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Full-Text Articles in Economics
The Optimality Of Upgrade Pricing, Dirk Bergemann, Alessandro Bonatti, Andreas Haupt, Alex Smolin
The Optimality Of Upgrade Pricing, Dirk Bergemann, Alessandro Bonatti, Andreas Haupt, Alex Smolin
Cowles Foundation Discussion Papers
We consider a multiproduct monopoly pricing model. We provide sufficient conditions under which the optimal mechanism can be implemented via upgrade pricing—a menu of product bundles that are nested in the strong set order. Our approach exploits duality methods to identify conditions on the distribution of consumer types under which (a) each product is purchased by the same set of buyers as under separate monopoly pricing (though the transfers can be different), and (b) these sets are nested. We exhibit two distinct sets of sufficient conditions. The first set of conditions is given by a weak version of monotonicity of …
The Optimality Of Upgrade Pricing, Dirk Bergemann, Alessandro Bonatti, Andreas Haupt, Alex Smolin
The Optimality Of Upgrade Pricing, Dirk Bergemann, Alessandro Bonatti, Andreas Haupt, Alex Smolin
Cowles Foundation Discussion Papers
We consider a multiproduct monopoly pricing model. We provide sufficient conditions under which the optimal mechanism can be implemented via upgrade pricing—a menu of product bundles that are nested in the strong set order. Our approach exploits duality methods to identify conditions on the distribution of consumer types under which (a) each product is purchased by the same set of buyers as under separate monopoly pricing (though the transfers can be different), and (b) these sets are nested. We exhibit two distinct sets of sufficient conditions. The first set of conditions weakens the monotonicity requirement of types and virtual values …
Selling Consumer Data For Profit: Optimal Market-Segmentation Design And Its Consequences, Kai Hao Yang
Selling Consumer Data For Profit: Optimal Market-Segmentation Design And Its Consequences, Kai Hao Yang
Cowles Foundation Discussion Papers
A data broker sells market segmentations created by consumer data to a producer with private production cost who sells a product to a unit mass of consumers with heterogeneous values. In this setting, I completely characterize the revenue-maximizing mechanisms for the data broker. In particular, every optimal mechanism induces quasi-perfect price discrimination. That is, the data broker sells the producer a market segmentation described by a cost-dependent cutoff, such that all the consumers with values above the cutoff end up buying and paying their values while the rest of consumers do not buy. The characterization of optimal mechanisms leads to …
Strategically Simple Mechanisms, Tilman Börgers, Jiangtao Li
Strategically Simple Mechanisms, Tilman Börgers, Jiangtao Li
Cowles Foundation Discussion Papers
We define and investigate a property of mechanisms that we call “strategic simplicity,” and that is meant to capture the idea that, in strategically simple mechanisms, strategic choices require limited strategic sophistication. We define a mechanism to be strategically simple if choices can be based on first-order beliefs about the other agents’ preferences and first-order certainty about the other agents’ rationality alone, and there is no need for agents to form higher-order beliefs, because such beliefs are irrelevant to the optimal strategies. All dominant strategy mechanisms are strategically simple. But many more mechanisms are strategically simple. In particular, strategically simple …
Markets For Information: An Introduction, Dirk Bergemann, Alessandro Bonatti
Markets For Information: An Introduction, Dirk Bergemann, Alessandro Bonatti
Cowles Foundation Discussion Papers
We survey a recent and growing literature on markets for information. We offer a comprehensive view of information markets through an integrated model of consumers, information intermediaries, and firms. The model embeds a large set of applications ranging from sponsored search advertising to credit scores to information sharing among competitors. We then review a mechanism design approach to selling information in greater detail. We distinguish between ex ante sales of information (the buyer acquires an information structure) and ex post sales (the buyer pays for specific realizations). We relate this distinction to the different products that brokers, advertisers, and publishers …
The Design And Price Of Information, Dirk Bergemann, Alessandro Bonatti, Alex Smolin
The Design And Price Of Information, Dirk Bergemann, Alessandro Bonatti, Alex Smolin
Cowles Foundation Discussion Papers
This paper analyzes the trade of information between a data buyer and a data seller. The data buyer faces a decision problem under uncertainty and seeks to augment his initial private information with supplemental data. The data seller is uncertain about the willingness-to-pay of the data buyer due to this private information. The data seller optimally offers a menu of (Blackwell) experiments as statistical tests to the data buyer. The seller exploits differences in the beliefs of the buyer’s types to reduce information rents while limiting the surplus that must be sacrificed to provide incentives.
