{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T13:26:20Z","timestamp":1775741180804,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T00:00:00Z","timestamp":1758672000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T00:00:00Z","timestamp":1758672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100015963","name":"Scuola IMT Alti Studi Lucca","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100015963","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Ann Oper Res"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Environmental pollution represents a significant negative externality for society by imposing substantial costs on it. By incentivizing eco-friendly production, one can mitigate these costs and promote sustainable development. Given this framework, the present article formulates and solves a multi-stage game-theoretical pricing model involving two firms: one commercializes a non-green product; the other one is involved in the production of a green product. The model also involves the State, which can influence the two firms through a dual mechanism based on imposing an excise duty and promoting an advertising campaign. Firms\u2019 profits and social welfare at the subgame perfect equilibrium are obtained in closed form, and their dependence on exogenous parameters is analyzed both theoretically and numerically. An interesting outcome of our analysis is that, in certain cases, the excise duty is found to be negative at the equilibrium and therefore corresponds to a subsidy rather than a green tax. Finally, possible future developments of the proposed model are discussed.<\/jats:p>","DOI":"10.1007\/s10479-025-06838-8","type":"journal-article","created":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T08:34:23Z","timestamp":1758702863000},"page":"2431-2456","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A game-theoretical pricing model for green\/non-green products"],"prefix":"10.1007","volume":"359","author":[{"given":"Francesco","family":"Biancalani","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5427-4328","authenticated-orcid":false,"given":"Giorgio","family":"Gnecco","sequence":"additional","affiliation":[]},{"given":"Leonardo","family":"Badia","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,24]]},"reference":[{"issue":"4","key":"6838_CR1","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1257\/jep.15.4.69","volume":"15","author":"JD Angrist","year":"2001","unstructured":"Angrist, J. D., & Krueger, A. B. (2001). Instrumental variables and the search for identification: from supply and demand to natural experiments. Journal of Economic Perspectives, 15(4), 69\u201385.","journal-title":"Journal of Economic perspectives"},{"key":"6838_CR2","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s10551-014-2425-z","volume":"134","author":"C Barbarossa","year":"2016","unstructured":"Barbarossa, C., & De Pelsmacker, P. (2016). Positive and negative antecedents of purchasing eco-friendly products: a comparison between green and non-green consumers. Journal of Business Ethics, 134, 229\u2013247.","journal-title":"Journal of Business Ethics"},{"issue":"2","key":"6838_CR3","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1108\/EMJB-09-2018-0055","volume":"14","author":"V Barros","year":"2019","unstructured":"Barros, V., & P\u00e1dua, H. (2019). Can green taxation trigger plug-in hybrid electric vehicle acquisition? EuroMed Journal of Business, 14(2), 168\u2013186.","journal-title":"EuroMed Journal of Business"},{"key":"6838_CR4","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1016\/j.jclepro.2017.01.060","volume":"145","author":"Z Basiri","year":"2017","unstructured":"Basiri, Z., & Heydari, J. (2017). A mathematical model for green supply chain coordination with substitutable products. Journal of Cleaner Production, 145, 232\u2013249.","journal-title":"Journal of Cleaner Production"},{"issue":"3","key":"6838_CR5","first-page":"307","volume":"62","author":"WJ Baumol","year":"1972","unstructured":"Baumol, W. J. (1972). On taxation and the control of externalities. The American Economic Review, 62(3), 307\u2013322.","journal-title":"The American Economic Review"},{"key":"6838_CR6","doi-asserted-by":"publisher","first-page":"19676","DOI":"10.1038\/s41598-024-70260-6","volume":"14","author":"F Biancalani","year":"2024","unstructured":"Biancalani, F., Gnecco, G., Metulini, R., & Riccaboni, M. (2024). The impact of the European Union emissions trading system on carbon dioxide emissions: a matrix completion analysis. Scientific Reports, 14, 19676.","journal-title":"Scientific Reports"},{"issue":"1","key":"6838_CR7","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/j.ejor.2021.10.050","volume":"300","author":"F Biancalani","year":"2022","unstructured":"Biancalani, F., Gnecco, G., & Riccaboni, M. (2022). Price-volume agreements: a one