Systematic analysis of multiple voting rules

For a class of voting rules, which includes Approval and Cumulative Voting, it is shown how to find and analyze all possible outcomes that arise with a specified profile, and, conversely, how to start with a potential region and determine whether there exist supporting profiles. The geometry of these regions is determined by the “Reversal symmetry” portion of a profile; i.e., components of the Asucc Bsucc C, Csucc Bsucc A{Asucc Bsucc C, Csucc Bsucc A} type.