The Power-Dependence Solution to Five Exchange Networks

C's share

B's share

4-line network

kite network

5-line network

tee network

hourglass network

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.

Details

[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.

Permanent Citation

Phillip Bonacich

"The Power-Dependence Solution to Five Exchange Networks"
http://demonstrations.wolfram.com/ThePowerDependenceSolutionToFiveExchangeNetworks/
Wolfram Demonstrations Project
Published: March 7, 2011