Two-dimensional price discrimination with uncertainty about willingness-to-pay

This paper discusses price discrimination in a duopoly in two-dimensional consumer characteristics space. We assume that there is uncertainty about a single unobservable characteristic, willingness-to-pay. We show that the subgame perfect equilibrium supports that both firms choose separate pricing policies, uniform pricing and discriminatory pricing, and the Prisoner's Dilemma does not occur. This result is maintained when the uncertainty is sufficiently high, and differs from Thisse and Vives's one-dimensional model in which discriminatory pricing is a dominant strategy. As the uncertainty decreases, the pricing structure converges to Thisse and Vives's model in which the Prisoner's Dilemma and all-out competition emerge.