Essays on game theory

This dissertation consists of four essays on game theory. The first essay investigates whether a community can sustain cooperation in the repeated prisoner's dilemma by having cheaters sanctioned not by their victims but by third parties. Motivated by systems of credit history recording, online feedback systems, and some experimental settings, I assume that players can access information about their partners' past play for free, but that acquiring information about their partners' past partners' past play is prohibitively costly. In this setting, even though players cannot distinguish cheaters from those who punish cheaters, I show that any level of cooperation can be sustained by an equilibrium. The second essay investigates absorption and global accessibility under perfect foresight dynamics in games with linear incentives. Using time symmetry of the dynamics, I show that every absorbing strict Nash equilibrium, if it exists, is globally accessible under zero rate of time preference. With the additional assumption of supermodularity, I prove that there generically exists an absorbing strict Nash equilibrium. The third essay, co-written with Drew Fudenberg and David K. Levine, provides a characterization of the limit set of perfect public equilibrium payoffs of repeated games with imperfect public monitoring as the discount factor goes to one. Our result covers general stage games including those that fail a "full-dimensionality" condition that had been imposed in past work. It also provides a characterization of the limit set when the strategies are restricted in a way that endogenously makes the full-dimensionality condition fail, as in the strongly symmetric equilibrium studied. Finally, we use our characterization to give a sufficient condition for the exact achievability of first-best outcomes. The fourth essay, co-written with Attila Ambrus, analyzes multi-sender cheap talk in multidimensional environments. Battaglini (2002) shows that if the state space is a multidimensional Euclidean space, then generically there exists a fully revealing equilibrium. We show that if the state space is restricted, then Battaglini's equilibrium construction is in general not valid. We provide a necessary and sufficient condition for the existence of fully revealing equilibrium.