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Just a side note:  If you aren't familiar with the various electoral (voting) systems [around the world], please do not read this as the wording may confuse you.

An electoral (i.e., voting) process in which full representation of all parties who have received votes is achieved by closely matching the percentage of party votes to the percentage of seats allocated in legislative assemblies.

There are 3 known ways of achieving proportional representation (PR) in electoral systems.

1.  Party-List PR:  There's a list of pre-determined candidates (closed-list) or candidates that the voters can rank (open-list), along with some [mathematical] formula of allocating the seats (D'Hondt or Sainte-Lague).  Countries which use Party-List PR include Israel (where the country is one closed-list constituency) and [the Netherlands] (open-list).
2.  Additional-Member System (AMS), Mixed-Member System (MMS):  Two votes, one vote for a legislator (MP -- Member of Parliament -- in places like the UK and New Zealand) to [represent] a single-member constituency (under plurality voting), the other vote for a party (under party-list PR).  In places like Germany, a certain number of seats are blockaded off for party-based legislators.  Besides Germany, New Zealand uses this (along with calculating party-list seats via the Sainte-Lague method) for its House of Representatives.
3.  Single Transferable Vote (STV) (in a multi-member constituency):  Usually 3 to 6 candidates per constituency.  Voters number their ballot according to their [preferences].  The first [preferences] are calculated first and candidates must achieve a quota (determined by the number of votes and the number of vacant seats) in order to be elected; if none of them meet the quota, the lowest-voted candidate gets eliminated and his/her 2nd [preferences] allocated to the next candidate, etc., until all the constituency seats are filled.  Australia uses this to elect its Senate (upper house).

The 1998 Jenkins Commission in the UK also suggested a broadly-PR type of voting system called [Alternative] Vote Top-Up, a variant of Additional-Member/Mixed-Member where voters number their ballot according to [preferences] to determine an electorate MP in single-member constituencies (just like Australia's preferential voting for its House of Representatives).  The other vote is a party vote, candidates on that list for each county.

A few notes on proportional representation:
1.  Parties are less likely to gain majorities in legislative assemblies, very likely resulting in coalition governments.
2.  Third parties, which are often disadvantaged under FPTP (first-past-the-post), often want PR so they can get more seats but not necessarily a majority.  Examples being the UK's [Liberal Democrats] and Canada's NDP.

For more on PR, just type "proportional representation voting" in your favorite search engine.

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