Researchers introduce a new class of auctions that let a monopolist sell goods while sharing the firm’s profits with consumers by selling them ownership stakes during the bidding. The auctions are designed so that the final price, quantity and how profit is divided all emerge from the bids. Ownership shares are created inside the auction, which both reveals consumer demand and splits revenue between the producer and buyers.

The paper frames this idea as a tool for a regulator who and the monopolist both know the cost structure of production but do not know buyers’ demand. The authors fix three policy goals for any acceptable auction: the outcome must be efficient (the items go to those who value them most), it must be fair across consumers in the sense of “no envy,” and it must be strategy-proof so that truthful bidding is each buyer’s best move without complicated tactics. Under these axioms they characterize a broad family they call endogenous subsidy auctions and a narrower family they call endogenous shareholding auctions.

At a high level, an endogenous shareholding auction works by picking a competitive equilibrium in which consumers receive common ownership shares of the seller. Those shares determine how much of the auction’s profit is returned to consumers. The paper studies formal assumptions such as unit-demand buyers (each buyer wants at most one unit), quasilinear preferences (money adds linearly to utility), and convex production costs summarized by a supply curve S(q). The authors give a concrete construction called the “myopic subsidy curve” that defines how subsidies to buyers change with prices. That construction helps build examples of auctions in the class and to show which ones are best for different goals.

Why this matters: the design gives regulators a principled way to run auctions that both uncover demand and share the gains from a monopoly sale in a transparent, non-manipulable way. The authors show several optimality results. For example, Vickrey–Clarke–Groves (VCG) style mechanisms are optimal for the producer in a prior-free sense (they do not depend on beliefs about demand). By contrast, the auctions that best protect consumers form a new subclass the authors call valvular auctions. They also prove a general existence result: for any compact set of beliefs and any continuous social-welfare measure, there exists a mechanism in this family that maximizes the regulator’s expected welfare.