91Ó°ÊÓ

Three candidates, A, B, and \(\mathrm{C}\), are running for mayor. Election rules stipulate that the pairwise comparison method will determine the winner. In the event that the pairwise comparison method leads to a tie, the Borda count method will decide the winner. The election results are summarized in the following preference table. Under these rules, which candidate becomes the new mayor? $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{6 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{4 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} & \mathbf{2 0 , 0 0 0} \\ \hline \text { First Choice } & \text { A } & \text { C } & \text { B } & \text { A } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { A } & \text { C } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { B } & \text { A } & \text { B } & \text { A } \\ \hline \end{array} $$

Step by step solution

01

Identify the Pairwise Comparison Results

First, you need to evaluate the pairwise comparisons. Comparing A to B, A gets the preference of 60,000 and 20,000 voters, for a total of 80,000 while B has the preference of 40,000 voters. Comparing A to C, A gets preference from 60,000 voters while C from 40,000 + 20,000 + 20,000 = 80,000 voters. Comparing B to C, B gets preference from 60,000 + 40,000 = 100,000 voters while C gets it from 20,000 voters.

02

Evaluate Pairwise Comparison for Tie

Next, compare the number of preferences each candidate gained from the pairwise comparison. Here A won one time, B also won one time and C won once. Hence, because all three candidates have won one head-to-head race, there is a tie.

03

Use Borda Count Method for Tie

Whenever there is a tie, use the Borda count method. To get the points, the Borda count assigns points to each position (1 point for third choice; 2 points for the second; 3 points for the first). Therefore, you need to calculate the total Borda count for each candidate: A gets 3*(60000+20000)+2*40000+1*20000= 240,000 points. B gets 3*40000+2*(60000+20000)=240,000 points. C gets 3*(40000+20000)+2*20000+1*(60000+20000)=240,000 points.

04

Identify the Winner

Finally, identify the winner. Because all three candidates have the same Borda count, it would be reasonable to consider the election to be tied.

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