{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:37:01Z","timestamp":1760243821041,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,10,27]],"date-time":"2022-10-27T00:00:00Z","timestamp":1666828800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>Almost every supplier faces uncertain and time-varying demand. E-commerce and online shopping have given suppliers unprecedented access to data on customers\u2019 behavior, which sheds light on demand uncertainty. The main purpose of this research project is to provide an analytic tool for decentralized supply channel members to devise optimal long-term (multi-period) supply, pricing, and timing strategies while catering to stochastic demand in a diverse set of market scenarios. Despite its ubiquity in potential applications, the time-dependent channel optimization problem in its general form has received limited attention in the literature due to its complexity and the highly nested structure of its ensuing equilibrium problems. However, there are many scenarios where a single-period channel optimization solution may turn out to be myopic as it does not consider the after-effects of current pricing on future demand. To remedy this typical shortcoming, using general memory functions, we include the strategic customers\u2019 cognitive bias toward pricing history in the supply channel equilibrium problem. In the form of two constructive theorems, we provide explicit solution algorithms for the ensuing Nash\u2013Stackelberg equilibrium problems. In particular, we prove that our recursive solution algorithm can find equilibria in the multi-periodic variation of many standard supply channel contracts such as wholesale, buyback, and revenue-sharing contracts.<\/jats:p>","DOI":"10.3390\/g13060070","type":"journal-article","created":{"date-parts":[[2022,10,27]],"date-time":"2022-10-27T20:37:58Z","timestamp":1666903078000},"page":"70","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Construction of Equilibria in Strategic Stackelberg Games in Multi-Period Supply Chain Contracts"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9132-9650","authenticated-orcid":false,"given":"Reza","family":"Azad Gholami","sequence":"first","affiliation":[{"name":"Department of Business and Management Science, Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway"},{"name":"Department of Mathematics, University of Bergen, Allegaten 41, 5007 Bergen, Norway"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5384-0316","authenticated-orcid":false,"given":"Leif Kristoffer","family":"Sandal","sequence":"additional","affiliation":[{"name":"Department of Business and Management Science, Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway"}]},{"given":"Jan","family":"Ub\u00f8e","sequence":"additional","affiliation":[{"name":"Department of Business and Management Science, Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,27]]},"reference":[{"key":"ref_1","unstructured":"de Kok, A.G., and Graves, S.C. (2003). Supply chain coordination with contracts. The Handbook of Operations Research and Management Science: Supply Chain Management: Design, Coordination and Operation, Elsevier. Chapter 6."},{"key":"ref_2","unstructured":"Holmes, B. (2022, August 25). The Economics of Freemium, Wall Street Jorunal, Available online: http:\/\/blogs.wsj.com\/accelerators\/2013\/03\/03\/the-economics-of-freemium\/."},{"key":"ref_3","unstructured":"Chowdhry, A. (2022, August 25). Samsung\u2019s \u2018Ultimate Test Drive\u2019 Offers Free Trial For Note 5, S6 Edge+ And S6 Edge To iPhone Users, Forbes, Available online: http:\/\/www.forbes.com\/sites\/amitchowdhry\/2015\/08\/21\/samsung-ultimate-test\/."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"102002","DOI":"10.1016\/j.omega.2018.11.011","article-title":"To sell or not to sell to an off-price retailer in the presence of strategic consumers","volume":"90","author":"Khouja","year":"2020","journal-title":"Omega"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1287\/mnsc.1080.0936","article-title":"Dynamic pricing in the presence of strategic consumers and oligopolistic competition","volume":"55","author":"Levin","year":"2009","journal-title":"Manag. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1287\/mnsc.15.5.215","article-title":"A new product growth for model consumer durables","volume":"15","author":"Bass","year":"1969","journal-title":"Manag. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1177\/002224299005400101","article-title":"New product diffusion models in marketing: A review and directions for research","volume":"54","author":"Mahajan","year":"1990","journal-title":"J. Mark."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1150","DOI":"10.1287\/mnsc.31.9.1150","article-title":"Bayes solution to dynamic inventory models under unknown demand distribution","volume":"31","author":"Azoury","year":"1985","journal-title":"Manag. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1287\/mnsc.45.3.346","article-title":"Stalking information: Bayesian inventory management with unobserved lost sales","volume":"45","author":"Lariviere","year":"1999","journal-title":"Manag. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1287\/moor.1060.0236","article-title":"A multiperiod newsvendor problem with partially observed demand","volume":"32","author":"Bensoussan","year":"2007","journal-title":"Math. Oper. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1287\/moor.1070.0272","article-title":"Provably near-optimal sampling-based policies for stochastic inventory control models","volume":"32","author":"Levi","year":"2007","journal-title":"Math. Oper. