Pyramid Comment

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Saturday, May 01, 2010

Voting Mathematics, Unfairness And Absurdity

The electoral system in the UQ (aka UK) Ltd can be manipulated and massaged by the boundary movement ‘device'. The result of a democratically assessed vote can be very misleading when it comes to the majority of the non-represented by a minority government that has been elected to control. This is democracy in action.

Anarchy In Democracy Or Democratic Anarchy
Electoral Engineering And Snake Oil
Political (Democratic) Choice For The Gang

Donald Saari about election math

  • Imagine that there is to be a party and that a committee of 15 will select which beverages should be available. Suppose 6 of them choose milk with wine in second place and beer the least preferred [M-W-B]. The next 5 select beer as first choice with wine as second and milk as third choice [B-W-M]. The last 4 regard wine as favourite followed by beer and lastly milk [W-B-M]. So, at the vote:
Milk = 6
Beer = 5
Wine = 4

In reality, 9 of 15 people (60%) preferred last-placed wine to first-placed milk. These represent 'landslide'-type proportions and comparing second-place beer to first-place milk the same thing applies: 9 (60%)  preferred beer over milk. But milk is still in first place. These same voters strongly prefer any other beverage to milk. When beer is compared to wine, two-thirds (10 of 15) of these voters prefer wine to beer. Wine came in last place, yet is these voters' beverage of choice. The plurality system, where only the first preference counts, results in the exact reverse of actual preference and in the example would be officially recorded as [M-B-W] and not [W-B-M] as evidenced by the real numbers preferring 'alternative' outcomes.

Milk appears to be the winner, but with just 40% of the vote, yet out of 15 voters, 9 prefer beer to milk and 9 would rather wine to milk. Both cases show a clear majority. The preference of wine over beer stands at 10 of 15 (two-thirds): 6 = MWB and 4 = WBM.

This is a typical scenario in many real-world examples and is a problem everywhere. Simply substitute milk, beer and wine for the three parties: Labour, Tory and Liberal Democrats. It demonstrates the unfairness and absurdity of first-past-the-post (plurality system) politics. The majority is completely unrepresented.

Boundary 'Tampering'

Results can be manipulated by simply redrawing the boundaries of a constituency. The way this works may be very simple, but it can have a major effect. It could alter the election result.

Consider a square of 3 x 3  that constitutes a region broken up into 9  roughly equal areas:
1  2  3
4  5  6
7  8  9

The three rows (1  2  3)  = constituency A, (4  5  6) = constituency B and (7  8  9) = constituency C each represent an electorate population of, say, 300,000  voters (or 100,000 + 100,000 + 100,000) and a boundary exists that separates the top from the middle from the bottom constituency. The total voting population in these three constituencies amounts to 900,000.

1  2  3  = 300,000
4  5  6  = 300,000
7  8  9  = 300,000

It is quite common that opinions within closely related neighbourhoods are similar and so the 9 areas translates into 9 voting opinions though the choice may still only be two (right or wrong) in each area, so each of the three constituencies will comprise three separate choices within the constituency. For the sake of this hypothetical argument, assume that the voting preferences are as follows:

1 = right
2 = wrong
3 = wrong
4 = right
5 = wrong
6 = right
7 = wrong
8 = wrong
9 = right

This translates to a new grid, but still has 900,000 constituents in the three constituencies:

1  2  3     constituency A
5      constituency B
7  8  9     constituency C

Constituency A = right, wrong, wrong
  • Votes: 100,000  100,000 +100,000
    • Constituency A = wrong WINS

Constituency B = right, wrong, right 
  • Votes: 100,000  100,000    100,000
    • Constituency B = right WINS

Constituency C = wrong, wrong, right 
  • Votes:  100,000 + 100,000  100,000
    • Constituency C = wrong WINS

right WINS 1 seat
wrong WINS 2 seats


Moving the boundaries around within the 9 areas, yet still retaining three groups and an overall voting population of 900,000, major consequences will result.
2  3
4  6
8  9

Constituency A = 1, 4, 7
Constituency B = 2, 3, 5
Constituency C = 6, 8, 9

Voting of the constituents in each area remains unchanged, but the boundary changes affect the overall result:

1 = right
2 = wrong
3 = wrong
4 = right
5 = wrong
6 = right
7 = wrong
8 = wrong
9 = right

Constituency A = right, right, wrong
  • Votes: 100,000 + 100,000   100,000
    • Constituency A = right WINS
Constituency B = wrong, wrong, wrong
  • Votes: 100,000  + 100,000 + 100,000
    • Constituency B = wrong WINS
Constituency C = right,, wrong, right,
  • Votes:  100,000 100,000 100,000
    • Constituency C = right WINS
right WINS 2 seats
wrong WINS 1 seat


So, by changing boundaries, this places people in different constituencies, Even though voting preferences may be unchanged, the overall result can be reversed as the above example illustrates. It's tampering, but quite legal.

Note: the selection of colour has no significance and the example is based on an article published in New Scientist No 2758