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Global Robust Stability in a General Price and Assortment Competition Model

We analyze a general but parsimonious price competition model for an oligopoly in which each firm offers any number of products. The demand volumes are general piecewise affine functions of the full price vector, generated as the "regular" extension of a base set of affine functions. The model specifies a product assortment, along with their prices and demand volumes, in contrast to most commonly used demand models such as the MNL model or any of its variants. We show that a special equilibrium in this model has global robust stability. This means that, from any starting point, the market converges to this equilibrium when firms use a particular response mapping to dynamically adjust their own prices in response to their competitors' prices. The mapping involves each firm optimizing its own prices over a limited subset of possible prices and requires each firm to only know the demand function and cost structure for its own products (but not for other firms' products).