D’Hondt/Jefferson Allocation

# D’Hondt/Jefferson Allocation
## flipbook | made with Xaringan
### <br><br>Gina Reynolds, July 2019
### <br>

---

# Introduction

When awarding seats to parties in proportional representation systems, a dilemma is faced, as the seats available in a representative body are unlikely to able to be perfectly apportioned --- only if fractions of seats were apportioned would that be possible.

---

# Plan of the book

This book walks through the D'Hondt/Jefferson seat allocation method using dplyr and tidyr, using simulated data.

Then we create an animation of the steps of the method using ggplot and gganimate.

```r
library(tidyverse)
library(animation)
set.seed(2020)
```

---

# Step one: Collecting the facts

To determine how many seats each party wins we need to know two things.

- The number of seats available
- The number of votes won by each party

---

## Number of seats available

We need to know the number of seats available.  Let's say 6.

```r
num_seats <- 6
```

---

## Number of votes won by each party

We also need to know how many votes were won by each party.  We'll simulate a number of seats won by 3 parties.

```r
num_parties <- 3
party <- c("equality", "justice", "peace", "liberty")[1:num_parties]
votes <- sample(1000:10000, num_parties)
party_votes <- tibble(party, votes)
```

---

##  Number of votes won by each party

Let's check out the table of party and votes won.

```r
party_votes
```

```
  # A tibble: 3 x 2
    party    votes
    <chr>    <int>
  1 equality  6822
  2 justice   4548
  3 peace     6565
```

---

# Step 2: Calculate the "D'Hondt Outcome"

What I call "the D'Hondt Outcome"  is the number of votes won divided by 1, and 2, and 3, up to the total number of seats available for each party.

Notice how the function `crossing()` is used to set ourselves up to be able to do this in the next slide.

---

```r
*party_votes
```
]]
.column[.content[

```
  # A tibble: 3 x 2
    party    votes
    <chr>    <int>
  1 equality  6822
  2 justice   4548
  3 peace     6565
```
]]
---
class: split-40
count: false
.column[.content[

```r
party_votes %>%
* crossing(divisor = 1:num_seats)
```
]]
.column[.content[

```
  # A tibble: 18 x 3
     party    votes divisor
     <chr>    <int>   <int>
   1 equality  6822       1
   2 equality  6822       2
   3 equality  6822       3
   4 equality  6822       4
   5 equality  6822       5
   6 equality  6822       6
   7 justice   4548       1
   8 justice   4548       2
   9 justice   4548       3
  10 justice   4548       4
  11 justice   4548       5
  12 justice   4548       6
  13 peace     6565       1
  14 peace     6565       2
  15 peace     6565       3
  16 peace     6565       4
  17 peace     6565       5
  18 peace     6565       6
```
]]
---
class: split-40
count: false
.column[.content[

```r
party_votes %>%
  crossing(divisor = 1:num_seats) %>%
* mutate(dh_outcome = votes/divisor)
```
]]
.column[.content[

```
  # A tibble: 18 x 4
     party    votes divisor dh_outcome
     <chr>    <int>   <int>      <dbl>
   1 equality  6822       1      6822 
   2 equality  6822       2      3411 
   3 equality  6822       3      2274 
   4 equality  6822       4      1706.
   5 equality  6822       5      1364.
   6 equality  6822       6      1137 
   7 justice   4548       1      4548 
   8 justice   4548       2      2274 
   9 justice   4548       3      1516 
  10 justice   4548       4      1137 
  11 justice   4548       5       910.
  12 justice   4548       6       758 
  13 peace     6565       1      6565 
  14 peace     6565       2      3282.
  15 peace     6565       3      2188.
  16 peace     6565       4      1641.
  17 peace     6565       5      1313 
  18 peace     6565       6      1094.
```
]]
---
class: split-40
count: false
.column[.content[

```r
party_votes %>%
  crossing(divisor = 1:num_seats) %>%
  mutate(dh_outcome = votes/divisor) ->
*dhondt_calc
```
]]
.column[.content[

]]

---

# Step 3: Allocating seats

Finally allocate seats using the D’Hondt/Jefferson Allocation method, we rank our "D'Hondt Outcomes", and hand out all available seats to the parties with the top D'Hondt outcomes.

