Manipulation of social choice functions under incomplete information

We propose a general framework to analyse how vulnerable to manipulation social choice functions are when a limited amount of information about individual preferences is available. We focus then on two properties called WMG-strategy-proofness and group WMG-strategy-proofness. A social choice function is [group] WMG-strategy-proof if it cannot be manipulated by an individual [a coalition of individuals] whose information about the preferences of the others is limited to the knowledge, for every pair of alternatives, of the number of people preferring the first alternative to the second one. We prove that there are Pareto optimal, WMG-strategy-proof and non-dictatorial social choice functions, that every Pareto optimal and WMG-strategy-proof social choice function is not anonymous if the alternatives are at least three, and that every Pareto optimal and group WMG-strategy-proof social choice function is dictatorial if the alternatives are three.