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Full-Text Articles in Economics
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We describe a methodology for making counterfactual predictions when the information held by strategic agents is a latent parameter. The analyst observes behavior which is rationalized by a Bayesian model, in which agents maximize expected utility, given partial and differential information about payoff-relevant states of the world, represented as an information structure. A counterfactual prediction is desired about behavior in another strategic setting, under the hypothesis that the distribution of the state and agents’ information about the state are held fixed. When the data and the desired counterfactual prediction pertain to environments with finitely many states, players, and actions, there …
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We describe a methodology for making counterfactual predictions in settings where the information held by strategic agents is unknown. The analyst observes behavior assumed to be rationalized by a Bayesian model, in which agents maximize expected utility, given partial and differential information about payoff-relevant states of the world. A counterfactual prediction is desired about behavior in another strategic setting, under the hypothesis that the distribution of the state and agents’ information about the state are held fixed. When the data and the desired counterfactual prediction pertain to environments with finitely many states, players, and actions, the counterfactual prediction is described …
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We describe a methodology for making counterfactual predictions when the information held by strategic agents is a latent parameter. The analyst observes behavior which is rationalized by a Bayesian model in which agents maximize expected utility, given partial and differential information about payoff-relevant states of the world. A counterfactual prediction is desired about behavior in another strategic setting, under the hypothesis that the distribution of and agents’ information about the state are held fixed. When the data and the desired counterfactual prediction pertain to environments with finitely many states, players, and actions, there is a finite dimensional description of the …
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We describe a methodology for making counterfactual predictions when the information held by strategic agents is a latent parameter. The analyst observes behavior which is rationalized by a Bayesian model in which agents maximize expected utility given partial and differential information about payoff-relevant states of the world. A counterfactual prediction is desired about behavior in another strategic setting, under the hypothesis that the distribution of and agents’ information about the state are held fixed. When the data and the desired counterfactual prediction pertain to environments with finitely many states, players, and actions, there is a finite dimensional description of the …
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Counterfactuals With Latent Information, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We describe a methodology for making counterfactual predictions in settings where the information held by strategic agents and the distribution of payoff-relevant states of the world are unknown. The analyst observes behavior assumed to be rationalized by a Bayesian model, in which agents maximize expected utility, given partial and differential information about the state. A counterfactual prediction is desired about behavior in another strategic setting, under the hypothesis that the distribution of the state and agents’ information about the state are held fixed. When the data and the desired counterfactual prediction pertain to environments with finitely many states, players, and …
First-Price Auctions With General Information Structures: A Short Introduction, Dirk Bergemann, Benjamin Brooks, Stephen Morris
First-Price Auctions With General Information Structures: A Short Introduction, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
In a recent paper, [Bergemann et al. 2017a], we derive results about equilibrium behavior in the first-price auction that hold across all common-prior information structures. The purpose of this letter is to give an informal introduction into the results. At the end we offer a brief discussion of related work.
Informationally Robust Optimal Auction Design, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Informationally Robust Optimal Auction Design, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
A single unit of a good is to be sold by auction to one of two buyers. The good has either a high value or a low value, with known prior probabilities. The designer of the auction knows the prior over values but is uncertain about the correct model of the buyers’ beliefs. The designer evaluates a given auction design by the lowest expected revenue that would be generated across all models of buyers’ information that are consistent with the common prior and across all Bayesian equilibria. An optimal auction for such a seller is constructed, as is a worst-case …
Information Design, Bayesian Persuasion And Bayes Correlated Equilibrium, Dirk Bergemann, Stephen Morris
Information Design, Bayesian Persuasion And Bayes Correlated Equilibrium, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
A set of players have preferences over a set of outcomes. We consider the problem of an “information designer” who can choose an information structure for the players to serve his ends, but has no ability to change the mechanism (or force the players to make particular action choices). We describe a unifying perspective for information design. We consider a simple example of Bayesian persuasion with both an uninformed and informed receiver. We extend information design to many players and relate it to the literature on incomplete information correlated equilibrium.
