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The Power-Dependence Solution to Five Exchange Networks

The power-dependence solution to exchange networks assumes that both members of every exchange have equally good alternatives outside the exchange; this solution is an application of the game-theoretic concept of the kernel [2] to exchange networks. All the illustrated networks have four or five nodes and all exchanges are worth 24 points. In every network, nodes and divide 24 points, as do nodes and . You can try to equalize dependence within exchanging pairs by controlling the amounts earned by nodes and .
For example, in the 4-line network let earn 16 in a trade with , who earns 8, and let earn 16 in a trade with , who earns 8. If leaves for , his only alternative partner, he will have to offer at least 16 points and will earn the remaining 8, 8 less than he was earning. , who has no alternative partner, will receive nothing, 8 less than what he was earning. and , then, are equally harmed by a change, as are and . This is the power-dependence solution.

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[1] K. S. Cook and T. Tamagishi, "Power in Exchange Networks: A Power-Dependence Formulation," Social Networks, 14(3-4), 1992 pp. 245–265.
[2] R. B. Myerson, Game Theory: Analysis of Conflict, Cambridge: Harvard University Press, 1991 p. 454.
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