Jointly reproducing the pull-to-center effect and the non-linear ordering curve in a prospect-theory newsvendor model: extending the loss-averse reference-dependent (LR) framework of Vipin and Amit (2019)

Laboratory newsvendor experiments show a pull-to-center (PTC) effect — average orders lie between the profit-maximizing quantity and mean demand — and, in the treatment-dense experiment of Ockenfels and Selten (2014) with eleven cost treatments, a significantly non-linear aggregate ordering curve whose precise shape (including concavity in low-profit-margin regions) is not captured by existing models. Vipin and Amit (2019) proposed the loss averse–reference dependent (LR) newsvendor model, a prospect-theory model with a stochastic-subjective reference point (pessimism, perception, and loss-aversion parameters) that reproduces PTC, its asymmetry, boundary behavior, and qualitative non-linearity; but its best fit to the eleven Ockenfels–Selten aggregate points (loss neutrality, RMSE 4.576) cannot simultaneously capture the curve's concavity in the low-profit region and PTC. The open problem is to extend the prospect-theory reference-dependent framework so that one model with one estimated parameterization jointly reproduces the full aggregate experimental record while retaining analytical tractability; supply-chain coordination and competition with such a behavioral retailer are downstream applications that motivate the modeling goal. No prospect-theory reference-dependent model is known to achieve the joint curve-shape/PTC fit within this family: later literature developed competing prospect-theory models (heterogeneity-focused), contract analysis with behavioral retailers, competition among behavioral newsvendors under other specifications, and empirical results that partially destabilize the target, but the joint curve-shape/PTC fit for the LR framework remains open.

Background

The newsvendor problem and its behavioral record. A newsvendor must choose an order quantity $Q$ of a perishable good before demand is realized: each unit costs $w$, sells at price $p>w$, and unsold units salvage at $v<w$. Under a known demand distribution $F$, the expected-profit-maximizing order is the critical fractile solution $Q^*=F^{-1}((p-w)/(p-v))$. Since Schweitzer and Cachon (2000), laboratory experiments robustly find a pull-to-center (PTC) effect: average orders lie between $Q^*$ and mean demand — subjects order too much in low-profit (high-cost) conditions and too little in high-profit conditions — with a larger deviation in the low-profit condition (the PTC asymmetry). Ockenfels and Selten (2014) ran a treatment-dense design — 340 subjects, 200 rounds each, demand uniform on $\{0,\dots,100\}$, price fixed at 10, unit cost varied across eleven treatments $c=0,1,\dots,10$ — and showed that the aggregate ordering curve, plotted against the cost ratio $z=c/p$, is significantly non-linear (a Ramsey RESET regression-specification test rejects linearity with $p<0.0001$) and behaves non-trivially at both extremes: average orders approach maximum demand as $z\to 0$ but remain well above zero as $z\to 1$. They also proposed impulse balance equilibrium (IBE), a parameter-free account in which the tendencies to adjust orders upward after stockouts and downward after leftovers balance; IBE predicts this non-linear central tendency ex ante, though not individual heterogeneity or adjustment dynamics.

Prospect-theory models of the bias. Prospect theory (PT) models choice over gains and losses relative to a reference point, with a value function that is concave over gains, convex over losses, and steeper over losses (loss aversion, coefficient $\lambda>1$). Schweitzer and Cachon's PT model with a status-quo (zero-payoff) reference point cannot produce PTC. Long and Nasiry (2015) showed that a PT model with a reference point built from salient newsvendor payoffs can. Vipin and Amit (2019) then proposed the loss averse–reference dependent (LR) newsvendor: the reference point is a stochastic, subjective weighted average of the payoffs at the maximum and minimum demand realizations, shaped by a pessimism parameter $\phi\in[0,1]$ and a perception parameter $\eta\ge 0$ that scale with the margin parameter $z=(w-v)/(p-v)$ (equal to the cost ratio $c/p$ when the salvage value is zero, as in the Ockenfels–Selten design), together with loss aversion $\lambda$. Under uniform demand they prove strict concavity of expected utility (hence a unique optimal order), derive a closed-form threshold in $z$ separating under-ordering from over-ordering, and show the model reproduces PTC, its asymmetry, non-linearity of the ordering curve in $z$, the boundary behavior at both margin extremes, and — by varying $\phi$ and $\eta$ — the spectrum of individual-level ordering strategies. Fitting $\lambda,\phi,\eta$ by nonlinear least squares to the eleven Ockenfels–Selten aggregate points yields $\lambda=1$ (loss neutrality), $\phi=0.8670$, $\eta=0.6430$, RMSE $=4.576$ (the loss-averse specification $\lambda=2.25$ fits worse, RMSE $=5.748$), so the descriptive success rests on the reference-point parameters rather than loss aversion.