principal\/two agents model. European Journal of Operational Research, 300(1), 296\u2013309.","journal-title":"European Journal of Operational Research"},{"key":"6838_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.eneco.2021.105164","volume":"96","author":"D Buccella","year":"2021","unstructured":"Buccella, D., Fanti, L., & Gori, L. (2021). To abate, or not to abate? A strategic approach on green production in cournot and bertrand duopolies. Energy Economics, 96, Article 105164.","journal-title":"Energy Economics"},{"key":"6838_CR9","doi-asserted-by":"crossref","unstructured":"Cappelletti, M., Cibin, N., & Badia, L. (2021). Game-theoretic economic models of duopolies applied to green ICT design., In Proceedings of the IEEE Electric Power and Energy Conference (EPEC), pages 267\u2013272.","DOI":"10.1109\/EPEC52095.2021.9621643"},{"key":"6838_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jedc.2022.104589","volume":"146","author":"E Ciola","year":"2023","unstructured":"Ciola, E., Turco, E., Gurgone, A., Bazzana, D., Vergalli, S., & Menoncin, F. (2023). Enter the matrix model: a multi-agent model for transition risks with application to energy shocks. Journal of Economic Dynamics and Control, 146, Article 104589.","journal-title":"Journal of Economic Dynamics and Control"},{"key":"6838_CR11","doi-asserted-by":"publisher","DOI":"10.1016\/j.eneco.2022.106115","volume":"112","author":"T Compernolle","year":"2022","unstructured":"Compernolle, T., Kort, P. M., & Thijssen, J. J. J. (2022). The effectiveness of carbon pricing: the role of diversification in a firm\u2019s investment decision. Energy Economics, 112, Article 106115.","journal-title":"Energy Economics"},{"issue":"1","key":"6838_CR12","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/s10640-021-00616-3","volume":"83","author":"T Compernolle","year":"2022","unstructured":"Compernolle, T., & Thijssen, J. J. J. (2022). The role of industrial and market symbiosis in stimulating $$\\text{ co}_2$$ emission reductions. Environmental and Resource Economics, 83(1), 171\u2013197.","journal-title":"Environmental and Resource Economics"},{"issue":"5","key":"6838_CR13","first-page":"826","volume":"44","author":"R Dorfman","year":"1954","unstructured":"Dorfman, R., & Steiner, P. O. (1954). Optimal advertising and optimal quality. The American Economic Review, 44(5), 826\u2013836.","journal-title":"The American Economic Review"},{"issue":"3","key":"6838_CR14","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1108\/10309611011092574","volume":"23","author":"JM Drew","year":"2010","unstructured":"Drew, J. M., & Drew, M. E. (2010). Establishing additionality: fraud vulnerabilities in the clean development mechanism. Accounting Research Journal, 23(3), 243\u2013253.","journal-title":"Accounting Research Journal"},{"key":"6838_CR15","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1007\/s10726-017-9545-7","volume":"27","author":"F El Ouardighi","year":"2018","unstructured":"El Ouardighi, F., Kogan, K., Gnecco, G., & Sanguineti, M. (2018). Commitment-based equilibrium environmental strategies under time-dependent absorption efficiency. Group Decision and Negotiation, 27, 235\u2013249.","journal-title":"Group Decision and Negotiation"},{"key":"6838_CR16","doi-asserted-by":"publisher","first-page":"653","DOI":"10.1007\/s10479-018-2927-7","volume":"287","author":"F El Ouardighi","year":"2020","unstructured":"El Ouardighi, F., Kogan, K., Gnecco, G., & Sanguineti, M. (2020). Transboundary pollution control and environmental absorption efficiency management. Annals of Operations Research, 287, 653\u2013681.","journal-title":"Annals of Operations Research"},{"key":"6838_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijpe.2022.108758","volume":"257","author":"L Fang","year":"2023","unstructured":"Fang, L., & Zhao, S. (2023). On the green subsidies in a differentiated market. International Journal of Production Economics, 257, Article 108758.","journal-title":"International Journal of Production Economics"},{"issue":"3","key":"6838_CR18","doi-asserted-by":"publisher","first-page":"1100","DOI":"10.1016\/j.ejor.2022.06.024","volume":"305","author":"K Fu","year":"2023","unstructured":"Fu, K., Li, Y., Mao, H., & Miao, Z. (2023). Firms\u2019 production and green technology strategies: the role of emission asymmetry and carbon taxes. European Journal of Operational Research, 305(3), 1100\u20131112.","journal-title":"European Journal of Operational Research"},{"issue":"1","key":"6838_CR19","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1007\/s10479-018-2903-2","volume":"290","author":"D Ghosh","year":"2020","unstructured":"Ghosh, D., Shah, J., & Swami, S. (2020). Product greening and pricing strategies of firms under green sensitive consumer demand and environmental regulations. Annals of Operations Research, 290(1), 491\u2013520.","journal-title":"Annals