Investor Sentiments

We consider a general class of games that have been used to model many economic problems where players' sentiments are believed to play an important role. Dropping the common-prior assumption, we identify the relevant notion of sentiments for strategic behavior in these games. This notion is tied to how likely a player thinks that some other player has a more optimistic outlook than himself when they obtain their private information. Under this notion, we show that sentiments have a profound effect on strategic outcomes - even with vanishing uncertainty.

Public Disagreement

Members of different social groups often hold widely divergent public beliefs regarding the nature of the world in which they live. We develop a model that can accommodate such public disagreement, and use it to explore questions concerning the aggregation of distributed information and the consequences of social integration. The model involves heterogeneous priors, private information, and repeated communication until beliefs become public information. We show that when priors are correlated, all private information is eventually aggregated and public beliefs are identical to those arising under observable priors. When priors are independently distributed, however, some private information is never revealed and the expected value of public disagreement is greater when priors are unobservable than when they are observable. If the number of individuals is large, communication breaks down entirely in the sense that disagreement in public beliefs is approximately equal to disagreement in prior beliefs. Interpreting integration in terms of the observability of priors, we show how increases in social integration can give rise to less divergent public beliefs on average. Communication in segregated societies can cause initial biases to be amplified and new biases to emerge where none previously existed. Even though all announcements are public and all signals equally precise, minority group members face a disadvantage in the interpretation of public information that results in medium run beliefs that are less closely aligned with the true state.

Waiting to persuade

October 200

Nash meets Rubinstein in final-offer arbitration

I consider a final-offer arbitration model in which the offers are submitted sequentially, the parties are allowed to accept offers, and the arbitrator maximizes Nash's social welfare function. I show that backwards induction in this three-period model leads to the subgame-perfect equilibrium outcome of Rubinstein's infinite-horizon alternating-offer bargaining game

Learning and Disagreement in an Uncertain World

Most economic analyses presume that there are limited differences in the prior beliefs of individuals, as assumption most often justified by the argument that sufficient common experiences and observations will eliminate disagreements. We investigate this claim using a simple model of Bayesian learning. Two individuals with different priors observe the same infinite sequence of signals about some underlying parameter. Existing results in the literature establish that when individuals are certain about the interpretation of signals, under very mild conditions there will be asymptotic agreement---their assessments will eventually agree. In contrast, we look at an environment in which individuals are uncertain about the interpretation of signals, meaning that they have non-degenerate probability distributions over the conditional distribution of signals given the underlying parameter. When priors on the parameter and the conditional distribution of signals have full support, we prove the following results: (1) Individuals will never agree, even after observing the same infinite sequence of signals. (2) Before observing the signals, they believe with probability 1 that their posteriors about the underlying parameter will fail to converge. (3) Observing the same sequence of signals may lead to a divergence of opinion rather than the typically presumed convergence. We then characterize the conditions for asymptotic agreement under "approximate certainty"---i.e., as we look at the limit where uncertainty about the interpretation of the signals disappears. When the family of probability distributions of signals given the parameter has "rapidly-varying tails" (such as the normal or exponential distributions), approximate certainty restores asymptotic agreement. However, when the family of probability distributions has "regularly-varying tails" (such as the Pareto, the log-normal, and the t-distributions), asymptotic agreement does not obtain even in the limit as the amount of uncertainty disappears. Lack of common priors has important implications for economic behavior in a range of circumstances. We illustrate how the type of learning outlined in this paper interacts with economic behavior in various different situations, including games of common interest, coordination, asset trading and bargaining.

Learning and Disagreement in an Uncertain World

Most economic analyses presume that there are limited differences in the prior beliefs of individuals, an assumption most often justified by the argument that sufficient common experiences and observations will eliminate disagreements. We investigate this claim using a simple model of Bayesian learning. Two individuals with di.erent priors observe the same infinite sequence of signals about some underlying parameter. Existing results in the liter- ature establish that when individuals know the interpretation of signals, under very mild conditions, there will be asymptotic agreementtheir assessments will eventually agree. In contrast, we look at an environment in which individuals are uncertain about the inter- pretation of signals, meaning that they have non-degenerate probability distributions over the conditional distribution of signals given the underlying parameter. When priors on the parameter and the conditional distribution of signals have full support, we show the following: (1) Individuals will never agree, even after observing the same infinite sequence of signals. (2) Before observing the signals, they believe with probability 1 that their posteri- ors about the underlying parameter will fail to converge. (3) Observing the same (infinite) sequence of signals may lead to a divergence of opinion rather than the typically-presumed convergence. We then characterize the conditions for asymptotic agreement under “approx- imate certainty”–i.e., as we look at the limit where uncertainty about the interpretation of the signals disappears. When the family of probability distributions of signals given the parameter has rapidly-varying tails (such as the normal or the exponential distributions), approximate certainty restores asymptotic agreement. However, when the family of proba- bility distributions has regularly-varying tails (such as the Pareto, the log-normal, and the t-distributions), asymptotic agreement does not obtain even in the limit as the amount of uncertainty disappears. We also discuss how lack of common priors implied by the type of learning in this paper interacts with economic behavior in various different situations, including games of common interest, coordination, asset trading and bargaining.asymptotic disagreement, Bayesian learning, merging of opinions.

Public disagreement

February 15, 200

Robust Predictions in Infinite-Horizon Games--an Unrefinable Folk Theorem

We show that in any game that is continuous at infinity, if a plan of action ai is played by a type ti in a Bayesian Nash equilibrium, then there are perturbations of ti for which ai is the only rationalizable plan and whose unique rationalizable belief regarding the play of the game is arbitrarily close to the equilibrium belief of ti. As an application to repeated games, we prove an unrefinable folk theorem: any individually rational and feasible payoff is the unique rationalizable payoff vector for some perturbed type profile. This is true even if perturbed types are restricted to believe that the repeated-game payoff structure and the discount factor are common knowledge.Institute for Advanced Study (Princeton, N.J.