Conceeding fights is part of the new Fate rebuild, introduced in Spirit of the Century.
The FATE system runs primarily at a character-focused rather than narrative-focused level, and so neither needs nor has a concession system to encourage players not to press conflicts until they are taken out. Nevertheless, being taken out is an established part of the system, and the rules around that are:
Taken Out is not killed. Killing usually should occur after the fight, be it by cutting throats or by leaving
opponents to die. Characters tend to be sufficiently willing to kill themselves through their own enthusiasm
that there's no need to help it along with a bad throw of the dice.
(source, page 40)
FATE encompasses both 1st and 2nd edition FATE, at least in the minds of the developers-- there are no notable design-goal differences between 1st and 2nd edition FATE, most of the difference being due to learning how to make a good PDF rulebook.
Instead of conceding in exchange for greater narrative control, in FATE, a player concedes a fight by having their character concede the fight; running away after realizing one is outmatched to avoid getting killed is a decent example. It's still generally true that getting taken out lets your opponents decide what happens to you, including if you live or die, and conceeding a fight before that point if you're not going to win is how you live to fight another day, but that's a consequence of what is likely to narratively happen if you e.g. lose a swordfight with someone who wants to kill you, rather than an explicit mechanic narration and characterization is expected to conform to.
Starting in Spirit of the Century, conceding is a thing
The Spirit of the Century SRD says, in part, about this:
Concessions
Any time a character takes a consequence, he also has the option of offering a concession. A concession is essentially equivalent to surrendering, and is the best way to end a fight before someone is taken out (short of moving away and ending the conflict). The character inflicting the damage can always opt to not take the concession, but doing so is a clear indication that the fight will be a bloody one (literally or metaphorically). If the GM declares that the concession was a reasonable offer, then the character who offered it gains one fate point, and the character who refused it loses one.
The concession is an offer of the terms under which the character is taken out. If the concession is accepted, the conceding character is immediately taken out, but rather than letting the victor determine the manner of his defeat, he is defeated according to the terms of his concession.
Many conflicts end with a concession when one party or the other simply does not want to risk taking moderate or severe consequences as a result of the conflict, or when neither party wants to risk a taken out result that might come at too high a price.
This is a fairly drastic departure from FATE's previous narrative paradigm; while the options previously present for losing a fight without being taken out are still present, "(short of moving away and ending the conflict)", they now take a back seat to concessions, which are the expected way fights that don't have people taken out end. Concessions differ from being taken out only in who is responsible for narrating the character taken out's fate, which is not even a coherent idea in the previous edition of the system, where control was essentially character-based and narration always negotiated.
Indeed, the primary purpose of SOTC concessions-- to avoid being taken out-- isn't really a need that exists in FATE; "Taken out", in FATE, means something very different.
In SotC, though, you can always refuse concessions, which, if everyone did, would lead to more-or-less the same situation as in FATE, but with extra Fate Points if you lose a fight.
Modern Fate
Fate Core says, about conflicts:
You know the conflict is over when everyone on one of the sides has conceded or been taken out.
No longer do characters have the option to walk away from a fight. Once a fight starts, walking away from it must be modeled within the 'four actions' framework (possibly "create an advantage" for the aspect "Not There" or "Got Away" or something) or as a concession. More realistically, though, Fate core characters don't ever concede conflicts. Instead, Fate core players concede conflicts-- indeed, the Fate Core SRD site includes a forum post from the author of the section in question as an explanation about conceding that says that:
Concession, as a mechanic, specifically faces outward to the real people playing. It's explicitly about the parsing of narrative authority over the fate of the character post-conflict, and nothing else.
Characters do not concede. People do.
This is mostly the same as SotC style concessions, except that instead of being a predominant option for how to end conflicts, this is now the only way conflicts end, and also the system is rather more doubled-down on narrative negotiation taking priority
over characterization, losing several of the 'but make sure characterization makes sense too' caveats from SotC, including the option to refuse concessions.
Fate Accelerated is less intense, but more or less the same
FAE says:
The conflict is over when only one side has characters still in the fight.
and
If things look grim for you, you can give in (or concede the fight)—but you have to say that’s what you’re going to do beforeyour opponent rolls their dice.
This is different than being taken out, because you get a say in what happens to you. Your opponent gets some major concession from you—talk about what makes sense in your situation—but it beats getting taken out and having no say at all.
Additionally, you get one fate point for conceding, and one fate point for each consequence you took in this conflict. This is your chance to say, “You win this round, but I’ll get you next time!” and get a tall stack of fate points to back it up.
That's a lot less strongly worded than 'conflicts only end when everyone on one side has conceded or been taken out', but it basically amounts to the same thing; there aren't really rules or actions available for leaving fights otherwise and zone-based movement plus well-defined action options doesn't leave much space for leaving fights any other way.
It's basically the same as Fate Core, just with the FAE tone and wording, and has basically the same changes from SotC.
