Freemium Model for Information Provision

Abstract

The paper explores a theoretical freemium model for the sale of information, drawing on mathematical tools used in the study of repeated zero-sum games and Bayesian persuasion. Unlike standard Bayesian persuasion models, the information seller (IS) is indifferent to the actions taken by the information buyer (IB) and is concerned solely with maximizing the revenue from selling information. Offering some information for free may increase the IB's willingness to pay for additional information. The information that the IB seeks is about the state of the world. Initially, the IB only knows the prior distribution over possible states. The IS supplies both free and paid information through signals whose state-dependent distributions determine the IB's posterior via Bayes' rule. The IB's utility is a function of the posterior. An optimal free signal is one that maximizes the IS's expected revenue from the subsequent paid signal. That revenue is equal to the IB's expected utility gain when moving from the posterior induced by the free signal to that induced by the paid signal. The paper characterizes the optimal free and paid signals and derives a formula for the maximal revenue in terms of the IB's utility function. It shows that a revenue gain for the IS from the provision of free information is accompanied by a loss to the IB. Whether free information can increase the IS's revenue depends on the form of the IB's utility function. In the two-state case, that dependence is fully characterized. In the general case, only necessary conditions are obtained. In particular, if the IB's utility function is convex, the IS can never profit from providing free information. This occurs, in particular, when the IB uses the information to solve a decision problem. By contrast, when the IB is engaged in a strategic interaction with a third party, the IS may benefit from providing free information.