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A Sensitive Flexible Network Approach
Abstract
This paper takes an axiomatic approach to find rules for allocating the value of a network when the externalities generated across components are identifiable. Two new, and different, allocation rules are defined and characterized in this context. The first one is an extension of the player-based flexible-network allocation rule (Jackson (2005)). The second one follows the flexible network approach from a component-wise point of view, where the notion of network flexibility is adjusted with a flavor of core stability. Furthermore, two other allocation rules are proposed by relaxing the axiom of equal treatment of vital players. These collapse into the player-based flexible-network allocation rule (Jackson (2005)) for zero-normalized value functions with no externalities across components.
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Shapley's value axioms are adapted to games in partition function form, and the natural extension of theShapley value for partition function games is derived.
We investigate the uniqueness of stable coalition structures in a simple coalition formation model, for which specific coalition formation games, such as the marriage and roommate models, are special cases that are obtained by restricting the coalitions that may form. The main result is a characterization of collections of permissible coalitions which ensure that there is a unique stable coalition structure in the corresponding coalition formation model. In particular, we show that only single-lapping coalition formation models have a unique stable coalition structure for each preference profile, where single-lapping means that two coalitions cannot have more than one member in common, and coalitions do not form cycles. We also give another characterization using a graph representation, explore the implications of our results for matching models, and examine the existence of strategyproof coalition formation rules.
Previous allocation rules for network games, such as the Myerson value, implicitly or explicitly take the network structure as fixed. In many situations, however, the network structure can be altered by players. This means that the value of alternative network structures (not just sub-networks) can and should influence the allocation of value among players on any given network structure. I present a family of allocation rules that incorporate information about alternative network structures when allocating value.
Thesis (Ph. D.)--State University of New York at Stony Brook, 1996. Includes bibliographical references (leaves 105-108).
Graph-theoretic ideas are used to analyze cooperation structures in games. Allocation rules, selecting a payoff for every possible cooperation structure, are studied for games in characteristic function form. Fair allocation rules are defined, and these are proven to be unique, closely related to the Shapley value, and stable for a wide class of games.
We study the stability and efficiency of social and economic networks, when self-interested individuals have the discretion to form or sever links. First, in the context of two stylized models, we characterize the sets of stable networkds (immune to incentives to form or sever links) and the sets of efficient networks (those which maximize total production or utility). The sets of stable networks and efficients networks do not always intersect. Next, we show that this tension is not unique to these models, but persists generally. In order to assure that there is always at least one efficient graph which is stable, one is forced to allocate resources to nodes (players) who are not responsible for any of the production. We characterize one such allocations rule: the equal split rule. We characterize another rule which fails to assure that efficient graphs are stable, but arises naturally if the allocations result from the bargaining of players.
I prove existence and uniqueness of a component efficient and fair allocation rule when the value of the network is allowed to exhibit any type of externalities across its components. This is done by means of a new specification of the value function, generalizing partial results appearing in Myerson [Myerson, R.B., 1977a. Graphs and cooperation in games. Math. Operations Res. 2, 225–229], Feldman [Feldman, B.E., 1996. Bargaining, coalition formation and value. PhD dissertation. State University of New York at Stony Brook] and Jackson and Wolinsky [Jackson, M.O., Wolinsky, A., 1996. A strategic model of social and economic networks. J. Econ. Theory 71, 44–74]. This component efficient and fair allocation rule is found closely related to an extension of the Shapley value to TU-games in partition function form proposed by Myerson [Myerson, R.B., 1977b. Values of games in partition function form. Int. J. Game Theory 6 (1), 23–31].