Ordinal Bayesian Incentive Compatibility at the Boundary of Genericity

Under independent generic beliefs, unanimity and ordinal Bayesian incentive compatibility (OBIC) imply dictatorship. We study where this conclusion first fails as genericity is relaxed, in the minimal nontrivial environment of two voters, three alternatives, and the unrestricted domain of strict preferences. We obtain an exact classification of the exceptional balance relations and the mechanisms they support, without imposing any further symmetry on the ranking distribution. Among the 301 reduced genericity relations, only pairwise-balance relations-those equating the probabilities that two alternatives defeat a common benchmark-can support a non-dictatorial unanimous deterministic OBIC rule when exactly one marginal satisfies a single genericity relation and the other remains generic. In this one-sided case, exactly five rules are OBIC, including three non-dictatorial unilateral agenda mechanisms. Under i.i.d. beliefs, the only additional relation type is top balance-equality of first-choice probabilities for two alternatives-which supports a unique non-dictatorial agreement-or-default rule. These two exceptional relation types operate through distinct approval and agreement channels, respectively, and both channels persist in larger lectorates.