This is functionally identical to range voting

Mathematically, it is irrelevant which range you pick of a given size, be that 1 to 3, -1 to 1, or -997 to -995.

Let's run your sample election at two ranges (I'm assuming 3500 voters as it's the smallest number you need, but you can add more voters and it does not change the result):

-1 to 1

Gaullists:

2000*1 + 0*0 + 1500*-1 = 500

Socialists:

1500*1 + 1500*0 + 500*-1 = 1000

Communists:

100*1 + 3325*0 + 75*-1 = 25

National Front:

300*1 + 2800*0 + 400*-1 = -100

1 to 3

Gaullists:

2000*3 + 0*2 + 1500*1 = 7500

Socialists:

1500*3 + 1500*2 + 500*1 = 8000

Communists:

100*3 + 3325*2 + 75*1 = 7025

National Front:

300*3 + 2800*2 + 400*1 = 6900

As you can see, both the results and margins are the same.

While there may well be cognitive differences (voting against might seem more aggressive than simply giving the lowest score, people may be more likely give the middle score as it is inaction) there are no mathematical changes based on the range you choose.

This means that this method has the same advantages and drawbacks as range voting. While it may appear to violate Arrow's Impossibility Theorem, that is because it is a cardinal voting method, while the "universality" criterion of Arrow's theorem effectively restricts that result to ordinal voting methods.

It also fails both the Condorcet and Majority criteria.

Range voting has never been implemented for a national election.

What you're describing is a form of disapproval, negative, or anti-plurality voting, although this version (allowing up and down votes) doesn't seem to be widely used, if at all.

One possible example of being able to vote either for or against a candidate comes from the Soviet Union in the late 1980's, as described in this 1987 New York Times article:

According to the rules for the experimental multimember districts, the number of candidates in each district would exceed the number of seats allocated. Voters may withhold their vote from particular candidates by crossing out their names on the ballot.

Candidates backed by at least half of the registered voters in the constituency would be considered elected. If the number of elected candidates exceeds the number of seats allotted to the constituency, those with the least number of votes are declared stand-by members...

In essence, this voting system gave each candidate a +1 vote per ballot cast by default, voters could cast a -1 to cancel that out, and every candidate who didn't lose half of their votes got elected. The forced +1 to start with isn't really in the spirit of what you're describing, but it's close.

However, there are many voting reform proposals across the world with varying levels of traction, and at least some are bound to be similar to what you describe. One such proposal was detailed in this Desert Sun article from January:

If we want our civic life to be more positive, we might need to vote in the negative.

That’s the compelling case that Sam Chang, a retired banker who lives in Taipei, was making as I rode BART with him between meetings with California election experts.

...

His concept of “the negative vote” is straightforward. Today we can vote for one candidate in each race. Chang proposes to give voters the ability to use that one vote to cast a ballot against the candidate instead. In such a system, each candidate’s tally of votes would be a net — between the number of positive and negative votes.

More details can be found at what appears to be a page dedicated to Chang's proposal, but they don't really cover any of the Arrow theorem stuff and their main selling point seems to be that it potentially increases voter turnout because people like saying no to things.

Yes, this is called "Combined Approval Voting", "evaluative voting", or "dis&approval voting" and has been proposed and studied by a number of people, including exit poll tests in France. (I've also seen people say that the correct name is "Net Approval Voting", but the people who say that seem to be the only ones calling it that.)

It's mathematically identical to Score Voting (except that blanks are assigned to the middle of the scale instead of the bottom), but not conceptually identical, since it features explicit disapproval, which studies have shown changes the way people vote. (However, the French study has a discrepancy in that the research paper said that blanks get counted as 0, while the ballots themselves said that blanks get counted as -1. I'm not sure if that affected this conclusion.)

The Republic of Venice also supposedly used this type of ballot, except with balls placed in green and red urns to indicate approval or disapproval. It's hard to find a convincingly reliable source, though.

There's also another Taiwanese proposal with the same type of ballot, but different rules, called "Negative Vote". The difference is that it only allows you to vote for or against one candidate, and all other candidates must be left blank, so it still suffers from vote-splitting effects.

To address your title question, according to some sources, parts of the system to elect the Doges of Venice had a three-option voting scheme with choices either approval, disapproval or "doubt" for each candidate. Unfortunately, other sources disagree, and it's certain that the system changed over time. It also involved a very small electorate and multiple rounds of voting.

A certain kind of this voting actually happens in all elections in Latvia.

The Central Election Commission’s website seems to have been redesigned recently and I can’t find descriptions/infographics of this in the new design, so I’ll use old images and references to laws.

All candidates in an election are split into lists (corresponding to political parties or alliances). When a voter arrives at the polling station, they are given a whole pack of ballot sheets, one per candidate list. Each of these sheets looks like this:

It has the list’s name and number, an ordered list of candidates and a box next to each candidate’s name.

The voter chooses one sheet and votes for the corresponding list. (The one sheet is cast into the ballot box and the others discarded.) But beyond that, they may choose to vote for or against any individual candidate(s) on this list by either putting a plus into the box or drawing a line through their name:

Eventually when the votes are counted and seats are distributed, seats are distributed between candidate lists (ignoring the pluses and strikeouts), and within each list candidates are ordered by the individual votes they’ve got. Only in case of ties are candidates ordered the way they were listed in the ballot (which was decided by the corresponding party itself when it applied for the election).

Usually, parties put their most well-known and popular members at the top of the list to catch the voter’s eye and collect many pluses, but it’s not uncommon for candidates near the top to be struck out a lot and fall behind or for candidates who start several places behind to catch up and surpass the list’s leaders.

Unfortunately, the relevant laws don’t have official English translations, but for reference:

In the Parliament election law:

• §23.2–3 describes how a voter can add a “+” or strike out a candidate;


• §35 describes how these marks are added up for each individual candidate;


• §39 describes how candidates are rearranged in order of “number of ballots + number of pluses − number of strikeouts” (with ties resolved in the original order the candidates are listed on the ballot) and the top of this list are elected.

In the European Parliament election law:

• §24.2–3 describes how a voter can add a “+” or strike out a candidate;


• §40 describes how these marks are added up for each individual candidate;


• §39 describes how candidates are rearranged in order of “number of ballots + number of pluses − number of strikeouts” (with ties resolved in the original order the candidates are listed on the ballot) and the top of this list are elected.

In the local election law:

• §29.1–2 describes how a voter can add a “+” or strike out a candidate;


• §40² describes how these marks are added up for each individual candidate;


• §41.6–7 describes how candidates are rearranged in order of “number of ballots + number of pluses − number of strikeouts” (with ties resolved in the original order the candidates are listed on the ballot) and the top of this list are elected.

Check out Democracy 2.1. It is a proposed voting system whose advantage is that it produces less controversial and more consensual outcomes. The idea is that each voter has more votes than there are electable candidates, and for every 2 (or more) positive votes they cast, they may optionally cast a negative vote. In other words, voters have to select multiple acceptable outcomes if they also want to cast the negative vote. This encourages consensus building and hopefully rational thought, and the results indicate not just who won by the greatest number of votes, but also who is the most acceptable and who the most controversial or even not acceptable.

(I say "who" but it can just as well be "what". The system has for example been used for participatory budgeting in New York.)