Tuesday, 9 February 2010

Simpsons Paradox

Electoral Systems 102: Divide and miscount

Another problem with our electoral system is Simpsons' Paradox.

In a nutshell, Simpsons Paradox is where you can divide a set of results into chunks, assess each chunk, and then combine the results of each assessment to draw a conclusion about the whole which disagrees with a direct assessment of the whole.

Or, in less confusing terms, it is how someone can come second in each leg of the tour-de-france yet still come 1st overall: as long as no-one consistantly comes 1st in each leg, the best overall cyclist may well be the one who was always 2nd. Yet, the fact that he came 2nd in each leg makes it appear as if he'd be overall 2nd...

It can occur in more confusing ways, too.

Lets devise an experiment to figure out who is harder: Charlie Chaplin or Mike Tyson?

The experiment can be that we will pit each of them against two opponents — Bruce Lee and a 10-year-old girl — and see who wins most.

Stupid? Yes, but the results are:

vs. 10-year-old girl:

Charlie does suprisingly poorly against the 10-year-old girl, who manages to win nearly half her bouts against him; Mike, tho, predictably wins all of his bouts against the girl. The results are so obvious that not many fights are neccesary to establish that Mike Tyson is harder than a 10-year-old girl:

Charlie Chaplin: 50/90
Mike Tyson: 10/10
WINNER: Tyson, with 100% victory vs. 10-year-old girl, against Charlie's 55%

Against Bruce Lee, Tyson manages to win some, but nonetheless Bruce Lee proves the stronger; Chaplin, again, gets his ass handed to him, but at least this time it's not by a 10-year-old girl:

Charlie Chaplin: 0/10
Mike Tyson: 35/90
WINNER: Tyson, with a 38% success rate vs. Lee, against Chaplin's 0%.



Well, overall they both fought 100 fights: Chaplin won 50 of these, whilst Tyson won only 45, meaning that Chaplin wins more bouts, grabs the trophy and has proven himself in trial to be harder than Tyson, until Tyson decks him, grabs the trophy, and runs off shouting 'bloody Simpsons Paradox!'

Ok, that was all daft, but that's the essence of Simpsons Paradox. In this case, breaking the data down and assessing in chunks gives the correct interpretation (Tyson does best against a girl + Tyson does best against Lee = Tyson is better fighter), whilst in other times it can be the overall view that is correct whilst breaking it into chunks gives a false view.

A really quick example: this time, it's two fighters vs. Frank Bruno:

Trial one:

A: 65/100
B: 7/10
WINNER: B (70%, vs. A's 65%)

Trial two:

A: 4/10
B: 53/100
WINNER: B(53% vs. A's 40%)

OVERALL WINNER: A (69/110 vs. B's 60/110)

Unlike the first example, here the correct interpretation is the overall view, with the breaking-into-chunks-then-combining-results giving a false view.


So our country is divided into constituencies. The votes are tallied per constituency, and then these results are combined to give the national result. This is the situation that you need for Simpsons Paradox to arise, and is how the Labour party managed to 'win' the last election with 55% of the power (i.e., majority control) with only 35% of the vote[1]; hell, on two occasions the 'winner' actually had less votes than the party that came second[2] (on one of those occasions, the winner had enough for a majority control, i.e. they were in charge), and also explains the huge disparity between how many votes Lib Dem get and how much power they end up with:

That first result is appauling: just over a quarter of the vote translating to just over a thirtieth of the power...

In fact, Gerrymandering (see previous post) is essentially an attempt to force Simpsons Paradox in order to grant yourself even more disproportionate power.

Tl; dr?

Dividing the country up into constituencies is the fundamental cause for the disproportionality of our voting system: it's what causes some parties to get more power than their vote-share indicates they deserve, whilst others get less; it's one of the main reasons why our electoral system is not fit-for-purpose.

One of the repercussions of this is that gerrymandering is possible (see last post); the other is 'wasted votes' and the repercussions of wasted votes (see next post?)


BBC News election scoreboard
356 seats for Labour = 356/646*100 = 55.1%

Wikipedia: Jan '10 election results
Wikipedia: Feb '74 election results

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