First Price Auctions With General Information Structures: Implications For Bidding And Revenue, Dirk Bergemann, Benjamin Brooks, Stephen Morris
First Price Auctions With General Information Structures: Implications For Bidding And Revenue, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We explore the impact of private information in sealed-bid first-price auctions. For a given symmetric and arbitrarily correlated prior distribution over values, we characterize the lowest winning-bid distribution that can arise across all information structures and equilibria. The information and equilibrium attaining this minimum leave bidders indifferent between their equilibrium bids and all higher bids. Our results provide lower bounds for bids and revenue with asymmetric distributions over values. We also report further characterizations of revenue and bidder surplus including upper bounds on revenue. Our work has implications for the identification of value distributions from data on winning bids and …
Information And Market Power, Dirk Bergemann, Tibor Heumann, Stephen Morris
Information And Market Power, Dirk Bergemann, Tibor Heumann, Stephen Morris
Cowles Foundation Discussion Papers
We consider demand function competition with a finite number of agents and private information. We analyze how the structure of the private information shapes the market power of each agent and the price volatility. We show that any degree of market power can arise in the unique equilibrium under an information structure that is arbitrarily close to complete information. In particular, regardless of the number of agents and the correlation of payoff shocks, market power may be arbitrarily close to zero (so we obtain the competitive outcome) or arbitrarily large (so there is no trade in equilibrium). By contrast, price …
First Price Auctions With General Information Structures: Implications For Bidding And Revenue, Dirk Bergemann, Benjamin Brooks, Stephen Morris
First Price Auctions With General Information Structures: Implications For Bidding And Revenue, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We explore the impact of private information in sealed bid first price auctions. For a given symmetric and arbitrarily correlated prior distribution over values, we characterize the lowest winning bid distribution that can arise across all information structures and equilibria. The information and equilibrium attaining this minimum leave bidders uncertain whether they will win or lose and indifferent between their equilibrium bids and all higher bids. Our results provide lower bounds for bids and revenue with asymmetric distributions over values. We report further analytic and computational characterizations of revenue and bidder surplus including upper bounds on revenue. Our work has …
First Price Auctions With General Information Structures: Implications For Bidding And Revenue, Dirk Bergemann, Benjamin Brooks, Stephen Morris
First Price Auctions With General Information Structures: Implications For Bidding And Revenue, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We explore the impact of private information in sealed-bid first-price auctions. For a given symmetric and arbitrarily correlated prior distribution over values, we characterize the lowest winning-bid distribution that can arise across all information structures and equilibria. The information and equilibrium attaining this minimum leave bidders indifferent between their equilibrium bids and all higher bids. Our results provide lower bounds for bids and revenue with asymmetric distributions over values. We report further analytic and computational characterizations of revenue and bidder surplus including upper bounds on revenue. Our work has implications for the identification of value distributions from data on winning …
First Price Auctions With General Information Structures: Implications For Bidding And Revenue, Dirk Bergemann, Benjamin Brooks, Stephen Morris
First Price Auctions With General Information Structures: Implications For Bidding And Revenue, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
This paper explores the consequences of information in sealed bid first price auctions. For a given symmetric and arbitrarily correlated prior distribution over valuations, we characterize the set of possible outcomes that can arise in a Bayesian equilibrium for some information structure. In particular, we characterize maximum and minimum revenue across all information structures when bidders may not know their own values, and maximum revenue when they do know their values. Revenue is maximized when buyers know who has the highest valuation, but the highest valuation buyer has partial information about others’ values. Revenue is minimized when buyers are uncertain …
Information And Market Power, Dirk Bergemann, Tibor Heumann, Stephen Morris
Information And Market Power, Dirk Bergemann, Tibor Heumann, Stephen Morris
Cowles Foundation Discussion Papers
We analyze demand function competition with a finite number of agents and private information. We show that the nature of the private information determines the market power of the agents and thus price and volume of equilibrium trade. We establish our results by providing a characterization of the set of all joint distributions over demands and payoff states that can arise in equilibrium under any information structure. In demand function competition, the agents condition their demand on the endogenous information contained in the price. We compare the set of feasible outcomes under demand function to the feasible outcomes under Cournot …