The gap. The authors' own limitation statement concedes that the LR model cannot simultaneously characterize the precise shape of the non-linear ordering curve and the PTC effect: the fitted curve does not reproduce the concavity that the Ockenfels–Selten data exhibit in low-profit-margin (high-cost) regions while maintaining the PTC description. No parameterization is shown to achieve this joint fit; the estimation reports no standard errors or out-of-sample validation; and no head-to-head fit comparison against the Long–Nasiry or Uppari–Hasija models on the same data exists. Vipin and Amit list two normative applications: applying the behavioral retailer model to supply-chain coordination (which contracts coordinate a chain containing such a retailer) — a direction the authors themselves subsequently pursued (Vipin & Amit 2021) — and analyzing competition among behavioral retailers. Note that the curve-shape gap is specific to the prospect-theory reference-point family — Ockenfels and Selten's parameter-free IBE already tracks the non-linear central tendency — so the scientific question is whether a tractable PT reference-dependent model can do the same.

Problem Statement

As posed by Vipin and Amit (2019) for their LR framework: develop a prospect-theoretic, reference-dependent newsvendor model — extending or modifying the LR reference-point construction — such that one model, with one estimated parameterization, simultaneously reproduces on the Ockenfels and Selten (2014) eleven-treatment aggregate data:

1. the pull-to-center effect (average orders between the risk-neutral optimum and mean demand: over-ordering in low-profit conditions, under-ordering in high-profit conditions);
2. the asymmetry of this bias (stronger deviation in low-profit than in high-profit conditions);
3. the boundary behavior at extreme profit margins (average orders approaching maximum demand at the highest margin while remaining interior at the lowest margin); and
4. the precise non-linear shape of the aggregate ordering curve over the full cost range — in particular the concavity in low-profit-margin regions that the best-fitting LR parameterization ($\lambda=1$, $\phi=0.8670$, $\eta=0.6430$, RMSE 4.576) fails to reproduce while maintaining the pull-to-center description.

The extended model must remain an analytically tractable prospect-theory reference-dependent model in the spirit of the LR framework: the extension may add at most two parameters to the reference-point construction, each entering through a fixed, low-dimensional functional form of the margin parameter $z$ rather than a curve fitted pointwise to the eleven aggregate points, and must yield a unique, computable optimal order quantity. The joint fit means that one and the same estimated parameterization of the extended model satisfies all four conditions of the Problem Statement above — the pull-to-center pattern; its asymmetry; the boundary behavior at the profit-margin extremes; and a fitted ordering curve whose second difference is non-positive over the low-profit-margin (high-cost) treatments, the concavity that the best-fitting LR parameterization fails to reproduce — with the model estimated on the eleven Ockenfels–Selten points and the overall fit there at least as good as the LR baseline (RMSE no greater than 4.576); further experimental newsvendor datasets are supplementary to that eleven-point fit.

Supply-chain coordination with such a behavioral retailer — which contracts (e.g., wholesale price versus buyback) coordinate a chain whose retailer orders according to the model, a direction the source's authors subsequently pursued (Vipin & Amit 2021) — and competition among behavioral retailers so described are outside the scope of this question; they are downstream applications of the model.

