Models of Network Formation in Cooperative Games

Network structures play an important role in many economic situations. The types of networks considered in this chapter connect many individuals who each must establish and maintain their own links. I refer to such networks as communications networks; they describe the bilateral channels through which individuals can communicate and thereby coordinate their actions. The worth that coalitions of individuals can obtain by coordinating their actions is modeled by a coalitional game, which specifies for each coalition S of individuals a worth v(S). Suppose, for example, that there are three individuals: one seller who has one indivisible unit of a good for sale and two potential buyers. Suppose the value of the good is 0 to the seller (s), 1 to the first buyer (b1), and 2 to the second buyer (b2). This situation can be modeled as a coalitional game with player set {s, b1, b2} and v({s}) = v({b1}) = v({b2}) = v({b1, b2}) = 0, v({s, b1}) = 1, and v({s, b2}) = v({s, b1, b2}) = 2. If only the two buyers are linked (the only link formed is b1b2), then the seller cannot communicate with any of the buyers and thus no worth can be generated. If the seller is linked to the first buyer and the first buyer, in turn, is linked to the second buyer (the two links sb1 and b1b2 have been formed), then all three can communicate and coordinate their actions (the seller and the second buyer do so through the first buyer) and a worth of 2 can be generated by selling the good to the second buyer.