Quiz 1         MAT 118, Spring 2003

PRINT your Name: Solution

Nine 10-year-old boys were asked to rank their favorite T.V. shows. Three kids liked The Happy Bunny Mayhem Hour best, with Dragonball Z second, followed by Ed, Edd, and Eddie, and then Rabid Ninjas last. One child ranked had The Happy Bunny Mayhem Hour first, followed by Rabid Ninjas, Ed, Edd, and Eddie, and then The Happy Bunny Mayhem Hour. Three other kids ordered the shows as Rabid Ninjas, Dragonball Z, Ed, Edd, and Eddie, and then The Happy Bunny Mayhem Hour. The two remaining children ranked the shows Dragonball Z, The Happy Bunny Mayhem Hour, Ed, Edd, and Eddie, and finally Rabid Ninjas.

The results are summarized in the table below:

   3   1   3   2 
The Happy Bunny Mayhem Hour 1 1 4 2
Dragonball Z 2 4 2 1
Ed, Edd, and Eddie 3 3 3 3
Rabid Ninjas 4 2 1 4

  1. Which show would win in a plurality election? (circle your answer)

    Solution: The Happy Bunny Mayhem Hour is the plurality winner, since it is the choice of 4 voters. The second place winner is Rabid Ninjas, prefered by 3 boys, and Dragonball Z comes in third with the remaining 2 voters. No one took Ed, Edd, and Eddie as their first choice.

  2. Which show has the highest Borda count?

    Solution: Below are the calculations. Remember that we give 4 points for a first-place vote, 3 points for a second-place, 2 for a third, and 1 for a last-place showing.

    Happy Bunny Mayhem Hour:          $(3+1)\cdot 4 + 2\cdot 3 + 0\cdot 2 + 3\cdot 1 = 25$
    Dragonball Z:          $2\cdot 4 + (3+3)\cdot 3 + 0 \cdot 2 + 1\cdot 1 = 27$
    Ed, Edd, and Eddie:          $0\cdot4 + 0\cdot3 + (3+3+3+3)\cdot 2 + 0\cdot 1 = 18$
    Rabid Ninjas:          $3\cdot 4 + 1 \cdot 3 + 0\cdot 2 + (3+2)\cdot 1 = 20$

    So Dragonball Z has the highest Borda count, with 27.

  3. Which show, if any, is the Condorcet winner?

    Solution: My first guess is that it would either be the bunnies (the plurality winner) or Dragonball. Let's see who wins in the head-to-head with those two:

    The Happy Bunny Mayhem Hour gets only four votes (from the first two columns). The remaining five voters prefer Dragonball Z. So clearly the happy bunnies can't be our Condorcet winner. Perhaps it is DBZ.

    Now lets consider DBZ against the Eds: Here, all except one voter prefer Dragonball Z- this voter is the one who ranked Dragonball Z in 4th place. So again, DBZ wins.

    Finally, we should must compare DBZ to the Rabid Ninjas, and again it wins, this time with another close 5 to 4 race (the outer two columns prefer DBZ, the inner two like the ninjas.)

    Since it beat all the other candidates in a one-on-one race, Dragonball Z is the Condorcet winner.

    There is no need to check the three other possible contests.

Scott Sutherland 2003-01-31