Math 111 Review for Final Exam

Transitive Property

If a=b and b=c, then a=c

Majority criterion

If candidate X has a majority of the first place votes, then candidate X should be the winner of the election

Monotonicity Criterion

If candidate X is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor X (and only X), then X should remain a winner of the election

Condorcet Criterion

If candidate X is preferred by the voters over each of the other candidates in a head-to-head comparison, then candidate X should be the winner of the election

Majority of votes

If there are three or more candidates in an election, it often happens that no single candidate receives more than 50% of the first place votes

Explain why the method of pairwise comparisons satisfies the majority criterion

If x has a majority of the first-place votes, then x is ranked above all other candidates on more than half the ballots and is, therefore, the winner under the method of pairwise comparisons

Explain why the method of pairwise comparisons satisfies the condorcet criterion

If x is the Condorcet candidate, then by definition x wins every pairwise comparison. Therefore, it is the winner under the method of pairwise comparisons.

Borda Count method

One of the voting methods requires voters to rank all candidates from the most favorable to the least favorable. Each last place vote receives 1 point, each next to last place vote receives 2 points, and so on. The candidate with the most points is the winner.

15 x 15 = 25

The election with 5 candidates and 15 voters has how many total Borda points?

74000 x (.035+1)^t

The polulation of a town with an initial population of 74000 grows at a rate of 3.5% per year

It is impossible for any democratic voting system to satisfy each of the fairness criteria

The statement is true because it was proven by Arrow's Impossibility Theorem

A candidate received the majority of first place votes and lost the election

The statement makes sense because the Borda count method using preference ballots can result in a situation where this can happen

My candidate was favored when compared head-to-head with every other candidate and lost the election

The statement makes sense. The Head-to-head criterion does not ensure that all a candidate will win an election for certain voting systems

What is the total number of points that all candidates can earn in an election using the pairwise comparison method if there are five candidates?

The total number of points that all candidates can earn is 10

Pairwise Comparison Method

The voting method in which each candidate is compared with each of the other candidates

Plurality method

The voting method in which the candidate receiving the most 1st place votes is declared the winner

Plurality-with-elimination method

The voting method that may involve a series of elections or eliminations using a preference table until a candidate receives a majority of the votes

Briefly describe how exponential functions are useful for modeling inflation, environmental and resource issues, physiological processes, and radioactive decay.

There are cases where the absolute growth rate is not constant, but the relative growth rate is constant. Exponential growth and decay can be used to model these situations

It is impossible for any democratic voting system to satisfy each of the fairness criteria

This statement is true because it was proven by Arrow's impossibility theorem

Since the Pairwise comparisons method satisfies the majority criterion, then all elections using the pairwise comparison method and having a majority candidate will also satisfy the majority criterion

True

Explain why the plurality method satisfies the monotonicity

When a voter moves a candidate to first place in his or her ballot, that candidate's first-place votes increase

Explain why the Borda count method satisfies the monotonicity criterion

When a voter moves a candidate up in his or her ballot, that candidate's Borda points increase. If X had the most borda points and a voter changes his or her ballot to rank X higher, then X still has the most Borda points

Explain why the borda count method satisfies the monotonocity criterion

When a voter moves a candidate up in his or her ballot, that candidate's Borda points increase. If x had the most borda points and a voter change his or her ballot to rank x higher, then x still has the most borda points

Independence of Irrelevant Alternatives Criterion

a criterion that says that a candidate that would otherwise win an election should not lose the election merely because one of the losing candidates withdraws from the race

proportional

each party receives at least

equitable

each player values each share equally

If an election using the plurality with elimination method is found to have a condorcet candidate, then we are guaranteed the condorcet candidate will be the winner of the election

false

If one election use the pairwise comparison method satisfies a particular fairness criterion, then we are guaranteed that all elections using the pairwise comparison method will satisfy that particular fairness criterion, as long as the hypothesis has been met

false

Irrelevant Alternatives Criterion

if a candidate wins a 1st election and in a 2nd election one or more candidates are removed, the winner of the 1st election should still win

Since the initial total is more than the total number of seats, we should

increase the divisor so the quotients are smaller

pareto optimal

no other division exists

envy-free

no party would prefer

No, none of the original states' representation decreases

no quota rule violations or paradoxes any of the 5 apportionment method couldve been used

The price of a particular model car is 19000 today and rises with time at a constant rate of 920 per year. How much will a new car cost in 3.1 years?

p = 19000 + (1920 x t)

symmetry

players have equal rights in sharing the assets

the divider chooser method will always be

propiortional and envy free

In the eyes of the divider, the Lone-divider method is always

proportional, equitable and envy free

A snowplow has maximum speed of 60 mph on a dry highway. Its max speed = 1.8 mph

s = 60-(1.8 x d)

