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Exercises 11.9 Chapter Test

Exercise Group.

Use the preference table to answer the questions.

Table 11.9.1.
12 15 9 13
1st Choice A B C A
2nd Choice B C B D
3rd Choice C A D B
4th Choice D D A C

1.

How many voted in this election?

Answer.
49
2.

How many votes are needed for a majority?

Answer.
25
3.

Does anyone have a majority? If so, who?

Answer.
Yes, A has 25 first place votes.
4.

Determine the winner using the Borda Count method.

Answer.
B
Solution.
\begin{equation*} A = 12 \cdot 4 + 15 \cdot 2 + 9 \cdot 1 + 13 \cdot 4 = 139 \end{equation*}
\begin{equation*} B = 12 \cdot 3 + 15 \cdot 4 + 9 \cdot 3 + 13 \cdot 2 = 149 \end{equation*}
\begin{equation*} C = 12 \cdot 2 + 15 \cdot 3 + 9 \cdot 4 + 13 \cdot 1 = 118 \end{equation*}
\begin{equation*} D = 12 \cdot 1 + 15 \cdot 1 + 9 \cdot 2 + 13 \cdot 3 = 84 \end{equation*}

B wins with 149 pts

5.

Determine the winner using Copeland’s method (pairwise comparisons).

Answer.
A
Solution.

A (25) v B (24): A wins

A (25) v C (24): A wins

A (40) v D (9): A wins

B (40) v C (9): B wins

B (36) v D (13): B wins

C (36) v D (13): C wins

A = 3 pts; B = 2 pts; C = 1pt; D = 0 pts

A wins

Exercise Group.

Answer the following questions using the preference schedule

Table 11.9.2.
15 7 13 5 2
1st T E F P E
2nd P P G G F
3rd E G E F G
4th F F P E P
5th G T T T T

6.

Find the winner with the plurality method.

Answer.
T
Solution.

T = 15, E = 9, F = 13, P = 5, G =0

T has the most first place votes, T wins.

7.

Find the winner with the instant runoff method.

Answer.
F
Solution.

Round 1: G has 0 first place votes and is eliminated.

\begin{equation*} T = 15, E = 9, F = 13, P = 5 \end{equation*}

Round 2: P has the fewest votes and is eliminated. Those five votes go to F, since their 2nd choice (G) was already eliminated.

\begin{equation*} T = 15, E = 9, F = 18 \end{equation*}

Round 3: E is eliminated. All 9 votes go to F.

\begin{equation*} T = 15, F = 27 \end{equation*}

F is the winner.

Exercise Group.

Consider the following preference schedule:

Table 11.9.3.
80 70 50
1st A B C
2nd C A B
3rd B C A

8.

Is there a Condorcet candidate? If so, who? If not, why not?

Answer.
No. No one can beat all other candidates in a head-to-head comparison.
Solution.

A (80) v B (120): B wins

A (150) v C (50): A wins

B (70) v C (130): C wins

No one can beat all the other candidates, so there is no Condorcet candidate.

9.

Who wins using the Borda Count method?

Answer.
A
Solution.
\begin{equation*} A = 80 \cdot 3 + 70 \cdot 2 + 50 \cdot 1 = 430 \end{equation*}
\begin{equation*} B = 80 \cdot 1 + 70 \cdot 3 + 50 \cdot 2 = 390 \end{equation*}
\begin{equation*} C = 80 \cdot 2 + 70 \cdot 1 = 50 \cdot 3 = 380 \end{equation*}

A wins

10.

If C drops out of the race, who will using the Borda Count method? Was one of the fairness criteria violated? If so, which one?

Answer.
B wins. Since the winner changed when a losing candidate (C), dropped out of the race, the IIA (Independence of Irrelevant Alternatives) criterion was violated.
Solution.

The preference schedule is now:

Table 11.9.4.
80 70 50
1st Choice A B B
2nd Choice B A A

\begin{equation*} A = 80 \cdot 2 + 120 \cdot 1 = 280 \end{equation*}
\begin{equation*} B = 80 \cdot 1 + 120 \cdot 2 = 320 \end{equation*}

B wins

Since the winner changed when a losing candidate (C), dropped out of the race, the IIA (Independence of Irrelevant Alternatives) criterion was violated.