[EM] Lewis Carroll and limited vote

[Michael Rouse]

While I was surfing the net for pros and cons of different voting systems, I
came upon this blurb about Lewis Carroll:

"Writing in 1884, he praised the unpopular 'limited vote' which then
operated in the big English cities, in which each voter had fewer votes than
there were seats to fill. Most people thought that this was less democratic
than giving each voter as many votes as there were seats. Dodgson proved
that it was more democratic. To do so, he used concepts we would today label
'game theoretic', although such concepts were not formalised until decades
after his death."

This raised a few questions in my mind, such as:
1.) Is there a fairly simple description of this proof?
2.) Does it matter if the voting method is positional (like Condorcet) or
simply yes-no (Approval-style)?
3.) At what point does limiting the vote maximize the "democratic"
aspects -- 50% of the total seats? 75%? Or does it depend on the strength of
the party?
4.) What about methods that allow you to rank any number of candidates? For
example, if there were 50 seats and 100 candidates, and you ranked all 100
candidates? What about fifty candidates? What about only 25?

Thanks for any information.

Michael Rouse
mrouse at cdsnet.net