755 research outputs found
Iterated Expectations, Compact Spaces and Common Priors
Extending to infinite state spaces that are compact metric spaces a result previously attained by Dov Samet solely in the context of finite state spaces, a necessary and sufficient condition for the existence of a common prior for several players is given in terms of the players’ present beliefs only. A common prior exists if and only if for each random variable it is common knowledge that all its iterated expectations with respect to any permutation converge to the same value; this value is its expectation with respect to the common prior. It is further shown that the restriction to compact metric spaces is ‘natural’ when semantic type spaces are derived from syntactic models, and that compactness is a necessary condition. Many proofs are based on results from the theory of Markov chains
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Information Aggregation Under Ambiguity: Theory and Experimental Evidence
We study information aggregation in a dynamic trading model with partially informed and ambiguity averse traders. We show theoretically that separable securities, introduced by Ostrovsky (2012) in the context of Subjective Expected Utility, no longer
aggregate information if some traders have imprecise beliefs and are ambiguity averse. Moreover, these securities are prone to manipulation, as the degree of information aggregation can be influenced by the initial price, set by the uninformed market maker. These observations are also confirmed in our experiment, using prediction markets. We define a new class of strongly separable securities which are robust to the above considerations, and show that they characterize information aggregation in both strategic and non-strategic environments. We derive several theoretical predictions, which we are able to confirm in the lab
Contagion through Learning
We study learning in a large class of complete information normal form games. Players continually face new strategic situations and must form beliefs by extrapolation from similar past situations. We characterize the long-run outcomes of learning in terms of iterated dominance in a related incomplete information game with subjective priors. The use of extrapolations in learning may generate contagion of actions across games even if players learn only from games with payoffs very close to the current ones. Contagion may lead to unique long-run outcomes where multiplicity would occur if players learned through repeatedly playing the same game. The process of contagion through learning is formally related to contagion in global games, although the outcomes generally differ.Similarity, learning, contagion, case-based reasoning, global games, coordination, subjective priors.
Efficient posterior simulation in cointegration models with priors on the cointegration space
A message coming out of the recent Bayesian literature on cointegration is that it is important to elicit a prior on the space spanned by the cointegrating vectors (as opposed to a particular identi…ed choice for these vectors). In this note, we discuss a sensible way of eliciting such a prior. Furthermore, we develop a collapsed Gibbs sampling algorithm to carry out e¢ cient posterior simulation in cointegration models. The computational advantages of our algorithm are most pronounced with our model, since the form of our prior precludes simple posterior simulation using conventional methods (e.g. a Gibbs sampler involves non-standard posterior conditionals). However, the theory we draw upon implies our algorithm will be more e¢ cient even than the posterior simulation methods which are used with identi…ed versions of cointegration models
Coherent Price Systems and Uncertainty-Neutral Valuation
We consider fundamental questions of arbitrage pricing arising when the
uncertainty model is given by a set of possible mutually singular probability
measures. With a single probability model, essential equivalence between the
absence of arbitrage and the existence of an equivalent martingale measure is a
folk theorem, see Harrison and Kreps (1979). We establish a microeconomic
foundation of sublinear price systems and present an extension result. In this
context we introduce a prior dependent notion of marketed spaces and viable
price systems. We associate this extension with a canonically altered concept
of equivalent symmetric martingale measure sets, in a dynamic trading framework
under absence of prior depending arbitrage. We prove the existence of such sets
when volatility uncertainty is modeled by a stochastic differential equation,
driven by Peng's G-Brownian motions
Incomplete Information
In interactive contexts such as games and economies, it is important to take account not only of what the players believe about substantive matters (such as payoffs), but also of what they believe about the beliefs of other players. Two different but equivalent ways of dealing with this matter, the semantic and the syntactic, are set forth. Canonical and universal semantic systems are then defined and constructed, and the concepts of common knowledge and common priors formulated and characterized. The last two sections discuss relations with Bayesian games of incomplete information and their applications, and with interactive epistemology -- the theory of multi-agent knowledge and belief as formulated in mathematical logic
Approximate knowledge of rationality and correlated equilibria
We extend Aumann's [3] theorem deriving correlated equilibria as a consequence of common priors and common knowledge of rationality by explicitly allowing for non-rational behavior. We replace the
assumption of common knowledge of rationality with a substantially weaker notion, joint p-belief of rationality, where agents believe the other agents are rational with probabilities p = (pi)i2I or more.
We show that behavior in this case constitutes a constrained correlated equilibrium of a doubled game
satisfying certain p-belief constraints and characterize the topological structure of the resulting set of p-rational outcomes. We establish continuity in the parameters p and show that, for p su ciently close to
one, the p-rational outcomes are close to the correlated equilibria and, with high probability, supported on strategies that survive the iterated elimination of strictly dominated strategies. Finally, we extend
Aumann and Dreze's [4] theorem on rational expectations of interim types to the broader p-rational belief systems, and also discuss the case of non-common priors.Spanish Ministry of Science and Technology (Grants SEJ2007-64340 and ECO2011-28965) Spanish Ministry of Science and Technology (Grant ECO2009-11213
Stepwise Thinking in Strategic Games with Incomplete Information
This paper proposes a general incomplete information framework for studying behavior in strategic games with stepwise (viz. `level-k' or `cognitive hierarchy') thinking, which has been found to describe strategic behavior well in experiments involving players' initial responses to games. It is shown that there exist coherent stepwise beliefs, implied by step types, that have the potential to encode all relevant information. In the structure of stepwise beliefs, players are unaware of opponents doing at least as much thinking as themselves. As a result, there exists a Bayesian Nash equilibrium strategy profile in which any player at some step fixes the best responses of opponents at lower steps and then best responds herself.game theory; interactive epistemology; unawareness; Bayesian Nash equilibrium; bounded rationality; level-k; cognitive hierarchy
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