From Posteriors to Priors via Cycles: An Addendum

Rodrigues-Neto (2009) has shown that a given specification of posteriors of different players in an incomplete-information setting is compatible with a common prior if and only if the posteriors satisfy the so-called cycle equations. This note shows that, if, for any player, any element of the partition of the this player has a nonempty intersection with any element of the partition of any other player, then it suffices to verify the cycle equations for all cycles of length 4 or less.Belief systems, consistency, common priors, cycle equations

Incomplete-Information Models of Large Economies with Anonymity: Existence and Uniqueness of Common Priors

The paper provides a speci?cation of belief systems for models of large economies with anonymity in which aggregate states depend only on cross-section distributions of types. For belief systems satisfying certain conditions of mutual absolute continuity, the paper gives a necessary and sufficient condition for the existence of a common prior. Under the given conditions, the common prior is unique.Large Economy, Belief systems, consistency, common prior

Bayesian games with a continuum of states

We show that every Bayesian game with purely atomic types has a measurable Bayesian equilibrium when the common knowl- edge relation is smooth. Conversely, for any common knowledge rela- tion that is not smooth, there exists a type space that yields this common knowledge relation and payoffs such that the resulting Bayesian game will not have any Bayesian equilibrium. We show that our smoothness condition also rules out two paradoxes involving Bayesian games with a continuum of types: the impossibility of having a common prior on components when a common prior over the entire state space exists, and the possibility of interim betting/trade even when no such trade can be supported ex ante

The probability of nontrivial common knowledge

We study the probability that two or more agents can attain common knowledge of nontrivial events when the size of the state space grows large. We adopt the standard epistemic model where the knowledge of an agent is represented by a partition of the state space. Each agent is endowed with a partition generated by a random scheme. Assuming that agents' partitions are independently and identically distributed, we prove that the asymptotic probability of nontrivial common knowledge undergoes a phase transition. Regardless of the number of agents, when their cognitive capacity is sufficiently large, the probability goes to one; and when it is small, it goes to zero.Common knowledge; Epistemic game theory; Random partitions

The probability of nontrivial common knowledge

We study the probability that two or more agents can attain common knowledge of nontrivial events when the size of the state space grows large. We adopt the standard epistemic model where the knowledge of an agent is represented by a partition of the state space. Each agent is endowed with a partition generated by a random scheme consistent with his cognitive capacity. Assuming that agents' partitions are independently distributed, we prove that the asymptotic probability of nontrivial common knowledge undergoes a phase transition. Regardless of the number of agents, when their cognitive capacity is sufficiently large, the probability goes to one; and when it is small, it goes to zero. Our proofs rely on a graph-theoretic characterization of common knowledge that has independent interest

Structure‐preserving transformations of epistemic models

The prevailing approaches to modelling interactive uncertainty with epistemic models in economics are state-based and type-based. We explicitly formulate two general procedures that transform state models into type models and vice versa. Both transformation procedures preserve the belief hierarchies as well as the common prior assumption. By means of counterexamples it is shown that the two procedures are not inverse to each other. However, if attention is restricted to maximally reduced epistemic models, then isomorphisms can be constructed and an inverse relationship emerges

Essays on beliefs and knowledge

The unifying theme of all three chapters of this dissertation is incomplete information games. Each chapter investigates two essential components, namely beliefs and knowledge, of incomplete information games. In particular, the first two chapter studies an alternative equilibrium notion of Sakovics (2001)- mirage equilibrium- and the final chapter introduces a new notion of metric to measure the distance between partitions. All relevant notations and definitions are defined for each chapter so that any of them can be read independently. In the first chapter, I restudy the Purification theorem of Harsanyi (1973) by relaxing the common knowledge assumption on priors for 2 x 2 games. I show that the limit of the (Mirage) equilibrium points in perturbed games generically converge to a pure strategy of the original complete information. This result, unlike the original one in which the limit is a mixed equilibrium point, is reminiscent of risk dominance criterion of Carlsson and van Damme (1993). I also study the conditions for different hierarchy levels that yields risk dominant outcome for coordination games. That is, I give conditions (first order stochastic dominance and monotone likelihood ratio order) that yield the risk dominant outcome of a coordination game as the limit of perturbed game ´a la Harsanyi (1973). In the second chapter, I attempt to provide a generalization of mirage equilibrium for dynamic games in the context of Cournot duopoly in which costs are private information. The task of extending the definition of mirage equilibrium is a nontrivial issue since it is not clear on which level of finite hierarchies of beliefs the update takes place. I take a short-cut to tackle this problem and instead of working on beliefs (probability distributions) directly, I work on the support of them. Broadly speaking, players update their beliefs by eliminating the support of ”types” that do not explain the opponents’ behavior. I show that the limit of this update process converges to a Nash equilibrium of the corresponding complete information game. I also show that the rate of convergence is linear. In the third chapter, I define a new metric to measure the distance between the partitions of a given finite set. I compare the proposed metric with the ones in the literature through examples

How Common Are Common Priors?

To answer the question in the title we vary agents' beliefs against the background of a fixed knowledge space, that is, a state space with a partition for each agent. Beliefs are the posterior probabilities of agents, which we call type profiles. We then ask what is the topological size of the set of consistent type profiles, those that are derived from a common prior (or a common improper prior in the case of an infinite state space). The answer depends on what we term the tightness of the partition profile. A partition profile is tight if in some state it is common knowledge that any increase of any single agent's knowledge results in an increase in common knowledge. We show that for partition profiles which are tight the set of consistent type profiles is topologically large, while for partition profiles which are not tight this set is topologically small.