That is, the above conditions are deemed necessary and sufficient for linear pricing. The market of the consumers trading with each other and with the monopolist resolves in an efficient and individually rational manner. All the consumers are indifferent between the equilibrium and the no-trade outcome. This outcome can be supported as equilibrium by considering the selection of the equilibrium correspondence in the second stage to be the one which in case of indifference the consumers buy a positive amount. As of uniqueness of outcome, note that the monopolist profit cannot be higher than that of the above mentioned equilibrium due to individual rationality.
Conclusion Never trust textbooks.
A Course in Microeconomic Theory , Harvester Wheatsheaf, England.