A cardinal approach to straightforward probabilistic mechanisms

In every probabilistic mechanism, society selects an alternative, through a random device, out of a subset of indifferent alternatives. Consequently, in this context individuals face uncertainty and value the different lotteries on alternatives by their expected utility, so that they make use of a Von Neumann-Morgenstern cardinal utility function. Surprisingly, the social choice approach to probabilistic mechanisms assumes the use of ballots which preclude the complete expression of behaviour towards risk: individuals can only announce their ordinal preferences, or an approximation of their cardinal preferences, since in any case only a finite number of representations of preferences is available. This paper attempts to study voting systems in which individuals can express the cardinality of their preferences by assigning weights to the alternatives. It is shown that by voting with ballots which reflect weighting a new class of straightforward probabilistic mechanisms is defined, and that this class strictly contains the class of probabilistic straightforward mechanism designed by Gibbard.