Bertrand competition and captive customers

We study a Bertrand oligopoly with asymmetric costs in which each seller has some “captive” buyers. In the limit as captive buyers vanish, the lowest-cost firm sells to all buyers at a price equal to the second-lowest marginal cost. However, the closest competing price arises from non-degenerate mixed strategies, firms play exclusively undominated strategies, and with positive probability all but one firm sets the monopoly price. • A vanishing small set of captive buyers selects an equilibrium of classic Bertrand oligopoly. • The lowest-cost firm sells to all buyers at a price equal to the second-lowest marginal cost. • The closest competing price arises from non-degenerate mixed strategies. • Firms play exclusively undominated strategies. • With positive probability all but one firm sets the monopoly price.