Knowledge that Transforms

CONSIDER A REPEATED GAME with incomplete information in which a patient long run player, whose type is unknown, faces a sequence of short run opponents (as in Fudenberg and Levine (1989)). The standard result is that the patient long run player can obtain an average long payoff almost equal to the payoff of the stage game by consistently playing as a leader. In the analysis, one generally takes as fundamental an assumption that the players have common prior beliefs on the states of the world and that behavior is consistent with the concept of Bayesian Nash equilibrium. However, these related suppositions have been called into question as unrealistic and too stringent (cf. Gul (1991)). In many games of incomplete information a player's prior probabilities on types (or states of the world) are best regarded as purely subjective psychological parameters, unknown to the modeler and to this player's opponents. Therefore, it is important to understand whether the standard reputation results (among others) are implied by weaker assumptions on the knowledge and behavior of the players. In fact, as Watson (1993) demonstrates, the reputation result does not require equilibrium. It is implied by a weak notion of rationalizability with some restrictions on the beliefs of the players. Here we qualify Watson's (1993) study and extend the line of inquiry of Watson (1993) and Battigalli (1994) concerning settings in which reputations are effective. As Watson shows, two main conditions on the beliefs of the players, along with weak rationalizability, imply the reputation result. First, there must be a strictly positive and uniform lower bound on the subjective probability that players assign to the Stackelberg type. Second, the conditional beliefs of the short run players must not be too dispersed. Watson (1993) does not explicitly indicate on what the updated beliefs of the short run players are conditioned. We make this explicit and show that it is necessary to assume that the conditional beliefs of the short run players satisfy a stochastic independence property (cf. Battigalli (1996)). We also comment on the dispensability of equilibrium regarding the reputation result in games with two long run players.

Consiglio Nazionale delle Ricerche (CNR). Biblioteca Centrale / CNR - Consiglio Nazionale delle Richerche

We derive explicitly the exact density functions of two key mixed normal variates that arise from cointegration analysis. We also plot them, and analyze their analytical features and implications.

The author demonstrates that despite variables that are integrated, the fundamental issues on structural equation modeling raised by the Cowles Commission remain valid and standard estimation and testing procedures can still be applied. A basic framework linking the multiple time series model and the dynamic simultaneous equation model is provided and implications under the long-run cointegrating relations are discussed. Conditions for identifying both the short-run dynamics and long-run equilibrium conditions are given. Limiting properties of the least squares and simultaneous equation estimators under cointegration are derived. Implications for hypothesis testing are also discussed.

There are several solutions to the Nash bargaining problem in the literature. Since various authors have expressed preferences for one solution over another, we find it useful to study preferences over solutions in their own right. We identify a set of appealing axioms on such preferences that lead to unanimity in the choice of solution, which turns out to be the solution of Nash.

We analyze two-candidate elections in which voters are uncertain about the realization of a state variable that affects the utility of all voters. Each voter has noisy private information about the state variable. We show that the fraction of voters whose vote depends on their private information goes to zero as the size of the electorate goes to infinity. Nevertheless, elections fully aggregate information in the sense that the chosen candidate would not change if all private information were common knowledge. Equilibrium voting behavior is to a large extent determined by the electoral rule, i.e., if a candidate is required to get at least x percent of the vote in order to win the election, then in equilibrium this candidate gets very close to x percent of the vote with probability close to one. Finally, if the distribution from which preferences are drawn is uncertain, then elections will generally not satisfy full information equivalence and the fraction of voters who take informative action does not converge to zero.

A number of papers have shown that a strict Nash equilibrium action profile of a game may never be played if there is a small amount of incomplete information. The authors present a general approach to analyzing the robustness of equilibria to a small amount of incomplete information. A Nash equilibrium of a complete information game is said to be robust to incomplete information if every incomplete information game with payoffs almost always given by the complete information game has an equilibrium which generates behavior close to the Nash equilibrium. The authors show that many games with strict equilibria have no robust equilibrium and examine why they get such different results from existing refinements. If a game has a unique correlated equilibrium, it is robust. A natural many-player many-action generalization of risk dominance is a sufficient condition for robustness.

To obtain consistency and asymptotic normality, a generalized method of moments (GMM) estimator typically is defined to be an approximate global minimizer of a GMM criterion function. To compute such an estimator, however, can be problematic because of the difficulty of global optimization. In consequence, practitioners usually ignore the problem and take the GMM estimator to be the result of a local optimization algorithm. This yields an estimator that is not necessarily consistent and asymptotically normal. The use of a local optimization algorithm also can run into the problem of instability due to flats or ridges in the criterion function, which makes it difficult to know when to stop the algorithm. To alleviate these problems of global and local optimization, we propose a stopping-rule (SR) procedure for computing GMM estimators. The SR procedure eliminates the need for global search with high probability. And, it provides an explicit SR for problems of stability that may arise with local optimization problems.

This paper provides an empirical analysis of how the U.S. Social Security and Medicare insurance system affect the labor supply of older males in the presence of incomplete markets for loans, annuities, and insurance. We estimate a detailed dynamic programming (DP) model of the joint labor supply and Social Security acceptance decision, focusing on a sample of males in the low to middle income brackets whose only pension is Social Security. The DP model delivers a rich set of predictions about the dynamics of retirement behavior, and comparisons of actual vs. predicted behavior show that the DP model is able to account for wide variety of phenomena observed in the data, including the pronounced peaks in the distribution of retirement ages at 62 and 65 (the ages of early and normal eligibility for Social Security benefits, respectively). We identify a significant fraction of health insurance constrained individuals who have no form of retiree insurance other than Medicare, and who can only obtain fairly priced private insurance via their employer's group plan. The combination of significant individual risk aversion and a long tailed (Pareto) distribution of care expenditures implies that there is a significant security value for these individuals to remain employed until they are eligible for Medicare coverage at age 65. Overall, our model suggests that a number of heretofore puzzling aspects of retirement behavior can be viewed as artifacts of particular details of the Social Security rules, whose incentive effects are especially strong for lower income individuals and those who do not have access to fairly priced loans, annuities, and insurance.