Games of Private Information and Learning

This dissertation studies private information and learning in games. Chapter 1 considers a dynamic game of strategic adoption, in which a patient buyer faces a seller of privately known quality. The buyer learns about the seller gradually through an exogenous news process, and the seller is able to exit the market or privately upgrade its quality at any time. I discover two novel kinds of reputational dynamics in equilibrium: a resetting barrier, where low types upgrade at low states inducing discrete upward jumps in reputation, and skew Brownian motion, where low types exit continuously at intermediate states, creating a permeable barrier for reputation. Contrary to the classic lemons result, the VC prefers this private information environment to one with symmetric information. The rich strategic interaction between the startup and VC implies, somewhat surprisingly, that players may benefit from increases in their own cost parameters.

Chapter 2 applies a similar information structure to an adversarial setting of optimal market entry timing. A player of privately known strength chooses when to enter a market, and an incumbent chooses whether to compete or concede. Information about the potential entrant's type is revealed publicly according to an exogenous news process and the timing of entry. I analyze stationary equilibria using the public belief as a state variable. No equilibria in pure strategies exist, and smooth-pasting conditions need not hold. Under both D1 and a novel refinement, the informed player has nondecreasing value functions and her strategy has the following structure: for high states, both types enter with certainty; for a possibly empty interval of intermediate states, no type enters; and for low states, the high type enters while the low type mixes. I obtain closed form solutions and analyze comparative statics for such equilibria. The welfare effects of the presence of news, relative to no news, depend on the starting belief; however, for a fixed equilibrium, a marginal increase in news quality always helps the informed player regardless of her type and always hurts total welfare.

Chapter 3 explores private information in a delegation setting. We investigate competition in a delegation framework, with a coarsely

informed principal. Two imperfectly informed and biased experts

simultaneously propose action choices. A principal with a diffuse prior, and

only able to ordinally compare the two proposals, has to choose one of

them. In equilibrium, experts may exaggerate their biases, and moreover, such an equilibrium may maximize the principal's welfare. We show that having a second expert can benefit the principal, even if the two experts have the same biases or if one expert is known to be unbiased. In contrast with other models of expertise, in our setting the principal prefers experts with equal rather than opposite biases. The principal may also benefit from commitment to an ``element of surprise," making an ex post suboptimal choice with positive probability. A methodological contribution of our paper is characterizing restrictions on the set of strategies which allows a formal generalization of ex ante expected payoffs to games with diffuse prior.