Voting, from a mathematical perspective, is the process of aggregating the preferences of individuals in a way that attempts to describe the preferences of a whole group.
Arrow’s Impossibility Theorem
Types of Voting Systems
Apportionment Paradox(Most Common)
The Condorcet's Paradox Suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows (candidates being listed left-to-right for each voter in decreasing order of preference):VoterFirst preferenceSecond preferenceThird preferenceVoter 1ABCVoter 2BCAVoter 3CABIf C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion.
Arrow's Impossibility Theorem Three desirable features for a voting system are as follows:Unanimity: If everyone prefers A to B, A should win.No Dictators: There should not be anyone whose individual preferences always determine who wins.Independence of Irrelevant Alternatives (IIA): Adding extra options should not make existing relations change. That is, if A≥B, adding option C should not make B ≥A.
GerrymanderingIn the same way that a nation is divided into states, states are divided into districts, each of which votes on a particular candidate. Candidates are elected by counting the number of districts they win, under the assumption that winning the most districts is the same as winning the overall vote. However, it is possible for one candidate to win in most districts while still losing the popular vote. Gerrymandering refers to the practice of purposefully redrawing district lines so that one candidate is more likely to win.Also present in voting is Simpson's paradox in statistics, which says it is possible for variables to be positively correlated in subgroups despite being negatively correlated overall.