The Informational Basis of Social Choice under Model Uncertainty: The Transparency Paradox and the Impossibility of Robustness under Public Witnessing

Classical impossibility theorems in social choice are axiomatic: they derive infeasibility fromlogical incompatibilities among transitivity, Pareto, IIA, and non-dictatorship. This paperisolates a different obstruction: an informational incompatibility between Transparency andRobustness under model uncertainty with strategic frame control.We formalize institutional admissibility constraints as σ-fields Gxy governing pairwise decisions.Under spurious unanimity—expressed unanimity on (x, y) while the informed comparison flipsacross feasible frames—any expressed-Pareto rule misranks with probability at least the posteriormass of frames whose informed ordering disagrees with the forced output. Witness evidenceexcluded by Gxy appears to offer an escape: MAP triangulation of the latent frame achieves anexponential error decay controlled by the minimal Chernoff information Cmin.The main result reverses this conclusion once transparency is imposed on witnessing. Ina public witnessing regime, the strategic Sender observes the witness signal (or equivalentlyknows the test statistic voters will apply) before the downstream action. This expanded timingallows the Sender to solve a belief-space analogue of drift cancellation from Fathi [6]: the Senderselects frame mixtures conditional on the witness so that the voters’ posterior does not move,μh(· | E) = μh(·) a.s. Consequently the induced witness channel becomes statistically degenerate:for every pair of frames (f, g), the pairwise Chernoff information collapses to zero, C(f, g) = 0,and the minimax error exponent collapses to zero. Hence Transparency destroys Robustness:“sunshine” converts witnesses from measurements of truth into targets of stabilization.This extends the Bateson Game logic of dynamically stabilized certainty traps to socialchoice, and yields a Goodhartian principle for institutions: once the witness metric is known, itceases to be a measure of truth and becomes a control variable for the Sender.