A group of 12 students have to decide among three types of pizza: sausage (S), mushroom (M), and beef (B). Their preference rankings are shown below. Is there a Condorcet winner among the pizza types? $ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$\text { Number of students }$
$\begin{array} { l l l l l l }
& 3 & 3 & 2 & 2 & 2 \
\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } & \mathrm { S } \
\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } \
\text { Third choice } & \mathrm { S } & \mathrm { S } & \mathrm { B } & \mathrm { M } & \mathrm { M }
\end{array}$

Question

A group of 12 students have to decide among three types of pizza: sausage (S), mushroom (M), and beef (B). Their preference rankings are shown below. Is there a Condorcet winner among the pizza types? $ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$ \quad$$\text { Number of students }$
$\begin{array} { l l l l l l } 
& 3 & 3 & 2 & 2 & 2 \
\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } & \mathrm { S } \
\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } \
\text { Third choice } & \mathrm { S } & \mathrm { S } & \mathrm { B } & \mathrm { M } & \mathrm { M }
\end{array}$

Quizplus · Accepted Answer

A Condorcet winner is a choice that would beat any other choice in a head-to-head matchup. To determine if there is a Condorcet winner, we compare each pair of choices.

- Beef (B) vs. Mushroom (M): Beef is ranked higher by 7 students, while Mushroom is ranked higher by 5 students. Therefore, Beef beats Mushroom in a head-to-head matchup.
- Beef (B) vs. Sausage (S): Beef is ranked higher by 6 students, while Sausage is ranked higher by 6 students. There is no clear winner in this matchup.
- Mushroom (M) vs. Sausage (S): Mushroom is ranked higher by 8 students, while Sausage is ranked higher by 4 students. Therefore, Mushroom beats Sausage in a head-to-head matchup.

Since there is a choice (Beef) that beats every other choice in a head-to-head matchup, it is the Condorcet winner. Therefore, the best choice is C.

A Group of 12 Students Have to Decide Among Three \quad

A Group of 12 Students Have to Decide Among Three $\quad$