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Brain Teaser #89

  1. Apr 21, 2004 #1
    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?
    Last edited: Apr 21, 2004
  2. jcsd
  3. Apr 21, 2004 #2


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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 threshhold 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.
  4. Apr 21, 2004 #3
    42858 people.
  5. Apr 21, 2004 #4


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    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.
  6. Apr 22, 2004 #5
    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.
    Last edited: Apr 22, 2004
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