# And Also Adams

In my last post I showed you the seat allocation method, called Sainte-Laguë/Schepers method. I recommend you reading it, before you continue with this post, as this post about the D’Hondt method heavily builds upon it.

There are actually two seat allocation methods, that are pretty similar to Sainte-Laguë/Schepers. One of them being the D’Hondt 1 method. The other one being the Adam’s method2.
Both are really remarkably similar to the method from my last post. With really the only difference being the way they round the seats. While Sainte-Laguë/Schepers uses standard round, D’Hondt uses the floor function. Meaning that it always round to the next lower integer. And Adams uses the ceiling function, which round to the next higher integer.

Now immediately you should scream: “STOP! What?! Adams uses the ceiling function?! So does this mean, that every party, that gets votes, gets at least one seat?!” The answer would be yesish. Yes, if your election doesn’t have an election threshold, every party, that gets votes, would at least get one seat.
“Well isn’t that incredibly unfavorable?” Yepp… But there are cases, where an allocation method like this could make sense. Not for regular election in my opinion, but for elections in parliaments. Let’s say, you have 20 mandates in a parliament, you want to distribute in a parliament with 300 elected politicians. Then the consideration, that it would be fair, if every party will get at least one mandate, could be made.

However the Adams method is incredibly uncommon. This Wikipedia article, which also served as my source for the method, only mentions the French parliament as example.

The D’Hondt method on the other side is pretty common. It’s actually the most common one in this year’s EU election.
And my source was also the corresponding German Wikipedia article.

## Implementation D’Hondt Method

Luckily I don’t have to do much to implement those two methods. I just have to change a little bit about my function from last time.

seatAllocation <- function(votes, seats, roundMethod = round){
## calculate the initial divisor
divisor <- roundMethod(sum(votes) / seats)

## get the initial seats per party
seatsPerParty <- roundMethod(votes / divisor)

## if they already satisfy the seats to be assigned, return the seat allocation
if(sum(seatsPerParty) == seats){
return(seatsPerParty)
}

## otherwise increment or decrement the divisor until
## the result fits and then return it
if(sum(seatsPerParty) < seats){
while(sum(seatsPerParty) < seats){
divisor = divisor - 1
seatsPerParty <- roundMethod(votes / divisor)
}
return(seatsPerParty)
}else{
while(sum(seatsPerParty) > seats){
divisor = divisor + 1
seatsPerParty <- roundMethod(votes / divisor)
}
return(seatsPerParty = seatsPerParty)
}

}


You see, what I did there? And why I love functional programming? Now by default, it’s the Sainte-Laguë/Schepers method and through giving the parameter roundMethod either the floor or ceiling function, we can make the D’Hondt and respectively Adams method out of it.
And we could even come up with some other rounding function in the future and use it.

## Test and Compare The Methods

And without further a due let’s test and compare the methods on our previous example.

votes <- c(AP = 11345, CVP = 563342, EP = 618713, OSP = 305952, PDP = 95001)
seatsSLS <- seatAllocation(votes, seats = 310, roundMethod = round)
seatsDH <- seatAllocation(votes, seats = 310, roundMethod = floor)
seatsA <- seatAllocation(votes, seats = 310, roundMethod = ceiling)

library(data.table)
DT <- rbind(data.table(party = names(seatsA), seats = seatsA, method = "Adams"),
data.table(party = names(seatsSLS), seats = seatsSLS, method = "Sainte-Laguë/Schepers"),
data.table(party = names(seatsDH), seats = seatsDH, method = "D'Hondt"))

library(ggplot2)
g <- ggplot(DT, aes(x = party, y = seats, fill = method))
g <- g + geom_bar(stat = "identity", position = "dodge")
g <- g + geom_text(aes(label=seats), position=position_dodge(width=0.9), vjust=-0.25)
g


Thanks, stackoverflow!
And you see… The actual difference isn’t big at all. The only thing one could say, is that Adams give a bonus to the small parties. D’Hondt method favors the big ones a bit. And Sainte-Laguë/Schepers is somehow in the middle.

And for me at least it’s really hard to say, which one is favorable. Sainte-Laguë/Schepers seems like a good compromise. However the differences more or less only affect small parties. But for them the difference is important. This doesn’t mean, that there’s no difference for large parties. On seat could mean the difference between majority and well… Not majority. Especially if you factor coalitions into the mix.
Maybe we will talk about possible problems in one of my next posts. I’m beginning to like this topic. I’m already thinking about becoming a lobbyist… lol.

