Consistent Beliefs without Common Prior

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.

💡 Research Summary

The paper revisits the notion of a common prior in games of incomplete information and shows that the traditional requirement of a full‑support prior is unnecessary once the state space is infinite. Building on Morris’s (1991, 1994) duality— which states that a finite type space admits a full‑support common prior iff there is no “acceptable bet” — the authors extend this result to infinite type spaces. They first formalize a type space in the Heifetz‑Samet style, defining each player’s belief function t_i and the associated set of priors Π_i. A bet is a finite collection of bounded, measurable payoff functions that sum to zero everywhere; an “acceptable bet” is one that yields non‑negative expected payoff for every player at every state and a strictly positive payoff for at least one player at some state.

Lemma 4 shows that the existence of an acceptable bet is equivalent to the existence of a linear functional that properly separates two convex sets: the diagonal of the product of the Π_i’s (excluding one player i*) and the cone generated by the remaining Π_i’s. This geometric reformulation works both for σ‑additive probability measures and for finitely additive “charges,” demonstrating robustness across belief models.

The authors then introduce “strong consistency of beliefs” (Definition 5) as the infinite‑state‑space analogue of a full‑support common prior. In finite spaces, strong consistency coincides with the non‑emptiness of the intersection of the relative interiors of the Π_i’s (Lemma 6). In infinite spaces, however, this simple interior‑intersection condition fails. The main result, Theorem 7, establishes a dichotomy: either the players’ beliefs are strongly consistent, or there exists an acceptable bet in the type space. The proof proceeds by showing that strong consistency precludes proper separation (hence no acceptable bet) and that the absence of an acceptable bet forces the inclusion condition defining strong consistency.

Section 3 presents two illustrative examples where two closed convex sets of priors have identical intersections yet differ in proper separability. These examples demonstrate that, in infinite‑dimensional settings, a single probability distribution—or even a set of distributions lying in the intersection—cannot serve as a witness to proper separation. Consequently, the usual “common prior” representation becomes inadequate.

The paper concludes that the concept of a common prior, whether as a single distribution or a family, may be meaningless in infinite models. Instead, the authors advocate using “strong consistency of beliefs” as the appropriate epistemic notion. This reframing aligns with earlier debates by Aumann (1998) and Gul (1998) and underscores that belief consistency can be captured without invoking an ex‑ante common prior, especially when the state space is infinite. The robustness of the results across both σ‑additive and additive belief frameworks further strengthens the claim that strong consistency is a more fundamental and stable property than the traditional common‑prior assumption.