We develop a two-stage oligopoly model of price competition in markets with both informed and uninformed (captive) consumers. The model introduces a novel mechanism through which interfirm collaborative R D influences market outcomes. In particular, the second stage of the game where firms set prices is a supermodular game allowing us to analyze strategic complementarities in pricing behavior. We show that this type of market friction creates a new channel of influence for collaborative R D. Our analysis reveals how consumer heterogeneity and cost heterogeneity jointly shape the incentives for collaboration among firms, offering new insights into the design of efficient innovation networks in oligopolistic markets.

We examine the role of an environmental tax on R&D collaboration networks between firms in the context of developing cleaner production processes. First, we show that three types of innovation networks can emerge in equilibrium. Then, we examine the relationship between the environmental tax and the incentives for firms to add more collaborative links. We show that an increase in the environmental tax does not always incentivize firms to increase the number of R&D collaborations. Finally, we examine the optimal tax that the public authorities should set in order to maximize the social welfare. Given tractability issues, for this problem, we restrict our attention to the case of three firms. We establish that the optimal environmental tax depends on the cost of link formation. The higher the cost of link formation, the lower is the environmental tax. Similarly, we observe that the optimal environmental tax increases with the number of links that an efficient network contains.

We study the formation of multilayer networks where payoffs are determined by the degrees of players in each network. We begin by imposing either concavity or convexity in degree on the payoff function of the players. We then explore distinct network relationships that result from inter- and intra-network spillovers captured by the properties of supermodularity/submodularity and strategic complementarity respectively. We show the existence of equilibria and characterize them. Additionally, we establish both necessary and sufficient conditions for an equilibrium to occur. We also highlight the connection, in equilibrium, between inter-network externalities and the identity of linked players in one network given the identity of linked players in the other network. Furthermore, we analyze efficient multilayer networks. Finally, we extend our models to contexts with more than two layers, and scenarios where agents receive a bonus for being connected to the same individuals in both networks.

We consider a setting where players can form multiple links under different networks. This gives rise to a multi-relational network. First, we restrict our study to situations where the payoff of a player depends on the number of links, he or she has formed in the different networks, without any externalities providing by the links formed by the other players, but each link requires the consent of the two players involved in. We examine the existence of multi-relational networks, and provide properties of the set of equilibrium networks. Second, we introduce other players links externalities in the model. We examine the existence of pairwise equilibrium multirelational networks, as well as the properties of the pairwise equilibrium multi-relational networks, depending on the type of spillovers that occur within and across distinct networks. Moreover, we compare equilibrium networks obtained in this context and equilibrium networks obtained in the first framework. Université de Lyon ; CNRS, GATE Lyon St Etienne, France. email: pascal.billand@univ-st-etienne.fr. Université de Grenoble 2, GAEL, GATE Lyon St Etienne. Email: christophe.bravard@univ-st-etienne.fr. Georges Washington University, Department of Economics, Monroe Hall, Suite 34

We consider an oligopoly setting in which firms form pairwise collaborative links in research and development with other firms. Each collaboration generates a value that depends on the identity of the firms that collaborate. First, we provide properties satisfied by pairwise equilibrium networks and efficient networks. Second, we use these properties in two types of situation: (1) there are two groups of firms, and the value of a collaboration is higher when firms belong to the same group; (2) some firms have more innovative capabilities than others. These two situations provide clear insights about how firms' heterogeneity affects both equilibrium and efficient networks. We also show that the most valuable collaborative links do not always appear in equilibrium, and a public policy that increases the value of the most valuable links may lead to a loss of social welfare. ( JEL C70, L13, L20)

We introduce a price oligopoly model with informed and uninformed consumers, thus creating a new channel of influence for collaborative R&D. Firms can establish pair-wise collaborative research links with other firms to lower production costs. Informed consumers buy from the lowest cost firms while uninformed consumers buy from any firm whose price does not exceed their reservation prices. In contrast to earlier models of price setting oligopolies which result in no R&D, our model can lead to positive levels of R&D collaborations. It also leads to increased social welfare, since increased R&D collaborations lower productions costs. Although uninformed consumers pay higher prices, informed consumers are better off as higher R&D leads to lower marginal costs and prices. We then allow for heterogeneity in consumers reservation prices or firm profits and again find higher levels of R&D. Interestingly, under such heterogeneity, the size of the stable collaborative group increases as R&D costs increase. Finally, we extend the model by introducing different markets where several firms compete for the uninformed consumers. We now find an interesting pattern of R&D activity; in equilibrium, each market has only one active firm and firms collaborate only with (active) firms that do not operate in their own markets.

