Simple Voting System Comparison

Algebra Level 4

Mathland, a state of Brilliantopia, is holding a senate election.

Last election, states in Brilliantopia used a system called plurality voting or first-past-the-post voting. Under this system, a voter can only vote once for a candidate and the candidate with the most votes wins.

However, Brilliantopia recently switched to using a system called preferential voting or ranked-choice voting. Under preferential voting, voters can rank the candidates however they choose. The counting process happens in stages. At each stage, we check if a candidate has received a majority; if not, we eliminate the candidate with the least votes and redistribute the ballots based on the voters' next preference. For example, consider the ballot {A, D, B} where A is most preferred by this voter and B is least preferred. If A is eliminated, then we would give this voter's vote to D, the second-preferred candidate. If D was eliminated next, the vote would then go to B. If at any stage the voter's preferred candidate has been eliminated, proceed to the next preferred candidate. If all of a voter's preferred candidates has been eliminated, discard their vote.

All the polls in Mathland have closed. Here are the ballots:

3,000 people ranked Candidate A first followed by Candidate B.
1,500 people ranked Candidate D first, followed by Candidate C, followed by Candidate A.
1,750 people ranked Candidate C first, followed by Candidate B, followed by Candidate D.
1,250 people ranked Candidate D first, followed by Candidate B, followed by Candidate C.
2,500 people voted for Candidate B first followed by Candidate A.
200 people voted only for Candidate D.

Let us give each candidate a number as follows: A = 1, B = 2, C = 3, and D = 4. Determine who the winner of the election is under plurality voting and under preferential voting and denote the winners as $X$ and $Y$ respectively. In the preferential voting election, compute the amount of points the winner won by and record this value as $Z$ .

What is $\displaystyle Z(X + Y)$ ?

The answer is 3000.

1 solution

Chris Lewis
Nov 9, 2020

Assuming* that the most-preferred candidate on each voter's ballot is the one they would vote for in a plurality contest, that contest would be won by candidate A (with $3000$ votes), so $X=1$ .

In the preferential system, the first round looks like this:

1st	2nd	3rd	num votes
A	B	-	$3000$
D	C	A	$1500$
C	B	D	$1750$
D	B	C	$1250$
B	A	-	$2500$
D	-	-	$200$

The total 1st place votes for each candidate are: A: $3000$ B: $2500$ C: $1750$ D: $2950$

So, no candidate has the majority of votes (more votes than the others combined), and candidate C is eliminated. Redistributing C votes for the second round:

1st	2nd	3rd	num votes
A	B	-	$3000$
D	A	-	$1500$
B	D	-	$1750$
D	B	-	$1250$
B	A	-	$2500$
D	-	-	$200$

The total 1st place votes for each candidate are: A: $3000$ B: $4250$ D: $2950$

Still no candidate has the more votes, and D is eliminated.

For the 3rd round, we must have a winner (or a tie):

1st	2nd	3rd	num votes
A	B	-	$3000$
A	-	-	$1500$
B	-	-	$1750$
B	-	-	$1250$
B	A	-	$2500$
-	-	-	$200$

The total 1st place votes for each candidate are: A: $4500$ B: $5500$

So candidate B is the winner, with a margin of $1000$ votes; that is $Y=2$ , $Z=1000$ .

Hence $Z(X+Y)=\boxed{3000}$ .

this assumption isn't actually a given. People's voting habits would likely change given the different voting systems; plurality encourages tactical voting (for example, to avoid a split vote), whereas preferential systems don't to the same extent.

Simple Voting System Comparison

The answer is 3000.

1 solution

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