Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. C wins in a Borda count. Is the outcome different in a rank system that assigns five points for first choice, two points for second, and one point for third? $ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$\text { Number of voters}$
$\begin{array} { l c c c c }
& 12 & 8 & 6 & 3 \
\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { C } \
\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { B } \
\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { A }
\end{array}$

Question

Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. C wins in a Borda count. Is the outcome different in a rank system that assigns five points for first choice, two points for second, and one point for third? $ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$\text { Number of voters}$
$\begin{array} { l c c c c } 
& 12 & 8 & 6 & 3 \
\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { C } \
\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { B } \
\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { A }
\end{array}$

Quizplus · Accepted Answer

In the Borda count system, each rank is typically assigned a point value (e.g., 2 points for first place, 1 point for second, and 0 for third in a three-choice scenario). However, the question proposes a different point system: 5 points for first choice, 2 points for second, and 1 point for third. Calculating the points for each alternative under this new system:- **For B**: $12 \times 5 + 6 \times 2 + 3 \times 2 = 60 + 12 + 6 = 78$- **For C**: $8 \times 5 + 12 \times 2 + 3 \times 5 = 40 + 24 + 15 = 79$- **For A**: $6 \times 5 + 8 \times 2 + 3 \times 1 = 30 + 16 + 3 = 49$However, my calculation for C was incorrect in the explanation; let's correct that:- **For C correctly**: $8 \times 5 + 12 \times 2 + 3 \times 5 = 40 + 24 + 15 = 79$ should actually be calculated with the correct distribution of voters for C, which is $8 \times 5 + 12 \times 2 + 3 \times 5 = 40 + 24 + 15 = 79$, but considering the correct allocation of points for each group of voters:  - For C from the first group (12 voters as second choice): $12 \times 2 = 24$  - For C from the second group (8 voters as first choice): $8 \times 5 = 40$  - For C from the third group (3 voters as first choice): $3 \times 5 = 15$  So, the correct calculation for C should indeed total $24 + 40 + 15 = 79$, which was initially correctly stated but questioned in the explanation process.Given the correct point allocations and the mistake in questioning the calculation for C (which was actually correct), the outcome based on the provided point system (5 for first, 2 for second, 1 for third) would indeed change the winner compared to a standard Borda count. However, the mistake was in doubting the calculation for C, not in the outcome based on the provided point system.Upon recalculating correctly for all candidates with the specified point system:- **B**: $12 \times 5 + 6 \times 2 + 3 \times 2 = 60 + 12 + 6 = 78$- **C**: $8 \times 5 + 12 \times 2 + 3 \times 5 = 40 + 24 + 15 = 79$ (as initially correctly calculated)- **A**: $6 \times 5 + 8 \times 2 + 3 \times 1 = 30 + 16 + 3 = 49$The correct interpretation should focus on the recalculated totals under the specified point system, which does not change the outcome for C winning, thus indicating an error in the process of questioning the calculation rather than in the initial outcome. Therefore, the initial assertion that the outcome changes based on the point system needs to be corrected to reflect that the outcome does not change; C still wins under the specified point system, which contradicts the initial conclusion drawn. The mistake lies in the interpretation of the question's requirement and the calculation process, not in the calculation for C itself.

Twenty-Nine Voters Must Choose from Among Three Alternatives: A, B \quad

Twenty-Nine Voters Must Choose from Among Three Alternatives: A, B $\quad$