---

```r
*dhondt_calc
```
]]
.column[.content[

```r
dhondt_calc %>%
* arrange(-dh_outcome)
```
]]
.column[.content[

```
  # A tibble: 18 x 4
     party    votes divisor dh_outcome
     <chr>    <int>   <int>      <dbl>
   1 equality  6822       1      6822 
   2 peace     6565       1      6565 
   3 justice   4548       1      4548 
   4 equality  6822       2      3411 
   5 peace     6565       2      3282.
   6 equality  6822       3      2274 
   7 justice   4548       2      2274 
   8 peace     6565       3      2188.
   9 equality  6822       4      1706.
  10 peace     6565       4      1641.
  11 justice   4548       3      1516 
  12 equality  6822       5      1364.
  13 peace     6565       5      1313 
  14 equality  6822       6      1137 
  15 justice   4548       4      1137 
  16 peace     6565       6      1094.
  17 justice   4548       5       910.
  18 justice   4548       6       758
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  arrange(-dh_outcome) %>%
* slice(1:num_seats)
```
]]
.column[.content[

```
  # A tibble: 6 x 4
    party    votes divisor dh_outcome
    <chr>    <int>   <int>      <dbl>
  1 equality  6822       1      6822 
  2 peace     6565       1      6565 
  3 justice   4548       1      4548 
  4 equality  6822       2      3411 
  5 peace     6565       2      3282.
  6 equality  6822       3      2274
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  arrange(-dh_outcome) %>%
  slice(1:num_seats) %>%
* count(party)
```
]]
.column[.content[

```
  # A tibble: 3 x 2
    party        n
    <chr>    <int>
  1 equality     3
  2 justice      1
  3 peace        2
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  arrange(-dh_outcome) %>%
  slice(1:num_seats) %>%
  count(party) %>%
* rename(seats_won = n)   # Done!
```
]]
.column[.content[

```
  # A tibble: 3 x 2
    party    seats_won
    <chr>        <int>
  1 equality         3
  2 justice          1
  3 peace            2
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  arrange(-dh_outcome) %>%
  slice(1:num_seats) %>%
  count(party) %>%
  rename(seats_won = n) ->  # Done!
*party_seats
```
]]
.column[.content[

]]

---

# Visualization

Now we have calculated the seat allocation for the election.    But it might be fun to also visualize this.

We'll represent the data associated with the parties with colored columns, the height of which is the "D'Hondt outcome".

For each of the steps used in the method, specify positions for each of the calculation steps. The variables t1 through t4 are the positions at different steps.

- t1 will just show the total vote counts leaving space for those counts to be divided through by 1, 2,... [total number of votes].
- t2 shows the resultant "D'Hondt outcomes", grouped by party
- t3 shows the rank among all the "D'Hondt outcomes"
- t4 will show only the "D'Hondt outcomes" that are ranked high enough to earn a seat

