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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 $${A\succ B\succ C, C\succ B\succ A}$$ type.

Systematic analysis of multiple voting rules