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A Closed-Form Characterization of Buyer Signaling Schemes in Monopoly Pricing

We consider a setting where a revenue maximizing monopolist sells a single item to a buyer. A mediator first collects the buyer's value and can reveal extra information about the buyer's value by sending signals. Mathematically, a signal scheme can be thought of as a decomposition of the prior value distribution into a linear combination of posterior value distributions, and based on each of them, the monopolist separately posts a price. According to the theory of Bayesian persuasion, a well-designed signal scheme can lead to utility improvements for both the monopolist and the buyer. We put forward a novel technique to analyze the effects of signal schemes of the mediator. Using this technique, we are able to construct explicitly a closed-form solution, and thus characterize the set of seller-buyer utility pairs achievable by any signal scheme, for any prior type distribution. Our result generalizes a well-known result by Bergemann et. al., who derive a characterization for the same problem but only restricted to the discrete distribution case. Similar to the result derived by Bergermann et. al., we show that the set of seller and buyer utility pairs achievable form a triangle: any point within the triangle can be achieved by an explicitly constructed signal scheme and any point outside the triangle cannot be achievable by any such scheme. Our result is obtained by establishing the endpoints of the triangle: one corresponds to the point where the buyer obtains the highest utility among all schemes, another corresponds to the point where the buyer obtains zero utility and the seller has the lowest possible revenue, and the third corresponds to the point where the buyer has zero utility while the seller extracts full social surplus. We then prove that the triangle described fully characterizes all possible signal schemes.