This paper develops new recursive, set based methods for studying repeated games with private monitoring. For any finite-state strategy profile, we find necessary and sufficient conditions for whether there exists a distribution over initial states such that the strategy, together with this distribution, form a correlated sequential equilibrium (CSE). Also, for any given correlation device for determining initial states (including degenerate cases where players' initial states are common knowledge), we provide necessary and sufficient conditions for the correlation device and strategy to be a CSE, or in the case of a degenerate correlation device, for the strategy to be a sequential equilibrium. We also consider several applications. In these, we show that the methods are computationally feasible, and how to construct and verify equilibria in a secret price-setting game.

We propose a new inverse probability weighting (IPW) estimator for moment condition models with missing data. Our estimator is easy to implement and compares favourably with existing IPW estimators, including augmented IPW estimators, in terms of efficiency, robustness, and higher-order bias. We illustrate our method with a study of the relationship between early Black–White differences in cognitive achievement and subsequent differences in adult earnings. In our data set, the early childhood achievement measure, the main regressor of interest, is missing for many units.

In the context of a “beauty-contest” coordination game (in which pay-offs depend on the quadratic distance of actions from an unobserved state variable and from the average action), players choose how much costly attention to pay to various informative signals. Each signal has an underlying accuracy (how precisely it identifies the state) and a clarity (how easy it is to understand). The unique linear equilibrium has interesting properties: the signals which receive attention are the clearest available, even if they have poor underlying accuracy; the number of signals observed falls as the complementarity of players' actions rises; and, if actions are more complementary, the information endogenously acquired in equilibrium is more public in nature. The consequences of “rational-inattention” constraints on information transmission and processing are also studied.

We study efficient non-linear taxation of labour and capital in a dynamic Mirrleesian model incorporating political economy constraints. Policies are chosen sequentially over time, without commitment. Our main result is that the marginal tax on capital income is progressive, in the sense that richer agents face higher marginal tax rates.