QuestionVoters choose between designs $A, B, C$ , and $D$ . A. Which design has most first-place votes? B. Which wins by Borda count? Majority?

Studdy Solution

STEP 1

Assumptions1. The number of votes for each preference order is given in the table. . The Borda count method is used for determining the winning design. In this method, each voter ranks the options, and points are assigned to each rank. The option with the most points wins.
3. The majority criterion is satisfied if the option that has a majority of first-place votes wins.

STEP 2

First, let's answer part A of the question. We need to determine which design has a majority of first-place votes. We can do this by looking at the first row of the preference table.

STEP 3

From the preference table, we can see that design D has125 first-place votes, design C has88 first-place votes (50+38), and design A has25 first-place votes. Design B has no first-place votes.

STEP 4

The total number of votes is the sum of the votes in each column.
$Total\, votes =125 +50 +38 +25$

STEP 5

Calculate the total number of votes.
$Total\, votes =125 +50 +38 +25 =238$

STEP 6

A majority of votes is more than half of the total votes. So, we need to calculate half of the total votes.
$Majority\, votes = Total\, votes /2$

STEP 7

Calculate the majority votes.
$Majority\, votes =238 /2 =119$

STEP 8

From the preference table, we can see that design D has more than119 votes. So, design D has a majority of first-place votes.
Now, let's move on to part B of the question. We need to use the Borda count method to determine the winning design. In this method, each rank is assigned a point value. The first choice gets4 points, the second choice gets3 points, the third choice gets2 points, and the fourth choice gets1 point.

STEP 9

First, let's calculate the Borda count for design A. From the preference table, we can see that design A is the2nd choice for125 voters, the2nd choice for50 voters, the2nd choice for38 voters, and thest choice for25 voters.
$Borda\, count\, for\, A = (2nd\, choice\, votes \\times3\, points) + (st\, choice\, votes \\times4\, points)$

STEP 10

Plug in the values for the2nd choice votes andst choice votes to calculate the Borda count for design A.
$Borda\, count\, for\, A = (125 \\times3) + (25 \\times4)$

STEP 11

Calculate the Borda count for design A.
$Borda\, count\, for\, A = (125 \\times3) + (25 \\times4) =375 +100 =475$

STEP 12

Repeat steps9 to11 for designs B, C, and D to calculate their Borda counts.

STEP 13

From the preference table, we can see that design B is the3rd choice for125 voters, the3rd choice for50 voters, theth choice for38 voters, and the3rd choice for25 voters.
$Borda\, count\, for\, B = (3rd\, choice\, votes \\times2\, points) + (th\, choice\, votes \\times\, point)$

STEP 14

Plug in the values for the3rd choice votes and4th choice votes to calculate the Borda count for design B.
$Borda\, count\, for\, B = (125 \\times2) + (50 \\times2) + (38 \\times) + (25 \\times2)$

STEP 15

Calculate the Borda count for design B.
$Borda\, count\, for\, B = (125 \\times2) + (50 \\times2) + (38 \\times) + (25 \\times2) =250 +100 +38 +50 =438$

STEP 16

From the preference table, we can see that design C is the4th choice for125 voters, thest choice for50 voters, thest choice for38 voters, and the4th choice for25 voters.
$Borda\, count\, for\, C = (4th\, choice\, votes \\times\, point) + (st\, choice\, votes \\times4\, points)$

STEP 17

Plug in the values for the4th choice votes andst choice votes to calculate the Borda count for design C.
$Borda\, count\, for\, C = (125 \\times) + (50 \\times4) + (38 \\times4) + (25 \\times)$

STEP 18

Calculate the Borda count for design C.
$Borda\, count\, for\, C = (125 \\times) + (50 \\times4) + (38 \\times4) + (25 \\times) =125 +200 +152 +25 =502$

STEP 19

From the preference table, we can see that design D is the1st choice for125 voters, the4th choice for50 voters, the3rd choice for38 voters, and thend choice for25 voters.
$Borda\, count\, for\, D = (1st\, choice\, votes \\times4\, points) + (nd\, choice\, votes \\times3\, points) + (3rd\, choice\, votes \\times\, points) + (4th\, choice\, votes \\times1\, point)$

STEP 20

Plug in the values for thest choice votes,nd choice votes,3rd choice votes, and4th choice votes to calculate the Borda count for design D.
$Borda\, count\, for\, D = (125 \\times4) + (25 \\times3) + (38 \\times) + (50 \\times)$

STEP 21

Calculate the Borda count for design D.
$Borda\, count\, for\, D = (125 \\times4) + (25 \\times3) + (38 \\times) + (50 \\times1) =500 +75 +76 +50 =701$

STEP 22

The design with the highest Borda count will be used for the affordable housing. From the Borda counts calculated, we can see that design D has the highest Borda count.

STEP 23

The majority criterion is satisfied if the option that has a majority of first-place votes wins. From the Borda counts calculated, we can see that design D, which has a majority of first-place votes, also has the highest Borda count. So, the majority criterion is satisfied.
The design that has a majority of first-place votes is design D. Using the Borda count method, design D will be used for the affordable housing. The majority criterion is satisfied because the design with a majority of first-place votes also has the highest Borda count.

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