The Diagonal Agreement

Two people receive similar information about whether a bank is solvent. Standard measures — correlation, mutual information — say their signals are similar. Yet they may take opposite actions: one withdraws, the other does not. The Silicon Valley Bank run demonstrated that algorithmically homogenized information can trigger coordinated panic, but the mechanism was not the similarity of information itself — it was something more specific about the structure of shared beliefs.

Basak, Deb, and Kuvalekar (arXiv:2603.28190, March 2026) identify that structure. Standard similarity measures fail to predict strategic behavior in coordination games because they do not constrain conditional beliefs — what each agent believes the other will do, given their own signal. Two agents can have highly correlated information yet hold divergent conditional beliefs about each other's actions. Correlation measures the joint distribution; coordination depends on the conditional.

The authors introduce Concentration Along the Diagonal (CAD), a stochastic ordering based on conditional beliefs. In binary-action coordination games, greater CAD-similarity is both necessary and sufficient for strategic similarity — for agents to choose identical strategies. The diagonal in question is the set of states where both agents receive the same signal. When the joint distribution of signals concentrates along this diagonal, each agent's conditional belief about the other's signal is tight, and their strategic responses align. When the distribution spreads off-diagonal — even if it maintains high correlation — conditional beliefs diverge, and so do actions.

The distinction matters because algorithmic content targeting increases diagonal concentration specifically: it pushes everyone toward the same information at the same time. This is not merely high correlation (which could involve offsetting private signals). It is diagonal concentration — the geometric property that guarantees strategic alignment. The algorithm does not just make information similar; it makes conditional beliefs about others' information tight, which is the property that triggers coordinated action.

The structural observation: the right similarity measure for strategic interaction is not a property of the marginal distributions or even their correlation — it is a geometric property of the joint belief structure. The question is not "how similar is our information?" but "how concentrated is our shared belief along the set of agreement?"