On strategy-proofness and single peakedness

This paper investigates one of the possible weakening of the (too demanding) assumptions of the Gibbard-Satterthwaite theorem. Namely we deal with a class of voting schemes where at the same time the domain of possible preference preordering of any agent is limited to single-peaked preferences, and the message that this agent sends to the central authority is simply its ‘peak’ — his best preferred alternative. In this context we have shown that strategic considerations justify the central role given to the Condorcet procedure which amounts to elect the ‘median’ peak: namely all strategy-proof anonymous and efficient voting schemes can be derived from the Condorcet procedure by simply adding some fixed ballots to the agent's ballots (with the only restriction that the number of fixed ballots is strictly less than the number of agents). Therefore, as long as the alternatives can be ordered along the real line with the preferences of the agents being single-peaked, it makes little sense to object against the Condorcet procedure, or one of its variants that we display in our characterization theorem. An obvious topic for further research would be to investigate reasonable restrictions of the domain of admissible preferences such that a characterization of strategy-proof voting schemes can be found. The single-peaked context is obviously the simplest one, allowing very complete characterizations. When we go on on to the two-dimensional state of alternatives the concept of single peakedness itself is not directly extended and a generalization of our one-dimensional results seems to us to be a difficult but motivating goal. Copyright Martinus Nijhoff Publishers bv 1980