of Operations Research"},{"key":"6838_CR20","doi-asserted-by":"crossref","unstructured":"Gori, L., Purificato, F., & Sodini, M. (2024). Green quality choice in a duopoly. Metroeconomica.","DOI":"10.1111\/meca.12460"},{"issue":"2","key":"6838_CR21","doi-asserted-by":"publisher","first-page":"15","DOI":"10.2753\/JOA0091-3367360201","volume":"36","author":"LM Hassan","year":"2007","unstructured":"Hassan, L. M., Walsh, G., Shiu, E. M., Hastings, G., & Harris, F. (2007). Modeling persuasion in social advertising: a study of responsible thinking in antismoking promotion in eight Eastern EU (European Union) member states. Journal of Advertising, 36(2), 15\u201331.","journal-title":"Journal of Advertising"},{"key":"6838_CR22","doi-asserted-by":"publisher","first-page":"150","DOI":"10.1016\/j.jclepro.2018.08.109","volume":"202","author":"J-W Ho","year":"2018","unstructured":"Ho, J.-W., Huang, Y.-S., & Hsu, C.-L. (2018). Pricing under internal and external competition for remanufacturing firms with green consumers. Journal of Cleaner Production, 202, 150\u2013159.","journal-title":"Journal of Cleaner Production"},{"issue":"2","key":"6838_CR23","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1007\/s13235-021-00390-3","volume":"12","author":"Z-H Hu","year":"2022","unstructured":"Hu, Z.-H., & Wang, S.-W. (2022). An evolutionary game model between governments and manufacturers considering carbon taxes, subsidies, and consumers\u2019 low-carbon preference. Dynamic Games and Applications, 12(2), 513\u2013551.","journal-title":"Dynamic Games and Applications"},{"issue":"2","key":"6838_CR24","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1002\/sys.21731","volume":"27","author":"C Li","year":"2024","unstructured":"Li, C., Gu, X., Li, Z., & Lai, Y. (2024). Government-enterprise collusion and public oversight in the green transformation of resource-based enterprises: a principal-agent perspective. Systems Engineering, 27(2), 417\u2013429.","journal-title":"Systems Engineering"},{"key":"6838_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijpe.2024.109491","volume":"280","author":"Y Liang","year":"2025","unstructured":"Liang, Y., & Zhang, Q. (2025). Esg equity or green credit: financing strategies for green transformation in the supply chain under consumption subsidies. International Journal of Production Economics, 280, Article 109491.","journal-title":"International Journal of Production Economics"},{"key":"6838_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.jclepro.2021.130028","volume":"338","author":"Y Ling","year":"2022","unstructured":"Ling, Y., Xu, J., & \u00dclk\u00fc, M. A. (2022). A game-theoretic analysis of the impact of government subsidy on optimal product greening and pricing decisions in a duopolistic market. Journal of Cleaner Production, 338, Article 130028.","journal-title":"Journal of Cleaner Production"},{"issue":"3","key":"6838_CR27","doi-asserted-by":"publisher","first-page":"602","DOI":"10.1016\/j.ejor.2011.11.027","volume":"218","author":"ZL Liu","year":"2012","unstructured":"Liu, Z. L., Anderson, T. D., & Cruz, J. M. (2012). Consumer environmental awareness and competition in two-stage supply chains. European Journal of Operational Research, 218(3), 602\u2013613.","journal-title":"European Journal of Operational Research"},{"key":"6838_CR28","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1007\/s43069-022-00175-3","volume":"3","author":"A Mondal","year":"2022","unstructured":"Mondal, A., Jana, D. K., & Jana, R. K. (2022). Competition of forward and reverse supply chain for selling two substitutable products: novel game theory approach. Operations Research Forum, 3, 66.","journal-title":"Operations Research Forum"},{"key":"6838_CR29","volume-title":"A Course in Game Theory","author":"MJ Osborne","year":"1994","unstructured":"Osborne, M. J., & Rubinstein, A. (1994). A Course in Game Theory. MIT Press."},{"key":"6838_CR30","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1016\/j.ecolecon.2018.02.010","volume":"148","author":"EC Ricci","year":"2018","unstructured":"Ricci, E. C., Banterle, A., & Stranieri, S. (2018). Trust to go green: an exploration of consumer intentions for eco-friendly convenience food. Ecological Economics, 148, 54\u201365.","journal-title":"Ecological Economics"},{"key":"6838_CR31","doi-asserted-by":"publisher","DOI":"10.1016\/j.jretconser.2020.102118","volume":"55","author":"SS Sana","year":"2020","unstructured":"Sana, S. S. (2020). Price competition between green and non green products under corporate social responsible firm. Journal of Retailing and Consumer Services, 55, Article 102118.","journal-title":"Journal of Retailing and Consumer Services"},{"issue":"2","key":"6838_CR32","doi-asserted-by":"publisher","first-page":"1997","DOI":"10.1007\/s10479-020-03895-z","volume":"315","author":"SS Sana","year":"2022","unstructured":"Sana, S. S. (2022). A