When Knowing Early Matters: Gossip, Percolation and Nash Equilibria
Continually arriving information is communicated through a network of n agents, with the value of information to the j’th recipient being a decreasing function of j ∕ n, and communication costs paid by recipient. Regardless of details of network and commu
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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 function 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 ! 1 limit) may be efficient (Nash payoff D social optimum payoff) or wasteful (0 < Nash payoff < social optimum payoff) or totally wasteful (Nash payoff D 0). We study the cases of the complete network (constant communication costs between all agents), the grid with only nearest-neighbor 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 N torus with short and long range interactions; Sect. 6.2) has now been studied rigorously by Chatterjee and Durrett [4]. Keywords First passage percolation • Gossip • Information • Nash equilibrium • Rank based • Social network
Mathematics Subject Classification (2010): 60K35, 91A28
D.J. Aldous () Department of Statistics, University of California, 367 Evans Hall, Berkeley, CA 94720-3860, USA e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 1, © Springer-Verlag Berlin Heidelberg 2013
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D.J. Aldous
1 Introduction A topic which one might loosely call “random percolation of information through networks” arises in many different contexts, from epidemic models [2] and computer virus models [10] to gossip algorithms [8] designed to keep nodes of a decentralized network updated about information needed to maintain the network. This topic differs from communication networks in that we envisage information as having a definite source but no definite destination. In this paper we study an aspect where the vertices of the network are agents, and where there are costs and benefits associated with the different choices that agents may make in communicating information. In such “economic game theory” settings one anticipates a social optimum strategy that maximizes the total net payoff to all agents combined, and an (often different) Nash equilibrium characterized by the property that no one agent can benefit from deviating from the Nash equilibrium strategy followed by all other agents (so one anticipates that any reasonable process of agents adjusting strategies in a selfish way will lead to some Nash equilibrium). Of course a huge number of different models of costs, benefits and choices could f
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