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Decentralized dynamics for finite opinion games. (English) Zbl 1414.91072
Summary: Game theory studies situations in which strategic players can modify the state of a given system, in the absence of a central authority. Solution concepts, such as Nash equilibrium, have been defined in order to predict the outcome of such situations. In multi-player settings, it has been pointed out that to be realistic, a solution concept should be obtainable via processes that are decentralized and reasonably simple. Accordingly we look at the computation of solution concepts by means of decentralized dynamics. These are algorithms in which players move in turns to decrease their own cost and the hope is that the system reaches an “equilibrium” quickly.
We study these dynamics for the class of opinion games, recently introduced by D. Bindel et al. [in: Proceedings of the 2011 IEEE 52nd annual symposium on foundations of computer science – FOCS 2011, Palm Springs, CA, USA, October 22–25. Los Alamitos, CA: IEEE Computer Society. 57–66 (2011; Zbl 1292.91148)]. These are games, important in economics and sociology, that model the formation of an opinion in a social network. We study best-response dynamics and show upper and lower bounds on the convergence to Nash equilibria. We also study a noisy version of best-response dynamics, called logit dynamics, and prove a host of results about its convergence rate as the noise in the system varies. To get these results, we use a variety of techniques developed to bound the mixing time of Markov chains, including coupling, spectral characterizations and bottleneck ratio.

##### MSC:
 91A43 Games involving graphs 91A06 $$n$$-person games, $$n>2$$ 91D30 Social networks; opinion dynamics
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