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

R. A. Edwards () and R. R. Routledge ()
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R. A. Edwards: University of Nottingham
R. R. Routledge: University of Liverpool

International Journal of Game Theory, 2023, vol. 52, issue 2, No 10, 569-586

Abstract: 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.

Keywords: Discontinuous payoffs; Existence; Uniqueness; Bertrand competition; Necessary and sufficient conditions (search for similar items in EconPapers)
JEL-codes: C62 C72 D43 L11 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s00182-022-00830-3

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