Exercise 1: How 0.41% of the Vote Bought Real Seats — Possible Solution
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This exercise is really about understanding what "no minimum vote
threshold" actually means in mechanical terms, not just repeating the
real 1930 figures.

HOW THRESHOLD-FREE PR ACTUALLY WORKS
Under proportional representation, seats are allocated in direct
proportion to a party's own share of the national vote. Under a
system WITH a minimum threshold - many real modern systems set this
somewhere around 5% - a party has to clear that floor before it
receives any seats at all, no matter how proportional its raw vote
share might otherwise be. Weimar's own system, per Chapter 4, set no
such floor. That meant the proportional formula was applied
completely literally: whatever share of the vote a party won, it
received the mathematically equivalent share of seats, with no minimum
cutoff filtering out very small parties first.

WHY THIS LET THE GERMAN-HANOVERIAN PARTY WIN SEATS
The German-Hanoverian Party won 0.41% of the national vote in the 1930
election - a genuinely tiny sliver of the electorate, concentrated in
a specific region with its own specific regional identity and
interests. Under a system with a 5% threshold, that 0.41% share would
have earned the party nothing at all - it would have fallen well
below the cutoff and been excluded from the Reichstag entirely, its
votes effectively redistributed among the larger parties that did
clear the threshold. But because Weimar's own system applied the
proportional formula with no floor, that 0.41% share still translated
directly into the party's own three real Reichstag seats.

WHY THIS COULDN'T HAPPEN UNDER A THRESHOLD SYSTEM
A threshold system is specifically designed to filter out parties
below a certain minimum size, on the theory that representation for
extremely small, narrow-interest parties comes at the real cost of
making it harder to form stable governing majorities - exactly the
problem this chapter goes on to describe at the national level. Any
threshold set higher than 0.41% - and virtually every real threshold
system in use anywhere sets it well above that level - would have
excluded the German-Hanoverian Party from the Reichstag entirely, no
matter how genuine or locally significant its own regional support
was.

ANSWER: The German-Hanoverian Party could win real Reichstag seats on
just 0.41% of the vote because Weimar's proportional representation
system applied the proportional formula with no minimum vote threshold
at all - every vote share, however small, translated directly and
mathematically into a corresponding share of seats. This couldn't
happen under a threshold-based system, because any real minimum cutoff
(virtually all of which are set well above 0.41%) would have excluded
the party from Reichstag representation entirely before the
proportional formula was ever applied.

WHY THIS WORKS AS AN ANSWER
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It explains the actual mechanical difference between threshold-free and
threshold-based PR - filtering happens before allocation in one system
and never happens at all in the other - rather than simply restating
the 0.41% figure without explaining why that number was still enough
to win seats.