Reaching Optimal Distributed Estimation Through Myopic Self-Confidence Adaptation

Consider discrete-time linear distributed averaging dynamics, whereby a finite number of agents in a network start with uncorrelated and unbiased noisy measurements of a common state of the world modeled as a scalar parameter, and iteratively update their estimates following a non-Bayesian learning rule. Specifically, let every agent update her estimate to a convex combination of her own current estimate and those of her neighbors in the network (this procedure is also known as the French-DeGroot model, or the consensus algorithm). As a result of this iterative averaging process, each agent obtains an asymptotic estimate of the state of the world, and the variance of this individual estimate depends on the matrix of weights the agents assign to themselves and to the others. We study a game-theoretic multi-objective optimization problem whereby every agent seeks to choose her self-confidence value in the convex combination in such a way to minimize the variance of her asymptotic estimate of the state of the world. Assuming that the relative influence weights assigned by the agents to their neighbors in the network remain fixed and form an irreducible relative influence matrix, we characterize the Pareto frontier of the problem, as well as the set of Nash equilibria in the resulting game.