A policymaker discloses public information to interacting agents who also acquire costly private information. More precise public information reduces the precision and cost of acquired private information. Considering this effect, what disclosure rule should the policymaker adopt? We address this question under two alternative assumptions using a linear-quadratic-Gaussian game with arbitrary quadratic material welfare and convex information costs. First, the policymaker knows the cost of private information and adopts an optimal disclosure rule to maximize the expected welfare. Second, the policymaker is uncertain about the cost and adopts a robust disclosure rule to maximize the worst-case welfare. Depending on the elasticity of marginal cost, an optimal rule is qualitatively the same as in the case of either a linear information cost or exogenous private information. The worst-case welfare is strictly increasing if and only if full disclosure is optimal under some information costs, which provides a new rationale for central bank transparency.
Consider a policymaker (such as a central bank) who discloses public information to interacting agents (such as firms and consumers) who also acquire costly private information. The policymaker’s concern is social welfare, including the agents’ cost of information acquisition. When the policymaker provides more precise public information, the agents have less incentive to acquire private information, reducing its precision and cost. This effect of public information is referred to as the crowding-out effect (Colombo et al., 2014). Less private information can be harmful to welfare, but less information cost is beneficial; that is, the welfare implication of the crowding-out effect is unclear. Then, what disclosure rule should the policymaker adopt?
WP039
This paper proposes normative criteria for voting rules under uncertainty about individual preferences to characterize a weighted majority rule (WMR). The criteria stress the significance of responsiveness, i.e., the probability that the social outcome coincides with the realized individual preferences. A voting rule is said to be robust if, for any probability distribution of preferences, the responsiveness of at least one individual is greater than one-half. This condition is equivalent to the seemingly stronger condition requiring that, for any probability distribution of preferences and any deterministic voting rule, the responsiveness of at least one individual is greater than that under the deterministic voting rule. Our main result establishes that a voting rule is robust if and only if it is a WMR without ties. This characterization of a WMR avoiding the worst possible outcomes provides a new complement to the well-known characterization of a WMR achieving the optimal outcomes, i.e., efficiency in the set of all random voting rules.
Consider the choice of a voting rule on a succession of two alternatives (such as “yes” or “no”) by a group of individuals. When a voting rule is chosen, the alternatives to come in the future are unknown, and the individuals are uncertain about their future preferences. An individual votes sincerely being concerned with the probability that the outcome agrees with his or her preference, which is referred to as responsiveness (Rae, 1969). More specifically, an individual prefers a voting rule with higher responsiveness because he or she can expect that a favorable alternative is more likely to be chosen. For example, if an individual has a von NeumannMorgenstern (VNM) utility function such that the utility from the passage of a favorable issue is one and that of an unfavorable issue is zero, then the expected utility equals the responsiveness.
WP038
A decision problem under uncertainty is often given with a piece of objective but imprecise information about the states of the world such as in the Ellsberg urn. By incorporating such information into the smooth ambiguity model of Seo (2009), we characterize a class of smooth ambiguity representations whose second-order beliefs are consistent with the objective information. As a corollary, we provide an axiomatization for the second-order expected utility, which has been studied by Nau (2001), Neilson (2009), Grant, Polak, and Strzalecki (2009), Strzalecki (2011), and Ghirardato and Pennesi (2019). In our model, attitude toward uncertainty can be disentangled from a perception about uncertainty and connected with attitude toward reduction of compound lotteries.
Choice under uncertainty is an important aspect of decision making. Since Ellsberg’s seminal work, it is admitted that a decision maker’s behavior may be inconsistent with a probabilistic belief (or a subjective probability measure) about the states of the world. Such a situation is called ambiguity and has been studied by several models of decision making, such as the Choquet expected utility (Schmeidler [31]), the maxmin expected utility (Gilboa and Schmeidler [12]), and the smooth ambiguity model (Klibanoff, Marinacci, and Mukerji [22] and Seo [33]).
WP037
In incomplete information games with ambiguous information, rational behavior depends on fundamental ambiguity (ambiguity about states) and strategic ambiguity (ambiguity about others’ actions). We study the impact of strategic ambiguity in global games, which is evident when one of the actions yields a constant payoff. Ambiguous-quality information makes more players choose this action, whereas (unambiguous) low-quality information makes more players choose an ex-ante best response to the uniform belief over the opponents’ actions. If the ex-ante best-response action yields a constant payoff, sufficiently ambiguous-quality information makes most players choose this action, thus inducing a unique equilibrium, whereas sufficiently low-quality information generates multiple equilibria. In applications to financial crises, we demonstrate that news of more ambiguous quality triggers a debt rollover crisis, whereas news of less ambiguous quality triggers a currency crisis.
Consider an incomplete information game with players who have ambiguous beliefs about a payoff-relevant state. Players receive signals about a state, but they do not exactly know the true joint distribution of signals and a state. In this game, players’ beliefs about the opponents’ actions are also ambiguous even if players know the opponents’ strategies, which assign an action to each signal, because their beliefs about the opponents’ signals are ambiguous. Thus, rational behavior depends not only on fundamental ambiguity (ambiguity about states) but also on strategic ambiguity (ambiguity about others’ actions).
