + wf η(r + w q F and q EU(q = f(q ηf(q (λ η(r + wf (x wq rdf (x + η(( wq r F (q. ( ( r + w ληr F + η F (q = 0, (3 ηr F < 0. The inequality follows because r (q + w = β > 0 and r (q = 0. Substituting r(q into (3 and recalling that demand is uniform, we obtain the desired result. w+η( β w +(λ ηβ +η = η( β ( w [(λ ηβ +η] (ii F ( q F (q = > 0 if and only if w > (λ β +β. (λ ηβ ++η (λ β + Proof of Proosition. By Proosition, F ( q F (q = β ( w/[(λ β +]. Therefore, B( a = and B( + a = β (/ a[(λ β +]. β (/+a[(λ β +] = β /[(λ β +] (i If t(λ, β >, then by Proosition, F ( q F (q w = a = β /[(λ β +] a [(λ β +] > β /[(λ β +] + a [(λ β +] holds because β /[(λ β +] < 0, λ, and a > 0. (ii The roof is similar to that of art (i. Aendix B: General Value Functions < 0. It follows that B( = B( + a. The inequality In this aendix, we show that our results in Proosition do not deend on the linearity of the value function. Assume for examle that { ηy d if y 0, v(y = λη( y d if y < 0, where 0 < d < and λ > (see Tversky and Kahneman 99. This value function incororates the risk attitudes of the subjects as secified by PT. In short, decision makers are risk-averse in the domain of gains but are risk-seeking in the domain of losses. The resulting threshold attern is similar to that described in Proosition. Proosition 3. Suose that demand is uniformly distributed on [x, x] and that the newsvendor s reference oint is given by equation (. The newsvendor overorders if w > otherwise. λβ d+ +β( β d d( β d +λβ d+ +β( β d and underorders Proof. The newsvendor s exected utility is EU(q = q x +η (x wqdf (x + ( wq F (q λη q r+wq r+wq (x wq r d df (x + η(( wq r d F (q. x (r x + wq d df (x

8 Long and Nasiry: Reference Points in Prosect Theory and the Newsvendor Problem 8 Article submitted to Management Science; manuscrit no. (Please, rovide the mansucrit number! Differentiating twice with resect to q and considering that x is uniformly distributed, we have EU(q q (β(q xd (( β(q xd = w F (q ληβ ηβ + ηd (( β(q x d ( β x x x x F (q, and EU(q q = f(q ληβ (β(q xd (( β(q xd d ηβ( βd x x x x +ηd(d (( β(q x d ( β F (q ηd (( β(q x d ( βf(q < 0. The inequality follows because 0 < d <. Therefore, EU(q is concave. Substituting q into EU(q q now yields EU(q = ( x x ( q q=q d ληβ d+ ( w d ηβ( β d ( w d + ηd( β d w( w d, which is ositive if and only if w > λβ d+ +β( β d d( β d +λβ d+ +β( β d.