social choice midterm

Plurality Method

method that selects a winner based on whichever
candidate received the highest number of (first choice) votes.

majority candidate

one who is placed first by more than half the voters

has plurality, would be a Condorcet

Spoilers (overall)

Candidates that wouldn’t win and affect the vote for other candidates

(or: candidates excluded from a runoff election)

Runoff election

Secondary election off of a plurality election with two leading candidates

Elimination method

(4+ candidates, works for multiple rounds, one candidate eliminated each round, lowest vote-getter, ends up with last two or a majority) 

Reduced preference schedule

• Summarizes/combines preferences by number of ballots with same rankings, helps to find answers to different questions about the results

• Make scompact all the information you have, compress voters who vote the exact same way

• Works with the Principle of One Person, One Vote

Principle of one person, one vote

If the outcome of an election method only depends on the reduced preference schedule (RPS), then we say it satisfies the principle of one person one vote

Borda Score method

• Uses Borda Points (i.e. 4 possible candidates, 4 points per preference ballot if ranked first on the ballot, 3 points if ranked second, etc.) 

  • Relies on reduced pref. schedules, averages relative rank among all voters

  • Satisfies OPOV principle

  • However many candidates there are = # of points for first place

• Candidate with greatest Borda score wins

pairwise comparison graph

• Draw line from winner to loser (arrow away means that candidate won) 

• Represents societal preferences, candidates score points based on the number of candidates they would beat head to head 

• Greatest possible number of head to head battles a candidate can win in the PWC is 1 less than the number of candidates

Condorcet candidate

• a candidate that would win a head-to-head
  competition against any/every other candidate

(comparison between the only themselves and one other candidate for highest number of votes)

• majority candidate is the Condorcet, you can have CC without plurality

there can only be one CC!

majority fair criterion

IIf a winner selection method would always choose only the majority winner (if there is one), then such a method is considered majority fair

• Majority is not the same as plurality – majority is if you have more than 50% of the vote

Condorcet fair implies majority fair (not reversible)

majority fair methods

• plurality — if a candidate gets a majority of the votes, they are the plurality winner

• elimination

• runoff — both need majority, left with majority winner

• pairwise comparison — majority candidate would win every head to head

NOT BORDA SCORE

Condorcet-fair criterion

A winner selection method satisfies the Condorcet criterion if it makes any Condorcet candidate—if there is one—the sole winner.

• more strict than majority criterion (CF is MF, not vice versa)

Condorcet fair methods

• pairwise comparison — shows the Condorcet candidate, would win every head to head

NOT PLURALITY, RUNOFF, ELIMINATION (majority but not condorcet fair), BORDA

Circular Collective Preferences

No Condorcet — when there is no CC, societal preferences work circularly and no candidate wins universally over the others

the unanimity criterion

A winner selection method satisfies the unanimity criterion if it guarantees that candidate Y is not among the winners of the election if there is a candidate X who is preferred to Y by every voter.

if everyone hates one candidate and they are always ranked below another candidate, the mega loser should never win

Unanimity fair methods

• plurality

• runoff

• elimination

• Borda Score

• Pairwise comparison

NOT SEQUENTIAL COMPARISON

Sequential comparison

• does not satisfy unanimity criterion, but is Condorcet-fair

  Like a bracket – two candidates go head to head, one gets eliminated and the other moves on to the next round 

• Very unstable method – could get vastly different outcomes depending on which pair you start with 

The Independence of Candidate Name principle

• For sequential comparison – the first two choices change the outcome, which we do not want 

• An election that is not affected by order satisfies the ICNP (independent of candidate name) – created to fix the variability of the sequential comparison 

If Candidate X joins the ballot right before the election, Candidate X is not a spoiler if and only if:

1. Candidate X wins the election

   1. (WA = X)

2. Candidate X joins the set of winners, in a system with multiple winners (this winners’ set is otherwise the same as before)

   1. (WA = WB U X)

3. The set of winners is unaffected by Candidate X

   1. (WB = WA)

No spoiler criterion

A method satisfies this criterion if:

In the case of a new candidate being added to the ballot, it guarantees the only possibilities are:

1. the new candidate wins (avoids losing spoilers)

2. the new candidate joins the set of winners (avoids winning spoilers)

3. or the new candidate does not change the set of winners from before (avoids losing spoilers)