We study network formation in a situation where the network allows players to obtain information (signals) about other players. This information is important for making a payoff relevant decision. However, not all information is reliable and so players may have an incentive to check it. By obtaining multiple messages about the same player through the network, a player learns whether his information is reliable for making the payoff relevant decision. We study the existence and architecture of strict Nash networks. We find that players who are involved in at least three links sponsor all links they are involved in. These players are similar to the central players in center sponsored stars. We show that strict Nash networks can be over-connected as well as under-connected as compared to efficient networks. Finally, we extend the basic model to study heterogeneous populations. In the first scenario, we allow for the co-existence of players who only value checked information and players who also value information with unknown reliability. In the second scenario, players who do not care about checking their information co-exist with players who do. Our results are robust to both types of heterogeneity, with one exception: the presence of a single player who cares only about checked information is enough to ensure that center sponsored stars are no longer stable.

We consider a multimarket framework where a set of firms compete on two oligopolistic markets. The cost of production of each firm allows for spillovers across markets, ensuring that output decisions for both markets have to be made jointly. Prior to competing in these markets, firms can establish business intelligence gathering links with other firms. A link formed by a firm generates two types of externalities for competitors and consumers. We characterize the business intelligence equilibrium networks and networks that maximize social welfare. By contrast with single-market competition, we show that in multimarket competition there exist situations where intelligence-gathering activities are underdeveloped with regard to social welfare and should be tolerated, if not encouraged, by public authorities.

We examine a non-cooperative model of network formation where players may stop functioning. We identify conditions under which Nash and efficient networks will remain connected after the loss of k nodes by introducing the notion of k-Node Super Connectivity.

We consider an oligopoly setting in which firms form pairwise collaborative links in RD (2) some firms have more innovative capabilities than others. These two situations provide clear insights about how uncertainty affects both equilibrium and efficient networks. We also show that a public policy that targets the most valuable collaborative links by an increase of their probabilities of success may lead to a loss of expected welfare. JEL classification: C70, L13, L20.

We study formation of mutual insurance networks in a model where agents who obtain more resources share a xed amount of resources with all directly linked agents that obtain fewer resources. We identify the pairwise stable networks and ecient networks in a basic model where agents are identical. Then, we introduce in the model two types of heterogeneity: an exogenous one, where agents diers in their income or in their preferences over the transfer scheme, and an endogenous heterogeneity where the costs of linking to an agent depends on the number of links the latter has already formed in the network. We examine the impact of these heterogeneities on stability and eciency.

In this paper we examine efficient networks in network formation games with global spillovers that satisfy convexity and sub-modularity properties. Unlike the previous literature we impose these properties on individual payoff functions. We establish that efficient networks of this class of games are nested split graphs. This allows us to complete the work of Goyal and Joshi (2006) and Westbrock (2010) on collaborative oligopoly networks.

We study formation of mutual insurance networks in a model where agents who obtain more resources share a xed amount of resources with all directly linked agents that obtain fewer resources. We identify the pairwise stable networks and ecient networks in a basic model where agents are identical. Then, we introduce in the model two types of heterogeneity: an exogenous one, where agents diers in their income or in their preferences over the transfer scheme, and an endogenous heterogeneity where the costs of linking to an agent depends on the number of links the latter has already formed in the network. We examine the impact of these heterogeneities on stability and eciency.

We consider an oligopoly setting in which firms form pair-wise collaborative links in R&D with other firms, and then compete in an oligopoly. Each collaborative link allows firms involved in to obtain process innovation with some idiosyncratic probability. First, we assume that the process innovation impacts identically the two firms involved in a collaborative link. We provide a condition satisfied by any equilibrium network. Second, we deal with situations where the process innovation associated with a collaborative link between two firms affects these firms in a different way. We highlight two results. The first is called the tyranny of the weakest: firms which are the most able to use the process innovation cannot form links while firms which are the least able to use the process innovation are linked together. The second is called positive assortative matching with regard to the ability to take advantage of a process innovation. JEL classification: C70, L13, L20.

In this note, we extend the Goyal and Joshi's model of collaboration networks in oligopoly to multi-market situations. We examine the incentive of firms to form links and the architectures of the resulting equilibrium networks in this setting. We then present some results on efficient networks.

Social networks, be it on the internet or in real life, facilitate information flows. We model this by giving agents incentives to link with others and receive information through those links. We consider networks where agents have an incentive to confirm the information they receive from others. Our paper analyzes the social networks that are formed. We first study the existence of Nash equilibria and then characterize the set of strict Nash networks. Next, we characterize the set of strictly efficient networks and discuss the relationship between strictly efficient networks and strict Nash networks. Finally, we check the robustness of our results by allowing for heterogeneity among agents, possibility of bilateral deviations of agents, and decay in the network.

In this article, we examine the role played by heterogeneity in the popular “connections model” of Jackson and Wolinsky (J Econ Theory 71(1):355–365, 1996). We prove that when heterogeneity with respect to the values of resources, or the information decay parameter, depends on the identity of the player who forms the link, and the player who receives the link, all networks can be supported as Nash. Moreover, we show that Nash networks may not always exist. Interestingly, in the absence of decay, neither result occurs in a model with value heterogeneity. Finally, we show that when reducing heterogeneity, all networks cannot be supported as Nash, and the non-existence problem also disappears.