Informational Robustness And Solution Concepts, Dirk Bergemann, Stephen Morris
Informational Robustness And Solution Concepts, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
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 …
Informational Robustness And Solution Concepts, Dirk Bergemann, Stephen Morris
Informational Robustness And Solution Concepts, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
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 …
Information And Volatility, Dirk Bergemann, Tibor Heumann, Stephen Morris
Information And Volatility, Dirk Bergemann, Tibor Heumann, Stephen Morris
Cowles Foundation Discussion Papers
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 …
Information And Volatility, Dirk Bergemann, Tibor Heumann, Stephen Morris
Information And Volatility, Dirk Bergemann, Tibor Heumann, Stephen Morris
Cowles Foundation Discussion Papers
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 …
Information, Interdependence, And Interaction: Where Does The Volatility Come From?, Dirk Bergemann, Tibor Heumann, Stephen Morris
Information, Interdependence, And Interaction: Where Does The Volatility Come From?, Dirk Bergemann, Tibor Heumann, Stephen Morris
Cowles Foundation Discussion Papers
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 …
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 …
Correlated Equilibrium And The Comparison Of Information Structures In Games, Dirk Bergemann, Stephen Morris
Correlated Equilibrium And The Comparison Of Information Structures In Games, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
A game of incomplete information can be decomposed into a basic game and an information structure. The basic game defines the set of actions, the set of payoff states the payoff functions and the common prior over the payoff states. The information structure refers to the signals that the players receive in the game. We characterize the set of outcomes that can arise in Bayes Nash equilibrium if players observe the given information structure but may also observe additional signals. The characterization corresponds to the set of (a version of) incomplete information correlated equilibria which we dub Bayes correlated equilibria. …
The Limits Of Price Discrimination, Dirk Bergemann, Benjamin Brooks, Stephen Morris
The Limits Of Price Discrimination, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We analyze the welfare consequences of a monopolist having additional information about consumers’ tastes, beyond the prior distribution; the additional information can be used to charge different prices to different segments of the market, i.e., carry out “third degree price discrimination.” We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that: (i) consumer surplus is non-negative, (ii) producer surplus is at least as high as profits under the uniform monopoly price, and (iii) total surplus does not exceed the efficient gains from trade. As well as characterizing …
The Limits Of Price Discrimination, Dirk Bergemann, Benjamin Brooks, Stephen Morris
The Limits Of Price Discrimination, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We analyze the welfare consequences of a monopolist having additional information about consumers’ tastes, beyond the prior distribution; the additional information can be used to charge different prices to different segments of the market, i.e., carry out “third degree price discrimination.” We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that: (i) consumer surplus is non-negative, (ii) producer surplus is at least as high as profits under the uniform monopoly price, and (iii) total surplus does not exceed the surplus generated by efficient trade.
The Limits Of Price Discrimination, Dirk Bergemann, Benjamin Brooks, Stephen Morris
The Limits Of Price Discrimination, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We analyze the welfare consequences of a monopolist having additional information about consumers’ tastes, beyond the prior distribution; the additional information can be used to charge different prices to different segments of the market, i.e., carry out “third degree price discrimination.” We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that: (i) consumer surplus is non-negative, (ii) producer surplus is at least as high as profits under the uniform monopoly price, and (iii) total surplus does not exceed the efficient gains from trade. As well as characterizing …
The Limits Of Price Discrimination, Dirk Bergemann, Benjamin Brooks, Stephen Morris
The Limits Of Price Discrimination, Dirk Bergemann, Benjamin Brooks, Stephen Morris
Cowles Foundation Discussion Papers
We analyze the welfare consequences of a monopolist having additional information about consumers’ tastes, beyond the prior distribution; the additional information can be used to charge different prices to different segments of the market, i.e., carry out “third degree price discrimination.” We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that: (i) consumer surplus is non-negative, (ii) producer surplus is at least as high as profits under the uniform monopoly price, and (iii) total surplus does not exceed the surplus generated by efficient trade.