Nonlinearduopolygameswithpositivecostexternalitiesdueto spillover eects Gian-Italo Bischi a, * , Fabio Lamantia b a Istituto di Scienze Economiche, University of Urbino, via Sa, 61026 Urbino, Italy b Facolt a di Economia, University of Catania, Italy Accepted 3 January 2001 Abstract ACournotduopolygameisproposedwheretheinterdependencebetweenthequantity-setting®rmsisnotonlyrelatedtotheselling price,determinedbythetotalproductionthroughagivendemandfunction,butalsooncost-reductioneectsrelatedtothepresenceof thecompetitor.Suchcostreductionsareintroducedtomodeltheeectsofknow-howspillovers,causedbytheabilityofa®rmtotake advantage, for free, of the results of competitors' Research and Development R&D) results, due to the diculties to protect intel- lectualpropertiesortoavoidthemovementsofskilledworkersamongcompeting®rms.Theseeectsmaybeparticularlyimportantin themodelingofhigh-techmarkets,wherecostsaremainlyrelatedtoR&Dandworkers'training.Theresultsofthispaperconcernthe existence and uniqueness of the Cournot±Nash equilibrium, located at the intersection of non-monotonic reaction curves, and its stabilityundertwodierentkindsofboundedrationalityadjustmentmechanisms.TheeectsofspilloversontheexistenceoftheNash equilibrium are discussed, as well as their in¯uence on the kind of attractors arising when the Nash equilibrium is unstable. Methods for the global analysis of two-dimensional discrete dynamical systems are used to study the structure of the basins of attrac- tion. Ó 2001 Elsevier Science Ltd. All rights reserved. 1. Introduction An oligopoly isamarketstructurewhereafewproducers,eachofappreciablesize,manufacturethesame commodity,orhomogeneouscommoditiesi.e.perfectsubstitutablegoods).Thefewnessof®rmsgivesrise to interdependencies, that is, each producer must take into account the actions of the competitors in choosing its own action. In the industrial organization literature, one of the most widely used mathematical representations of oligopolycompetitionistheCournotmodel,®rstintroducedbytheFrenchMathematicianCournotabout 160 years ago, which describes a market where N quantity-setting ®rms, producing homogeneous goods, update their production strategies in order to maximize their pro®ts. In the original work of Cournot, as well as in many of the subsequent papers, the above-mentioned interdependence only depends on the fact thattheretailprice p isdeterminedbythetotalsupplyonthemarket, Q q 1 q 2 q N ,accordingto agiveninversedemandfunction, p f Q.Butalsoothersourcesofinterdependenciescanbeconsidered, forexampleoriginatedbypositivecostexternalities,i.e.areductionofproductioncostduetothepresence of competitors. This may appear rather paradoxical, but several reasons can be given to support such a cost-reduction eect, due to technological and intellectual spillovers between companies, related to ex- changes of information on technological innovations, skilled labor, results of Research and Development Chaos, Solitons and Fractals 13 2002) 701±721 www.elsevier.com/locate/chaos * Corresponding author. E-mail addresses: bischi@econ.uniurb.it G.-I. Bischi), lamantia@unict.it F. Lamantia). 0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0960-077901)00006-6