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Existence and uniqueness of Nash equilibrium in discontinuous Bertrand games: a complete characterization

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  • R. A. Edwards

    (University of Nottingham)

  • R. R. Routledge

    (University of Liverpool)

Abstract

Since (Reny in Econometrica 67:1029–1056, 1999) a substantial body of research has considered what conditions are sufficient for the existence of a pure strategy Nash equilibrium in games with discontinuous payoffs. This work analyzes a general Bertrand game, with convex costs and an arbitrary sharing rule at price ties, in which tied payoffs may be greater than non-tied payoffs when both are positive. On this domain, necessary and sufficient conditions for (i) the existence of equilibrium (ii) the uniqueness of equilibrium are presented. The conditions are intuitively easy to understand and centre around the relationships between intervals of real numbers determined by the primitives of the model.

Suggested Citation

  • R. A. Edwards & R. R. Routledge, 2023. "Existence and uniqueness of Nash equilibrium in discontinuous Bertrand games: a complete characterization," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(2), pages 569-586, June.
    • Handle: RePEc:spr:jogath:v:52:y:2023:i:2:d:10.1007_s00182-022-00830-3
    DOI: 10.1007/s00182-022-00830-3
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    References listed on IDEAS

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    More about this item

    Keywords

    Discontinuous payoffs; Existence; Uniqueness; Bertrand competition; Necessary and sufficient conditions;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • L11 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Production, Pricing, and Market Structure; Size Distribution of Firms

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