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Consistent Beliefs without Common Prior

Ziv Hellman, Miklós Pintér

TL;DR

The paper addresses whether a single common prior can underpin belief consistency when the state space is infinite. It extends Morris's epistemic–behavioral duality from finite to infinite type spaces by defining strong consistency of beliefs in terms of acceptable bets, and proves robustness to whether beliefs are modeled as sigma-additive probabilities or probability charges. The key contribution is showing that strong consistency corresponds to the absence of acceptable bets, and that in infinite spaces the notion cannot generally be captured by a single distribution or a set of distributions. The results prompt rethinking the ex-ante stage in games with incomplete or asymmetric information and advocate using consistency of beliefs rather than a real common prior.

Abstract

In a strand of the literature, it is assumed that the common prior has full support; that is, every type of every player is assigned positive probability. Morris (1991,1994) established an epistemological-behavioral duality characterisation of the common prior with full support, showing that a finite type space admits such a prior if and only if it contains no acceptable bet. This result forms the basis of the present paper. The paper makes three contributions: (1) The characterisation of Morris (1991,Morris1994) is extended to infinite type spaces. (2) The extension is robust: it does not depend on whether the infinite model applies countably additive or purely additive probabilities as beliefs. (3) The analysis implies that the notion of a real common prior-understood as a single probability distribution or a set of probability distributions-is not necessarily meaningful.

Consistent Beliefs without Common Prior

TL;DR

The paper addresses whether a single common prior can underpin belief consistency when the state space is infinite. It extends Morris's epistemic–behavioral duality from finite to infinite type spaces by defining strong consistency of beliefs in terms of acceptable bets, and proves robustness to whether beliefs are modeled as sigma-additive probabilities or probability charges. The key contribution is showing that strong consistency corresponds to the absence of acceptable bets, and that in infinite spaces the notion cannot generally be captured by a single distribution or a set of distributions. The results prompt rethinking the ex-ante stage in games with incomplete or asymmetric information and advocate using consistency of beliefs rather than a real common prior.

Abstract

In a strand of the literature, it is assumed that the common prior has full support; that is, every type of every player is assigned positive probability. Morris (1991,1994) established an epistemological-behavioral duality characterisation of the common prior with full support, showing that a finite type space admits such a prior if and only if it contains no acceptable bet. This result forms the basis of the present paper. The paper makes three contributions: (1) The characterisation of Morris (1991,Morris1994) is extended to infinite type spaces. (2) The extension is robust: it does not depend on whether the infinite model applies countably additive or purely additive probabilities as beliefs. (3) The analysis implies that the notion of a real common prior-understood as a single probability distribution or a set of probability distributions-is not necessarily meaningful.
Paper Structure (7 sections, 5 theorems, 17 equations)

This paper contains 7 sections, 5 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Outline of the Paper
  3. Main result
  4. Examples
  5. Conclusions
  6. Notions and Notations
  7. Separation

Key Result

Lemma 4

A bet $f=(f_{i})_{i \in I}$ is acceptable if and only if there exists $i^\ast \in I$ such that $(f_i)_{i \in I \setminus \{i^\ast\}}$ properly separates the sets $\mathop{\rm diag}\nolimits \Pi_{i^\ast}^{|I|-1}$ and $\mathop{\rm cone}\nolimits (\prod_{i \in I \setminus \{i^\ast\}} \Pi_i)$, where for all $i \in I$, here $^\ast$ denotes the weak* closure, $\mathop{\rm diag}\nolimits$ denotes the

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • proof
  • Definition 5
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • ...and 6 more