Mathematics of Voting Systems. Tanya Leise Mathematics & Statistics Amherst College
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1 Mathematics of Voting Systems Tanya Leise Mathematics & Statistics Amherst College
2 Arrow s Impossibility Theorem 1) No special treatment of particular voters or candidates 2) Transitivity A>B and B>C implies A>C No cycles C A B 3) Monotonicity A voter changing their ballot in a way favoring cannot cause that candidate s overall ranking to go down. 4) Independence of irrelevant alternatives Overall relative ranking of two candidates depends on only their relative ranking on voter ballots
3 Why independence of irrelevant alternatives matters: 1995 Figure Skating World Championship Rankings prior to Michelle Kwan skating: 1 st place: Chen Lu (China) 2 nd place: Nicole Bobek (USA) 3 rd place: Surya Bonaly (France) Rankings after judging of Michelle Kwan: 1 st place: Chen Lu (China) 2 nd place: Surya Bonaly (France) 3 rd place: Nicole Bobek (USA) 4 th place: Michelle Kwan (USA)
4 Plurality: whoever gets the most votes wins Strengths Simple ballot to fill out Transparent results Easy to understand Monotonic Weaknesses Vote splitting Spoilers Tactical voting Negative campaigning 1860 US Presidential Election Abraham Lincoln Stephen Douglas John Breckinridge John Bell
5 Borda count Point system for field of N candidates, e.g., N-1 points for 1 st place N-2 points for 2 nd place 0 points for last place (or other point scheme, for instance, weighting 1st place more heavily) Strengths Takes into account full set of preferences Can promote compromise candidates Monotonic Weaknesses Vulnerable to strategic voting, such as burying favorite s main rivals
6 Borda count: 1999 baseball MVP elections 28 voters 14 points for 1 st place 9 points for 2 nd place 8 points for 3 rd place 7 points for 4 th place awards_1999.shtml
7 Approval voting Vote for all candidates you find acceptable May reduce vote splitting and support third parties Not as expressive as ranked methods Saari s example: 9,999 voters strongly support A, find B marginally acceptable, and strongly oppose C 1 voter strongly supports C, finds B marginally acceptable, and strongly opposes A
8 Pairwise comparison/condorcet method Winner based on head-to-head matches of all possible pairings of candidates Beatpath/CSSD takes into account margins of victory using a weighted directed graph calculation Condorcet winner: candidate who wins all head-to-head matches Condorcet winner criterion: when a Condorcet winner exists, that candidate should win the election.
9 Instant runoff voting (IRV)/ranked choice Eliminate candidate with least 1 st place votes Move up candidates and repeat until single winner left Burlington, VT 2009 mayoral race used IRV IRV winner was Kiss, followed by Wright then Montroll Montroll was Condorcet winner If Kiss had won more 1 st place votes, he would have lost IRV is not monotonic IRV doesn t satisfy Condorcet winner criterion
10 Gibbard-Satterthwaite Theorem Tactical voting: dishonest voting to improve ranking of your preferred candidate. All ranked voting systems with no special treatment of particular voters or candidates are susceptible to tactical voting.
11 Gerrymandering Incumbent (sweetheart) Ruled OK by court Racial Voting Rights Act of 1965 Partisan No clear measure Packing and cracking
12 Baker vs Carr, 1962 Supreme Court case One person, one vote Each individual is weighted equally in apportionment (doesn t matter whether legally able to vote or not) Established right of federal courts to review redistricting maps (redrawn every 10 years after census) Found Tennessee district map unconstitutional github.com/jeffreyblewis/congressional-district-boundaries Districts did not reflect movement of population to cities 2/3 of representatives elected by 1/3 of the state population
13 Cooper vs Harris: North Carolina district map Supreme Court ruled 5-3 earlier this week that Districts 1 and 12 exhibit unconstitutional racial gerrymandering District 12 elected African-American-favored candidates with 64-72% of vote New map increased packing of African-American voters map: 7 Dem to 6 Rep seats in map: 10 Rep to 3 Dem seats in 2015
14 Quantifying partisan gerrymandering Efficiency gap Stephanopoulos and McGhee Assesses wasted votes in 2-party election If a party loses the election, all of that party s votes are wasted. If a party wins the election, the votes past 50% are wasted. Sum wasted votes for each party across the districts in that state Find difference in total wasted votes between the 2 parties, divided by total # of votes 6 Red : 4 Blue 1 vs 4 wasted Efficiency gap of 30% 1 Red : 9 Blue 1 vs 4 wasted
15 Quantifying partisan gerrymandering Efficiency gap of zero doesn t imply proportional representation District Red Blue Winner Wasted votes Red 1 vs Red 1 vs Red 1 vs Blue 4 vs Blue 3 vs Blue 3 vs Blue 3 vs Blue 3 vs Blue 3 vs Blue 3 vs 2 Total vs voters in 10 districts 40 total Red voters 60 total Blue voters Red wins 3 districts Blue wins 7 districts Efficiency gap = 0 Biased toward Blue
16 Felony disenfranchisement in the US Depends on state laws Overall in US, 7.7% of black adults disenfranchised, compared to 1.8% of non-black adults. Large prison populations also used as form of gerrymandering (count as population but can t vote) States with most severe laws: Florida (21% of African-Americans disenfranchised) Kentucky (26%) Virginia (22%) Up to 40% of black men disenfranchised in these states
17 Thank you for listening!
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