A committee is trying to award a scholarship to one of four students, Anna (A), Brian (B), Carlos (C), and Dimitry (D). The votes are shown below:
Solution
Making the comparisons:
\(\begin{array} {ll} {\text{A vs B: }10\text{ votes to }10\text{ votes}} & {\text{A gets }\frac{1}{2}\text{ point, B gets }\frac{1}{2}\text{ point}} \\ {\text{A vs C: }14\text{ votes to }6\text{ votes:}} & {\text{A gets }1\text{ point}} \\ {\text{A vs D: }5\text{ votes to }15\text{ votes:}} & {\text{D gets }1\text{ point}} \\ {\text{B vs C: }4\text{ votes to }16\text{ votes:}} & {\text{C gets }1\text{ point}} \\ {\text{B vs D: }15\text{ votes to }5\text{ votes:}} & {\text{B gets }1\text{ point}} \\ {\text{C vs D: }11\text{ votes to }9\text{ votes:}} & {\text{C gets }1\text{ point}} \end{array} \)
Totaling:
\(\begin{array} {ll} {\text{A has }1\frac{1}{2}\text{ points}} & {\text{B has }1 \frac{1}{2}\text{ points}} \\ {\text{C has }2\text{ points}} & {\text{D has }1\text{ point}} \end{array} \)
So Carlos is awarded the scholarship. However, the committee then discovers that Dimitry was not eligible for the scholarship (he failed his last math class). Even though this seems like it shouldn’t affect the outcome, the committee decides to recount the vote, removing Dimitry from consideration. This reduces the preference schedule to:
\(\begin{array}{|l|l|l|l|l|}
\hline & 5 & 5 & 6 & 4 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{A} & \mathrm{A} & \mathrm{C} & \mathrm{B} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{C} & \mathrm{C} & \mathrm{B} & \mathrm{A} \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{B} & \mathrm{B} & \mathrm{A} & \mathrm{C} \\
\hline
\end{array}\)
\(\begin{array} {ll} {\text{A vs B: }10\text{ votes to }10\text{ votes}} & {\text{A gets }\frac{1}{2}\text{ point, B gets }\frac{1}{2}\text{ point}} \\ {\text{A vs C: }14\text{ votes to }6\text{ votes}} & {\text{A gets }1\text{ point}} \\ {\text{B vs C: }4\text{ votes to }16\text{ votes}} & {\text{C gets }1\text{ point}} \end{array} \)
Totaling:
\(\begin{array} {ll} {\text{A has }1 \frac{1}{2}\text{ points}} & {\text{B has }\frac{1}{2}\text{ point}} \\ {\text{C has }1\text{ point}} & { } \end{array} \)
Suddenly Anna is the winner! This leads us to another fairness criterion.