---

```r
*dhondt_calc
```
]]
.column[.content[

```r
dhondt_calc %>%
* mutate(position_t2 = 1:n())
```
]]
.column[.content[

```
  # A tibble: 18 x 5
     party    votes divisor dh_outcome position_t2
     <chr>    <int>   <int>      <dbl>       <int>
   1 equality  6822       1      6822            1
   2 equality  6822       2      3411            2
   3 equality  6822       3      2274            3
   4 equality  6822       4      1706.           4
   5 equality  6822       5      1364.           5
   6 equality  6822       6      1137            6
   7 justice   4548       1      4548            7
   8 justice   4548       2      2274            8
   9 justice   4548       3      1516            9
  10 justice   4548       4      1137           10
  11 justice   4548       5       910.          11
  12 justice   4548       6       758           12
  13 peace     6565       1      6565           13
  14 peace     6565       2      3282.          14
  15 peace     6565       3      2188.          15
  16 peace     6565       4      1641.          16
  17 peace     6565       5      1313           17
  18 peace     6565       6      1094.          18
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  mutate(position_t2 = 1:n()) %>%
* group_by(party)
```
]]
.column[.content[

```
  # A tibble: 18 x 5
  # Groups:   party [3]
     party    votes divisor dh_outcome position_t2
     <chr>    <int>   <int>      <dbl>       <int>
   1 equality  6822       1      6822            1
   2 equality  6822       2      3411            2
   3 equality  6822       3      2274            3
   4 equality  6822       4      1706.           4
   5 equality  6822       5      1364.           5
   6 equality  6822       6      1137            6
   7 justice   4548       1      4548            7
   8 justice   4548       2      2274            8
   9 justice   4548       3      1516            9
  10 justice   4548       4      1137           10
  11 justice   4548       5       910.          11
  12 justice   4548       6       758           12
  13 peace     6565       1      6565           13
  14 peace     6565       2      3282.          14
  15 peace     6565       3      2188.          15
  16 peace     6565       4      1641.          16
  17 peace     6565       5      1313           17
  18 peace     6565       6      1094.          18
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  mutate(position_t2 = 1:n()) %>%
  group_by(party) %>%
* mutate(position_t1 = min(position_t2))
```
]]
.column[.content[

```
  # A tibble: 18 x 6
  # Groups:   party [3]
     party    votes divisor dh_outcome position_t2 position_t1
     <chr>    <int>   <int>      <dbl>       <int>       <dbl>
   1 equality  6822       1      6822            1           1
   2 equality  6822       2      3411            2           1
   3 equality  6822       3      2274            3           1
   4 equality  6822       4      1706.           4           1
   5 equality  6822       5      1364.           5           1
   6 equality  6822       6      1137            6           1
   7 justice   4548       1      4548            7           7
   8 justice   4548       2      2274            8           7
   9 justice   4548       3      1516            9           7
  10 justice   4548       4      1137           10           7
  11 justice   4548       5       910.          11           7
  12 justice   4548       6       758           12           7
  13 peace     6565       1      6565           13          13
  14 peace     6565       2      3282.          14          13
  15 peace     6565       3      2188.          15          13
  16 peace     6565       4      1641.          16          13
  17 peace     6565       5      1313           17          13
  18 peace     6565       6      1094.          18          13
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  mutate(position_t2 = 1:n()) %>%
  group_by(party) %>%
  mutate(position_t1 = min(position_t2)) %>%
* ungroup()
```
]]
.column[.content[

```
  # A tibble: 18 x 6
     party    votes divisor dh_outcome position_t2 position_t1
     <chr>    <int>   <int>      <dbl>       <int>       <dbl>
   1 equality  6822       1      6822            1           1
   2 equality  6822       2      3411            2           1
   3 equality  6822       3      2274            3           1
