Cardinal Bayesian allocation mechanisms without transfers

In allocating goods with no use of monetary transfers, random allocation mechanisms can be designed in order to elicit information on preference intensities. I study the nontransfer allocation of two exante identical objects under Bayesian incentive compatibility, with symmetric agents and independent private valuations. I find the exante utilitarianoptimal mechanism, in which the probability of receiving a specified object is used as "numeraire" to purchase probability units of the other object. I characterize this mechanism as an appropriate combination of lotteries, auctions and insurance. The latter element ensures that efficient auctions are feasible. If the problem is constrained to guarantee exactly one object per agent, then the optimal mechanism uses no information other than the agents' ordinal preferences.