Although the literature in network games is relatively new, the equilibrium concept as suggested by Jackson and Wolinsky (1996) has already been applied in many economic and social fields. Some applications can be found in the context of buyer and seller networks, Kranton and Minehart (2001) and Corominas-Bosch (2004), in social contact and labor market networks, Armengol (2004) and Calvo-Armengol and Jackson (2004), and in industrial organization as for example in R&D networks, Goyal and Joshi (2003), and firm collaboration networks, Goyal and Moraga-Gonzalez (2001)32. In the following we will consider how models of network have been applied to these fields to investigate incentives of players to strategically form and sever links.
Kranton and Minehart (2001) analyze a model with a fixed number of buyers and sellers and an exogenously given network structure. To exchange goods a buyer and a seller have to be linked such that trading possibilities are represented by the lin-king structure. They elaborate on a player’s influence that depends on its position in a given network and investigate the equilibrium prices under a given network structure. Competition in networks takes place by means of the English auction and assigns the unit of good from a seller to the buyer with the highest valuation. A player’s valuation is random and a buyer knows its own valuation for the unit of good but not the valuation of the others. Sellers announce prices simultaneously and buyers drop out of the auction as soon as the price exceeds their valuation. They obtain that buyers tend to elaborate a large number of links with different sellers to pool demand uncertainties. The allocation rule is given by a player’s payoff that is determined by the overall network of buyers and sellers. Corominas-Bosch (2004) investigates a model similar to the one in Kranton and Minehart (2001) where the prices for goods are determined by an alternation offers bargaining process rather than through an auction. A link is necessary for a buyer and seller to bargain over the unit of good hold by the seller. They model the bargaining process as a variation of the Rubinstein bargaining model where the expected payoff of a buyer and a seller can be calculated as a function of the network structure where each players
bargai-32See a survey of this literature in Goyal and Moraga-Gonzalez (2003).
ning power depends on its position in the network. First sellers announce prices and each buyer receives offers from those sellers to whom he is linked. Buyers can choose whether to accept one of the offers or not. Afterwards the pairs of sellers and buyers that have successfully traded the goods are cleared from the market and in the next round the buyers are to call out prices. Each period the role of the proposer switches as long as there are no linked pairs of buyers and sellers left in the market, whereas each period a buyer’s and a seller’s valuation of the good is discounted by a common discount factor δ ∈ (0, 1). She defines a player’s expected payoff as a function of the link pattern between all buyers and sellers.
Calvo-Armengol (2004) analyzes a model in which initially all workers are employ-ed and may randomly lose their jobs with a certain probability. They hear about job opportunities through their links to other workers. He investigates the relati-on between the density of the persrelati-onal crelati-ontact network and the success in finding new jobs. He shows that each worker’s information flow of new job opportunities is determined by the shape of direct and two-links-away contacts given by the social network structure in the way that direct links are always beneficial for the informa-tion flow whereas two-links-away contacts are detrimental for the informainforma-tion flow.
When one of a worker’s direct contacts becomes better connected the direct contact might pass his information on to one of his other direct contacts which decreases the probability for a player to receive the information. Other indirect contacts have no influence on the probability of a worker to obtain a new job. He defines a worker’s payoff as a function of the direct and indirect contacts in the network and the total value of a network is given by the sum of all workers’ payoffs.
Calvo-Armengol und Jackson (2004) introduced a model in which workers are linked through a network of social contacts and obtain information about job opportuni-ties through their linking structure. Over time workers randomly lose their jobs and new job opportunities randomly arrive. They investigate two different aspects. First, they investigate the correlation between the employment status of workers in the same network. They show that there is a positive correlation between the employ-ment status of connected workers. Furthermore they investigate how the duration of unemployment affects future employment in a given network structure. They find
that with a higher density of the network a longer duration of unemployment lowers the probability of getting a job. The intuition of the result is the following: The longer the duration of unemployment, the more likely it is that the neighbours of an unemployed are also unemployed and therefore do not pass on information about job opportunities. A workers probability of finding a new job is determined by the current employment situation of the other workers and the overall network of social contacts.
Goyal and Joshi (2003) analyze firms’ incentives to form collaborative links to re-duce their cost of production. They find that under different market competitions firms have an incentive to form collaborative alliances. For a given network structure they can calculate equilibrium quantities in the Cournot competition case supplied by each firm and equilibrium prices in the Bertrand competition case, respectively.
Given the quantities of all firms one can calculate each firm’s payoff as a function of the network of collaboration alliances and the quantities supplied by each firm in the network. One ends up with a well defined allocation rule and value function, which is given by the overall payoff of all firms. Further, they can be adopted to analyze incentives for firms to form such collaboration alliances and to investigate stability and efficiency of network structures.
Goyal and Moraga-Gonzalez (2001) study incentives of competing firms to form col-laboration alliances to invest in R&D. Firms can form pairwise colcol-laboration links to share R&D knowledge about a cost-reducing technology such that the set of pairwise collaboration links defines a network. Given the network of collaboration links each firm chooses a level of R&D effort (costs to invest in R&D). After firms chose their effort level, each firm chooses a quantity to supply in the market. They investigate different network structures with respect to effort level of the firms. They define a firm’s payoff as its profit that derives from the network of collaboration agreements and the effort level of all firms in the network. Goyal and Moraga-Gonzales (2001) are now able to investigate firms’ incentives to strategically form R&D collaboration links.
These and many other applications to network games in economic and social
en-vironments have been applied that show the potential of network games to describe economic situations. Still relatively few attempts have been made to model strategic link formation in international trade by means of network formation games.33 In the following we will discuss the connection model as an introducing example in network formation which has often been discussed in the literature. In the connec-tion model players obtain a payoff from direct and indirect connecconnec-tions, whereas the payoff from an indirect connection decreases with the distance between the two players. Players have to weight the costs of forming new links with the returns from direct and indirect links.