Cosine Similarity Measures Agreement in Signal Interpretation

Disagreement in economic models often comes from different priors or different payoffs. This paper studies a narrower but common case: agents share the same states, actions, utilities, and prior, yet read the same signal differently because they assign different likelihoods to signals conditional on states. The question is not whether one agent has more information in the Blackwell sense. It is whether two agents’ interpretations are more compatible for joint decision-making.

The authors propose a behavioral answer. A pair of subjective signal models is more agreeable than another pair if, across decision problems, it supports a larger set of signal-contingent plans that both agents weakly prefer ex ante to a common reservation payoff. Agreement is therefore defined through implementable cooperation, not through a distance chosen in advance.

Core Contribution

The main contribution is an order over pairs of subjective models that separates interpretive agreement from information quality. The paper shows that this behavioral comparison is equivalent to a geometric inclusion condition: each model in the more agreeable pair must be expressible as a convex combination of the two models in the less agreeable pair. In plain terms, the more agreeable pair lies closer together in the space of interpretations because neither agent’s model sits outside the span created by the less agreeable pair.

That representation matters because it removes two possible sources of arbitrariness. First, the order is prior independent: although the decision problems have a common prior, the comparison of interpretive agreement does not depend on which prior is used. Second, the order is not a disguised version of Blackwell informativeness. The paper proves independence from Blackwell dominance, so two models can be more mutually agreeable without being more informative, and vice versa.

Technical Approach

The authors start from finite decision problems with a common reservation payoff. A signal-contingent protocol generates a surplus vector over states and signals. For each pair of subjective models, the plans acceptable to both agents form a cone: the intersection of the two agents’ ex-ante participation half-spaces. Comparing agreement becomes comparing these cones by inclusion.

The representation theorem links that cone inclusion to convex mixtures of likelihood models. If pair $(m, m')$ is less agreeable than pair $(\hat m, \hat m')$, then each of $\hat m$ and $\hat m'$ must lie on the line segment between $m$ and $m'$. This gives the order a concrete interpretation: disagreement shrinks when the two subjective models move toward each other within the same affine direction.

The scalar part of the paper asks which single-number completion is compatible with this order. Under rotation invariance, the answer is cosine similarity. The appendix proof shows that any rotation-invariant Bregman divergence on the sphere with dimension $d \geq 3$ must take the form

For nonzero vectors, this implies a dependence on $1-\cos(a,b)$. That result is why the paper favors quadratic geometry over KL-type Bregman divergences for this agreement concept.

Results and Analysis

Because this is a theory paper, the evidence is a chain of definitions, representation theorems, and applications rather than simulations or field data. The core result is strong within its model: the behavioral preorder, the convex-combination representation, and the cosine-similarity scalar completion line up cleanly. The paper also applies the order to speculative trade wedges, normalized ex-ante Pareto frontiers, and single-model rationalizations. In each case, greater agreement has a sharper behavioral implication: it reduces the wedge generated by different interpretations, expands the normalized feasible frontier, and makes it easier to rationalize behavior with one shared model.

The most useful feature is that the measure is not just another divergence between likelihood matrices. It is anchored in what two agents can jointly agree to do before signals are realized. That makes it well suited for economic settings where the same evidence is observed but interpreted differently: trading, persuasion, organizational forecasts, or disagreement over diagnostic signals.

The limits are also clear. The main comparison relies on finite decision problems and a fixed reservation payoff. Appendix B.2 shows that if the reservation payoff must instead come from an action that is prior-optimal under the common prior, the full-space cone inclusion remains sufficient but no longer necessary; the exact comparison has to be restricted to a subcone $K_p$. Appendix B.1 also clarifies that the main model does not require an objective signal-generating process. When one exists, the authors can impose introspection-proofness so agents match objective signal frequencies, but that is a refinement rather than the baseline. The result is a precise theory of interpretive agreement, not an empirical claim about how people form interpretations.