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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.