On Threshold Behavior In Query Incentive Networks

Motivated by the role of incentives in large-scale information systems, Kleinberg and Raghavan (FOCS 2005) studied strategic games in decentralized information networks. Given a branching process that specifies the network, the rarity of answers to a specific question, and a desired probability of success, how much reward does the root node need to offer so that it receives an answer with this probability, when all of the nodes are playing strategically? For a specific family of branching processes and a constant failure probability, they showed that the reward function exhibited a threshold behavior that depends on the branching parameter b.In this paper We Study two factors that can contribute to this transition behavior, namely, the branching process itself and the failure probability. On one hand we show that the threshold behavior is robust with respect to the branching process: for all branching processes and any constant failure probability, if b > 2 then the required reward is linear in the expected depth of the search tree, and if b < 2 then the required reward is exponential in that depth. On the other hand we show that the threshold behavior is fragile with respect to the failure probability sigma: if sigma is inversely polynomial in the rarity of the answer, then all branching processes require rewards exponential in the depth of the search tree.