What does it mean to say that a function is linear

the function has a constant rate of change the function has a straight line graph the function can be described by an equation of the form y = mx + b

rationality

the parties act in their own best interest and do not make emotional decisiona

cooperation

the players are willing participants and will accept the outcome of the division without outside arbitration (lawyers) or intervention

privacy

the players have no knowledge of the value system of other players

It's just a matter of time until mathematicians devise an ideal apportionment method that satisfies the quota rule and does not produce any paradoxes

the statement does not make sense. It has been proven that all apportionment methods must produce paradoxes or violate the quota rule

There is no perfect apportionment method that satisfies the quota rule and avoids any paradox

the statement is true because any appoortionment method that does not violate the quota rule must produce paradoxes, and any apportionment method that does not produce paradoxes must violate the quota rule

annuity

the value of annuity is the sum of all deposits plus interest

If a particular voting method satisfies the majority criterion, then we are guaranteed that all elections using that voting method will satisfy the majority, as long as the hypothesis has been met.

true

Since the Plurality Method satisfies the Majority Criterion, then all elections using the Plurality method and having a majority candidate will also satisfy the majority criterion

true

a 960 washing machine is depreciated for tax purposes at 80$ per year

v = 960 - 80t

Explain why the plurality method satisfies the monotonocity criterion

when a voter moves a candidate to first place in his or her ballot, that candidates first place votes increase

Most points =?

winner

Preference Table

A table that shows how often each particular outcome occurred and that summarizes an election's results

5(5+1)/2 = 15

Any single ballot with 5 candidates has how many Borda points

preference ballots

Ballots in which voters are asked to rank all the candidates

Number of comparisons formula

C = n(n-1) / 2

An election is held using the plurality with elimination methof and candidate x is declared the winner. In a re-election, the only ballots that changed moved candiodate x to 1st place. What do we know about the re-election?

Candidate X may not be the winner sinse the plurality with elimination method does not satisfy the monotonicity criterion

An election is held using the Borda Count Method and Candidate X is declared the winner. After the election, Candidate Y is found to be ineligible. Candidate Y is dropped from the preference schedule and the winner is recalculated. What do we know about the new winner?

Candidate X may not win the election since the Independence of Irrelevant Alternatives Criterion does apply here.

An election is held using the Plurality method and candidate X is declared the winner. In a re-election, the only ballots that changed moved candidate X to first place. What do we know about the winner of the re-election?

Candidate X will be the winner since the Plurality method satisfies the Monotonicity Criterion.

An election is held using the plurality method and candidate x is declared the winner. After the election, candidate y is found to be ineligible. Candidate y is dropped from the preference schedule and the winner is re-calculated. What do we know about the winner?

Candidate x may not win the election since the I of IAC doesn't apply here

If a particular voting method violates the Independence of Irrelevant Alternatives Criterion, then we are guaranteed that all elections using that voting method will violate the Independence of Irrelevant Alternatives, as long as the hypothesis has been met

False

If an election using the Borda count method is found to have a Condorcet candidate, then we are guaranteed the Condorcet candidate will be the winner of the election

False

If one election using the Borda Count Method violates a particular Fairness Criterion, then we are guarenteed that all elections using the Borda Count method will violate that particular Fairness Criterion, as long as the hypothesis has been met

False

1

How many Borda points does a last place candidate receive from a single ballot?

5

How many Borda points would a first place candidate receive from a single ballot?

Explain why the method of pairwise comparisons satisfies the majority criterion

If X has a majority of first place votes, then X is ranked above all other candidates on more than half the ballots and is, therefore, the winner under the method of pairwise comparisons

Explain why the method of Pairwise Comparisons satisfied the Condorcet criterion

If X is the Condorcet candidate, then by definition X wins every pairwise comparison. Therefore, it is the winner under the method of pairwise comparisons.

5 x 15 = 75

If a candidate got all first place votes, how many Borda Points would that candidate receive?

1 x 15 = 15

If a candidate got all last place votes, how many Borda points would that candidate receive?

The preferred candidate in each comparison receives how many points?

1

If the candidates tie, then each receives how many points?

1/2 points

equitable division

1/2 veggie 1/2 sausage and 1/2 veggie 1/2 sausage

Suppose that the pairwise comparison method is used to determine the winner in an election. If there are 7 candidates, how many comparisons must be made?

21 comparisons

The total number of distinct pairwise comparisons (head-to-head competitions) that must be made in an election with 3 candidates would be what?

3 candidates. It would be 2 to be declared a Condorcet Candidate

If we list each possible pairwise comparison (head-to-head) between the 3 candidates, what would be the total number of possible pairs?

6