   4 equality  6822       4      1706.           4           1
   5 equality  6822       5      1364.           5           1
   6 equality  6822       6      1137            6           1
   7 justice   4548       1      4548            7           7
   8 justice   4548       2      2274            8           7
   9 justice   4548       3      1516            9           7
  10 justice   4548       4      1137           10           7
  11 justice   4548       5       910.          11           7
  12 justice   4548       6       758           12           7
  13 peace     6565       1      6565           13          13
  14 peace     6565       2      3282.          14          13
  15 peace     6565       3      2188.          15          13
  16 peace     6565       4      1641.          16          13
  17 peace     6565       5      1313           17          13
  18 peace     6565       6      1094.          18          13
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  mutate(position_t2 = 1:n()) %>%
  group_by(party) %>%
  mutate(position_t1 = min(position_t2)) %>%
  ungroup() %>%
* arrange(-dh_outcome)
```
]]
.column[.content[

```
  # A tibble: 18 x 6
     party    votes divisor dh_outcome position_t2 position_t1
     <chr>    <int>   <int>      <dbl>       <int>       <dbl>
   1 equality  6822       1      6822            1           1
   2 peace     6565       1      6565           13          13
   3 justice   4548       1      4548            7           7
   4 equality  6822       2      3411            2           1
   5 peace     6565       2      3282.          14          13
   6 equality  6822       3      2274            3           1
   7 justice   4548       2      2274            8           7
   8 peace     6565       3      2188.          15          13
   9 equality  6822       4      1706.           4           1
  10 peace     6565       4      1641.          16          13
  11 justice   4548       3      1516            9           7
  12 equality  6822       5      1364.           5           1
  13 peace     6565       5      1313           17          13
  14 equality  6822       6      1137            6           1
  15 justice   4548       4      1137           10           7
  16 peace     6565       6      1094.          18          13
  17 justice   4548       5       910.          11           7
  18 justice   4548       6       758           12           7
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  mutate(position_t2 = 1:n()) %>%
  group_by(party) %>%
  mutate(position_t1 = min(position_t2)) %>%
  ungroup() %>%
  arrange(-dh_outcome) %>%
* mutate(position_t3 = 1:n())
```
]]
.column[.content[

```
  # A tibble: 18 x 7
     party    votes divisor dh_outcome position_t2 position_t1 position_t3
     <chr>    <int>   <int>      <dbl>       <int>       <dbl>       <int>
   1 equality  6822       1      6822            1           1           1
   2 peace     6565       1      6565           13          13           2
   3 justice   4548       1      4548            7           7           3
   4 equality  6822       2      3411            2           1           4
   5 peace     6565       2      3282.          14          13           5
   6 equality  6822       3      2274            3           1           6
   7 justice   4548       2      2274            8           7           7
   8 peace     6565       3      2188.          15          13           8
   9 equality  6822       4      1706.           4           1           9
  10 peace     6565       4      1641.          16          13          10
  11 justice   4548       3      1516            9           7          11
  12 equality  6822       5      1364.           5           1          12
  13 peace     6565       5      1313           17          13          13
  14 equality  6822       6      1137            6           1          14
  15 justice   4548       4      1137           10           7          15
  16 peace     6565       6      1094.          18          13          16
  17 justice   4548       5       910.          11           7          17
  18 justice   4548       6       758           12           7          18
```
]]
---
class: split-40
count: false
.column[.content[

```
  # A tibble: 18 x 8
     party votes divisor dh_outcome position_t2 position_t1 position_t3
