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There are few models of price competition in a homogeneous-good market which permit general asymmetries of information amongst the sellers. This work studies a price game with discontinuous payoffs in which both costs and market demand are ex ante uncertain. The sellers evaluate uncertain profits with maximin expected utilities exhibiting ambiguity aversion. The buyers in the market are permitted to split between sellers tieing at the minimum price in arbitrary ways which may be deterministic or random. The role of the primitives in determining equilibrium prices in the market is analyzed in detail.

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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.

Résumé Cet article porte sur la formation des prix dans deux approches : (i) celle de Bertrand dans laquelle il est supposé que les vendeurs entrent sur le marché avec l’engagement de satisfaire la demande et (ii) celle d’Edgeworth dans laquelle les vendeurs fixent les prix sans engagement quant au fait d’offrir plus que leur offre concurrentielle. Ces deux types de contrat marchand rivalisent pour donner des fondements stratégiques à la concurrence parfaite. Dans cet article, nous proposons une approche alternative à la formation des contrats qui combine les vues de Bertrand [1883] sur la concurrence en prix avec la notion de recontracting à la Edgeworth [1881]. Il est montré un résultat de convergence analogue à celui de Debreu-Scarf [1963] lorsque les marchés comprennent de nombreux agents. JEL classification: C72, D43

Bertrand equilibrium with arbitrary sharing rules were provided. Here, sufficient conditions for the existence of a coalition-proof Bertrand equilibrium with an arbitrary tie-breaking rule are provided. A classical Bertrand game in which sellers have convex costs is analyzed and sufficient conditions for the existence of coalition-proof Bertrand equilibrium which admit discontinuities in tied payoffs and a general class of tie-breaking rules are stated. Finally, an example is provided where one of the conditions is violated and there is non-existence of coalition-proof Bertrand equilibrium. This work generalizes an existence result of Chowdhury and Sengupta (Econ Theory 24:307-324, 2004).

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