The Design And Price Of Information, Dirk Bergemann, Alessandro Bonatti, Alex Smolin
The Design And Price Of Information, Dirk Bergemann, Alessandro Bonatti, Alex Smolin
Cowles Foundation Discussion Papers
A data buyer faces a decision problem under uncertainty. He can augment his initial private information with supplemental data from a data seller. His willingness to pay for supplemental data is determined by the quality of his initial private information. The data seller optimally offers a menu of statistical experiments. We establish the properties that any revenue-maximizing menu of experiments must satisfy. Every experiment is a non-dispersed stochastic matrix, and every menu contains a fully informative experiment. In the cases of binary states and actions, or binary types, we provide an explicit construction of the optimal menu of experiments.
Optimal Design For Social Learning, Yeon-Koo Che, Johannes Hörner
Optimal Design For Social Learning, Yeon-Koo Che, Johannes Hörner
Cowles Foundation Discussion Papers
This paper studies the design of a recommender system for organizing social learning on a product. To improve incentives for early experimentation, the optimal design trades off fully transparent social learning by over-recommending a product (or “spamming”) to a fraction of agents in the early phase of the product cycle. Under the optimal scheme, the designer spams very little about a product right after its release but gradually increases the frequency of spamming and stops it altogether when the product is deemed sufficiently unworthy of recommendation. The optimal recommender system involves randomly triggered spamming when recommendations are public — as …
Dynamic Mechanisms Without Money, Yingni Guo, Johannes Hörner
Dynamic Mechanisms Without Money, Yingni Guo, Johannes Hörner
Cowles Foundation Discussion Papers
We analyze the optimal design of dynamic mechanisms in the absence of transfers. The designer uses future allocation decisions as a way of eliciting private information. Values evolve according to a two-state Markov chain. We solve for the optimal allocation rule, which admits a simple implementation. Unlike with transfers, efficiency decreases over time, and both immiseration and its polar opposite are possible long-run outcomes. Considering the limiting environment in which time is continuous, we show that persistence hurts.
Nonlinear Pricing With Finite Information, Dirk Bergemann, Ji Shen, Yun Xu, Edmund M. Yeh
Nonlinear Pricing With Finite Information, Dirk Bergemann, Ji Shen, Yun Xu, Edmund M. Yeh
Cowles Foundation Discussion Papers
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 = n 2 . We …
Dynamic Revenue Maximization: A Continuous Time Approach, Dirk Bergemann, Philipp Strack
Dynamic Revenue Maximization: A Continuous Time Approach, Dirk Bergemann, Philipp Strack
Cowles Foundation Discussion Papers
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 …
Dynamic Revenue Maximization: A Continuous Time Approach, Dirk Bergemann, Philipp Strack
Dynamic Revenue Maximization: A Continuous Time Approach, Dirk Bergemann, Philipp Strack
Cowles Foundation Discussion Papers
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 …
Selling Experiments: Menu Pricing Of Information, Dirk Bergemann, Alessandro Bonatti, Alex Smolin
Selling Experiments: Menu Pricing Of Information, Dirk Bergemann, Alessandro Bonatti, Alex Smolin
Cowles Foundation Discussion Papers
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.
