One - Memory in Repeated Games

We study the extent to which equilibrium payo®s of discounted repeated games can be obtained by 1 { memory strategies. First, we present robust examples of games in which there is a subgame perfect equilibrium payo® pro¯le that cannot be obtained by any 1 { memory subgame perfect equilibrium. Then, a complete characterization of 1 { memory simple strategies is provided, and it is employed to establish the following in games with more than two players each having connected action spaces: 1. all subgame perfect equilibrium payo®s can be approximately supported by an " { sub- game perfect equilibrium strategy of 1 { memory, 2. all strictly enforceable subgame perfect equilibrium payo®s can be approximately sup- ported by a 1 { memory subgame equilibrium, and 3. the subgame perfect Folk Theorem holds for 1 { memory strategies. While no further restrictions are needed for the third result to hold in 2 { player games, an additional restriction is needed for the ¯rst two: players must have common punishments.

Business Cycles and Oligopoly Supergames: Some Empirical Evidence on Prices and Margins

There has been a significant interest on a theoretical level in the application of supergames to oligopoly behavior. Implications for pricing behavior in trigger-strategy models in response to aggregate demand are of particular importance for public policy considerations. We contrast the predictions for the movements of industry prices over the business cycle of two such models -- put forth by Edward Green and Robert Porter and by Julio Rotemberg and Garth Saloner -- and test the predictions using a panel data set of U.S. manufacturing industries. Our principal findings are four. First, the levels of price-cost margins of concentrated, homogeneous-goods industries, while higher than those of unconcentrated counterparts, appear to be closer to those predicted by a single-period Cournot-Nash equilibrium than monopoly. Second, there is little evidence to support the idea that price-cost margins of these industries have different cyclical patterns from other industries apart from effects by level of industry concentration. Maximum price declines for concentrated industries give little support for the occurrence of price wars during either recessions or booms. Finally, consistent with the predictions of the Rotemberg-Saloner model, the industries with high price-cost margins have more countercyclical price movements than those exhibited by other industries. That gradual price adjustment is quantitatively important for those industries, suggests, however, that other factors may lie behind the apparent rigidity of prices.

One - Memory in Repeated Games

We study the extent to which equilibrium payoffs of discounted repeated games can be obtained by 1 – memory strategies. First, we present robust examples of games in which there is a subgame perfect equilibrium payoff profile that cannot be obtained by any 1 – memory subgame perfect equilibrium. Then, a complete characterization of 1 – memory simple strategies is provided, and it is employed to establish the following in games with more than two players each having connected action spaces: 1. all subgame perfect equilibrium payoffs can be approximately supported by an ε – subgame perfect equilibrium strategy of 1 – memory, 2. all strictly enforceable subgame perfect equilibrium payoffs can be approximately supported by a 1 – memory subgame equilibrium, and 3. the subgame perfect Folk Theorem holds for 1 – memory strategies. While no further restrictions are needed for the third result to hold in 2 – player games, an additional restriction is needed for the first two: players must have common punishments.N/

Decision Making in Uncertain and Changing Environments

We consider an agent who has to repeatedly make choices in an uncertain and changing environment, who has full information of the past, who discounts future payoffs, but who has no prior. We provide a learning algorithm that performs almost as well as the best of a given finite number of experts or benchmark strategies and does so at any point in time, provided the agent is sufficiently patient. The key is to find the appropriate degree of forgetting distant past. Standard learning algorithms that treat recent and distant past equally do not have the sequential epsilon optimality property.Adaptive learning, experts, distribution-free, epsilon-optimality, Hannan regret

Decision making in uncertain and changing environments

We consider an agent who has to repeatedly make choices in an uncertain and changing environment, who has full information of the past, who discounts future payoffs, but who has no prior. We provide a learning algorithm that performs almost as well as the best of a given finite number of experts or benchmark strategies and does so at any point in time, provided the agent is sufficiently patient. The key is to find the appropriate degree of forgetting distant past. Standard learning algorithms that treat recent and distant past equally do not have the sequential epsilon optimality property.Adaptive learning, experts, distribution-free, e-optimality, Hannan regret

Acknowledgement Misspecification in Macroeconomic Theory

We explore methods for confronting model misspecification in macroeconomics. We construct dynamic equilibria in which private agents and policy makers recognize that models are approximations. We explore two generalizations of rational expectations equilibria. In one of these equilibria, decision makers use dynamic evolution equations that are imperfect statistical approximations, and in the other misspecification is impossible to detect even from infinite samples of time-series data. In the first of these equilibria, decision rules are tailored to be robust to the allowable statistical discrepancies. Using frequency domain methods, we show that robust decision makers treat model misspecification like time-series econometricians.

Learning Hyperinflations

Emprical studies of hyperinflations reveal that the rational expectations hypothesis fails to hold. To address this issue, we study a model of hyperinflation and learning in an attempt to better understand the volatility in movements of expectations, money, and prices. The findings surprisingly imply that the dynamics under neural network learning appear to support the outcome achieved under least squares learning reported in the earlier literature. Relaxing the assumption that inflationary expectations are rational, however, is essential since it improves the fit of the model to actual data from episodes of severe hyperinflation. Simulations provide ample evidence that if equilibrium in the model exists, then the inflation rate converges to the low inflation rational expectations equilibrium. This suggests a classical result: a permanent increase in the government deficit raises the stationary inflation rate (Marcet and Sargent, 1989)Hyperinflation, Learning, Rational Expectations Equlibria, Neural Networks

Learning Hyperinflations

Emprical studies of hyperinflations reveal that the rational expectations hypothesis fails to hold. To address this issue, we study a model of hyperinflation and learning in an attempt to better understand the volatility in movements of expectations, money, and prices. The findings surprisingly imply that the dynamics under neural network learning appear to support the outcome achieved under least squares learning reported in the earlier literature. Relaxing the assumption that inflationary expectations are rational, however, is essential since it improves the fit of the model to actual data from episodes of severe hyperinflation. Simulations provide ample evidence that if equilibrium in the model exists, then the inflation rate converges to the low inflation rational expectations equilibrium. This suggests a classical result: a permanent increase in the government deficit raises the stationary inflation rate (Marcet and Sargent, 1989)