Five Essays in Economic Theory

Abstract

This thesis consists of five chapters. The first three chapters form an entity and deal with stochastic contest models. The fourth chapter analyzes a problem of strategic experimentation with private payoffs. In the final chapter the behavior of agents with prospect theory preferences in optimal stopping problems is analyzed.<br /> Contests and tournaments appear in many real world situations, like sports, politics, patent races, relative reward schemes in firms, or (public) procurement. In a contest multiple agents compete for a fixed prize. The first two chapters of this dissertation analyze contest models in continuous time. Both are based upon joint work with Christian Seel.<br /> In the first two chapters we introduce a new type of contest model. In our model each agent decides when to stop a privately observed Brownian motion with drift. The processes of the different players are uncorrelated. The player who stops with the highest value wins a fixed prize. Each agent observes only his own progress, but not the progress or the stopping decision of the other players. As no deviations are observable we use Nash equilibrium as the solution concept. The model is similar to an all-pay auction with complete information as the payoffs of an agent depend only on the distribution of the stopped values of the Brownian motion of the other players. Hence the question which distributions can be implemented by stopping a Brownian motion naturally arises. This question is known in the probability theory literature as the Skorokhod embedding problem. Using insights from this literature we are able to construct an equilibrium and show uniqueness of the equilibrium outcome. Using a bound on stopping times derived in ? we are able to show in the second chapter that the equilibrium we construct is also the unique equilibrium if the contest ends at a fixed point in time – given this deadline is not to short. This result is important as many real world contests have a fixed deadline. From a technical point of view we provide a method to show that equilibria in an infinite horizon game are also equilibria in the finite horizon game given the time horizon is long enough. The next paragraph describes the applications of the model.<br /> The model of the first chapter can be used to analyze the competition between fund managers. Each fund manager decides when to sell a risky asset. The performance of the asset follows a Brownian motion with drift. Assets are uncorrelated among fund managers and might go bankrupt. The fund manager who makes the highest profit by selling his asset wins a fixed prize. We show that the model has unique Nash equilibrium outcome. In this equilibrium the fund managers hold the risky asset even if it yields losses in expectation, i.e. the drift of the Brownian motion is negative. We show that the expected losses incurred in equilibrium by the fund managers are non-monotone in the expected return of the risky asset. Losses are highest if the underlying assets yields only moderate losses. In the last step we analyze the asymmetric situation where two fund managers hold assets with different expected return (drift of the Brownian motion) and riskiness (variance of the Brownian motion). We prove that in the unique Nash equilibrium outcome the weaker fund manager makes up for his disadvantage by using more risky strategies.<br /> 1The second chapter focusses on Research and Development Tournaments. Each agent decides for how long to do research and pay the associated cost. The outcome of research is uncertain and described by a Brownian motion with drift. The research progress of different agents is uncorrelated. When all agents stop, the one who made the most progress, wins a prize. All agent have to pay the cost of their research. We assume that there exists a finite deadline at which all agents are forced to stop doing research. We prove that the game has a unique Nash equilibrium outcome. If costs per time are constant, and if the riskiness of the research process (variance of the Brownian motion) converges to zero the equilibrium outcome converges to the symmetric equilibrium outcome of the symmetric all-pay auction. As the symmetric all-pay auction has multiple equilibria, this result provides an equilibrium selection criterion in favor of the symmetric equilibrium.<br /> If the variance of the Brownian motion is strictly greater zero, i.e. the outcome of research is uncertain the predictions of our model are different from those of the all-pay auction. In equilibrium each agent makes positive profits, while in the all-pay auction the agents make zero profits. Furthermore we show that the profits of the agents are increasing in their costs and variance and decreasing in the drift. Hence the agents prefer to be in a situation where research is very costly, inefficient, and risky. This has possibly implications for many real world situations as the agents have incentives to make the designer choose inefficient research targets. The third chapter of this dissertation is joint work with Matthias Lang and Christian Seel. It provides a micro-foundation of the discrete-bid all-pay auction using a continuous time contest model. More precisely we analyze the model where each agent decides when to stop a privately observed Poisson process. The processes of the agents are uncorrelated and the stopping decision of other players is not observable. As long as an agent did not stop she pays constant flow cost. There exists a finite deadline where each agent is forced to stop. We show that if the deadline is long enough the distribution over final outcomes in any Nash equilibrium equals the distribution of bids in a discrete-bid all-pay auction.<br /> Chapter four is joint work with Paul Heidhues and Sven Rady and deals with a problem of strategic experimentation. Precisely, an optimal stopping game with an infinite time horizon is analyzed in which multiple players face an identical two-armed bandit problem. At all points in time, agents choose between the deterministic payoff of zero and a risky payoff whose distribution depends on the state of the world. In the bad state of the world the risky payoff is always negative, representing the cost of experimentation. If the world is in the good state, with a small probability an experiment is successful and yields a high payoff.<br /> Because a success can only happen in the good state of the world, it fully reveals the state of the world. Similar models analyzed in literature show a free-rider problem, which means that less than the socially efficient amount of information is acquired in all Markov perfect equilibria. This free-rider problem is a consequence of the ability of agents to observe each others payoffs, which results in information being a public good.<br /> We show that when payoffs are public information even in subgame-perfect equilibria, the ensuing free- rider problem is so severe that the number of experiments is at most one plus the number of experiments that a single agent would perform.<br /> When payoffs are private information and players can communicate via cheap talk, the social opti- mal symmetric experimentation profile can be supported as a perfect Bayesian equilibrium for sufficiently optimistic prior beliefs.<br /> 2Chapter five deals with optimal stopping under prospect theory preferences and is based upon joint work with Sebastian Ebert. While expected utility theory is the leading normative theory of decision making under risk, cumulative prospect theory is a popular descriptive theory. Expected utility theory is well-studied in both static and dynamic settings, ranging from game theory over investment problems to institutional economics. In contrast, for cumulative prospect theory most research so far has focused on the static case. In this chapter, we investigate cumulative prospect theory’s predictions in the dynamic context and point out fundamental properties of cumulative prospect theory. We show that already a small amount of probability weighting has strong implications for the application of prospect theory in the dynamic context. More precisely we analyze the behavior of a naive agent with cumulative prospect theory preferences in continuous time optimal stopping problems. We prove that naive agent will never stop a non-degenerate diffusion process that represents his wealth. This holds for a very large class of processes, and independently of the reference point and the curvatures of the value and weighting functions.<br /> This dynamic result is a consequence of a static result that we call skewness preference in the small: At any wealth level there exists an arbitrarily small right-skewed gamble that a prospect theory agent wants to take. By choosing a proper stopping strategy the agent can always implement such a gamble and thus never stops. We illustrate the implications for dynamic decision problems such as irreversible investment, casino gambling, and the disposition effect.