Bayesian Polarization (2509.02513v1)

Abstract: We study belief polarization among Bayesian agents observing public information about a multidimensional state. Baliga et al. (2013) show that divergence in the sense of first-order stochastic dominance is impossible for one dimensional beliefs, but we find that in multidimensional settings it can occur for all marginal beliefs, even with infinitely many signals. At the same time, we extend their impossibility result: divergence in the sense of multidimensional stochastic dominance is impossible. For an intermediate stochastic order, polarization may arise only in the short-run. We provide necessary and sufficient conditions on signal structures for persistent polarization and discuss implications for polarization in actions.

• The paper demonstrates that rational Bayesian updating achieves coordinatewise polarization in multidimensional settings, contrary to the impossibility in one-dimensional cases.
• It employs stochastic orders and a geometric characterization of identified sets to distinguish one-shot and limit polarization outcomes.
• The study links belief polarization to action polarization, showing how separability in utility functions interacts with signal structures to drive divergent opinions.

Bayesian Polarization: Belief Divergence in Multidimensional Bayesian Updating

Introduction and Motivation

The paper "Bayesian Polarization" (2509.02513) rigorously investigates the phenomenon of belief polarization among Bayesian agents who observe public signals about a multidimensional state. The central question is whether rational Bayesian updating can produce polarization—defined as beliefs moving further apart and in opposite directions—when agents with different priors observe the same information. Previous work, notably Baliga, Hanany, and Klibanoff (BHK), established an impossibility result for polarization in one-dimensional state spaces under first-order stochastic dominance. This paper extends the analysis to multidimensional settings, revealing that the impossibility result does not generalize: strong forms of polarization are possible in multidimensional spaces, but only under specific stochastic orders and signal structures.

Stochastic Orders and Polarization Notions

The paper formalizes polarization using three stochastic orders:

• Coordinatewise Stochastic Dominance ($\preceq_{cw}$): Marginal beliefs diverge on each dimension separately.
• Upper Orthant Dominance ($\preceq_{uo}$): Divergence on events where multiple dimensions are simultaneously high (or low).
• Multidimensional Stochastic Dominance ($\preceq_{st}$): The strongest order, requiring divergence for all increasing functions over the state space.

The main results are summarized as follows:

Stochastic Order	One-shot Polarization	Limit Polarization
Coordinatewise	✓	✓
Upper Orthant	✓	✗
Stochastic Dominance	✗	✗

This table encapsulates the nuanced landscape: while coordinatewise polarization is possible both in the short run and asymptotically, stronger forms are precluded by Bayesian updating.

Model and Formal Definitions

The model considers two Bayesian agents, $L$ and $H$, with full-support priors over a finite multidimensional state space $\Theta = \Theta_1 \times \cdots \times \Theta_d$, $d \geq 2$. Agents observe a public signal $X$ with known likelihoods $\ell(\theta) = \Pr[X = x | \theta]$. Upon observing $x$, agents update their beliefs via Bayes' rule.

One-shot polarization occurs if posteriors satisfy $Q^L \prec P^L \prec P^H \prec Q^H$ under a given stochastic order. Limit polarization is defined analogously for posteriors after an infinite sequence of i.i.d. signals, with the limiting posterior determined by the identified set of states observationally equivalent to the true state.

Possibility and Impossibility Results

Coordinatewise Polarization

The paper proves that both one-shot and limit coordinatewise polarization are possible. For example, in a $2 \times 2$ state space, if the identified set is the diagonal $\{(x_1, y_1), (x_2, y_2)\}$, agents' marginal beliefs on each dimension can diverge in the sense of first-order stochastic dominance. This result generalizes to higher dimensions and larger state spaces.

Upper Orthant and Stochastic Dominance

For upper orthant dominance, one-shot polarization is possible but limit polarization is impossible. For full multidimensional stochastic dominance, polarization is impossible in both cases. The impossibility proofs rely on the structure of Bayesian updating: after observing a partitional signal (i.e., one that reveals the identified set), agents' posteriors place more mass on the identified set, and zero elsewhere, precluding divergence on singleton events required for stronger orders.

Channels of Polarization

The analysis distinguishes two channels:

1. State-level polarization: Beliefs can diverge on all but two states (minimum and maximum likelihood).
2. Event-level polarization: Even if beliefs move in the same direction on every state, they may diverge on events due to differences in prior weights.

In the long run, only event-level polarization remains possible for coordinatewise order.

Characterization of Polarizing Signals

A major contribution is the geometric characterization of identified sets that can produce limit coordinatewise polarization in two-dimensional state spaces. The necessary and sufficient conditions are:

1. Spanning: Both the set and its complement must attain minimal and maximal values on every dimension.
2. Balanced: The set cannot be biased upward or downward (i.e., cannot contain all states strictly above or below a point while excluding all weakly below or above).
3. Non-compensatory: The set cannot be such that high values on one dimension are always paired with low values on the other.

In $2 \times 2$ spaces, only the diagonal set satisfies these conditions, corresponding to signals that reveal perfect correlation between dimensions. This result links the mathematical structure of polarization to empirical observations: rising ideological consistency in political polarization is mirrored by the requirement that polarizing signals must tie dimensions together.

Implications for Action Polarization

The paper extends the analysis to actions, considering agents with utility functions over candidates and multidimensional states. If utility is additively separable and increasing in each dimension, coordinatewise belief polarization translates directly into action polarization. However, if utility functions exhibit strong complementarities, impossibility results for belief polarization under stronger orders imply that action polarization is precluded for some utility functions.

Utility Class (Generating Functions)	One-shot Action Polarization	Limit Action Polarization
Sums of univariate increasing	✓	✓
Products of nonnegative univariate	✓	✗
Increasing functions	✗	✗

Thus, separability in preferences is a key condition for robust action polarization.

Probability and Magnitude of Polarization

The paper demonstrates a tradeoff between the probability and magnitude of polarization. In symmetric examples, as the prior probability of the identified set increases, the magnitude of polarization decreases, and vice versa. This quantifies the conditions under which polarization is both likely and substantial.

Theoretical and Practical Implications

The results challenge the view that polarization is evidence of irrationality or bias. In multidimensional settings, significant polarization is fully consistent with Bayesian rationality, provided the signal structure and stochastic order are appropriate. The geometric characterization of polarizing signals provides a framework for understanding how public information can drive polarization, especially when signals reveal correlations among dimensions.

Empirically, the findings suggest that observed polarization on separate issues (e.g., climate, trade, culture) is compatible with rational updating, while polarization on joint events (combinations of issues) would be harder to reconcile with Bayesian models. The link to ideological consistency and the role of balanced signals (as in media coverage) offers new perspectives on the mechanisms underlying polarization.

Conclusion

"Bayesian Polarization" rigorously delineates the conditions under which belief polarization is possible among Bayesian agents in multidimensional settings. The paper establishes that coordinatewise polarization is generically possible, even asymptotically, while stronger forms are ruled out. The geometric characterization of polarizing signals and the mapping to action polarization provide a comprehensive framework for understanding polarization in rational learning models. These results have significant implications for the interpretation of polarization in empirical settings and for the design of information structures in political and economic environments. Future research may extend these results to continuous state spaces, richer signal structures, and dynamic models of polarization.