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Cooperative Differential Games with Transferable Payoffs

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Handbook of Dynamic Game Theory

Abstract

In many instances, players find it individually and collectively rational to sign a long-term cooperative agreement. A major concern in such a setting is how to ensure that each player will abide by her commitment as time goes by. This will occur if each player still finds it individually rational at any intermediate instant of time to continue to implement her cooperative control rather than switch to a noncooperative control. If this condition is satisfied for all players, then we say that the agreement is time consistent. This chapter deals with the design of schemes that guarantee time consistency in deterministic differential games with transferable payoffs.

This work was supported by the Saint Petersburg State University under research grant No. 9.38.245.2014, and SSHRC, Canada, grant 435-2013-0532.

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Notes

  1. 1.

    This property has also been referred to as the stand-alone test.

  2. 2.

    This chapter focuses on how a cooperative solution in a differential game can be sustained over time. There is a large literature that looks at the dynamics of a cooperative solution, especially the core, in different game settings, but not in differential games. Here, the environment of the game changes when a coalition deviates; for instance, the set of players may vary over time. We refrain from reviewing this literature and direct the interested reader to Lehrer and Scarsini (2013) and the references therein.

  3. 3.

    There are (rare) cases in which a cooperative outcome “by construction” is in equilibrium.This occurs if a game has a Nash equilibrium that is also an efficient outcome. However, very few differential games have this property. The fishery game of Chiarella et al. (1984) is an example. Martín-Herrán and Rincón-Zapatero (2005) and Rincón-Zapatero et al. (2000) state conditions for Markov-perfect equilibria to be Pareto optimal in a special class of differential games.

  4. 4.

    Such games have the following two characteristics: (i) The instantaneous-payoff function and the salvage-value function are quadratic with no linear terms in the state and control variables; (ii) the state dynamics are linear in the state and control variables.

  5. 5.

    Typically, one considers \(x(t)\in X\subseteq \mathbb {R}^{n}\), where X is the set of admissible states. To avoid unecessary complications for what we are attempting to achieve here, we assume that the state space is \( \mathbb {R}^{n}\).

  6. 6.

    We have assumed the initial stock to be zero, which is not a severe simplifying assumption. Indeed, if this was not the case, then \(x\left ( 0\right ) =0\) can be imposed and everything can be rescaled consequently.

  7. 7.

    A cooperative game is convex if

    $$\displaystyle \begin{aligned} v(K\cup L)+v\left( K\cap L\right) \geq v(K)+v(L),\ \ \ \ \forall K,L\subseteq I. \end{aligned}$$
  8. 8.

    For an analysis of an optimal control problem with random duration, see, e.g., Boukas et al. (1990) and Chang (2004).

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Authors and Affiliations

  1. Faculty of Applied Mathematics-Control Processes, St Petersburg State University, St Petersburg, Russia

    Leon A. Petrosyan

  2. Department of Decision Sciences, GERAD, HEC Montréal, Montreal, QC, Canada

    Georges Zaccour

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  1. Leon A. Petrosyan
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  2. Georges Zaccour
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Correspondence to Leon A. Petrosyan .

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Editors and Affiliations

  1. Coordinated Science Laboratory and Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    Tamer Başar

  2. Department of Decision Sciences GERAD, HEC Montréal, Montreal, Québec, Canada

    Georges Zaccour

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Petrosyan, L.A., Zaccour, G. (2018). Cooperative Differential Games with Transferable Payoffs. In: Başar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-44374-4_12

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