Solved on Oct 23, 2023
Find the winner of the election with 2, 15, and 14 voters, where the 1st, 2nd, and 3rd choices are , , and , using Pairwise Comparison (Copeland's) method.
STEP 1
Assumptions1. The number of voters for each choice is given in the first row of the table.
. The1st,nd and3rd choices of the voters are given in the following rows.
3. The Pairwise Comparison method (Copeland's Method) is used to determine the winner.
STEP 2
In the Pairwise Comparison method, each candidate is compared with every other candidate in a head-to-head competition. The candidate who is preferred by the majority of voters wins the competition.
STEP 3
First, let's create a matrix to keep track of the head-to-head competitions. The matrix will have rows and columns representing each candidate (A, B, and C).
\begin{array}{c|ccc}
& A & B & C \\
\hlineA & - & - & - \\
B & - & - & - \\
C & - & - & - \\
\end{array}
STEP 4
Now, let's fill in the matrix with the results of the head-to-head competitions. We'll start with A vs B.Looking at the table, we see that14 voters prefer A to B (since A is their1st choice and B is their3rd choice), and15 voters prefer B to A (since B is their1st choice and A is their3rd choice).Since more voters prefer B to A, we'll put a1 in the B vs A cell and a0 in the A vs B cell.
\begin{array}{c|ccc}
& A & B & C \\
\hlineA & - &0 & - \\
B &1 & - & - \\
C & - & - & - \\
\end{array}
STEP 5
Next, let's compare A vs C.Looking at the table, we see that14 voters prefer A to C (since A is their1st choice and C is their2nd choice), and2 voters prefer C to A (since C is their1st choice and A is their2nd choice).Since more voters prefer A to C, we'll put a1 in the A vs C cell and a0 in the C vs A cell.
\begin{array}{c|ccc}
& A & B & C \\
\hlineA & - &0 &1 \\
B &1 & - & - \\
C &0 & - & - \\
\end{array}
STEP 6
Finally, let's compare B vs C.Looking at the table, we see that15 voters prefer B to C (since B is their1st choice and C is their2nd choice), and2 voters prefer C to B (since C is their1st choice and B is their3rd choice).Since more voters prefer B to C, we'll put a1 in the B vs C cell and a0 in the C vs B cell.
\begin{array}{c|ccc}
& A & B & C \\
\hlineA & - &0 &1 \\
B &1 & - &1 \\
C &0 &0 & - \\
\end{array}
STEP 7
Now, let's add up the scores for each candidate. The score for a candidate is the sum of the values in their row.
\begin{array}{c|ccc|c}
& A & B & C & Score \\
\hlineA & - &0 &1 &1 \\
B &1 & - &1 &2 \\
C &0 &0 & - &0 \\
\end{array}
STEP 8
The candidate with the highest score wins the election. In this case, candidate B has the highest score, so B is the winner of the election.
Winner = B
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