     <chr> <int>   <int>      <dbl>       <int>       <dbl>       <int>
   1 equa…  6822       1      6822            1           1           1
   2 peace  6565       1      6565           13          13           2
   3 just…  4548       1      4548            7           7           3
   4 equa…  6822       2      3411            2           1           4
   5 peace  6565       2      3282.          14          13           5
   6 equa…  6822       3      2274            3           1           6
   7 just…  4548       2      2274            8           7           7
   8 peace  6565       3      2188.          15          13           8
   9 equa…  6822       4      1706.           4           1           9
  10 peace  6565       4      1641.          16          13          10
  11 just…  4548       3      1516            9           7          11
  12 equa…  6822       5      1364.           5           1          12
  13 peace  6565       5      1313           17          13          13
  14 equa…  6822       6      1137            6           1          14
  15 just…  4548       4      1137           10           7          15
  16 peace  6565       6      1094.          18          13          16
  17 just…  4548       5       910.          11           7          17
  18 just…  4548       6       758           12           7          18
  # … with 1 more variable: position_t4 <int>
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  mutate(position_t2 = 1:n()) %>%
  group_by(party) %>%
  mutate(position_t1 = min(position_t2)) %>%
  ungroup() %>%
  arrange(-dh_outcome) %>%
  mutate(position_t3 = 1:n()) %>%
  mutate(position_t4 = ifelse(position_t3 > num_seats,
                              NA, position_t3)) %>%
* tidyr::gather(key = "step",
*               value = "position",
*               position_t2:position_t4)
```
]]
.column[.content[

```
  # A tibble: 72 x 6
     party    votes divisor dh_outcome step        position
     <chr>    <int>   <int>      <dbl> <chr>          <dbl>
   1 equality  6822       1      6822  position_t2        1
   2 peace     6565       1      6565  position_t2       13
   3 justice   4548       1      4548  position_t2        7
   4 equality  6822       2      3411  position_t2        2
   5 peace     6565       2      3282. position_t2       14
   6 equality  6822       3      2274  position_t2        3
   7 justice   4548       2      2274  position_t2        8
   8 peace     6565       3      2188. position_t2       15
   9 equality  6822       4      1706. position_t2        4
  10 peace     6565       4      1641. position_t2       16
  # … with 62 more rows
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  mutate(position_t2 = 1:n()) %>%
  group_by(party) %>%
  mutate(position_t1 = min(position_t2)) %>%
  ungroup() %>%
  arrange(-dh_outcome) %>%
  mutate(position_t3 = 1:n()) %>%
  mutate(position_t4 = ifelse(position_t3 > num_seats,
                              NA, position_t3)) %>%
  tidyr::gather(key = "step",
                value = "position",
                position_t2:position_t4) %>%
* mutate(step = as.numeric(as.factor(step)))
```
]]
.column[.content[

```
  # A tibble: 72 x 6
     party    votes divisor dh_outcome  step position
     <chr>    <int>   <int>      <dbl> <dbl>    <dbl>
   1 equality  6822       1      6822      2        1
   2 peace     6565       1      6565      2       13
   3 justice   4548       1      4548      2        7
   4 equality  6822       2      3411      2        2
   5 peace     6565       2      3282.     2       14
   6 equality  6822       3      2274      2        3
   7 justice   4548       2      2274      2        8
   8 peace     6565       3      2188.     2       15
   9 equality  6822       4      1706.     2        4
  10 peace     6565       4      1641.     2       16
  # … with 62 more rows
```
]]
---
class: split-40
count: false
.column[.content[

```r
dhondt_calc %>%
  mutate(position_t2 = 1:n()) %>%
  group_by(party) %>%
  mutate(position_t1 = min(position_t2)) %>%
  ungroup() %>%
  arrange(-dh_outcome) %>%
  mutate(position_t3 = 1:n()) %>%
  mutate(position_t4 = ifelse(position_t3 > num_seats,
                              NA, position_t3)) %>%
  tidyr::gather(key = "step",
                value = "position",
                position_t2:position_t4) %>%
  mutate(step = as.numeric(as.factor(step))) ->
*animation_prep
```
]]
.column[.content[

