Math 130A Bing exam 1

What are the five elements of an election?

• Candidates

• Voters

• Ballots

• Outcomes

• Voting methods

Plurality method

the person with more first place votes than any other candidate wins

Borda count

in a race with n candidates, each last place vote that a person receives is worth one point, each second to last place vote is worth two points, and so on until you award each first place vote with n points. The points are tallied for each candidate and the person with the most points wins. In a modified Borda count, some candidates may not be ranked at all, which
counts for zero points

Plurality with elimination (PWE)

the first place votes are tallied. If any candidate has a majority, they are declared the winner. If not, the person with the fewest first place votes is eliminated from the race and their votes are transferred to the next highest ranked option. The first place votes are retallied and the process repeats until someone receives a majority of votes.

Pairwise comparison

the results of every head-to-head matchup are calculated. Any head-to-head matchup that you win is worth one point. Any matchup that you tie is worth 0.5 points. Any matchup that you lose is worth no points. The points from all matchups are tallied and the candidate with the most points is declared the winner.

Condorcet Loser

a candidate who loses in every head-to-head matchup they are put in

Condorcet Winner

a candidate who wins in every head-to-head matchup they are put in.

What is insincere voting?

Insincere voting is when a voter does not vote for their true preferences, but instead votes
strategically, often for a less desirable option, in an effort to change the outcome of an election.
ex. 1st choice candidate is not likely to win so they change to a more likely candidate who might be their 2nd favorite

Choose one voting method we’ve discussed. In complete sentences, describe how this election method might be impacted by insincere voting strategies

Plurality where two candidates from the same party are running, voters may decide as a party to all vote for Candidate A instead of B so they don’t split the vote and let the other party win. This wouldn’t be an issue in a Borda count where people could rank A and B as their first and second choice and make clear their preference that the other party not be elected.

Majority criterion

if any candidate receives a majority (more than 50%) of first place votes that candidate should be the winner.

• This does not mean that to win you must have the majority vote, just that if they do have the majority then they win.

• Satisfies plurality

Condorcet criterion

if a candidate wins in every head-to-head comparison, then that candidate should be the winner of the election.

• It could be that a specific plurality example could satisfy the Condorcet criterion, but if any example fails then that voting theory fails the criterion

Monotonicity criterion

if candidate X is the winner of an election, then X should still be the winner if a voter changes their vote to rank X higher in their preferences

• Gaining votes should not hurt the winner Plurality only looks at voters first choice votes; if some voters rank A higher than before, then the number of first choice votes for A will never decrease

Independence of irrelevant alternatives (IIA)

the election results should not change if a losing candidate drops out of the race

• To check if IIA is violated, we need to remove a losing candidate and re-tally the votes to see if the winner changed.

Describe Arrow’s Theorem

Arrow’s Theorem says that no voting method will satisfy all four fairness criteria at once; in other words, it’s impossible to design a completely fair voting system.

What are the four legal guidelines for drawing congressional districts?

• Districts need to have roughly equal population
  ● Maps cannot discriminate on the basis of race
  ● Districts need to be connected/contiguous
  ● Districts should be compact

Explain what it means for a districting map to be proportional and what the U.S. Supreme Court has said about using proportionality as an indicator of gerrymandering

In a proportional map, the percentage of votes that you win (your vote share) should be roughly the same as the percentage of seats that you win (your seat share).

• The Supreme Court ruled that there is no constitutional right to proportional maps, so this cannot be used as evidence of gerrymandering

Describe the two strategies for gerrymandering another party out of power

• You can separate your opponents into many districts so they’ll never hold a majority in any of them. This is called cracking.

• You can also jam your opponents into a single district so they’ll win that one by a large margin and lose every other district. This is called packing

Explain what votes are considered “wasted” in the efficiency gap

Consider district di won by Party A. Any votes cast for the loser count as wasted - so every vote for Party B. If we find the total number of people who voted in di and cut that in half, we can subtract that number from the Party A votes to find the wasted votes cast for the winner (those over the 50% mark)

Explain the difference between a majority winner and a plurality winner.

• A majority winner is a candidate who wins more than 50% of all the first choice votes cast.

• A plurality winner is a candidate who wins more first choice votes than any other candidate.

Is the plurality winner always a majority winner?

A plurality winner is not always a majority winner - for example, in a 3 candidate race with
Candidate A getting 40% of the vote, Candidate B getting 30% and Candidate C getting 10%, A
will be a plurality winner but nobody has earned more than 50% so there is no majority winner.

Is the majority winner always a plurality winner?

A majority winner will always be the plurality winner since winning more than 50% of the votes means that you’ve won more than any other candidate. This is why the plurality method satisfies the majority fairness criterion

How many voters voted in this election? Did any candidate receive a majority?

74 voters in total. No - a majority would be 38 votes.

Use the plurality method to determine the winner of the election.

A wins

Use the Borda count to determine the winner of the election.

A wins
A - 238 points
B - 200 points
C - 156 points
D - 146 points

Use pairwise comparison to determine the winner of the election.

A wins

What percent of votes did Republicans win statewide? What about Democrats? What is the
vote share for this election?

R: 344 votes out of 700 --> 49.14%
D: 356 votes out of 700 --> 50.86%
vote share τ = (356-344)/700 = 0.02

What percent of seats did Republicans win statewide? What about Democrats? What is the
seat share for this election?

R :3/7 --> 43%
D: 4/7 --> 57%

seat share σ=(4-3)/7 = 0.14

Calculate the total number of wasted Republican votes in this state.

District 1: 2
District 2: 43
District 3: 31
District 4: 26
District 5: 36
District 6: 26
District 7: 30
Total: WR = 194

Calculate the total number of wasted Democrat votes in this state

District 1: 48
District 2: 7
District 3:19
District 4: 24
District 5: 14
District 6: 24
District 7: 20
Total: WD = 156

What is the efficiency gap in this election?

EG = (WR - WD)/T = (194 - 156) /700 = 0.05