John Doe
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Math question
Parliament has 100 seats. Only parties with strictly more than 5% of votes can qualify to get seats. There are 12 parties participating. Seats are distributed to the parties that qualify to get seats, proportionally to their votes. Votes are strictly for 1 party (no voting for multiple parties or no party).
What is the maximum number of seats that a party with 25% of the votes can get?
What is the maximum number of seats that a party with 25% of the votes can get?
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10 parties with 5% each, 2 parties with 25% each
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Thanks!
Hi there,
The answer is 50% perhaps.
In a parliament of sufficient size, the logic is easy to show that the party would maximise their seats by having as many of the small parties as possible at the 5% cutoff.
For all the other parties above 5% - needed to make up the 100% - having the total vote (including the fixed 25%) as low as possible would maximise the number of seats.
Given these provisions the 50% becomes the answer, ...However ....
In an actual parliament of the number of seats must be an integer, so that it is likely that some type of rounding rules would be in force. What is the number allocated when, for example, the proportion of votes translated into seats comes to 78.52 seats? If the other party(ies) were rounded down and the party of concern rounded up then the proportion of seats could be greater than 50%.
With a three seat parliament with parties getting 25%, 24.9%, 5.1% and the other 9 getting 5% the seat allocation would be
1.36364 1.35818 0.278182 for the first 3 parties. With what may be seen as natural rounding (rounding up the fractions in size order) to ensure the full complement of seats allocated, this would mean that seat allocation was 2, 1 and 0 for the top 3 parties. Hence the party with 25% of the votes could get 66.7% of the seats. In fact in a 1 seat parliament the seat fractions would be
0.454545 0.452727 0.0927273 resulting in (after rounding) 1, 0, 0 delivering 100% of the seats.
Additionally if (as often occurs) unless a party is entitled to at least 1 seat, rounding will not occur for that party, we would have->
for a five seat chamber fractions of 2.27231 2.26322 0.464461 delivering (after rounding) 3, 2, 0 - 60% of seats; and
for a seven seat chamber fractions of 3.18124 3.16851 0.650245 delivering 4, 3, 0 seats - 57.14% of seats; and finally
for a nine seat chamber fractions of 4.09017 4.0738 0.83603 delivering 5, 4, 0 seats - 55.55% of seats.
If you don't think that is possible to have these ridiculously small parliaments you haven't studied what dictators are liable to do to get their own way, the party with 25% of the vote could gain the clear majority in the parliament.
Other rounding rules are possible (and in existence) like rounding up (regardless of fraction size) on largest to smallest basis (unfair to small parties) or rounding of fraction size modified by total seat count. or rounding all cases down and having a parliament slightly smaller than the maximum size.
As is always the case, actual results must take into count rules of humans and rules of mathematics - in that the solution must be an integer!
Ian
The answer is 50% perhaps.
In a parliament of sufficient size, the logic is easy to show that the party would maximise their seats by having as many of the small parties as possible at the 5% cutoff.
For all the other parties above 5% - needed to make up the 100% - having the total vote (including the fixed 25%) as low as possible would maximise the number of seats.
Given these provisions the 50% becomes the answer, ...However ....
In an actual parliament of the number of seats must be an integer, so that it is likely that some type of rounding rules would be in force. What is the number allocated when, for example, the proportion of votes translated into seats comes to 78.52 seats? If the other party(ies) were rounded down and the party of concern rounded up then the proportion of seats could be greater than 50%.
With a three seat parliament with parties getting 25%, 24.9%, 5.1% and the other 9 getting 5% the seat allocation would be
1.36364 1.35818 0.278182 for the first 3 parties. With what may be seen as natural rounding (rounding up the fractions in size order) to ensure the full complement of seats allocated, this would mean that seat allocation was 2, 1 and 0 for the top 3 parties. Hence the party with 25% of the votes could get 66.7% of the seats. In fact in a 1 seat parliament the seat fractions would be
0.454545 0.452727 0.0927273 resulting in (after rounding) 1, 0, 0 delivering 100% of the seats.