Dynamic Revenue Maximization: A Continuous Time Approach, Dirk Bergemann, Philipp Strack
Dynamic Revenue Maximization: A Continuous Time Approach, Dirk Bergemann, Philipp Strack
Cowles Foundation Discussion Papers
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 …
Extremal Information Structures In The First Price Auction, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Extremal Information Structures In The First Price Auction, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
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 …
Multi-Dimensional Mechanism Design With Limited Information, Dirk Bergemann, Ji Shen, Yun Xu, Edmund M. Yeh
Multi-Dimensional Mechanism Design With Limited Information, Dirk Bergemann, Ji Shen, Yun Xu, Edmund M. Yeh
Cowles Foundation Discussion Papers
We analyze a nonlinear pricing model with limited information. Each buyer can purchase a large variety, d , of goods. His preference for each good is represented by a scalar and his preference over d goods is represented by a d -dimensional vector. The type space of each buyer is given by a compact subset of R d + with a continuum of possible types. By contrast, the seller is limited to offer a finite number M of d -dimensional choices. We provide necessary conditions that the optimal finite menu of the social welfare maximizing problem has to satisfy. We …
Robust Mechanism Design: An Introduction, Dirk Bergemann, Stephen Morris
Robust Mechanism Design: An Introduction, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
This essay is the introduction for a collection of papers by the two of us on “Robust Mechanism Design” to be published by World Scientific Publishing. The appendix of this essay lists the chapters of the book. The objective of this introductory essay is to provide the reader with an overview of the research agenda pursued in the collected papers. The introduction selectively presents the main results of the papers, and attempts to illustrate many of them in terms of a common and canonical example, the single unit auction with interdependent values. In addition, we include an extended discussion about …
Efficient Search By Committee, Dirk Bergemann, Juuso Välimäki
Efficient Search By Committee, Dirk Bergemann, Juuso Välimäki
Cowles Foundation Discussion Papers
This note constructs an efficient mechanism for finding the best candidate for a committee from a sequence of potential candidates. Committee members have independent private values information about the quality of the candidate. The mechanism selects the best candidate according to the standard utilitarian welfare criterion. Furthermore, the mechanism can be modified to have a balanced budget.
Mechanism Design With Limited Information: The Case Of Nonlinear Pricing, Dirk Bergemann, Ji Shen, Yun Xu, Edmund M. Yeh
Mechanism Design With Limited Information: The Case Of Nonlinear Pricing, Dirk Bergemann, Ji Shen, Yun Xu, Edmund M. Yeh
Cowles Foundation Discussion Papers
We analyze the canonical nonlinear pricing model with limited information. A seller offers a menu with a finite number of choices to a continuum of buyers with a continuum of possible valuations. By revealing an underlying connection to quantization theory, we derive the optimal finite menu for the socially efficient and the revenue-maximizing mechanism. In both cases, we provide an estimate of the loss resulting from the usage of a finite n-class menu. We show that the losses converge to zero at a rate proportional to 1/n2 asn becomes large.
The Role Of Commitment In Bilateral Trade, Dino Gerardi, Johannes Hörner, Lucas Maestri
The Role Of Commitment In Bilateral Trade, Dino Gerardi, Johannes Hörner, Lucas Maestri
Cowles Foundation Discussion Papers
This paper solves for the set of equilibrium payoffs in bargaining with interdependent values when the informed party makes all offers, as discounting vanishes. The seller of a good is informed of its quality, which affects both his cost and the buyer’s valuation, but the buyer is not. To characterize this payoff set, we derive an upper bound, using mechanism design with limited commitment. We then prove that this upper bound is tight, by showing that all its extreme points are equilibrium payoffs. Our results shed light on the role of different forms of commitment on the bargaining process. In …
The Role Of Commitment In Bilateral Trade, Dino Gerardi, Johannes Hörner, Lucas Maestri
The Role Of Commitment In Bilateral Trade, Dino Gerardi, Johannes Hörner, Lucas Maestri
Cowles Foundation Discussion Papers
We examine the buyer-seller problem under different levels of commitment. The seller is informed of the quality of the good, which affects both his cost and the buyer’s valuation, but the buyer is not. We characterize the allocations that can be achieved through mechanisms in which, unlike with full commitment, the buyer has the option to “walk away” after observing a given offer. We further characterize the equilibrium payoffs that can be achieved in the bargaining game in which the seller makes all the offers, as the discount factor goes to one. This allows us to identify how different levels …
Robust Implementation In General Mechanisms, Dirk Bergemann, Stephen Morris
Robust Implementation In General Mechanisms, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation. Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.
