Strategic basins of attraction, the path dominance core, and network formation games

Abstract

Given the preferences of players and the rules governing network formation, what networks are likely to emerge and persist? And how do individuals and coalitions evaluate possible consequences of their actions in forming networks? To address these questions we introduce a model of network formation whose primitives consist of a feasible set of networks, player preferences, the rules of network formation, and a dominance relation on feasible networks. The rules of network formation may range from noncooperative, where players may only act unilaterally, to cooperative, where coalitions of players may act in concert. The dominance relation over feasible networks incorporates not only player preferences and the rules of network formation but also assumptions concerning the degree of farsightedness of players. A specification of the primitives induces an abstract game consisting of (i) a feasible set of networks, and (ii) a path dominance relation defined on the feasible set of networks. Using this induced game we characterize sets of network outcomes that are likely to emerge and persist. Finally, we apply our approach and results to characterize the equilibrium of well known models and their rules of network formation, such as those of Jackson and Wolinsky (1996) and Jackson and van den Nouweland (2005).

Club networks with multiple memberships and noncooperative stability

“…In particular, we can conclude that no noncooperative improvement path forms a circuit and that each club network in K is either a Nash club network or is a network on a finite, noncooperative improvement path leading to a Nash club network (e.g., see Section 3.3.3 and Theorem 1 in Page and Wooders, 2009). Thus, all club network formation games satisfying (A-1) and (A-2) have singleton basins of attraction (i.e., basins containing only one network), and thus all such games have unique, nonempty noncooperative path dominance cores (see Page and Wooders, 2009, Theorem 4).…”

“…It is interesting to note that if the dominance relation is defined based on a notion of "possible replies", which can be thought of as "improving replies" (rather than best replies in the usual sense), then the admissible set is equivalent to the set of Nash equilibrium. The admissible set is related to "basins of attraction" for network formation games (Page andWooders, 2005, 2009). In the framework of the current paper, in part because of the finiteness of the strategy sets, each Nash equilibrium strategy profile is a basin of attraction and the union of all basins of attraction coincides with (the network rendition of) the admissible set.…”

An allocation rule for dynamic random network formation processes

“…They studied the notion of a supernetwork which is a collection of directed networks and represents coalitional preferences and rules governing network formation. Page and Wooders (2009) introduced a model of network formation with a set of feasible networks, player preferences, rules of network formation, and a dominance relation on feasible networks. The authors characterized sets of network outcomes that are likely to emerge and persist.…”

Networks and clubs

“…We could instead have preferences given by preference relations directly over club memberships; that is, we could equally well have had hedonic preferences (see Bogomolnaia and Jackson, 2002 and, for a formulation with networks, Page and Wooders, 2005).…”

“…A full treatment of Nash stability in club networks is beyond the scope of our paper. Moreover, it is treated in a more general setting allowing hedonic games as a special case in Page and Wooders (2005), where in fact we introduce a concept of farsighted Nash stability and provide some characterization results. 17 A paper successfully combining aspects of cooperative game theory and free entry is Bogomolnaia and Jackson (2002).…”