]]

---

# Build static plot for one time period, i.e. a single step in the method.

---

```r
*animation_prep
```
]]
.column[.content[

```r
animation_prep %>%
* # plotting single time period
* filter(step == 2)
```
]]
.column[.content[

```
  # A tibble: 18 x 6
     party    votes divisor dh_outcome  step position
     <chr>    <int>   <int>      <dbl> <dbl>    <dbl>
   1 equality  6822       1      6822      2        1
   2 peace     6565       1      6565      2       13
   3 justice   4548       1      4548      2        7
   4 equality  6822       2      3411      2        2
   5 peace     6565       2      3282.     2       14
   6 equality  6822       3      2274      2        3
   7 justice   4548       2      2274      2        8
   8 peace     6565       3      2188.     2       15
   9 equality  6822       4      1706.     2        4
  10 peace     6565       4      1641.     2       16
  11 justice   4548       3      1516      2        9
  12 equality  6822       5      1364.     2        5
  13 peace     6565       5      1313      2       17
  14 equality  6822       6      1137      2        6
  15 justice   4548       4      1137      2       10
  16 peace     6565       6      1094.     2       18
  17 justice   4548       5       910.     2       11
  18 justice   4548       6       758      2       12
```
]]
---
class: split-40
count: false
.column[.content[

```r
animation_prep %>%
  # plotting single time period
  filter(step == 2) %>%
* ggplot()
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_static_4-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
animation_prep %>%
  # plotting single time period
  filter(step == 2) %>%
  ggplot() +
* aes(xmin = position + .4,
*     xmax = position - .4,
*     ymin = 0,
*     ymax = dh_outcome,
*     fill = party,
*     group = paste(party, divisor),
*     alpha = as.numeric(step == 3))
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_static_11-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
animation_prep %>%
  # plotting single time period
  filter(step == 2) %>%
  ggplot() +
  aes(xmin = position + .4,
      xmax = position - .4,
      ymin = 0,
      ymax = dh_outcome,
      fill = party,
      group = paste(party, divisor),
      alpha = as.numeric(step == 3)) +
  scale_fill_viridis_d() +
  geom_rect(alpha = .7) +
  geom_vline(xintercept = 0.25,
             col = "slategray",
             size = 1.25) +
  geom_vline(xintercept = c(6.5, 12.5),
             col = "slategray",
             size = 1,
             linetype = "dotted") +
  labs(title = "D’Hondt/Jefferson Seat Allocation") +
  labs(subtitle = "Data: Hypothetical parties and results") +
* labs(caption = "Visualization: Gina Reynolds @EvaMaeRey in R with ggplot and gganimate")
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_static_23-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
animation_prep %>%
  # plotting single time period
  filter(step == 2) %>%
  ggplot() +
  aes(xmin = position + .4,
      xmax = position - .4,
      ymin = 0,
      ymax = dh_outcome,
      fill = party,
      group = paste(party, divisor),
      alpha = as.numeric(step == 3)) +
  scale_fill_viridis_d() +
  geom_rect(alpha = .7) +
  geom_vline(xintercept = 0.25,
             col = "slategray",
             size = 1.25) +
  geom_vline(xintercept = c(6.5, 12.5),
             col = "slategray",
             size = 1,
             linetype = "dotted") +
  labs(title = "D’Hondt/Jefferson Seat Allocation") +
  labs(subtitle = "Data: Hypothetical parties and results") +
  labs(caption = "Visualization: Gina Reynolds @EvaMaeRey in R with ggplot and gganimate") +
* labs(x = "Number of Seats (divisor)")
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_static_24-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
animation_prep %>%
  # plotting single time period
  filter(step == 2) %>%
  ggplot() +
  aes(xmin = position + .4,
      xmax = position - .4,
      ymin = 0,
      ymax = dh_outcome,
      fill = party,
      group = paste(party, divisor),
      alpha = as.numeric(step == 3)) +
  scale_fill_viridis_d() +
  geom_rect(alpha = .7) +
  geom_vline(xintercept = 0.25,
             col = "slategray",
             size = 1.25) +
  geom_vline(xintercept = c(6.5, 12.5),
             col = "slategray",
             size = 1,
             linetype = "dotted") +
  labs(title = "D’Hondt/Jefferson Seat Allocation") +
  labs(subtitle = "Data: Hypothetical parties and results") +
  labs(caption = "Visualization: Gina Reynolds @EvaMaeRey in R with ggplot and gganimate") +
  labs(x = "Number of Seats (divisor)") +
* labs(y = "Number of Votes\ndivided by 1, 2, ..., N")
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_static_25-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
