jefferson method example

Jefferson Method Example• Populations 56, 35, 26, 15, 6

– Total population p = 138

• House size h = 200– Standard divisor s = p/h = .69

pi pi/s ni = floor(pi/s)

56 81.159 81

35 50.725 50

26 37.681 37

15 21.739 21

6 8.696 8

Total: 197 ← must fill 3 seats

What happens to p5 as we lower s?

.667

109 p58

.690← s →

p5 bumped by 1 when s reaches 0.667

What happens to p4 as we lower s?

2322 p421

109 p58

.682.652

.667

.690← s →

p4 bumped up at these two s values

What happens to p3 as we lower s?

3938 p337

2322 p421

109 p58

.684.667

.682.652

.667

.690← s →

What happens to p2 as we lower s?

5251 p250

3938 p337

2322 p421

109 p58

.686.673

.684.667

.682.652

.667

.690← s →

What happens to p1 as we lower s?

8382 p181

5251 p250

3938 p337

2322 p421

109 p58

.683.675

.686.673

.684.667

.682.652

.667

.690← s →

Since we need 3 more seats, the first three that are bumped up get the seats

8382 p181

5251 p250

3938 p337

2322 p421

109 p58

.683.675

.686.673

.684.667

.682.652

.667

.690← s →

Final Result

• p2 bumped to 51

• p1 bumped to 82

• p3 bumped to 38

pi Initial Final

56 81 82

35 50 51

26 37 38

15 21 21

6 8 8

97 100

Need a quick way to determine these divisor values

8382 p181

5251 p250

3938 p337

2322 p421

109 p58

.683.675

.686.673

.684.667

.682.652

.667

.690← s →

p1, with population 56, gets a bump to 82 when s = 56/82

8382 p181

5251 p250

3938 p337

2322 p421

109 p58

.683.675

.686.673

.684.667

.682.652

.667

.690← s →

p1, with population 56, gets a bump to 82 when s = 56/82

8382 p181

5251 p250

3938 p337

2322 p421

109 p58

56/82.675

.686.673

.684.667

.682.652

.667

.690← s →

p1, with population 56, gets a bump to 83 when s = 56/83

8382 p181

5251 p250

3938 p337

2322 p421

109 p58

56/8256/83

.686.673

.684.667

.682.652

.667

.690← s →

p2, with population 35, gets a bump to 51 when s = 35/51

8382 p181

5251 p250

3938 p337

2322 p421

109 p58

56/8256/83

35/51.673

.684.667

.682.652

.667

.690← s →

and so on…8382 p181

5251 p250

3938 p337

2322 p421

109 p58

56/8256/83

35/5135/52

26/3826/39

15/2215/23

6/9

.690← s →

For a population pi with house seats ni, the divisor needed to get to ni+1 seats is

di =pi

ni + 1

To get to ni+2 seats:

di =pi

ni + 2

And so on …

In some cases, some populations get 2 seats before other can get 1. Consider the following example with four states, where p = 1012, h=100 and s = 10.12

pi ni = floor(pi/s)

708 69

201 19

66 6

37 3

Total: 97

7170 p169

2120 p219

7 p36

4 p43

10.119.97

10.059.57

9.43

9.25

10.12← s →

p1 bumped twice before p3, p4 bumped once

Again, need 3 seats, so the first three bumps get the seats

7170 p169

2120 p219

7 p36

4 p43

10.119.97

10.059.57

9.43

9.25

10.12← s →

In fact…

Again, need 3 seats, so the first three bumps get the seats

p1 would be bumped many times before p3, p4

7170 p169

2120 p219

7 p36

4 p43

10.119.97

10.059.57

9.43

9.25

10.12← s →

7273

9.839.69

Final Result

• p1 bumped twice to 71

• p2 bumped to 20

pi Initial Final

708 69 71

201 19 20

66 6 6

37 3 3

97 100

Another example with four states, where p = 1000, h=100 and s = 10

pi ni = floor(pi/s)

949 94

18 1

17 1

16 1

Total: 97

96 p19594

2 p21

2 p31

2 p41

In this case p1 is the only one that is bumped

9.999.89

9

8.5

8

10← s →

9798

9.789.68

…

Final Result

• p1 bumped three times to 97

pi Initial Final

949 94 97

18 1 1

17 1 1

16 1 1

97 100