References

1. Vipin, B., Amit, R.K. (2019). Describing decision bias in the newsvendor problem: A prospect theory model. Omega 82: 132–141. DOI: 10.1016/j.omega.2017.12.008. https://doi.org/10.1016/j.omega.2017.12.008
2. Ockenfels, A., Selten, R. (2014). Impulse balance in the newsvendor game. Games and Economic Behavior 86: 237–247. DOI: 10.1016/j.geb.2014.03.012. https://doi.org/10.1016/j.geb.2014.03.012
3. Ockenfels, A., Selten, R. (2015). Impulse Balance and Multiple-Period Feedback in the Newsvendor Game. Production and Operations Management 24(12): 1901–1906. DOI: 10.1111/poms.12399. https://doi.org/10.1111/poms.12399
4. Schweitzer, M.E., Cachon, G.P. (2000). Decision Bias in the Newsvendor Problem with a Known Demand Distribution: Experimental Evidence. Management Science 46(3): 404–420. DOI: 10.1287/mnsc.46.3.404.12070. https://doi.org/10.1287/mnsc.46.3.404.12070
5. Long, X., Nasiry, J. (2015). Prospect Theory Explains Newsvendor Behavior: The Role of Reference Points. Management Science 61(12): 3009–3012. DOI: 10.1287/mnsc.2014.2050. https://doi.org/10.1287/mnsc.2014.2050
6. Uppari, B.S., Hasija, S. (2019). Modeling Newsvendor Behavior: A Prospect Theory Approach. Manufacturing & Service Operations Management 21(3): 481–500. DOI: 10.1287/msom.2017.0701. https://doi.org/10.1287/msom.2017.0701
7. Gu, B., Zhang, X. (2020). Prospect Theory and the Newsvendor Problem with Mental Accounting. Journal of Systems Science and Systems Engineering 29(5): 525–536. DOI: 10.1007/s11518-019-5445-5. https://doi.org/10.1007/s11518-019-5445-5
8. Kirshner, S.N., Shao, L. (2019). The overconfident and optimistic price-setting newsvendor. European Journal of Operational Research 277(1): 166–173. DOI: 10.1016/j.ejor.2019.02.023. https://doi.org/10.1016/j.ejor.2019.02.023
9. Surti, C., Celani, A., Gajpal, Y. (2020). The newsvendor problem: The role of prospect theory and feedback. European Journal of Operational Research 287(1): 251–261. DOI: 10.1016/j.ejor.2020.05.013. https://doi.org/10.1016/j.ejor.2020.05.013
10. Vipin, B., Amit, R.K. (2021). Wholesale price versus buyback: A comparison of contracts in a supply chain with a behavioral retailer. Computers & Industrial Engineering 162: 107689. DOI: 10.1016/j.cie.2021.107689. https://doi.org/10.1016/j.cie.2021.107689
11. Zhang, Y., Yu, P., Feng, T. (2024). Full Quantity or Full Credit? Choosing the Right Buyback Policy for a Behavioral Retailer. Manufacturing & Service Operations Management 26(6): 2212–2230. DOI: 10.1287/msom.2023.0574. https://doi.org/10.1287/msom.2023.0574
12. Jammernegg, W., Kischka, P., Silbermayr, L. (2026). Supply chain coordination with a wholesale price contract for behavioral decision-makers. European Journal of Operational Research 329(1): 112–123. DOI: 10.1016/j.ejor.2025.06.029. https://doi.org/10.1016/j.ejor.2025.06.029
13. Xie, K., Guo, Y. (2021). The competitive loss-averse newsvendor problem with quantity-oriented reference point. Applied Stochastic Models in Business and Industry 37(3): 525–544 (published online 2020). DOI: 10.1002/asmb.2593. https://doi.org/10.1002/asmb.2593
14. Kirshner, S.N., Li, Z. (2022). Supply chain contracting with competing regretful retailers. Managerial and Decision Economics 43(6): 2196–2211 (published online 2021). DOI: 10.1002/mde.3518. https://doi.org/10.1002/mde.3518
15. Feng, T., Zhang, Y. (2017). Modeling Strategic Behavior in The Competitive Newsvendor Problem: An Experimental Investigation. Production and Operations Management 26(7): 1383–1398. DOI: 10.1111/poms.12683. https://doi.org/10.1111/poms.12683
16. Zheng, Y., Gou, Q., Li, J., Wei, F. (2025). Decision behaviour and performance in a paired newsvendor: The role of peer effects. Journal of the Operational Research Society 76(2): 375–392 (published online 2024). DOI: 10.1080/01605682.2024.2354896. https://doi.org/10.1080/01605682.2024.2354896
17. Zhang, Y., Siemsen, E. (2019). A Meta-Analysis of Newsvendor Experiments: Revisiting the Pull-to-Center Asymmetry. Production and Operations Management 28(1): 140–156. DOI: 10.1111/poms.12899. https://doi.org/10.1111/poms.12899
18. Brokešová, Z., Deck, C., Péliová, J. (2025). Experience Minimizes the Pull-to-Center Effect in Newsvendor Decisions. Manufacturing & Service Operations Management 27(2): 554–568. DOI: 10.1287/msom.2021.0439. https://doi.org/10.1287/msom.2021.0439
19. Khan, M.M., Vipin, B. (2025). Abundance Versus Scarcity: An Experimental Study of Financially Constrained Newsvendors. Journal of Operations Management 71(5): 630–650. DOI: 10.1002/joom.1361. https://doi.org/10.1002/joom.1361
20. Leppänen, I. (2025). Newsvendor stockouts and option discriminability. European Journal of Operational Research 326(1): 111–123. DOI: 10.1016/j.ejor.2025.04.002. https://doi.org/10.1016/j.ejor.2025.04.002
21. Dertwinkel-Kalt, M., Köster, M. (2017). Salient compromises in the newsvendor game. Journal of Economic Behavior & Organization 141: 301–315. DOI: 10.1016/j.jebo.2017.07.008. https://doi.org/10.1016/j.jebo.2017.07.008
22. Su, Y., Guha, S., Ulu, C., Seifert, M. (2026). Beyond Pull-to-Center: How System Neglect Shapes Ordering Decisions Under Shifting Demand. SSRN working paper. DOI: 10.2139/ssrn.6122346. https://doi.org/10.2139/ssrn.6122346
23. Cai, D., Kirshner, S.N., Wu, M. (2026). Newsvendor overconfidence and supply uncertainty: Pull-to- or push-from-center? European Journal of Operational Research 332(1): 115–131. DOI: 10.1016/j.ejor.2025.10.047. https://doi.org/10.1016/j.ejor.2025.10.047