Dynamic Advertising Games in Duopolies under One Step Ahead Optimal Control

Dynamic Advertising Games in Duopolies under One Step Ahead Optimal Control | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Dynamic Advertising Games in Duopolies under One Step Ahead Optimal Control Amit Bhaya, Eugenius Kaszkurewicz This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1898803/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 18 Jan, 2023 Read the published version in Dynamic Games and Applications → Version 1 posted 7 You are reading this latest preprint version Abstract Models of sales response to advertising, also referred to as market share dynamics under advertising have been the concern first of economists and later of researchers in marketing, starting from the late 1950s. Vidale and Wolfe, in 1957, proposed their now eponymous model based on sales data. Since then, most of the work in the field has centered around the application of optimal control methods, leading to open-loop control of the Vidale-Wolfe and variants, in order to achieve a given market share, while minimizing advertising expenditure. The extension to duopolies was made by Deal, in the continuous-time case, and is denominated the Vidale-Wolfe-Deal (VWD) model. In discrete-time VWD models and variants, two firms compete for market share, in a dynamic game setting , described by a pair of difference equations. This paper studies these dynamic games, using the natural concept of one step ahead optimal control, in which each firm optimizes its own performance index at the next step, and only has access to some information about its competitor. Two cases are studied: with and without stipulating target market shares for each firm. It is shown that when target market shares are not specified , for the VWD model, limit cycles of large period can occur when each firm uses linear performance indices, while multiple equilibria may arise when quadratic performance indices are used. Three other proposed models result in games that lead to equilibria and do not have limit cycle behavior. When target market shares are specified, convergence to an equilibrium occurs for all the models proposed in this paper. market dynamics advertising dynamic games discrete-time dynamical systems Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 18 Jan, 2023 Read the published version in Dynamic Games and Applications → Version 1 posted Editorial decision: Major revision 26 Sep, 2022 Reviews received at journal 22 Aug, 2022 Reviewers agreed at journal 07 Aug, 2022 Reviewers invited by journal 04 Aug, 2022 Editor assigned by journal 30 Jul, 2022 Submission checks completed at journal 27 Jul, 2022 First submitted to journal 26 Jul, 2022 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-1898803","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":124453409,"identity":"030803dc-5a28-4638-a054-9df665d70bea","order_by":0,"name":"Amit Bhaya","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0ElEQVRIiWNgGAWjYFACxgaGD2BGAgNDBQMbgwERWhobZ8C0nCFOCwNjMw9CCwNhLfzTDrc/tt1hJ8/fnsAmcXAPH4O59AH8WiRuJzY2555JNpxx5gGbxIFnbAyWfQkErAFraWNOYLiRwCb94QDQL2cI6JAHabFsq0+QB2qROECMFgOQFsa2wwkGRGsxBGqZ2dt23HDjmYfNFkAtPJY9BLTI3U5/8OFnW7W83PHkgzcOHDgmZ85DQAsSACYDBoZjJGiAghqSdYyCUTAKRsHwBwDy40bxaqQkngAAAABJRU5ErkJggg==","orcid":"","institution":"Federal University of Rio de Janeiro","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Amit","middleName":"","lastName":"Bhaya","suffix":""},{"id":124453410,"identity":"497face6-48cd-4182-956a-1ec136ddd5c7","order_by":1,"name":"Eugenius Kaszkurewicz","email":"","orcid":"","institution":"Federal University of Rio de Janeiro","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Eugenius","middleName":"","lastName":"Kaszkurewicz","suffix":""}],"badges":[],"createdAt":"2022-07-26 17:44:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1898803/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1898803/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s13235-023-00489-9","type":"published","date":"2023-01-19T00:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":24620938,"identity":"eaf13ba1-8e48-43de-b9a0-4bf4e00c17e7","added_by":"auto","created_at":"2022-08-01 18:30:27","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":8263229,"visible":true,"origin":"","legend":"","description":"","filename":"OnDynamicGames.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1898803/v1_covered.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Dynamic Advertising Games in Duopolies under One Step Ahead Optimal Control","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-1898803/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"dynamic-games-and-applications","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"dgaa","sideBox":"Learn more about [Dynamic Games and Applications](http://link.springer.com/journal/13235)","snPcode":"13235","submissionUrl":"https://submission.nature.com/new-submission/13235/3","title":"Dynamic Games and Applications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":" market dynamics, advertising, dynamic games, discrete-time dynamical systems","lastPublishedDoi":"10.21203/rs.3.rs-1898803/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1898803/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Models of sales response to advertising, also referred to as market share dynamics under advertising have been the concern first of economists and later of researchers in marketing, starting from the late 1950s. 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It is shown that when target market shares are not specified , for the VWD model, limit cycles of large period can occur when each firm uses linear performance indices, while multiple equilibria may arise when quadratic performance indices are used. Three other proposed models result in games that lead to equilibria and do not have limit cycle behavior. 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