Three essays on signaling games

Abstract

This collection of essays studies signaling games. In the first chapter, we study a model of Bayesian persuasion where the sender cannot commit to a signal for the receiver, so the sender can lie about the signal. The sender can choose an announced signal differently from a private real signal and should bear a cost of lying. The receiver only observes the announced signal and a signal realization drawn according to the real signal. We show that signal exaggeration in the downward direction arises in equilibrium when the sender can benefit from persuasion. Surprisingly, the sender may not lie if the lying cost is too low, in which case the equilibrium signal is uninformative. As the lying cost increases, the equilibrium real and announced signals converge to a subset (or a single signal under certain conditions) of optimal signals, as defined by Kamenica and Gentzkow [2011]. Thus our approach provides a selection guide for the case of multiple optimal signals in Kamenica and Gentzkow [2011]. If the optimal signal is unique and the lying cost is sufficiently high, the equilibrium real signal is less informative than the optimal signal. When the sender's value function is strictly convex with a sufficiently high lying cost, the sender chooses the unique optimal signal of Kamenica and Gentzkow [2011] as the announced signal. In the second chapter, we study a sender-receiver game in which a sender's type set consists of two elements: good and bad. The sender privately observes his type and chooses the complexity of a message describing the type. The receiver, in turn, chooses an analysis level for the message. The complexity and the analysis level affect what information the receiver acquires about the type and how much it costs. The receiver finally decides whether to take action. Under reasonable conditions, we show that a forward-induction-based refinement selects a good-type optimal equilibrium (GOE). By focusing on GOE, we find that the good-type sender‘s expected payoff can increase when the receiver's ability to interpret the message decreases. Additionally, we explore how the good-type sender's expected payoff is influenced by the receiver's prior belief, either by increasing or decreasing it. In the third chapter, we study firms' strategic communication of information about product attribute importance under competition. When consumers shop for products with multiple product attributes, conveying information about which attribute is more essential to consumers can be critical to engaging the consumers with the product category. Consumers are aware of product attributes and attribute levels but are uncertain about which attribute is the more important in shaping their valuation. Each of the two firms receives a noisy signal about which attribute is more essential and sends a cheap-talk message to consumers, highlighting one of the two product attributes. After observing the messages from the two firms, consumers decide whether to visit the stores at a positive cost. We show that there exists a truthful equilibrium in which both firms honestly announce the noisy signal they have received. In this equilibrium, the firm's messages collectively can credibly convey information about the more important attribute, especially when their messages coincide. Having partially resolved their uncertainty, consumers are encouraged to visit the stores and buy their preferred products. Interestingly, one of the firms is better off sending a truthful message even if the message highlights an attribute in which the firm has a competitive disadvantage. Moreover, we show that if only one firm could communicate with consumers, the truthful equilibrium would not exist. Thus, the presence of the competing firm enables credible communication of the more important attribute and makes both firms better off.