chalkdust A voting system is an algorithm that takes the voters’ preferences and outputs a relation between candidates, which we denote by “≻E ”. So c1 ≻E c2 would mean that the voting system rated c1 as preferable to c2 . The relation ≻E is also transitive. Formally, if T(C) denotes the set of total orderings of C then a voting system is a function F : T(C)n → T(C), where n is the number of voters. In the case of two candidates, May’s theorem (1952) provides some good news. Consider a voting system that satisfies the following sensible conditions: • Each voter is given the same weight. (For any permutation σ of the voters, F(E) = F(σ(E)).) • If we relabel the candidates then the result must be similarly relabelled. (For any permutation τ of the m candidates, if C is permuted to become τ (C) then F(E) becomes τ (F(E)).) • If c1 ≻E c2 or c1 ≈E c2 (the laer representing no preference between c1 and c2 ) and c1 increases in popularity then c1 must win. (If a voter vi changes their ranking from c2 ≻i c1 to c1 ≈i c2 or c1 ≻i c2 , or from c1 ≈i c2 to c1 ≻i c2 , then c1 ≻E c2 ). May’s theorem tells us that the only voting system satisfying these properties is plurality, where each voter casts a vote for their favourite outcome and whichever outcome is more popular is chosen. So in the case where there are only two options, there is a procedure that accurately reflects the wishes of the electorate. If there are more than two alternatives then Arrow’s theorem (1950) provides a negative counterpart to May’s theorem. Consider the following criteria for a voting system: 1. If every voter prefers candidate x to candidate y then x must be ranked higher than y by the voting system (∀E ∀x,y (∀i x ≻i y) =⇒ x ≻E y). If every voter is agreed on the ranking of two candidates, it is essential that the voting system reflects this. 2. Suppose that x and y are candidates and that x ≻E y. If some of the voters change their preferences for the other candidates (eg candidate z becomes more popular), but each voter preserves their preference between x and y, then x ≻E y must still hold. Consider a group that wants to rank candidates for a job position. They evaluate all the candidates and agree that they prefer candidate x to y. If they then receive new information about z (but no new information about x and y), it seems strange that they should reconsider their group preference x ≻E y. 3. There is no dictator, defined as a single voter vi who can always determine the election (∀E ∀x,y @vi (x ≻i y =⇒ x ≻E y)). This is an essential property for a voting system to have! Perhaps surprisingly, if there are more than two candiNo voting system exists that dates then Arrow’s theorem shows that no voting system satisfies the three criteria. exists that satisfies all three criteria. Proofs of this theorem usually use contradiction, assuming that the system satisfies the first two conditions and showing that it cannot satisfy the third. Most systems that are used in the world satisfy conditions 1 and 3 but do not satisfy condition 2. Condition 2 is known as independence of irrelevant alternatives. chalkdustmagazine.com

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