chalkdust Consistency is an important property for protecting against gerrymandering of electoral districts: if two areas both support the same candidate using the alternative vote, it is possible to merge the two areas and have the newly created area elect a diﬀerent candidate. A voting system satisfies the monotonicity criterion if the following holds: given an electorate E, if some voters improve their ranking of x without changing the rankings of Steven Nass, licensed under Creative Commons CC BY-SA 4.0 other candidates to create a new elecHow to steal an election by gerrymandering. torate E ′ , then for any y, x ≻E y =⇒ x ≻E ′ y, ie candidate x cannot be harmed if more voters favour x than before. As with consistency, plurality satisfies the monotonicity criterion but the alternative vote does not. Consider the following example from Johnson (2005), Voter preference Voters Voters′ shown in Table 3. Using the original electorate, z is elimiz ≻x ≻y ≻w 5 5 nated in the first round and z’s 5 votes pass to x, so x has y ≻ z ≻ x ≻ w 4 4 11 votes. Then y is eliminated and all 6 of y’s votes go y ≻x ≻z ≻w 2 0 to x because z is eliminated. So x now has 17 votes and x ≻y ≻z ≻w 6 8 w has 9, so x wins a substantial victory with an electoral w ≻z ≻x ≻y 9 9 preference of x ≻E w ≻E y ≻E z. Suppose the two voters with ranking y ≻ x ≻ z ≻ w now change their ranking to x ≻ y ≻ z ≻ w, making x even more popular than before. Table 3: An electorate showing that the alternative vote is not monotonic. Then under the new electorate (denoted Voters′ in Table 3), y is eliminated in the first round and y’s 4 votes pass to z, so z has 9 votes. Then x is eliminated and all 8 of x’s votes go to z. So the final result is that w has 9 votes and z has 17, and the new electorate preference is z ≻E ′ w ≻E ′ x ≻E ′ y. As a result of x’s increased popularity, x has been demoted from first choice to third choice! A related but diﬀerent property to monotonicity is the participation property. Under the alternative vote, it is possible for a voter to harm their preferred candidate by voting, whilst this is not possible with plurality.
Conclusion This is only a very brief introduction to this area, and there is much more to be said. What I find engaging is that mathematical devices and constructions such as theorems and counterexamples can be applied to voting systems in order to compare them. We can say with complete confidence that no preferential voting system will ever be devised that satisfies Arrow’s three criteria, and 15
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