The Price of Uncertainty for Social Consensus

How hard is it to achieve consensus in a social network under uncertainty? In this paper we model this problem as a social graph of agents where each vertex is initially colored red or blue. The goal of the agents is to achieve consensus, which is when the colors of all agents align. Agents attempt to do this locally through steps in which an agent changes their color to the color of the majority of their neighbors. In real life, agents may not know exactly how many of their neighbors are red or blue, which introduces uncertainty into this process. Modeling uncertainty as perturbations of relative magnitude $1+\varepsilon$ to these color neighbor counts, we show that even small values of $\varepsilon$ greatly hinder the ability to achieve consensus in a social network. We prove theoretically tight upper and lower bounds on the \emph{price of uncertainty}, a metric defined in previous work by Balcan et al. to quantify the effect of uncertainty in network games.

💡 Research Summary

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The paper studies consensus formation on a social network when agents have only approximate information about the colors of their neighbors. Each vertex of an undirected graph initially holds either red or blue. In the classic consensus game, an agent updates its color to the majority color among its neighbors, and the social cost is the number of edges whose endpoints have different colors (bad edges). The authors introduce observational uncertainty: an agent may mis‑count the number of red and blue neighbors by a multiplicative factor of at most (1+\varepsilon) (with (0<\varepsilon<1)). Under this model an agent’s decision rule becomes an uncertain best response – a color switch that would be optimal if the agent’s counts were exact, but because of the error it may actually increase the social cost.

Balcan et al. (2009, 2011) defined the price of uncertainty (PoU) as the worst‑case ratio between the final social cost after any sequence of uncertain best responses and the initial cost. Their bounds were (\Omega(\varepsilon^{3} n^{2})) (lower) and (O(\varepsilon n^{2})) (upper), leaving a large gap. This paper closes the gap by proving that for all (\varepsilon = \Omega(n^{-1/4})),

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