Brain Teaser #89 Brain Thumper #4: Inclusive PR System Key Threshold

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In summary, the conversation discusses the concept of blocs of voters and proportional representation systems. It explains that the key threshold for a bloc to guarantee a win is the fraction of voters they need to elect their top candidate regardless of how the remainder vote. The conversation also includes an example of how this works in a city council race and how the Common Denominator party needs to have a bloc of at least 3/7 of the total votes to guarantee the victory of their top 3 candidates.
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davilla
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Brain Thumper #4

First some definitions.

A bloc of voters is a group who cooperate to achieve their common goals. In other words, they have the same preferences and vote strategically, which means that they coordinate their votes so as to maximize the outcome. We will assume political party affiliation to be loyal so as to comprise blocs.

Generally speaking, a proportional representation (PR) system is a voting methodology that distributes a number of options evenly among voters. For instance, in an election for more than ten seats, if 10% of the voters rank a minority candidate highest, one would expect that the candidate should win a seat regardless of how the remainder vote.

Of course some PR systems are more representative than others. For instance, there may be imposed a minimum threshold for a candidate to win, to the favor of larger parties. Our interest is the threshold at which a bloc can force a win, that is, the fraction of voters required to guarantee that, voting strategically, they can elect their single most favorite candidate regardless of how the remainder vote. This is the key threshold.

How does this work? Consider a race for 5 candidates with a key threshold of at least 20%. If out of 100 thousand voters at least 20 thousand are members of the Elect Jim Scalawag Party, Mr. Scalawag is assured a seat. Of couse, he might win a seat with fewer supporters if, for instance, the remaining 80-plus thousand are bitterly divided, but he will be biting his fingernails as the results come in.

As another example, consider the majority voting system commonly used in the United States for city councils (and almost everything else). Since a majority is required regardless of the number of seats available, this system has a key threshold of more than 50%, and is quite rightly not considered to be proportional.

A PR system is inclusive if the key threshold is as low as theoretically possible.

Johnny Twoshoes is the campaign manager for the largest political party in his city, the Common Denominator. There is a city council race for 7 seats coming up and Johnny thinks his party has enough support to win 3 places. The government keeps the election formula a secret, which is eerily undemocratic, but anyways we are assured that it is an inclusive PR system. Mr. Twoshoes is aware of this, but he still isn't certain if he should run all 3 candidates because the key threshold applies to only a single candidate.

Suppose that other parties turn out exactly 100 thousand voters on the day of the election. (Any independent voters are included in that total.) Given that the voting system is inclusive, what size bloc does the Common Denominator need to guarantee the victory of its top 3 candidates?
 
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Since a bloc of voters co-ordinates its votes to maximum effect, we can assume that its voters would vote in equal numbers for the three candidates. Since the key threshold is as low as theoretically possible, it must be 1/7, multiplied by three for the three candidates = 3/7. The Common denominator therefore gets 3/7 of the votes, which leaves 4/7 other voters.

Since there are 100000 other voters, there are 100000 * 3/4 = 75000 Common Denominator voters.

Is that the answer? It seems very easy, so perhaps I misunderstood something.
 
  • #3
42858 people.
 
  • #4
Well, the most votes that each of 5 candidates could get in 100,000 votes is 20,000, so each of the twoshoes candidates would have to get more than 20,000
depending on whether votes can be split, that makes it 60,001 or 60,003 votes.
 
  • #5
NateTG said:
depending on whether votes can be split, that makes it 60,001 or 60,003 votes.
NateTG with the point!

Because it is inclusive, the key threshold is more than 1/8, as it is impossible to guarantee a win to more than 7 candidates.
 
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1. How does the "Inclusive PR System Key Threshold" work?

The "Inclusive PR System Key Threshold" is a mathematical formula used in a proportional representation voting system. It calculates a minimum percentage of votes that a party or candidate must receive in order to be eligible for a seat in the legislature.

2. What is the purpose of the "Inclusive PR System Key Threshold"?

The purpose of the "Inclusive PR System Key Threshold" is to ensure that only parties or candidates with significant support from the population are represented in the legislature. This helps to prevent small or fringe parties from gaining seats and potentially disrupting the functioning of the government.

3. How is the "Inclusive PR System Key Threshold" calculated?

The "Inclusive PR System Key Threshold" is calculated by taking the total number of valid votes cast in an election and dividing it by the number of seats available in the legislature. For example, if there are 100,000 valid votes and 10 seats available, the threshold would be 10,000 votes.

4. What happens if a party or candidate does not meet the "Inclusive PR System Key Threshold"?

If a party or candidate does not meet the "Inclusive PR System Key Threshold", they will not be eligible for a seat in the legislature. Their votes will not be counted towards the proportional representation system and they will not have any representation in the government.

5. Are there any drawbacks to using the "Inclusive PR System Key Threshold"?

One potential drawback of using the "Inclusive PR System Key Threshold" is that it can exclude smaller or newer political parties from having representation, even if they have a significant number of supporters. This can potentially limit the diversity of voices in the legislature. However, it also helps to prevent extremist or single-issue parties from gaining power.

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