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Sustainability of Cooperation in Dynamic Games Played over Event Trees

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Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science

Part of the book series: Fields Institute Communications ((FIC,volume 79))

Abstract

In this tutorial, we recall the main ingredients of the theory of dynamic games played over event trees and show step-by-step how to build a sustainable cooperative solution.

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Notes

  1. 1.

    To define a game in normal form, we need three elements: (a) a finite set of players M = {1, …, m}, (b) a strategy set S i of player i ∈ M, and (c) a payoff function \(\pi _{i}:\prod \limits _{i\in M}S_{i} \rightarrow \mathbb{R}\).

  2. 2.

    For a detailed treatment in the context of this class of games, see Haurie et al. [22].

  3. 3.

    We can easily extend our framework to the case where the players maximize a weighted sum of payoffs.

  4. 4.

    The implicit assumption here is that players’ utilities (gains) are comparable and transferable; otherwise side payments do not make sense.

  5. 5.

    The following example illustrates this statement. Consider a three-player cooperative game with characteristic function values given by

    $$\displaystyle\begin{array}{rcl} v(\left \{1\right \})& =& v(\left \{2\right \}) = v(\left \{3\right \}) = 0, {}\\ v(\left \{1, 2\right \})& =& v(\left \{1, 3\right \}) = v(\left \{2, 3\right \}) = a,\quad v(\left \{1, 2, 3\right \}) = 1 {}\\ \end{array}$$

    where 0 < a ≤ 1. It is easy to verify that three cases can occur: (i) If 0 < a < 2∕3, then the core contains all imputations satisfying \(y_{j} \geq 0,\sum \limits _{j\in G}y_{j} \geq a\) and \(\sum \limits _{j\in M}y_{j} = 1.\) (ii) If a = 2∕3, then the core is a singleton, that is, the only imputation belonging to the core is \(\left (1/3, 1/3, 1/3\right )\). (iii) If a > 2∕3, then the core is empty.

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Acknowledgements

This paper draws from my previous work on the subject, in particular Reddy, Shevkoplyas and Zaccour [52] and Parilina and Zaccour [38]. I would like to thank my co-authors in these papers, as well as Alain Haurie and Leon Petrosjan for many stimulating discussions over the last two decades or so on dynamic games played over event trees and time consistency in dynamic games. Research supported by SSHRC, Canada.

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  1. Chair in Game Theory and Management, GERAD, HEC Montréal, Montréal, QC, Canada

    Georges Zaccour

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  1. Georges Zaccour
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Correspondence to Georges Zaccour .

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  1. Department of Mathematics and MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roderick Melnik

  2. Department of Mathematics and MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roman Makarov

  3. Departement de Mathematiques, Universite de Montreal, Montreal, Québec, Canada

    Jacques Belair

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Zaccour, G. (2017). Sustainability of Cooperation in Dynamic Games Played over Event Trees. In: Melnik, R., Makarov, R., Belair, J. (eds) Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6969-2_14

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