structural mathematical model on two echelon supply chain system. Annals of Operations Research, 315(2), 1997\u20132025.","journal-title":"Annals of Operations Research"},{"key":"6838_CR33","doi-asserted-by":"publisher","DOI":"10.1016\/j.econlet.2023.111123","volume":"227","author":"JC Strandholm","year":"2023","unstructured":"Strandholm, J. C., Espinola-Arredondo, A., & Munoz-Garcia, F. (2023). Being green first: simultaneous vs. sequential abatement decisions. Economics Letters, 227, Article 111123.","journal-title":"Economics Letters"},{"issue":"2","key":"6838_CR34","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1016\/j.ejor.2020.07.030","volume":"289","author":"R Yang","year":"2021","unstructured":"Yang, R., Tang, W., & Zhang, J. (2021). Technology improvement strategy for green products under competition: the role of government subsidy. European Journal of Operational Research, 289(2), 553\u2013568.","journal-title":"European Journal of Operational Research"},{"key":"6838_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.jclepro.2020.124693","volume":"284","author":"Y Yi","year":"2021","unstructured":"Yi, Y., Wei, Z., & Fu, C. (2021). An optimal combination of emissions tax and green innovation subsidies for polluting oligopolies. Journal of Cleaner Production, 284, Article 124693.","journal-title":"Journal of Cleaner Production"},{"key":"6838_CR36","doi-asserted-by":"publisher","DOI":"10.1016\/j.rser.2023.113516","volume":"183","author":"P Zhang","year":"2023","unstructured":"Zhang, P., Jin, L., & Wang, Y. (2023). Optimizing mechanisms for promoting low-carbon manufacturing industries towards carbon neutrality. Renewable and Sustainable Energy Reviews, 183, Article 113516.","journal-title":"Renewable and Sustainable Energy Reviews"},{"key":"6838_CR37","doi-asserted-by":"publisher","DOI":"10.1016\/j.jclepro.2020.124032","volume":"276","author":"X Zhang","year":"2020","unstructured":"Zhang, X., Chen, T., & Shen, C. (2020). Green investment choice in a duopoly market with quality competition. Journal of Cleaner Production, 276, Article 124032.","journal-title":"Journal of Cleaner Production"},{"key":"6838_CR38","doi-asserted-by":"publisher","DOI":"10.1016\/j.jclepro.2022.135336","volume":"384","author":"J Zhu","year":"2023","unstructured":"Zhu, J., Lu, Y., Song, Z., Shao, X., & Yue, X.-G. (2023). The choice of green manufacturing modes under carbon tax and carbon quota. Journal of Cleaner Production, 384, Article 135336.","journal-title":"Journal of Cleaner Production"},{"issue":"1","key":"6838_CR39","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/j.ejor.2016.08.053","volume":"258","author":"W Zhu","year":"2017","unstructured":"Zhu, W., & He, Y. (2017). Green product design in supply chains under competition. European Journal of Operational Research, 258(1), 165\u2013180.","journal-title":"European Journal of Operational Research"},{"issue":"3","key":"6838_CR40","doi-asserted-by":"publisher","first-page":"1316","DOI":"10.1016\/j.ejor.2023.02.011","volume":"309","author":"H Zolfagharinia","year":"2023","unstructured":"Zolfagharinia, H., Zangiabadi, M., & Hafezi, M. (2023). How much is enough? government subsidies in supporting green product development. European Journal of Operational Research, 309(3), 1316\u20131333.","journal-title":"European Journal of Operational Research"},{"key":"6838_CR41","doi-asserted-by":"publisher","first-page":"4230","DOI":"10.1016\/j.jclepro.2016.11.050","volume":"142","author":"MA \u00dclk\u00fc","year":"2017","unstructured":"\u00dclk\u00fc, M. A., & Hsuan, J. (2017). Towards sustainable consumption and production: competitive pricing of modular products for green consumers. Journal of Cleaner Production, 142, 4230\u20134242.","journal-title":"Journal of Cleaner Production"}],"container-title":["Annals of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-025-06838-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10479-025-06838-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-025-06838-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T12:20:27Z","timestamp":1775737227000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10479-025-06838-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,24]]},"references-count":41,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["6838"],"URL":"https:\/\/doi.org\/10.1007\/s10479-025-06838-8","relation":{},"ISSN":["0254-5330","1572-9338"],"issn-type":[{"value":"0254-5330","type":"print"},{"value":"1572-9338","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,24]]},"assertion":[{"value":"8 May 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 September 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Generative AI was not used to prepare the manuscript.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Generative AI and AI-assisted Technologies in the Writing Process"}}]}}