WP032
Consider an analyst who models a strategic situation in terms of an incomplete information game and makes a prediction about players’ behavior. The analyst’s model approximately describes each player’s hierarchies of beliefs over payoff-relevant states, but the true incomplete information game may have correlated duplicated belief hierarchies, and the analyst has no information about the correlation. Under these circumstances, a natural candidate for the analyst’s prediction is the set of belief-invariant Bayes correlated equilibria (BIBCE) of the analyst’s incomplete information game. We introduce the concept of robustness for BIBCE: a subset of BIBCE is robust if every nearby incomplete information game has a BIBCE that is close to some BIBCE in this set. Our main result provides a sufficient condition for robustness by introducing a generalized potential function of an incomplete information game. A generalized potential function is a function on the Cartesian product of the set of states and a covering of the action space which incorporates some information about players’ preferences. It is associated with a belief-invariant correlating device such that a signal sent to a player is a subset of the player’s actions, which can be interpreted as a vague prescription to choose some action from this subset. We show that, for every belief-invariant correlating device that maximizes the expected value of a generalized potential function, there exists a BIBCE in which every player chooses an action from a subset of actions prescribed by the device, and that the set of such BIBCE is robust, which can differ from the set of potential maximizing BNE.
Consider an analyst who models a strategic situation in terms of an incomplete information game and makes a prediction about players’ behavior. He believes that his model correctly describes the probability distribution over the players’ Mertens-Zamir hierarchies of beliefs over payoff-relevant states (Mertens and Zamir, 1985). However, players may have observed signals generated by an individually uninformative correlating device (Liu, 2015), which allows the players to correlate their behavior. In other words, the true incomplete information game may have correlated duplicated belief hierarchies (Ely and Peski, 2006; Dekel et al., 2007). Then, a natural candidate for the analyst’s prediction is the set of outcomes that can arise in some Bayes Nash equilibrium (BNE) of some incomplete information game with the same distribution over belief hierarchies. Liu (2015) shows that this set of outcomes can be characterized as the set of belief-invariant Bayes correlated equilibria (BIBCE) of the analyst’s model. A BIBCE is a Bayes correlated equilibrium (BCE) in which a prescribed action does not reveal any additional information to the player about the opponents’ types and the payoff-relevant state, thus preserving the player’s belief hierarchy.
WP019
A linear-quadratic-Gaussian (LQG) game is an incomplete information game with quadratic payoff functions and Gaussian information structures. It has many applications such as a Cournot game, a Bertrand game, a beauty contest game, and a network game among others. LQG information design is a problem to find an information structure from a given collection of feasible Gaussian information structures that maximizes a quadratic objective function when players follow a Bayes Nash equilibrium. This paper studies LQG information design by formulating it as semidefinite programming, which is a natural generalization of linear programming. Using the formulation, we provide sufficient conditions for optimality and suboptimality of no and full information disclosure. In the case of symmetric LQG games, we characterize the optimal symmetric information structure, and in the case of asymmetric LQG games, we characterize the optimal public information structure, each of which is in a closed-form expression.
An equilibrium outcome in an incomplete information game depends not only upon a payoff structure, which consists of payoff functions together with a probability distribution of a payoff state, but also upon an information structure, which maps a payoff state to possibly stochastic signals of players. Information design analyzes the influence of an information structure on equilibrium outcomes, and in particular, characterizes an optimal information structure that induces an equilibrium outcome maximizing the expected value of an objective function of an information designer, who is assumed to be able to choose and commit to the information structure.1 General approaches to information design are presented by Bergemann and Morris (2013, 2016a,b, 2019), Taneva (2019), and Mathevet et al. (2020). A rapidly growing body of literature have investigated the economic application of information design in areas such as matching markets (Ostrovsky and Schwarz, 2010), voting games (Alonso and Camara, 2016), congestion games (Das et al., 2017), auctions (Bergemann et al., 2017), contests (Zhang and Zhou, 2016), and stress testing (Inostroza and Pavan, 2018), among others.
WP018
In the Lucas Imperfect Information model, output responds to unanticipated monetary shocks. We incorporate more general information structures into the Lucas model and demonstrate that output also responds to (dispersedly) anticipated monetary shocks if the information is imperfect common knowledge. Thus, the real effects of money consist of the unanticipated part and the anticipated part, and we decompose the latter into two effects, an imperfect common knowledge effect and a private information effect. We then consider an information structure composed of public and private signals. The real effects disappear when either signal reveals monetary shocks as common knowledge. However, when the precision of private information is fixed, the real effects are small not only when a public signal is very precise but also when it is very imprecise. This implies that a more precise public signal can amplify the real effects and make the economy more volatile.
In the Lucas Imperfect Information model (Lucas, 1972, 1973), which formalizes the idea of Phelps (1970), markets are decentralized and agents in each market have only limited information about prices in other markets. As a consequence, output responds to unanticipated monetary shocks; that is, imperfect information about prices generates real effects of money. However, if monetary shocks are anticipated, no real effects arise. This implies that monetary shocks cannot have lasting effects, which is considered to be a serious shortcoming of the Lucas model.
WP007