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# For Allocation of Seats in the EU Parliament

On Monday I had a talk over Discord with Boris Biba, who himself runs a blog. We wanted to do a cooperation for some time. The focus of his blog are philosophy and politics. And as I told him, that I’m interested in crunching numbers, the comming EU elections are the perfect opportunity for a cooperation.
First we talked about doing something regarding the Wahl-O-Mat. Now in hindsight it was probably good that we decided for something else, as the Wahl-O-Mat was taken offline just today.
Then Boris brought up that he wanted to a post about the seat allocation method, which is called Sainte-Laguë/Schepers method, for German votes in the EU election. And I thought to myself, that this is wonderful, as voting is basically a paradigm for statistics. So I would be able to implement a small algorithm.

So be also sure to check out the post, which you can find here, from Boris, if you’re able to read German!

What I’ll be doing in this post, is explain you the seat allocation method called Sainte-Laguë/Schepers and then give you a demonstrative example for it. And as an easteregg I throw in some election posters for the imaginary parties, I’ll use in the example. I created those posters with Adobe Spark.

As a main source for my post, I took the corresponding article from the German Wahl-Lexikon.

## Description of the Method

So there are basically three variants of this method, which all deliver the same result.
Two of them work by ranking the voting result. The other one by simple division, which is the one used for the German part of the EU election. It is either called iterative or divisor method.

The simple idea behind this divisor method is to find a divisor for the voting result, which delivers you the right amount of total seats, if you divide the voting results by it and then round them by standard rounding.

To find the right divisor, first the total amount of votes is divided by the number of seats to be assigned.

$$divisor = \frac{\#votesTotal}{\#seats}$$

The for each party the number of votes is divided by this divisor.

$$seatsParty_{i} = \frac{\#votesParty_{i}}{divisor}$$

And if the sum of the seats of all parties matches up with the amount to be assigned, we’re already done!
If not, we have to either increment or decrement the divisor depending on, if we have to few or to many seats.

Just think about that… If you increase the divisor, the amount of seats shrinks. And vice versa if you decrease the divisor, the amount of seats increases.

And so the divisor is adjusted and the final seats per party are obtained.

## Implementation of the Sainte-Laguë/Schepers method

And of course it wouldn’t be me, if I wouldn’t also implement the method.
Here we go…

seatAllocation <- function(votes, seats){
## calculate the initial divisor
divisor <- round(sum(votes) / seats)

## get the initial seats per party
seatsPerParty <- round(votes / divisor)

## if they already satisfy the seats to be assigned, return the seat allocation
if(sum(seatsPerParty) == seats){
return(list(divisor = divisor, seatsPerParty = seatsPerParty))
}

## otherwise increment or decrement the divisor until
## the result fits and then return it
if(sum(seatsPerParty) < seats){
while(sum(seatsPerParty) < seats){
divisor = divisor - 1
seatsPerParty <- round(votes / divisor)
}
return(list(divisor = divisor, seatsPerParty = seatsPerParty))
}else{
while(sum(seatsPerParty) > seats){
divisor = divisor + 1
seatsPerParty <- round(votes / divisor)
}
return(list(divisor = divisor, seatsPerParty = seatsPerParty))
}

}


The function is basically the same as what I described under the last point in plain text. As always, if you have some questions or remarks regarding my implementation feel free to write me a comment!

## Example with the Sainte-Laguë/Schepers method

Now to test the method, let’s just come up with some arbitrary voting result for our imaginary parties introduced earlier. And of course plot them as a pie chart!

votes <- c(AP = 11345, CVP = 563342, EP = 618713, OSP = 305952, PDP = 95001)
votesSorted <- sort(votes, decreasing = TRUE)
names(votesSorted) <- paste0(names(votesSorted), " (",
format(votesSorted/sum(votesSorted) * 100, digits = 2), "%)")
pie(votesSorted)


Subsequently, let’s test what result the method delivers and if the percentages match up approximately.

result <- seatAllocation(votes, 310)


OK, first let’s visualize the result. But let’s not use a pie chart again. Because to be honest they can be misleading. This time we will use a waffle chart, which displays the actual seats.
Of course we also need to do some preprocessing. We want the parties ordered after their size and we won’t their percentage of seats in the legend.

seatsPerParty <- result\$seatsPerParty
seatsPerParty <- sort(seatsPerParty, decreasing = TRUE)
names(seatsPerParty) <- paste0(names(seatsPerParty), " (",
format(seatsPerParty/sum(seatsPerParty) * 100, digits = 2), "%)")
waffle::waffle(seatsPerParty)


Well, there’s some difference in the percentage, but that’s to be expected as you can’t distribute fractions of seats between the parties.

## Outlook

Of course there are many other methods for allocating seats in an election. Some that are equivalent to this one and others that are not. And if you’re interesting in them, I would encourage you to write me.
If you like, we can look at a bunch of them an then compare them. And we could also take a look at things like overhang seat or different kinds of voting. I think it’s a nice topic for making plots.

By the way if you also wanna read this post in German, check the following link out!

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