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Math of Voting

Just count the votes and see who has the most, right? Voting systems can actually be quite complex, and the puzzling mathematical paradoxes that arise from them may surprise you!

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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