Strategies as states

We define rationality and equilibrium when states specify agents’ actions and agents have arbitrary partitions over these states. Although some suggest that this natural modeling step leads to paradox, we show that Bayesian equilibrium is well defined and puzzles can be circumvented. The main problem arises when player j's partition informs j of i's move and i knows j's strategy. Then i's inference about j's move will vary with i's own move, and i may consequently play a dominated action. Plausible conditions on partitions rule out these scenarios. Equilibria exist under the same conditions, and more generally ε equilibria usually exist.     

Bayesian learning and convergence to Nash equilibria without common priors

Summary. Consider an infinitely repeated game where each player is characterized by a “type” which may be unknown to the other players in the game. Suppose further that each player's belief about others is independent of that player's type. Impose an absolute continuity condition on the ex ante beliefs of players (weaker than mutual absolute continuity). Then any limit point of beliefs of players about the future of the game conditional on the past lies in the set of Nash or Subjective equilibria. Our assumption does not require common priors so is weaker than Jordan (1991); however our conclusion is weaker, we obtain convergence to subjective and not necessarily Nash equilibria. Our model is a generalization of the Kalai and Lehrer (1993) model. Our assumption is weaker than theirs. However, our conclusion is also weaker, and shows that limit points of beliefs, and not actual play, are subjective equilibria. Received: March 3, 1995; revised version: February 17, 1997     

Reshaping the schooling system: The role of immigration

This paper studies how the schooling system may be impacted by the number and skill type of immigrants. When the number of low-skilled immigrants is large, the education regime tends to become segregated. Wealthy locals are more likely to choose private schools and vote for a lower tax rate to finance public education. In contrast, high-skilled immigrants tend to reinforce the public system. The optimal immigration policy is highly skill-biased. The admission of high-skilled immigrants expedites redistribution toward the less-skilled local households through both a stronger fiscal support for public education and a reduction in the skill wage premium.     

Ergodic properties of conditional forecast functions of stable systems

Summary This paper analyzes the behavior of conditional forecast functions in stable systems. We study convergence of optimal forecast functions, of forecast functions obtained by conditioning on previous values, and conditional and joint densities.The research was done as part of the author's master thesis under Mordecai Kurz. The author thanks Mordecai Kurz, Carsten Nielsen, and Ho-Mou Wu for a careful reading of the paper and the Statistics Department of Stanford University for financial support. This research was supported in part by Fondazione ENI Enrico Mattei of Milan, Italy.     

Recursive preferences,learning and large deviations

We estimate the relative contribution of recursive preferences versus adaptive learning in accounting for the tail thickness of price–dividends/rents ratios. We find that both of these sources of volatility account for volatility in liquid (stocks) but not illiquid (housing) assets.