animation_prep %>%
  # plotting single time period
  filter(step == 2) %>%
  ggplot() +
  aes(xmin = position + .4,
      xmax = position - .4,
      ymin = 0,
      ymax = dh_outcome,
      fill = party,
      group = paste(party, divisor),
      alpha = as.numeric(step == 3)) +
  scale_fill_viridis_d() +
  geom_rect(alpha = .7) +
  geom_vline(xintercept = 0.25,
             col = "slategray",
             size = 1.25) +
  geom_vline(xintercept = c(6.5, 12.5),
             col = "slategray",
             size = 1,
             linetype = "dotted") +
  labs(title = "D’Hondt/Jefferson Seat Allocation") +
  labs(subtitle = "Data: Hypothetical parties and results") +
  labs(caption = "Visualization: Gina Reynolds @EvaMaeRey in R with ggplot and gganimate") +
  labs(x = "Number of Seats (divisor)") +
  labs(y = "Number of Votes\ndivided by 1, 2, ..., N") +
* scale_y_continuous(expand = c(.02, .02))
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_static_26-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
animation_prep %>%
  # plotting single time period
  filter(step == 2) %>%
  ggplot() +
  aes(xmin = position + .4,
      xmax = position - .4,
      ymin = 0,
      ymax = dh_outcome,
      fill = party,
      group = paste(party, divisor),
      alpha = as.numeric(step == 3)) +
  scale_fill_viridis_d() +
  geom_rect(alpha = .7) +
  geom_vline(xintercept = 0.25,
             col = "slategray",
             size = 1.25) +
  geom_vline(xintercept = c(6.5, 12.5),
             col = "slategray",
             size = 1,
             linetype = "dotted") +
  labs(title = "D’Hondt/Jefferson Seat Allocation") +
  labs(subtitle = "Data: Hypothetical parties and results") +
  labs(caption = "Visualization: Gina Reynolds @EvaMaeRey in R with ggplot and gganimate") +
  labs(x = "Number of Seats (divisor)") +
  labs(y = "Number of Votes\ndivided by 1, 2, ..., N") +
  scale_y_continuous(expand = c(.02, .02)) +
* scale_x_continuous(breaks = 1:(num_parties * num_seats),
*                    labels = rep(1:num_seats, num_parties),
*                    expand = c(.01, .01))
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_static_29-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
animation_prep %>%
  # plotting single time period
  filter(step == 2) %>%
  ggplot() +
  aes(xmin = position + .4,
      xmax = position - .4,
      ymin = 0,
      ymax = dh_outcome,
      fill = party,
      group = paste(party, divisor),
      alpha = as.numeric(step == 3)) +
  scale_fill_viridis_d() +
  geom_rect(alpha = .7) +
  geom_vline(xintercept = 0.25,
             col = "slategray",
             size = 1.25) +
  geom_vline(xintercept = c(6.5, 12.5),
             col = "slategray",
             size = 1,
             linetype = "dotted") +
  labs(title = "D’Hondt/Jefferson Seat Allocation") +
  labs(subtitle = "Data: Hypothetical parties and results") +
  labs(caption = "Visualization: Gina Reynolds @EvaMaeRey in R with ggplot and gganimate") +
  labs(x = "Number of Seats (divisor)") +
  labs(y = "Number of Votes\ndivided by 1, 2, ..., N") +
  scale_y_continuous(expand = c(.02, .02)) +
  scale_x_continuous(breaks = 1:(num_parties * num_seats),
                     labels = rep(1:num_seats, num_parties),
                     expand = c(.01, .01)) +
* scale_alpha_continuous(range = c(0, 1),
*                        guide = F)
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_static_31-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
animation_prep %>%
  # plotting single time period
  filter(step == 2) %>%
  ggplot() +
  aes(xmin = position + .4,
      xmax = position - .4,
      ymin = 0,
      ymax = dh_outcome,
      fill = party,
      group = paste(party, divisor),
      alpha = as.numeric(step == 3)) +
  scale_fill_viridis_d() +
  geom_rect(alpha = .7) +
  geom_vline(xintercept = 0.25,
             col = "slategray",
             size = 1.25) +
  geom_vline(xintercept = c(6.5, 12.5),
             col = "slategray",
             size = 1,
             linetype = "dotted") +
  labs(title = "D’Hondt/Jefferson Seat Allocation") +
  labs(subtitle = "Data: Hypothetical parties and results") +
  labs(caption = "Visualization: Gina Reynolds @EvaMaeRey in R with ggplot and gganimate") +
  labs(x = "Number of Seats (divisor)") +
  labs(y = "Number of Votes\ndivided by 1, 2, ..., N") +
  scale_y_continuous(expand = c(.02, .02)) +
  scale_x_continuous(breaks = 1:(num_parties * num_seats),
                     labels = rep(1:num_seats, num_parties),
                     expand = c(.01, .01)) +
  scale_alpha_continuous(range = c(0, 1),
                         guide = F) ->
*single_time_period_gg
```
]]
.column[.content[

