This paper provides a simple solution to the problem of non-existence of pure-strategy equilibria in Bertrand-Edgeworth models with strictly convex costs. The voluntary-trading constraint in standard Bertrand-Edgeworth models is generalized to allow for there being costs incurred when customers are turned away. So long as the industry is sufficiently large, the presence of such costs ensures that the competitive price will be an equilibrium. There will be other single price equilibria, but if the offense costs are small, all equilibria will be close to the competitive outcome.