This paper studies communication mechanisms for two players with symmetric single-peaked preferences. The peaks are privately known and drawn from a uniform distribution before the agents take a collective decision. While for the general setting Moulin [Moulin, H., 1980. On Strategy-Proofness and Single-Peakedness. Public Choice 35, 437-455.] characterized all strategy-proof mechanisms, much remains to be known in the Bayesian framework. The example consists of a dichotomic mechanism, that yields a strictly higher ex-ante expected utility than the best “min–max” rule. The properties of the mechanism are analyzed, then limits and possible directions for generalization are discussed.