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Full-Text Articles in Social and Behavioral Sciences
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 …
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 …
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 …
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. …
Matching With Incomplete Information, Quingmin Liu, George J. Mailath, Andrew Postlewaite, Larry Samuelson
Matching With Incomplete Information, Quingmin Liu, George J. Mailath, Andrew Postlewaite, Larry Samuelson
Cowles Foundation Discussion Papers
A large literature uses matching models to analyze markets with two-sided heterogeneity, studying problems such as the matching of students to schools, residents to hospitals, husbands to wives, and workers to firms. The analysis typically assumes that the agents have complete information, and examines core outcomes. We formulate a notion of stable outcomes in matching problems with one-sided asymmetric information. The key conceptual problem is to formulate a notion of a blocking pair that takes account of the inferences that the uninformed agent might make from the hypothesis that the current allocation is stable. We show that the set of …
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.
Robust Predictions In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Robust Predictions In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
We analyze games of incomplete information and offer equilibrium predictions which are valid for all possible private information structures that the agents may have. Our characterization of these robust predictions relies on an epistemic result which establishes a relationship between the set of Bayes Nash equilibria and the set of Bayes correlated equilibria. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior information of the analyst …
Robust Predictions In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Robust Predictions In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
We analyze games of incomplete information and offer equilibrium predictions which are valid for, and in this sense robust to, all possible private information structures that the agents may have. The set of outcomes that can arise in equilibrium for some information structure is equal to the set of Bayes correlated equilibria. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior knowledge about the private information …
Robust Predictions In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Robust Predictions In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
We analyze games of incomplete information and offer equilibrium predictions which are valid for, and in this sense robust to, all possible private information structures that the agents may have. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions. We consider information sharing among firms under demand uncertainty and find newly optimal information policies via …
Robust Predictions In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Robust Predictions In Games With Incomplete Information, Dirk Bergemann, Stephen Morris
Cowles Foundation Discussion Papers
We analyze games of incomplete information and offer equilibrium predictions which are valid for, and in this sense robust to, all possible private information structures that the agents may have. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions. We consider information sharing among firms under demand uncertainty and find newly optimal information policies via …
Uniform Topologies On Types, Yi-Chun Chen, Alfredo Di Tillio, Eduardo Faingold, Siyang Xiong
Uniform Topologies On Types, Yi-Chun Chen, Alfredo Di Tillio, Eduardo Faingold, Siyang Xiong
Cowles Foundation Discussion Papers
We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, called uniform weak topology, under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. This topology generalizes the now classic notion of proximity to common knowledge based on common p-beliefs (Monderer and Samet (1989)). We show that convergence in the uniform weak topology implies convergence in the …
Assortative Matching And Repubation, Axel Anderson, Lones Smith
Assortative Matching And Repubation, Axel Anderson, Lones Smith
Cowles Foundation Discussion Papers
Consider Becker’s classic 1963 matching model, with unobserved fixed types and stochastic publicly observed output. If types are complementary, then matching is assortative in the known Bayesian posteriors (the ‘reputations’). We discover a robust failure of Becker’s result in the simplest dynamic two type version of this world. Assortative matching is generally neither efficient nor an equilibrium for high discount factors. In a labor theoretic rationale, we show that assortative matching fails around the highest (lowest) reputation agents for ‘low-skill (high-skill) concealing’ technologies. We then find that as the number of production outcomes grows, almost all technologies are of either …
Generalized Potentials And Robust Sets Of Equilibria, Stephen Morris, Takashi Ui
Generalized Potentials And Robust Sets Of Equilibria, Stephen Morris, Takashi Ui
Cowles Foundation Discussion Papers
This paper introduces generalized potential functions of complete information games and studies the robustness of sets of equilibria to incomplete information. A set of equilibria of a complete information game is robust if every incomplete information game where payoffs are almost always given by the complete information game has an equilibrium which generates behavior close to some equilibrium in the set. This paper provides sufficient conditions for the robustness of sets of equilibria in terms of argmax sets of generalized potential functions and shows that the sufficient conditions generalize the existing sufficient conditions for the robustness of equilibria.