Abstract
A distributed Nash equilibrium seeking algorithm is presented for networked games. We assume an incomplete information available to each player about the other players’ actions. The players communicate over a strongly connected digraph to send/receive the estimates of the other players’ actions to/from the other local players according to a gossip communication protocol. Due to asymmetric information exchange between the players, a non-doubly (row) stochastic weight matrix is defined. We show that, due to the non-doubly stochastic property, there is no exact convergence. Then, we present an almost sure convergence proof of the algorithm to a Nash equilibrium of the game. Moreover, we extend the algorithm for graphical games in which all players’ cost functions are only dependent on the local neighboring players over an interference digraph. We design an assumption on the communication digraph such that the players are able to update all the estimates of the players who interfere with their cost functions. It is shown that the communication digraph needs to be a superset of a transitive reduction of the interference digraph. Finally, we verify the efficacy of the algorithm via a simulation on a social media behavioral case.
This work was supported by an NSERC Discovery Grant.
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Notes
- 1.
The same objective is followed by [9] to find a broadcast gossip algorithm (with non-doubly stochastic weight matrix) in the area of distributed optimization. However, in the proof of Lemma 2 ([9] page 1348) which is mainly dedicated to this discussion, the doubly stochasticity of W(k) is used right after Eq. (22) which violates the main assumption on W(k).
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Salehisadaghiani, F., Pavel, L. (2017). Nash Equilibrium Seeking with Non-doubly Stochastic Communication Weight Matrix. In: Duan, L., Sanjab, A., Li, H., Chen, X., Materassi, D., Elazouzi, R. (eds) Game Theory for Networks. GameNets 2017. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 212. Springer, Cham. https://doi.org/10.1007/978-3-319-67540-4_1
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