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/j.aei.2014.12.002","article-title":"Optimal inventory control in a multi-period newsvendor problem with non-stationary demand","volume":"29","author":"Kim","year":"2015","journal-title":"Adv. Eng. Inform."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1287\/mksc.4.2.166","article-title":"Optimal pricing and return policies for perishable commodities","volume":"4","author":"Pasternack","year":"1985","journal-title":"Mark. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1287\/mksc.2013.0784","article-title":"Returns policies between channel partners for durable products","volume":"32","author":"Ray","year":"2013","journal-title":"Mark. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Azad Gholami, R., Sandal, L.K., and Ub\u00f8e, J. (2016). Channel Coordination in a Multi-Period Newsvendor Model with Dynamic, Price-Dependent Stochastic Demand, Department of Business and Management Science NHH. NHH Dept. of Business and Management Science Discussion Paper.","DOI":"10.2139\/ssrn.2766931"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"102297","DOI":"10.1016\/j.omega.2020.102297","article-title":"A solution algorithm for multi-period bi-level channel optimization with dynamic price-dependent stochastic demand","volume":"102","author":"Gholami","year":"2021","journal-title":"Omega"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Azad Gholami, R., Sandal, L.K., and Ub\u00f8e, J. (2019). Markets With Memory: Dynamic Channel Optimization Models with Price-Dependent Stochastic Demand, Department of Business and Management Science NHH. NHH Dept. of Business and Management Science Discussion Paper.","DOI":"10.2139\/ssrn.3450493"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1287\/mnsc.2.1.61","article-title":"Inventory control and price theory","volume":"2","author":"Whitin","year":"1955","journal-title":"Manag. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.ejor.2020.05.013","article-title":"The Newsvendor Problem: The Role of Prospect Theory and Feedback","volume":"287","author":"Surti","year":"2020","journal-title":"Eur. J. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.omega.2017.12.008","article-title":"Describing decision bias in the newsvendor problem: A prospect theory model","volume":"82","author":"Vipin","year":"2019","journal-title":"Omega"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.omega.2017.10.003","article-title":"A loss averse competitive newsvendor problem with anchoring","volume":"81","author":"Wu","year":"2018","journal-title":"Omega"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1037\/h0048784","article-title":"Assimilation and contrast effects of anchoring stimuli on judgments","volume":"55","author":"Sherif","year":"1958","journal-title":"J. Exp. Psychol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.1126\/science.185.4157.1124","article-title":"Judgment under Uncertainty: Heuristics and Biases: Biases in judgments reveal some heuristics of thinking under uncertainty","volume":"185","author":"Tversky","year":"1974","journal-title":"Science"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Azad Gholami, R., Sandal, L.K., and Ub\u00f8e, J. (2022, August 25). Who is Benefiting from Postponement in Multi-Periodic Supply Channel Optimization?. Available online: https:\/\/ssrn.com\/abstract=4204363.","DOI":"10.2139\/ssrn.4204363"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"365","DOI":"10.2307\/2297564","article-title":"Introductory Price as a Signal of Cost in a Model of Repeat Business","volume":"54","author":"Bagwell","year":"1987","journal-title":"Rev. Econ. Stud."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1701","DOI":"10.1016\/S1573-448X(06)03028-7","article-title":"The Economic Analysis of Advertising","volume":"Volume 3","author":"Armstrong","year":"2007","journal-title":"Handbook of Industrial Organization"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1287\/opre.47.2.183","article-title":"Pricing and the newsvendor problem: A review with extensions","volume":"47","author":"Petruzzi","year":"1999","journal-title":"Oper. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"S39","DOI":"10.1086\/296235","article-title":"Pricing research in marketing: The state of the art","volume":"57","author":"Rao","year":"1984","journal-title":"J. Bus."},{"key":"ref_29","first-page":"259","article-title":"EDLP vs. Hi-Lo Pricing Strategies in Retailing\u2014A State of the Art Article","volume":"83","author":"Fassnacht","year":"2013","journal-title":"J. Bus. Econ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1284","DOI":"10.1016\/j.jedc.2013.02.010","article-title":"Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models","volume":"37","author":"Sandal","year":"2013","journal-title":"J. Econ. Dyn. Control."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"962","DOI":"10.1016\/j.ejor.2017.08.055","article-title":"On the unimodality of the price-setting newsvendor problem with additive demand under risk considerations","volume":"265","year":"2018","journal-title":"Eur. J. Oper. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1016\/j.ijpe.2011.05.013","article-title":"Unimodality of price setting newsvendor\u2019s objective function with multiplicative demand and its applications","volume":"133","author":"Xu","year":"2011","journal-title":"Int. J. Prod. Econ."}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/13\/6\/70\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:03:56Z","timestamp":1760144636000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/13\/6\/70"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,27]]},"references-count":32,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["g13060070"],"URL":"https:\/\/doi.org\/10.3390\/g13060070","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2022,10,27]]}}}