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We study the algorithmic problem faced by an information holder (seller) who wants to optimally sell such information to a budged-constrained decision maker (buyer) that has to undertake some action. Differently from previous works addressing this problem, we consider the case in which the seller is an interested party, as the action chosen by the buyer does not only influence their utility, but also seller's one. This happens in many real-world settings, where the way in which businesses use acquired information may positively or negatively affect the seller, due to the presence of externalities on the information market. The utilities of both the seller and the buyer depend on a random state of nature, which is revealed to the seller, but it is unknown to the buyer. Thus, the seller's goal is to (partially) sell their information about the state of nature to the buyer, so as to concurrently maximize revenue and induce the buyer to take a desirable action.We study settings in which buyer's budget and utilities are determined by a random buyer's type that is unknown to the seller. In such settings, an optimal protocol for the seller must propose to the buyer a menu of information-revelation policies to choose from, with the latter acquiring one of them by paying its corresponding price. Moreover, since in our model the seller is an interested party, an optimal protocol must also prescribe the seller to pay back the buyer contingently on their action.First, we show that the problem of computing a seller-optimal protocol can be solved in polynomial time. This result relies on a quadratic formulation of the problem, which we solve by means of a linear programming relaxation. Next, we switch the attention to the case in which a seller's protocol employs a single information-revelation policy, rather than proposing a menu. In such a setting, we show that computing a seller-optimal protocol is APX-hard, even when either the number of actions or that of states of nature is fixed. We complement such a negative result by providing a quasipolynomial-time approximation algorithm that, given any ρ > 0 and ǫ > 0 as input, provides a multiplicative approximation ρ of the optimal seller's expected utility, by only suffering a negligible 2 −Ω(1/ρ) + ǫ additive loss. Such an algorithm runs in polynomial time whenever either the number of buyer's actions or that of states of nature is fixed. In order to derive our results, we draw a connection between our information-selling problem and principal-agent problems with observable actions. Finally, we complete the picture of the computational complexity of finding seller-optimal protocols without menus by providing additional results for the specific setting in which the buyer has limited liability, and by designing a polynomial-time algorithm for the case in which buyer's types are fixed.

We study signaling in Bayesian ad auctions, in which bidders' valuations depend on a random, unknown state of nature. The auction mechanism has complete knowledge of the actual state of nature, and it can send signals to bidders so as to disclose information about the state and increase revenue. For instance, a state may collectively encode some features of the user that are known to the mechanism only, since the latter has access to data sources unaccessible to the bidders. We study the problem of computing how the mechanism should send signals to bidders in order to maximize revenue. While this problem has already been addressed in the easier setting of second-price auctions, to the best of our knowledge, our work is the first to explore ad auctions with more than one slot. In this paper, we focus on public signaling and VCG mechanisms, under which bidders truthfully report their valuations. We start with a negative result, showing that, in general, the problem does not admit a PTAS unless P = NP, even when bidders' valuations are known to the mechanism. The rest of the paper is devoted to settings in which such negative result can be circumvented. First, we prove that, with known valuations, the problem can indeed be solved in polynomial time when either the number of states d or the number of slots m is fixed. Moreover, in the same setting, we provide an FPTAS for the case in which bidders are single minded, but d and m can be arbitrary. Then, we switch to the random valuations setting, in which these are randomly drawn according to some probability distribution. In this case, we show that the problem admits an FPTAS, a PTAS, and a QPTAS, when, respectively, d is fixed, m is fixed, and bidders' valuations are bounded away from zero.