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On the non-uniqueness of linear Markov perfect equilibria in linear-quadratic differential games: a geometric approach

Author

Listed:
  • Markus Eigruber

    (University of Vienna)

  • Franz Wirl

    (University of Vienna)

Abstract

Although the possibility of multiple nonlinear equilibria in linear-quadratic differential games is extensively discussed, the literature on models with multiple linear Markov perfect equilibria (LMPEs) is scarce. Indeed, almost all papers confined to a single state (the vast majority of the application of differential games to economic problems) find a unique LMPE. This paper explains this finding and derives conditions for multiplicity based on the analysis of the phase plane in the state and the derivative of the value function. The resulting condition is applied to derive additional pathways different from the (two) known ones. All these examples, more precisely, their underlying pathways or the resulting outcomes, contradict usual assumptions in economic models. However, by extending the state space, we provide an economic setting (learning by doing) that gives rise to multiple LMPEs.

Suggested Citation

  • Markus Eigruber & Franz Wirl, 2025. "On the non-uniqueness of linear Markov perfect equilibria in linear-quadratic differential games: a geometric approach," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 79(3), pages 911-943, May.
  • Handle: RePEc:spr:joecth:v:79:y:2025:i:3:d:10.1007_s00199-024-01606-4
    DOI: 10.1007/s00199-024-01606-4
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    References listed on IDEAS

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    More about this item

    Keywords

    Linear-quadratic differential game; Uniqueness versus multiplicity; Learning by doing; Linear Markov perfect equilibrium;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D25 - Microeconomics - - Production and Organizations - - - Intertemporal Firm Choice: Investment, Capacity, and Financing
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • P48 - Political Economy and Comparative Economic Systems - - Other Economic Systems - - - Legal Institutions; Property Rights; Natural Resources; Energy; Environment; Regional Studies

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