The Informational Basis of Social Choice under Model Uncertainty: Reframing Arrow's Impossibility in the Presence of Strategic Obfuscation

Arrow’s theorem is traditionally read as a logical incompatibility among axioms given anordinal preference profile. This paper targets an upstream failure mode: when agents areuncertain not only about states but about the operative model mapping actions to consequences, expressed preferences become endogenous and can be strategically shaped. Crucially, we endogenize the frame set: we prove that even in a high-dimensional continuousbelief space, the optimal adversarial control policy is ’bang-bang, ’ collapsing reality into twodiscrete boundary solutions—the frames—which subsequently drive the Arrow impossibility. The core structural move is to represent any restriction on what a procedure may usewhen ranking a pair (x, y) as an admissible information constraint, formalized as a sub-σ-fieldGxy ⊆ σ (h, Re). We prove a sharp Bayes-risk floor: among all Gxy -admissible rules, the minimumachievable misranking probability on (x, y) equals the conditional Bayes error induced by therestricted posterior. Epistemic Independence of Irrelevant Alternatives is recovered as the specialcase Gxy = σ (h, Re|x, y). Upstream identification upgrade: We provide an identification theorem showing thatspurious unanimity arises generically in a broad class of large-electorate obfuscation games: underpayoff reversal, vanishing pivotality, and bounded residual evidence (bounded log-likelihoodratio), there exists an equilibrium with unanimous expressed support for x on (x, y) underboth endogenous frames while the informed unanimous comparison reverses across frames. Weendogenize obfuscation choice with costs and show equilibrium selects the minimal unanimity-guaranteeing obfuscation level. We then prove that maximal obfuscation is never optimal underdiminishing returns at the boundary, so equilibrium generically retains residual informativeness. Under a simple threshold-interiority condition, this residual informativeness implies endogenouswitness alternatives and hence diagnostic wedges. Canonical impossibility upgrade: We state and prove an Arrow-style impossibility underframe uncertainty: epistemic IIA, expressed Pareto-on-pair, and non-dictatorship imply a strictlypositive misranking floor under spurious unanimity. We then define a formal two-order domainof procedures, with independence as strict σ-field inclusion and robustness as strict Bayes-riskimprovement, and exhibit an explicit two-step mixed-order cycle. The cycle implies scalarevaluation is impossible: no scalar index can be strictly monotone in both independence androbustness. Disciplined evaluation must be vector-valued or must explicitly choose a weightingvia loss/cost parameters. Escape routes: We provide likelihood-ratio triangulation using witness evidence, with aChernoff error exponent, and audit-augmented procedures with an optimal audit trigger and anequilibrium discipline result showing rational obfuscation stops at the audit boundary.