]]

---

# Some theme fine tuning, if you please

---

```r
*single_time_period_gg
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_theme_1-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
single_time_period_gg +
* theme(axis.ticks = element_blank())
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_theme_2-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
single_time_period_gg +
  theme(axis.ticks = element_blank()) +
* theme(legend.background = element_rect(fill = "gainsboro"))
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_theme_3-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
single_time_period_gg +
  theme(axis.ticks = element_blank()) +
  theme(legend.background = element_rect(fill = "gainsboro")) +
* theme(plot.background = element_rect(fill = "gainsboro"))
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_theme_4-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
single_time_period_gg +
  theme(axis.ticks = element_blank()) +
  theme(legend.background = element_rect(fill = "gainsboro")) +
  theme(plot.background = element_rect(fill = "gainsboro")) +
  theme(panel.background = element_rect(fill = "gainsboro")) +
  theme(panel.grid = element_line(color = "plum4", size = .2)) +
  theme(panel.grid.minor = element_blank()) +
  theme(text = element_text(family = "Helvetica", size = 15)) +
  theme(plot.title = element_text(face = "bold")) +
* theme(legend.position = c(.85, .8))
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_theme_10-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
single_time_period_gg +
  theme(axis.ticks = element_blank()) +
  theme(legend.background = element_rect(fill = "gainsboro")) +
  theme(plot.background = element_rect(fill = "gainsboro")) +
  theme(panel.background = element_rect(fill = "gainsboro")) +
  theme(panel.grid = element_line(color = "plum4", size = .2)) +
  theme(panel.grid.minor = element_blank()) +
  theme(text = element_text(family = "Helvetica", size = 15)) +
  theme(plot.title = element_text(face = "bold")) +
  theme(legend.position = c(.85, .8)) +
* theme(legend.title = element_blank())
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_theme_11-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
single_time_period_gg +
  theme(axis.ticks = element_blank()) +
  theme(legend.background = element_rect(fill = "gainsboro")) +
  theme(plot.background = element_rect(fill = "gainsboro")) +
  theme(panel.background = element_rect(fill = "gainsboro")) +
  theme(panel.grid = element_line(color = "plum4", size = .2)) +
  theme(panel.grid.minor = element_blank()) +
  theme(text = element_text(family = "Helvetica", size = 15)) +
  theme(plot.title = element_text(face = "bold")) +
  theme(legend.position = c(.85, .8)) +
  theme(legend.title = element_blank()) +
* theme(legend.key = element_rect(color = "gainsboro"))
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_theme_12-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
single_time_period_gg +
  theme(axis.ticks = element_blank()) +
  theme(legend.background = element_rect(fill = "gainsboro")) +
  theme(plot.background = element_rect(fill = "gainsboro")) +
  theme(panel.background = element_rect(fill = "gainsboro")) +
  theme(panel.grid = element_line(color = "plum4", size = .2)) +
  theme(panel.grid.minor = element_blank()) +
  theme(text = element_text(family = "Helvetica", size = 15)) +
  theme(plot.title = element_text(face = "bold")) +
  theme(legend.position = c(.85, .8)) +
  theme(legend.title = element_blank()) +
  theme(legend.key = element_rect(color = "gainsboro")) +
* theme(axis.title.y = element_text(hjust = 1,
*                                   size = 10))
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_theme_14-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
single_time_period_gg +
  theme(axis.ticks = element_blank()) +
  theme(legend.background = element_rect(fill = "gainsboro")) +
  theme(plot.background = element_rect(fill = "gainsboro")) +
  theme(panel.background = element_rect(fill = "gainsboro")) +
  theme(panel.grid = element_line(color = "plum4", size = .2)) +
  theme(panel.grid.minor = element_blank()) +
  theme(text = element_text(family = "Helvetica", size = 15)) +
  theme(plot.title = element_text(face = "bold")) +
  theme(legend.position = c(.85, .8)) +
  theme(legend.title = element_blank()) +
  theme(legend.key = element_rect(color = "gainsboro")) +
  theme(axis.title.y = element_text(hjust = 1,
                                    size = 10))  +
* theme(axis.title.x = element_text(hjust = 0,
*                                   size = 10))
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_theme_16-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
single_time_period_gg +
  theme(axis.ticks = element_blank()) +
  theme(legend.background = element_rect(fill = "gainsboro")) +
  theme(plot.background = element_rect(fill = "gainsboro")) +
  theme(panel.background = element_rect(fill = "gainsboro")) +
  theme(panel.grid = element_line(color = "plum4", size = .2)) +
  theme(panel.grid.minor = element_blank()) +
  theme(text = element_text(family = "Helvetica", size = 15)) +
  theme(plot.title = element_text(face = "bold")) +
  theme(legend.position = c(.85, .8)) +
  theme(legend.title = element_blank()) +
  theme(legend.key = element_rect(color = "gainsboro")) +
  theme(axis.title.y = element_text(hjust = 1,
                                    size = 10))  +
  theme(axis.title.x = element_text(hjust = 0,
                                    size = 10))  ->
*static_w_theme
```
]]
.column[.content[

]]

---

# Animation

---

```r
*static_w_theme %+%
* # use ALL the data.
* # %+% is a special ggplot operator to
* # reset ggplot global data
* animation_prep
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_the_animation_5-1.png" width="100%" />
]]
---
class: split-40
count: false
.column[.content[

```r
static_w_theme %+%
  # use ALL the data.
  # %+% is a special ggplot operator to
  # reset ggplot global data
  animation_prep +
* # add animation declarations
* gganimate::transition_time(time = step) +
* gganimate::ease_aes("exponential-in-out")
```
]]
.column[.content[
<img src="dhondt_jefferson_allocation_files/figure-html/output_the_animation_8-1.gif" width="100%" />
]]

---

# Conclusion

A seat is awarded for each party column left in the final step of the visualization.

---

# Implications

In the opening lines of his 1973 paper, "The Relationship between Seats and Votes in Two-Party Systems", Edward Tufte makes the observation:

> *"An enduring fact of life in democratic electoral systems is that the party winning the largest share of the votes almost always receives a still larger share of the seats."*

While clearly qualified, Tufte "winner-takes-more" generalization is indeed true of D'Hondt seat allocation.  Let's compare the share of votes won in our example versus the share of seats won for each party in our hypothetical example.

---

# The End!

Thanks for joining!

The code for this work lives [here](https://github.com/EvaMaeRey/little_flipbooks_library).  Open to suggestions. Send me a pull request.

Interested in building your own flipbook? It is fun! The code is still underdevelopment, and we'd love to have your feedback. A minimal example is [here]( https://evamaerey.github.io/little_flipbooks_library/tidytuesday_minimal_example/tidytuesday_minimal_example#1).