Additionally if (as often occurs) unless a party is entitled to at least 1 seat, rounding will not occur for that party, we would have->
for a five seat chamber fractions of 2.27231 2.26322 0.464461 delivering (after rounding) 3, 2, 0 - 60% of seats; and
for a seven seat chamber fractions of 3.18124 3.16851 0.650245 delivering 4, 3, 0 seats - 57.14% of seats; and finally
for a nine seat chamber fractions of 4.09017 4.0738 0.83603 delivering 5, 4, 0 seats - 55.55% of seats.
If you don't think that is possible to have these ridiculously small parliaments you haven't studied what dictators are liable to do to get their own way, the party with 25% of the vote could gain the clear majority in the parliament.
Other rounding rules are possible (and in existence) like rounding up (regardless of fraction size) on largest to smallest basis (unfair to small parties) or rounding of fraction size modified by total seat count. or rounding all cases down and having a parliament slightly smaller than the maximum size.
As is always the case, actual results must take into count rules of humans and rules of mathematics - in that the solution must be an integer!
Ian
Under any reasonable rounding rules, with a 100 seat parliment, I'd expect any party with >5% to get at least 5 seats.
I'm not sure what rounding rule was being used in the above examples.
When people must be integers, the issue becomes which parties round down, and which parties round up.
I might suggest that after all parties take their integer floor, the remaining seats should be filled in order of the greatest remaining fractions.
Such a rule would produce
2, 1, 0 for 1.36364 1.35818 0.278182
1, 0, 0 for 0.454545 0.452727 0.0927273
2,2,1 for 2.27231 2.26322 0.46446,
3, 3, 1, for 3.18124 3.16851 0.650245
and
4, 4, 1 for 4.09017 4.0738 0.83603
Another possibility, which might arguably be more equitable, could be to let the rounded down votes be recast for the electors second choice.
I'm not sure what rounding rule was being used in the above examples.
When people must be integers, the issue becomes which parties round down, and which parties round up.
I might suggest that after all parties take their integer floor, the remaining seats should be filled in order of the greatest remaining fractions.
Such a rule would produce
2, 1, 0 for 1.36364 1.35818 0.278182
1, 0, 0 for 0.454545 0.452727 0.0927273
2,2,1 for 2.27231 2.26322 0.46446,
3, 3, 1, for 3.18124 3.16851 0.650245
and
4, 4, 1 for 4.09017 4.0738 0.83603
Another possibility, which might arguably be more equitable, could be to let the rounded down votes be recast for the electors second choice.
Dear Ozo,
You said, ...
I don't believe that such an animal exists in regard to the nitty gritty of human inspired laws and mathematics. How about 2 pages of a finance statute explaining how to round? You are likely to find that Electoral laws are full of subtle twists and turns and even some mathematical inconsistencies that surprise the unwary!
I agree that with a 100 seat chamber 5% (well in this case 5.00001% of the vote would be translated to at least 5 seats up to a maximum of 10.
It is the edge case of low parliament sizes where rounding will become an issue.
In the second stage of my example I stated that it is possible to have a rule that rounding will not apply unless the party passes the lower boundary of 1 seat. That would reduce all values BELOW 1.0 to ZERO. Other values above 1.0 will be rounded as necessary.
PS New Zealand has just finished the washup from their latest election. And they have a somewhat complicated system incorporating a portion of seats based on the proportion of votes obtained subject to minimum thresholds.
You said, ...
Under any reasonable rounding rules, ...
I don't believe that such an animal exists in regard to the nitty gritty of human inspired laws and mathematics. How about 2 pages of a finance statute explaining how to round? You are likely to find that Electoral laws are full of subtle twists and turns and even some mathematical inconsistencies that surprise the unwary!
I agree that with a 100 seat chamber 5% (well in this case 5.00001% of the vote would be translated to at least 5 seats up to a maximum of 10.
It is the edge case of low parliament sizes where rounding will become an issue.
In the second stage of my example I stated that it is possible to have a rule that rounding will not apply unless the party passes the lower boundary of 1 seat. That would reduce all values BELOW 1.0 to ZERO. Other values above 1.0 will be rounded as necessary.
PS New Zealand has just finished the washup from their latest election. And they have a somewhat complicated system incorporating a portion of seats based on the proportion of votes obtained subject to minimum thresholds.
ASKER