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Nonzero-Sum Differential Games

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Handbook of Dynamic Game Theory

Abstract

This chapter provides an overview of the theory of nonzero-sum differential games, describing the general framework for their formulation, the importance of information structures, and noncooperative solution concepts. Several special structures of such games are identified, which lead to closed-form solutions.

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Notes

  1. 1.

    See, e.g., the classical textbook on optimal control by Lee and Markus (1972) for examples of synthesis of state-feedback control laws.

  2. 2.

    This property will be discussed later in the chapter.

  3. 3.

    These are conditions which ensure that when all players’ strategies are implemented, then the differential equation (1) describing the evolution of the state admits a unique piecewise continuously differentiable solution for each initial condition x 0; see, e.g., Başar and Olsder (1999).

  4. 4.

    With a slight abuse of notation, we have included here also the pair (x 0, t 0) as an argument of \(\bar{J}_{j}\), since under the SF information γ does not have (x 0, t 0) as an argument for t > t 0.

  5. 5.

    We use \(\mathcal{T}\) instead of T because, in a general setting, T may be endogenously defined as the time when the target is reached.

  6. 6.

    We use the convention that λ j (t)f(⋅ , ⋅ , ⋅ ) is the scalar product of two n dimensional vectors λ j (t) and f(⋯ ).

  7. 7.

    This is a standard result in optimal control, which can be found in any standard text, such as Bryson et al. (1975).

  8. 8.

    The existence of a conjugate point in [0, T) implies that there exists a sequence of policies by the maximizer which can drive the value of the game arbitrarily large, that is, the upper value of the game is infinite.

  9. 9.

    The setup can be easily extended to the case of several followers. A standard assumption is then that the followers play a (Nash) simultaneous-move game vis-a-vis each other, and a sequential game vis-a-vis the leader (Başar and Olsder 1999).

  10. 10.

    This assumption allows us to use the strong-optimality concept and avoid introducing additional technicalities.

  11. 11.

    Here, and in the balance of this section, we depart from our earlier convention of state-time ordering (x, t), and use the reverse ordering (t, x).

  12. 12.

    In a linear-state differential game, the objective functional, the salvage value and the dynamics are linear in the state variables. For such games, it holds that a feedback strategy is constant, i.e., independent of the state and hence open-loop and state-feedback Nash equilibria coincide.

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Author information

Authors and Affiliations

  1. Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, 1308 West Main, 61801, Urbana, IL, USA

    Tamer Başar

  2. ORDECSYS and University of Geneva, Geneva, Switzerland

    Alain Haurie

  3. GERAD-HEC Montréal PQ, Montreal, QC, Canada

    Alain Haurie

  4. Département de sciences de la décision, GERAD, HEC Montréal, Montreal, QC, Canada

    Georges Zaccour

Authors
  1. Tamer Başar
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  2. Alain Haurie
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  3. Georges Zaccour
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Corresponding author

Correspondence to Tamer Başar .

Editor information

Editors and Affiliations

  1. Center for Advanced Study, University of Illinois Urbana-Champaign, Urbana, Illinois, USA

    Tamer Basar

  2. GERAD, HEC Montréal, Montreal, Québec, Canada

    Georges Zaccour

Section Editor information

  1. Coordinated Science Laboratory, University of Illinois, 1308 West Main, 61801, Urbana, Illinois, USA

    Tamer Basar

  2. Département de sciences de la décision, HEC Montreal, Montreal, Canada

    Georges Zaccour

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Başar, T., Haurie, A., Zaccour, G. (2016). Nonzero-Sum Differential Games. In: Basar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-27335-8_5-1

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  • DOI: https://doi.org/10.1007/978-3-319-27335-8_5-1

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