Robust Implementation In General Mechanisms, Dirk Bergemann, Stephen Morris
Robust Implementation In General Mechanisms, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation. Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.
The Role Of The Common Prior In Robust Implementation, Dirk Bergemann, Stephen Morris
The Role Of The Common Prior In Robust Implementation, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
We consider the role of the common prior for robust implementation in an environment with interdependent values. Specifically, we investigate a model of public good provision which allows for negative and positive informational externalities. In the corresponding direct mechanism, the agents’ reporting strategies are strategic complements with negative informational externalities and strategic substitutes with positive informational externalities. We derive the necessary and sufficient conditions for robust implementation in common prior type spaces and contrast this with our earlier results without the common prior. In the case of strategic complements the necessary and sufficient conditions for robust implementation do not depend …
Strategic Distinguishability And Robust Virtual Implementation, Dirk Bergemann, Stephen Morris
Strategic Distinguishability And Robust Virtual Implementation, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
In a general interdependent preference environment, we characterize when two payoff types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents. A social choice function — mapping payoff type profiles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an outcome …
Strategic Distinguishability With An Application To Robust Virtual Implementation, Dirk Bergemann, Stephen Morris
Strategic Distinguishability With An Application To Robust Virtual Implementation, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
In a general interdependent preference environment, we characterize when two payoff types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents. A social choice function — mapping payoff type profiles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an outcome …
Robust Virtual Implementation, Dirk Bergemann, Stephen Morris
Robust Virtual Implementation, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
In a general interdependent preference environment, we characterize when two payoff types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two payoff types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents. A social choice function — mapping payoff type profiles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an …
Robust Implementation In Direct Mechanisms, Dirk Bergemann, Stephen Morris
Robust Implementation In Direct Mechanisms, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
A social choice function is robustly implementable if there is a mechanism under which the process of iteratively eliminating strictly dominated messages leads to outcomes that agree with the social choice function for all beliefs at every type profile. In an interdependent value environment with single crossing preferences, we identify a contraction property on the preferences which together with strict ex post incentive compatibility is sufficient to guarantee robust implementation in the direct mechanism. Strict ex post incentive compatibility and the contraction property are also necessary for robust implementation in any mechanism, including indirect ones. The contraction property requires that …
Perfect Competition In A Bilateral Monopoly (In Honor Of Martin Shubik), Pradeep Dubey, Dieter Sondermann
Perfect Competition In A Bilateral Monopoly (In Honor Of Martin Shubik), Pradeep Dubey, Dieter Sondermann
Cowles Foundation Discussion Papers
We show that if limit orders are required to vary smoothly, then strategic (Nash) equilibria of the double auction mechanism yield competitive (Walras) allocations. It is not necessary to have competitors on any side of any market: smooth trading is a substitute for price wars. In particular, Nash equilibria are Walrasian even in a bilateral monopoly.
Robust Mechanism Design, Dirk Bergemann, Stephen Morris
Robust Mechanism Design, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
The mechanism design literature assumes too much common knowledge of the environment among the players and planner. We relax this assumption by studying implementation on richer type spaces, with more higher order uncertainty. We study the “ex post equivalence” question: when is interim implementation on all possible type spaces equivalent to requiring ex post implementation on the space of payoff types? We show that ex post equivalence holds when the social choice correspondence is a function and in simple quasi-linear environments. When ex post equivalence holds, we identify how large the type space must be to obtain the equivalence. We …