On the Existence of an Equilibrium in Bertrand Pricing Games with Discrete Choice Models of Demand

This note examines equilibrium existence of Nash-Bertrand pricing games with discrete-choice models of demand. We provide a sufficient and necessary condition on the distribution of firm idiosyncratic shocks for best response correspondences to exhibit monotonicity. We then use Tarski’s fixed point result to show that the condition we provide generates a unique pure-strategy equilibrium. Moreover, we conclude that any equilibrium existence results that employ Tarski’s fixed point argument must satisfy these error-term conditions.