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The Subgame-Consistent Shapley Value for Dynamic Network Games with Shock

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Abstract

In the paper, cooperative repeated network games containing network formation stages are studied. After the first network formation stage, a particular player with a given probability may stop influencing other players by removing all her links and receiving zero payoffs. This effect is called “shock.” The effect of shock may appear only once, and the stage number, at which shock appears, is chosen at random. In the cooperative scenario of the game, subgame consistency of the Shapley value, based on a characteristic function, which is constructed in a special way, is investigated. To prevent players from breaking the cooperative agreement, a mechanism of stage payments—so-called imputation distribution procedure—is designed.

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Notes

  1. Henceforth, to avoid confusion and simplify understanding, we will not add network formation stages to the number of stages in the game, whereas we will enumerate other stages from 1 to \(\ell \). The game will be considered as an \(\ell \)-stage dynamic game with at most two network formation stages.

  2. It means that starting this moment, all links involving player m are eliminated from the network. However, player m still belongs to set N, but receives zero stage payoffs.

  3. Behavior \((0,{\bar{{\bar{u}}}}_m)\) of player m, who leaves the game after shock, prescribes her to choose control \({\bar{{\bar{u}}}}_m\). However, according to (9), her payoff will be equal to 0 afterward for any of her controls \({\bar{{\bar{u}}}}_m \in U_m\).

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Acknowledgments

We thank three anonymous referees for their comments that have helped in the improvement of the paper. We also thank the audience of the 20th Conference of the International Federation of Operational Research Societies for helpful discussion and suggestions.

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Authors and Affiliations

  1. Faculty of Applied Mathematics and Control Processes, Saint Petersburg State University, Universitetsky Prospekt 35, Petergof, Saint Petersburg, 198504, Russia

    Leon Petrosyan & Artem Sedakov

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  1. Leon Petrosyan
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  2. Artem Sedakov
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Correspondence to Artem Sedakov.

Additional information

The reported study was supported by the Russian Foundation for Basic Research (Projects 13-01-91160, 14-01-31141) and Saint Petersburg State University (Project 9.38.245.2014).

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Petrosyan, L., Sedakov, A. The Subgame-Consistent Shapley Value for Dynamic Network Games with Shock. Dyn Games Appl 6, 520–537 (2016). https://doi.org/10.1007/s13235-015-0166-y

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