We analyze nonlinear pricing with finite information. A seller offers a menu to a continuum of buyers with a continuum of possible valuations. The menu is limited to offering a finite number of choices representing a finite communication capacity between buyer and seller.
We identify necessary conditions that the optimal finite menu must satisfy, either for the socially efficient or for the revenue-maximizing mechanism. These conditions require that information be bundled, or “quantized” optimally. We show that the loss resulting from using the n-item menu converges to zero at a rate proportional to 1 = n2.
We extend our model to a multi-product environment where each buyer has preferences over a d dimensional variety of goods. The seller is limited to offering a finite number n of d-dimensional choices. By using repeated scalar quantization, we show that the losses resulting from using the d-dimensional n-class menu converge to zero at a rate proportional to d = n2/d. We introduce vector quantization and establish that the losses due to finite menus are significantly reduced by offering optimally chosen bundles.
We discuss four solution concepts for games with incomplete information. We show how each solution concept can be viewed as encoding informational robustness. For a given type space, we consider expansions of the type space that provide players with additional signals. We distinguish between expansions along two dimensions. First, the signals can either convey payoff relevant information or only payoff irrelevant information. Second, the signals can be generated from a common (prior) distribution or not. We establish the equivalence between Bayes Nash equilibrium behavior under the resulting expansion of the type space and a corresponding more permissive solution concept under the original type space. This approach unifies some existing literature and, in the case of an expansion without a common prior and allowing for payoff relevant signals, leads us to a new solution concept that we dub belief-free rationalizability.
Consider the following “informational robustness” question: what can we say about the set of outcomes that may arise in equilibrium of a Bayesian game if players may observe some additional information? This set of outcomes will correspond to a solution concept that is weaker than equilibrium, with the solution concept depending on what restrictions are imposed on the additional information.
We describe a unified approach encompassing prior informational robustness results, as well as identifying the solution concept that corresponds to no restrictions on the additional information; this version of rationalizability depends only on the support of players’ beliefs and implies novel predictions in classic economic environments of coordination and trading games.
Our results generalize from complete to incomplete information the classical results in Aumann (1974, 1987) and Brandenburger and Dekel (1987) which can be (and were) given informational robustness interpretations. We discuss the relation between informational robustness and “epistemic” foundations of solution concepts.
We characterize the revenue-maximizing mechanism for time separable allocation problems in continuous time. The valuation of each agent is private information and changes over time. At the time of contracting every agent privately observes his initial type which influences the evolution of his valuation process. The leading example is the repeated sales of a good or a service.
We derive the optimal dynamic mechanism, analyze its qualitative structure and frequently derive its closed form solution. This enables us to compare the distortion in various settings. In particular, we discuss the cases where the type of each agent follows an arithmetic or geometric Brownian motion or a mean reverting process. We show that depending on the nature of the private information the distortion might increase or decrease over time.
We characterize the profit-maximizing mechanism for repeatedly selling a non-durable good in continuous time. The valuation of each agent is private information and changes over time. At the time of contracting every agent privately observes his initial type which influences the evolution of his valuation process. In the profit-maximizing mechanism the allocation is distorted in favor of agents with high initial types.
We derive the optimal mechanism in closed form, which enables us to compare the distortion in various examples. The case where the valuation of the agents follows an arithmetic/geometric Brownian motion, Ornstein-Uhlenbeck process, or is derived from a Bayesian learning model are discussed. We show that depending on the nature of the private information and the valuation process the distortion might increase or decrease over time.
We characterize the profit-maximizing mechanism for repeatedly selling a non-durable good in continuous time. The valuation of each agent is private information and changes over time. At the time of contracting every agent privately observes his initial type which influences the evolution of his valuation process. In the profit-maximizing mechanism the allocation is distorted in favor of agents with high initial types.
We derive the optimal mechanism in closed form, which enables us to compare the distortion in various examples. The case where the valuation of the agents follows an arithmetic/geometric Brownian motion, Ornstein-Uhlenbeck process, or is derived from a Bayesian learning model are discussed. We show that depending on the nature of the private information and the valuation process the distortion might increase or decrease over time.
A monopolist sells informative experiments to heterogeneous buyers. Buyers differ in their prior information, and hence in their willingness to pay for additional signals. The monopolist can profitably offer a menu of experiments. We show that, even under costless information acquisition and free degrading of information, the optimal menu is quite coarse. The seller offers at most two experiments, and we derive conditions under which at vs. discriminatory pricing is optimal.
We characterize the revenue-maximizing mechanism for time separable allocation problems in continuous time. The willingness-to-pay of each agent is private information and changes over time.
