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Sustainable Cooperation in Dynamic Games on Event Trees with Players’ Asymmetric Beliefs

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Abstract

We built a time-consistent cooperative solution for the class of dynamic games played over event trees in the context where the tree structure is given, but the players have different beliefs about the transition probabilities between nodes. Our three-step approach is as follows. First, we consider three alternative methods for aggregating the players’ beliefs and assume that the players agree to adopt one of them if they decide to cooperate. Second, we determine the Nash bargaining outcomes for the whole duration of the game. Finally, to insure sustainability of cooperation throughout the whole duration of the game, we propose two time-consistent decompositions over nodes of each player’s cooperative share, namely a proportion-consistent and a node-consistent allocation schemes. We illustrate our results with a simple Cournot oligopoly with capacity constraints.

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Notes

  1. In this paper, we do not consider imprecise probabilities and refer the interested reader to [31] for a discussion on their use in event trees.

  2. We call the strategy \( S \)-adapted to highlight that at each period we have a Sample of events corresponding to the nodes.

  3. We use term “belief aggregation” in Sect. 3 when describing the methods of aggregating probability distributions, whereas in the literature the term “opinion pooling” is used ([4, 26, 27]). To avoid confusion, we use the “opinion pooling” technique to aggregate players’ beliefs on transition probabilities but use the term “belief aggregation” to be in line with the theory of dynamic games with imperfect information.

  4. Updating the weights may be considered only in the cases of linear and geometric belief aggregations. In multiplicative belief aggregation the weights are not used. The “calibrating” probability function plays the role of weights.

  5. See [12] for a definition of the Nash bargaining solution.

  6. In the considered problem, only the regular case is meaningful. Therefore, the non-regular case, when \(\nu _{0}=0\), is omitted here.

  7. One can think of these additional properties as a refinement approach of the IDP.

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Acknowledgements

We would like to thank the editor and the two anonymous reviewers for their helpful comments. This research was partially conducted during the research stay of the first author at GERAD. The work of the first author was supported by the Shandong Province “Double-Hundred Talent Plan” (No. WST2017009). The work of the second author is supported by NSERC, Canada, Grant RGPIN-2016-04975.

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Authors and Affiliations

  1. Saint Petersburg State University, Saint Petersburg, Russia

    Elena M. Parilina

  2. Institute of Applied Mathematics of Shandong, School of Mathematics and Statistics, Qingdao University, Qingdao, People’s Republic of China

    Elena M. Parilina

  3. Chair in Game Theory and Management, GERAD, HEC Montréal, Montreal, Canada

    Georges Zaccour

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  1. Elena M. Parilina
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Correspondence to Elena M. Parilina.

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Communicated by Felix L. Chernousko.

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Parilina, E.M., Zaccour, G. Sustainable Cooperation in Dynamic Games on Event Trees with Players’ Asymmetric Beliefs. J Optim Theory Appl 194, 92–120 (2022). https://doi.org/10.1007/s10957-022-02010-5

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