Impossibility and Other Alternative Voting Methods MAT 105 Spring 2008 Impossibility and Other Alternative Voting Methods

Which Method to Use? We have seen many methods, all of them flawed in some way Which method should we use? Maybe we shouldn’t use any of them, and keep searching for a better way

Arrow’s Theorem In 1951, Kenneth Arrow proved that this search will be in vain Specifically, he proved that there is no voting system that satisfies all of the following conditions: is not a dictatorship voters rank their candidates in order independence of irrelevant alternatives Pareto condition

Getting Around Arrow’s Theorem Instead of giving up hope, we might look for ways around Arrow’s Theorem Since we know we can’t have a voting method with all of those conditions being true, we might look for one condition to drop We certainly don’t want a dictatorship or a method that doesn’t satisfy the Pareto condition If we drop IIA, we could use Borda Count

Dropping Preference Orders If we drop the preference order condition, we have two options: allow voters to express less information allow voters to express more information These choices lead to approval voting and range voting, respectively

Approval Voting Voters cast a single vote for each candidate they approve of The candidate who receives more votes than any other is the winner

Properties of Approval Voting We can assume that voters are still using (mental) preference lists, but simply have a “cutoff” above which they approve of a candidate and below which they do not For example, suppose we have two voters both with preference A>B>C>D One of these voters might approve of A, B, and C, and the other might approve only of A

Approval Conditions Voters Preference Order 1 A > B > C B > A > C C > A > B Approval voting does not satisfy the Condorcet winner criterion Consider the profile shown here, where red indicates that the voter approves of the candidate A is the Condorcet winner, but B wins the approval vote

Range Voting Voters rate each candidate on a scale Examples scale from 0 to 10 1 star to 5 stars -2 to 2 thumbs The candidate with the most points wins

Properties of Range Voting Again we assume that voters are using an internal ranking to cast votes If a voter likes A more than B, they will give A at least as many points as they give B However, there are lots of possibilities for how a voter with preference A>B>C>D could fill out a ballot

Range Conditions Range voting satisfies the Pareto condition If every voter prefers A over B, then every ballot will give A at least as many points as B, so A’s total will be at least B’s total The only way this could make B win is if every single voter gave A the same score as B, and A and B tied for the win

Range Conditions In fact, range voting satisfies many important conditions (though notably not Condorcet winner) Range voting appears to be the closest to “fair” that we may be able to get