Why the D’Hondt system is a balls-up

Given today’s European elections counted on mainland UK using the D’Hondt system (in Northern Ireland using STV), I thought I’d better update my earlier post on how democracy works.

Guinness

And the winner is: Guinness (but only proportionately, using the D’Hondt system)

If you remember, a rugby team (15 blokes) walk into a bar. The player holding the whip asks what everyone wants to drink. Five of them want be in the pub down the road drinking real ale, so don’t care what they’re going to drink in this bar and take no part (abstain).

Of those voting three want Guinness, two lager, two cider, two want rum and coke and one wants a glass of red wine.

Under the D’Hondt system, the round needs to have 15 drinks. Each time a drink is added to the round its votes are reduced by the number of times it has already been added, plus one.

Count 1

  • Guinness 3
  • Lager 2
  • Cider 2
  • Rum and coke 2
  • Red wine 1

Guinness gets added to the round. The initial total for Guinness is now divided by two (one drink elected to the round, plus one).

Count 2

  • Lager 2
  • Cider 2
  • Rum and coke 2
  • Guinness 1.5
  • Red wine 1

Lager, cider and rum and coke get added to the round (four drinks decided all one each). The totals of all those on the round are now divided by two (one on the round, plus one).

Count 3

  • Guinness 1.5
  • Lager 1
  • Cider 1
  • Rum and coke 1
  • Red wine 1

Another Guinness gets on the round (five drinks decided, two Guinness, one of the others). The totals votes for Guinness (3) is now divided by three (two on the round plus one).

Count 4

  • Guinness 1
  • Lager 1
  • Cider 1
  • Rum and coke 1
  • Red wine 1

Because there are enough spaces available, in the event of this kind of tie, all get added to the round (10 drinks, three Guinness, others all with two except red wine on one). Had there been only one space available the tied votes would have been listed based on the initial weight of votes, so Guinness would have been at the top and would have been the only one getting through.

The Guinness total vote is now divided by four (three on the round plus one). The initial votes for lager, cider and rum and coke are divided by three (two on the round plus one) and the initial vote for red wine is divided by two (one on the round plus one).

Count 5

  • Guinness 0.75
  • Lager 0.67
  • Cider 0.67
  • Rum and coke 0.67
  • Red wine 0.5

Another Guinness gets added to the round (11 drinks, Guinness with four, red wine one, rest two). The initial total vote for Guinness (3) is now divided by five (four elected to the round, plus one)

Count 6

  • Lager 0.67
  • Cider 0.67
  • Rum and coke 0.67
  • Guinness 0.6
  • Re wine 0.5

Lager, cider and rum and coke are added to the round (14 drinks, four Guinness, all others have three except red wine with one). Lager , cider and rum and coke’s initial votes are now divided by four (three on the round , plus one).

Count 7

  • Guinness 0.6
  • Lager 0.5
  • Cider 0.5
  • Rum and coke 0.5
  • Red wine 0.5

Another Guinness is added (15 drinks, five Guinness, three lagers, ciders and rum and cokes, plus one red wine).

The round:

  • 5 pints of Guinness
  • 3 pints of lager
  • 3 pints of cider
  • 3 rum and cokes
  • 1 red wine.

The rugby players still need to decide who drinks what. The five who wanted real ale will take the Guinness. One of those who wanted Guinness takes the red wine and the other two take a lager and cider each. The red wine voter takes a rum and coke. Eight of the team (53%) get the drink they actually wanted.

Obviously this is not realistic. In EU elections the turnout is normally about 40% not the 66% in this scenario, so usually an even small number of people disproportionately impose their will on everyone else.

Democracy is:

rugby balls

Balls!

Many thanks to  Christina Pagel, Professor of operational research at UCL, who wrote about the D’Hondt system much more sensibly on website The UK in a Changing Europe and allowed me to bounce back stupid questions by email this afternoon.

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