 
Summary: When Knowing Early Matters: Gossip,
Percolation and Nash Equilibria
David J. Aldous
Abstract Continually arriving information is communicated through a network of
n agents, with the value of information to the j'th recipient being a decreasing func
tion of j/n, and communication costs paid by recipient. Regardless of details of
network and communication costs, the social optimum policy is to communicate
arbitrarily slowly. But selfish agent behavior leads to Nash equilibria which (in the
n limit) may be efficient (Nash payoff = social optimum payoff) or wasteful
(0 < Nash payoff < social optimum payoff) or totally wasteful (Nash payoff = 0).
We study the cases of the complete network (constant communication costs between
all agents), the grid with only nearestneighbor communication, and the grid with
communication cost a function of distance. The main technical tool is analysis of
the associated first passage percolation process or SI epidemic (representing spread
of one item of information) and in particular its "window width", the time interval
during which most agents learn the item. In this version (written in July 2007) many
arguments are just outlined, not intended as complete rigorous proofs. One of the
topics herein (first passage percolation on the N ŚN torus with short and long range
interactions; section 6.2) has now been studied rigorously by Chatterjee and Durrett
[4].
