FULL NOTEBOOK FOR MATH130: MATHEMATICS IN ACTION. COURSE OFFERED AT BINGHAMTON UNIVERSITY FOR FALL 2023. INSTRUCTOR JAGDEEP SINGH. FRESHMAN/SOPHOMORE LEVEL COURSE.
"Emphasizes the real-world significance of mathematics and applications of several areas of mathematics. Topics covered include des...
voting: typically; candidates run, people vote, votes are counted, majority wins
two-candidate election: only 2 candidates
e.g. stockholders often get a number of votes proportional to the number of stocks
they own
voting methods, 4 categories:
- one person, one vote, two candidates
- one person, one vote, multiple candidates
- one person, multiple votes, two candidates
- one person, multiple votes, multiple candidates
* if there’s 8 votes for A, 5 votes for B, 7 votes for C, 0 votes for D, A wins but
majority voted against A/voted for someone else (12 people)
^ under the plurality method, the candidate with the largest number of votes
is declared the winner; winning candidate may or may not have a majority of votes
(majority: more than half of the votes)
drawbacks of the plurality method
• voters only get to express their first choice (e.g. maybe all 5 people who voted for
B would prefer C over A)
advantages of the plurality method
• if a candidate happens to get a majority, then that candidate will be the winner
under the plurality method
• the plurality method respects the majority criterion (if a candidate receives a
majority of first-place votes, then that candidate ought to be the winner of the
election)
* the majority criterion does not say that a candidate must have a majority in
order to win
,1.2.2: Preference ballots and intro to fairness criteria
preference ballots: the voters rank all or some of the candidates according to their
preferences
preference schedule: each distinct ballot is listed only once, with the number of
occurrences indicated on top
*there’s no simple way to pick a
winner, taking everything into
account and being fair; therefore
different methods have been designed
, What does “fair” mean?
• there are 4 basic fairness criteria that a method may or may not satisfy
• majority criterion was the first
• each voting method fails at least one of the four criteria, therefore there is no fair
voting method (Kenneth Arrow proved the General Impossibility Theorem in
1952, which says it’s impossible to have a voting method satisfying all 4 fairness
criteria)
Borda count method: each candidate gets a certain number of points, depending on
the ranking
• the points for all the ballots are added up
• candidate with the largest number of points win
the basic version of the Borda count method:
• first-choice candidate gets as many points as there are candidates
• second-choice gets one fewer point, and so on
e.g. points for A
4 x (8) + 3 x (0) + 2 x (1) + 1 x (11)
= 32 + 0 + 2 + 11
= 45
points for B, C, D: 57, 58, 40
* C is the winner.
* one of the drawbacks of the Borda count method is that it violates the majority
criterion.
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