Essays on communication and information in game theory

In this dissertation, I investigate the role of information and communication in game theory. The dissertation consists of three chapters. The first chapter is co-authored with Takuo Sugaya, and we prove a folk theorem in two-player repeated games with public monitoring using the individual full-rank condition. The individual full-rank condition allows players to statistically distinguish a player's action holding fixed the other player's action. Under this condition, we may not be able to directly identify the pair of actions played in each period and punish the player who is likely to have deviated. We build an equilibrium where the players first coordinate on their actions, individually review the other player using noisy signals, and reward or punish the other player based on the history of signals and her own actions. One of the challenges of this construction is the problem of higher-order belief. The coordination and reviews are conducted through a noisy signal, and a player's optimal strategy depends in turn on the other player's belief about the coordinated strategy and her history of actions. This chain of beliefs can be extended indefinitely, and a strategy profile can easily be intractable, if not infeasible as an equilibrium. In this chapter, we use the idea of ``keeping each other in the dark" to build strategies where the players can coordinate on their actions and review each other based on their own histories. The players intentionally mix noisy actions with their prescribed actions to cause the other player to think that miscoordination is most likely to have happened because of this noise. A player makes the other player indifferent among all actions after playing the noisy action too often, so the other player does not need to change his action although he detects miscoordination. In this setup, we only use the players' actions without public randomization or cheap-talk communication. The second chapter is about refinement in signaling games with cheap-talk communication. In many applications of signaling games, the sender can both play costly actions and send costless messages to the receiver. Introducing cheap-talk messages to the model can expand the set of equilibria in this game. Given a finite signaling game, I define a signaling game with cheap talk by adding costless messages to the sender's action space. I first find a finite message space that allows all possible equilibrium outcomes to be found in a tractable way, then define a forward-induction refinement of equilibria based on cheap-talk messages. Many forward-induction refinements use the idea that a type of sender can play an action and simultaneously make a speech about the subset of types to which she belongs. Different refinements arise based on different assumptions about which messages are credible to the receiver. I define a message called a strongly self-signaling set, which defines a binary partition over the type space such that for every type, the worst-case equilibrium payoff when the receiver believes that the type belongs to its own subset is larger than the best-case equilibrium payoff when the receiver believes otherwise. I argue that this message is credible when all types in the subset can be better off by sending this message because types in the other subset have no incentive to imitate this message. When there is no such strongly self-signaling subset, we say that the equilibrium is not vulnerable to a partitional message. I prove that, in every finite signaling game with cheap talk, there exists an equilibrium that is not vulnerable to a partitional message. We apply this refinement to various signaling games and compare them to other refinements, especially those implied by stability. The last chapter is about the repeated signaling game between a long-lived sender and a series of short-lived receivers. In the stage game, there is an equilibrium that maximizes the sender's ex-ante payoff, which I call a payoff-maximizing equilibrium. I consider the setting where states are independently realized at each period and the sender can use transfers. In this case, the repeated interaction does not increase the sender's maximum payoff and I argue that a payoff-maximizing equilibrium is a reasonable outcome of the game. However, a payoff-maximizing equilibrium does not necessarily satisfy the intuitive criterion in the stage game. In other words, after the sender learns about her type, she can deviate by playing an off-path action and simultaneously making a speech that if the receiver believes that she is in one of the states where she can be weakly better off from playing this action, then she will be strictly better-off even in the worst case equilibrium. I view it as a viable deviation that the sender can make, and investigate how the sender can deter this through repeated interaction. I define a signaling game induced by an equilibrium in a history by defining the sender's payoff as a weighted sum of the current-period payoff and the continuation payoff from her action. Because the receiver of that period is short-lived, his payoff stays the same. An equilibrium satisfies the dynamic intuitive criterion if it satisfies the intuitive criterion in the induced signaling game in each history. I show that if the sender is sufficiently patient, there exists an equilibrium that achieves the maximum payoff and satisfies the dynamic intuitive criterion. As in repeated games, the sender can vary the continuation payoffs to deter her deviation in each period. Throughout this chapter, I mainly consider the sender's deviation based on forward-induction reasoning

Keywords: Humanities and Sciences.
Economics.
Advisor: Segal Ilya
Jackson Matthew O.
Sugaya Takuo