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First some definitions.

Ablocof 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 thekey 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 isinclusiveif 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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# Brain Teaser #89

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