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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.
Information Design: A Unified Perspective, Dirk Bergemann, Stephen Morris
Information Design: A Unified Perspective, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
Fixing a game with uncertain payoffs, information design identifies the information structure and equilibrium that maximizes the payoff of an information designer. We show how this perspective unifies existing work, including that on communication in games (Myerson (1991)), Bayesian persuasion (Kamenica and Gentzkow (2011)) and some of our own recent work. Information design has a literal interpretation, under which there is a real information designer who can commit to the choice of the best information structure (from her perspective) for a set of participants in a game. We emphasize a metaphorical interpretation, under which the information design problem is used …
Information Design: A Unified Perspective, Dirk Bergemann, Stephen Morris
Information Design: A Unified Perspective, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
Fixing a game with uncertain payoffs, information design identi.es the information structure and equilibrium that maximizes the payoff of an information designer. We show how this perspective unifies existing work, including that on communication in games (Myerson (1991)), Bayesian persuasion (Kamenica and Gentzkow (2011)) and some of our own recent work. Information design has a literal interpretation, under which there is a real information designer who can commit to the choice of the best information structure (from her perspective) for a set of participants in a game. We emphasize a metaphorical interpretation, under which the information design problem is used …
Information Design: A Unified Perspective, Dirk Bergemann, Stephen Morris
Information Design: A Unified Perspective, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
Given a game with uncertain payoffs, information design analyzes the extent to which the provision of information alone can influence the behavior of the players. Information design has a literal interpretation, under which there is a real information designer who can commit to the choice of the best information structure (from her perspective) for a set of participants in a game. We emphasize a metaphorical interpretation, under which the information design problem is used by the analyst to characterize play in the game under many different information structures. We provide an introduction into the basic issues and insights of a …
Information Design: A Unified Perspective, Dirk Bergemann, Stephen Morris
Information Design: A Unified Perspective, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
Given a game with uncertain payoffs, information design analyzes the extent to which the provision of information alone can influence the behavior of the players. Information design has a literal interpretation, under which there is a real information designer who can commit to the choice of the best information structure (from her perspective) for a set of participants in a game. We emphasize a metaphorical interpretation, under which the information design problem is used by the analyst to characterize play in the game under many different information structures. We provide an introduction into the basic issues and insights of a …
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 …
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 …
The Comparison Of Information Structures In Games: Bayes Correlated Equilibrium And Individual Sufficiency, Dirk Bergemann, Stephen Morris
The Comparison Of Information Structures In Games: Bayes Correlated Equilibrium And Individual Sufficiency, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
We define and characterize a notion of correlated equilibrium for games with incomplete information, which we call Bayes correlated equilibrium: The set of outcomes that can arise in Bayes Nash equilibria of an incomplete information game where players may have access to additional signals beyond the information structure is characterized and shown to be equivalent to the set of Bayes correlated equilibria. A game of incomplete information can be decomposed into a basic game, given by actions sets and payoff functions, and an information structure. We introduce a partial order on many player information structures — which we call individual …
The Comparison Of Information Structures In Games: Bayes Correlated Equilibrium And Individual Sufficiency, Dirk Bergemann, Stephen Morris
The Comparison Of Information Structures In Games: Bayes Correlated Equilibrium And Individual Sufficiency, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
The set of outcomes that can arise in Bayes Nash equilibria of an incomplete information game where players may have access to additional signals beyond the given information structure is characterized and shown to be equivalent to the set of a version of incomplete information correlated equilibria which we dub Bayes correlated equilibria. A game of incomplete information can be decomposed into a basic game, given by actions sets and payoff functions, and an information structure. We introduce a partial order on many player information structures — which we call individual sufficiency — under which more information shrinks the set …
Bayes Correlated Equilibrium And The Comparison Of Information Structures In Games, Dirk Bergemann, Stephen Morris
Bayes Correlated Equilibrium And The Comparison Of Information Structures In Games, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
The set of outcomes that can arise in Bayes Nash equilibria of an incomplete information game where players may have access to additional signals beyond the given information structure is equivalent to the set of a version of incomplete information correlated equilibrium which we dub Bayes correlated equilibrium . A game of incomplete information can be decomposed into a basic game, given by actions sets and payoff functions, and an information structure. We identify a partial order on many player information structures (individual sufficiency) under which more information shrinks the set of Bayes correlated equilibria.
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. …
Sequential Information Disclosure In Auctions, Dirk Bergemann, Achim Wambach
Sequential Information Disclosure In Auctions, Dirk Bergemann, Achim Wambach
Cowles Foundation Discussion Papers
We consider the design of an optimal auction in which the seller can determine the allocation and the disclosure rule of the mechanism. Thus, in contrast to the standard analysis of a optimal auctions, the seller can explicitly design the disclosure of the information received by each bidder as his private information. We show that the optimal disclosure rule is a sequential disclosure rule, implemented in an ascending price auction. In the optimal disclosure mechanism, each losing bidder learns his true valuation, but the winning bidder only learns that his valuation is sufficiently high to win the auction. We show …
Sequential Information Disclosure In Auctions, Dirk Bergemann, Achim Wambach
Sequential Information Disclosure In Auctions, Dirk Bergemann, Achim Wambach
Cowles Foundation Discussion Papers
We propose a sequential auction mechanism for a single object in which the seller jointly determines the allocation and the disclosure policy. A sequential disclosure rule is shown to implement an ascending price auction in which each losing bidder learns his true valuation, but the winning bidder’s information is truncated from below. As the auction ends, the winning bidder only has limited information, namely that his valuation is sufficiently high to win the auction. The sequential mechanism implements the allocation of the handicap auction of Esö and Szentes [10] but strengthens the participation constraints of the bidders from interim to …
Bayes Correlated Equilibrium And The Comparison Of Information Structures, Dirk Bergemann, Stephen Morris
Bayes Correlated Equilibrium And The Comparison Of Information Structures, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
The set of outcomes that can arise in Bayes Nash equilibria of an incomplete information game where players may or may not have access to more private information is characterized and shown to be equivalent to the set of an incomplete information version of correlated equilibrium, which we call Bayes correlated equilibrium. We describe a partial order on many player information structures — which we call individual sufficiency — under which more information shrinks the set of Bayes correlated equilibria. We discuss the relation of the solution concept to alternative definitions of correlated equilibrium in incomplete information games and of …
Correlated Equilibrium In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Correlated Equilibrium In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
We define a notion of correlated equilibrium for games with incomplete information in a general setting with finite players, finite actions, and finite states, which we call Bayes correlated equilibrium. The set of Bayes correlated equilibria of a fixed incomplete information game equals the set of probability distributions over actions, states and types that might arise in any Bayes Nash equilibrium where players observed additional information. We show that more information always shrinks the set of Bayes correlated equilibria.