We derive the dynamic revenue-maximizing mechanism, analyze its qualitative structure and frequently derive its closed form solution. In the leading example of repeat sales of a good or service, we establish that commonly observed contract features such as at rates, free consumption units and two-part tariffs emerge as part of the optimal contract. We investigate in detail the environments in which the type of each agent follows an arithmetic or geometric Brownian motion or a mean-reverting process. We analyze the allocative distortions and show that depending on the nature of the private information the distortion might increase or decrease over time.
In an economy of interacting agents with both idiosyncratic and aggregate shocks, we examine how the information structure determines aggregate volatility. We show that the maximal aggregate volatility is attained in a noise free information structure in which the agents confound idiosyncratic and common components of the payoff state, and display excess response to the common component, as in Lucas (1972). The upper bound on aggregate volatility is linearly increasing in the variance of idiosyncratic shocks, for any given variance of aggregate shocks. Our results hold in a setting of symmetric agents with linear best responses and normal uncertainty. We show our results by providing a characterization of the set of all joint distributions over actions and states that can arise in equilibrium under any information structure. This tractable characterization, extending results in Bergemann and Morris (2013b), can be used to address a wide variety of questions.
We analyze a class of games with interdependent values and linear best responses. The payoff uncertainty is described by a multivariate normal distribution that includes the pure common and pure private value environment as special cases. We characterize the set of joint distributions over actions and states that can arise as Bayes Nash equilibrium distributions under any multivariate normally distributed signals about the payoff states. We characterize maximum aggregate volatility for a given distribution of the payoff states. We show that the maximal aggregate volatility is attained in a noise-free equilibrium in which the agents confound idiosyncratic and common components of the payoff state, and display excess response to the common component. We use a general approach to identify the critical information structures for the Bayes Nash equilibrium via the notion of Bayes correlated equilibrium, as introduced by Bergemann and Morris (2013b).
In an economy of interacting agents with both idiosyncratic and aggregate shocks, we examine how the structure of private information influences aggregate volatility. The maximal aggregate volatility is attained in a noise free information structure in which the agents confound idiosyncratic and aggregate shocks, and display excess response to the aggregate shocks, as in Lucas [14]. For any given variance of aggregate shocks, the upper bound on aggregate volatility is linearly increasing in the variance of the idiosyncratic shocks. Our results hold in a setting of symmetric agents with linear best responses and normal uncertainty. We establish our results by providing a characterization of the set of all joint distributions over actions and states that can arise in equilibrium under any information structure. This tractable characterization, extending results in Bergemann and Morris [8], can be used to address a wide variety of questions linking information with the statistical moments of the economy.
We study how the outcomes of a private-value first price auction can vary with bidders’ information, for a fixed distribution of private values. In a two bidder, two value, setting, we characterize all combinations of bidder surplus and revenue that can arise, and identify the information structure that minimizes revenue. The extremal information structure that minimizes revenue entails each bidder observing a noisy and correlated signal about the other bidder’s value.
In the general environment with many bidders and many values, we characterize the minimum bidder surplus of each bidder and maximum revenue across all information structures. The extremal information structure that simultaneously attains these bounds entails an efficient allocation, bidders knowing whether they will win or lose, losers bidding their true value and winners being induced to bid high by partial information about the highest losing bid. Our analysis uses a linear algebraic characterization of equilibria across all information structures, and we report simulations of properties of the set of all equilibria.
We analyze data pricing and targeted advertising. Advertisers seek to tailor their spending to the value of each consumer. A monopolistic data provider sells cookies. informative signals about individual consumers.preferences. We characterize the set of consumers targeted by the advertisers and the optimal monopoly price of cookies. The ability to influence the composition of the targeted set provides incentives to lower prices. Thus, the price of data decreases with the reach of the database and increases with the fragmentation of data sales. We characterize the optimal policy for selling information and its implementation through nonlinear pricing of cookies.
We develop a model of data pricing and targeted advertising. A monopolistic data provider determines the price to access “cookies,” i.e., informative signals about individual consumers’ preferences. The demand for information is generated by advertisers who seek to tailor their spending to the value of each consumer. We characterize the set of consumers targeted by the advertisers and the optimal monopoly price of cookies. The ability to influence the composition of the set of targeted consumers provides incentives to lower prices. Thus, the monopoly price of data is decreasing in the reach of the database and increasing in the number of competing sellers of exclusive data. Finally, we explore the implications of nonlinear pricing of information and characterize the exclusive data sales that emerge as part of the optimal mechanism.
We propose a model of data provision and data pricing. A single data provider controls a large database that contains information about the match value between individual consumers and individual firms (advertisers). Advertisers seek to tailor their spending to the individual match value. The data provider prices queries about individual consumers’ characteristics (cookies). We determine the equilibrium data acquisition and pricing policies. Advertisers choose positive and/or negative targeting policies. The optimal query price influences the composition of the targeted set. The price of data decreases with the reach of the database and increases with the fragmentation of data sales.