Math of Voting

Voting Paradoxes


Suppose Aimee, Borodin, and Callisto are choosing a place to host a party. They can go to Xeno's, Yoselin's, or Zippo's. They rank their preferences:

A: wants X more than Y more than Z

B: wants Z more than X more than Y

C: wants Y more than Z more than X

Is this an instance of Condorcet's Paradox?

Suppose three people rank the ice cream flavors C, S, and V (chocolate, strawberry, and vanilla) in the following way:

Chet: S > C > V

Omari: C > V > S

Taj: V > S > C

Only one flavor can be bought for a party, so they're going up for a series of runoff votes. That is, two of the flavors will be pitted against each other in a vote, and the winner will face the remaining flavor in a final vote.

For example, if C faces V in a vote, Chet and Omari prefer C to V so that C will win. Then C faces off against S in a vote; since Chet and Taj prefer S to C, S wins the final vote.

If the runoffs can happen in any order, is there a way for chocolate (C) to win the final vote?

There are 10 seats in a legislative body and three states represented:

State A, Population 6
State B, Population 6
State C, Population 2

Since there is a total population of 14, a state will get 1 seat for each \(14 \div 10 = 1.4\) people it has. The remaining seat(s) will be given to the state(s) with the greatest remaining population. Since \[6 = 4 \times 1.4 + 0.4 \quad \text{and} \quad 2 = 1 \times 1.4 + 0.6,\] States A and B will get 4 seats each, while State C will get 2 (1 for the 1.4 people, and the 1 remaining seat since \(0.6 > 0.4\)).

If the number of seats is increased to 11, what will happen to the number of seats \(C\) gets?

Gillis, Quintana, and Xenia are trying to decide on a movie. They can vote between A Darker World (A), Boom Goes the Dynamite (B) and Crystal Dragon (C).

Gillis prefers B more than A more than C. ( \( B > A > C \) )

Quintana prefers A more than C more than B. ( \( A > C > B \) )

What would Xenia's preferences be to cause Condorcet's Paradox?

True or False: If two political groups have fifty seats in a State House allocated according to percentage of the vote won, it's impossible to decide how many seats each gets without having the Apportion Paradox.


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