Yes; apology is an underrated consequentialist tool among nerds.

Some of the social function of apology can be understood game theoretically: Apology explicitly disavows a past action, allowing the one to whom the apology was made to leverage that confession in future: If someone apologises for something then does it again, then response can escalate because we have evidence that they are doing it even knowing that it's 'wrong'. The person who apologised knows this, and often the implicit threat of escalation if they do the same thing checks their future behaviour. Therefore apology is (possibly among other things) a signal, where the cost to apologising is the greater susceptibility to escalation in future cases.

Apology falls into a class--along with other things such as forgiving misdeeds, forgetting misdeeds, retribution, punishing an agent against its will, compensation for misdeeds--of things that would make no sense among sufficiently advanced and cooperative rationalists. Some things in that class (e.g. forgiveness) might already have been transcended by LW, and others (e.g. apology) are probably not possible to transcend even on LW, because the knowledge of other participants (e.g. confidence of their cooperativeness) required to transcend apology is probably too high for an online community of this size.

I would guess that the Bay Area rationalist set and its associates--which as far as I can tell is by far the most advanced community in the world in terms of how consummately instrumental x-rationality is forged into their swords--apologizes way, way, way more than the average LW'er, just like they talk about/express their feelings way more than people on LW typically do, and win because they're willing to confront that prospect of 'being vulnerable'.

HPMoR status:

"Well," said the boy. His eyes had not wavered from the Defense Professor's. "I certainly regret hurting you, Professor. But I do not think the situation calls for me to submit to you. I never really did understand the concept of apology, still less as it applies to a situation like this; if you have my regrets, but not my submission, does that count as saying sorry?"

Again that cold, cold laugh, darker than the void between the stars.

"I wouldn't know," said the Defense Professor, "I, too, never understood the concept of apology. That ploy would be futile between us, it seems, with both of us knowing it for a lie. Let us speak no more of it, then. Debts will be settled between us in time."

Two mistakes in thinking that my past self made a lot and others might also:

(1) Refusing to apologize if another party was 'more wrong'. Even if you're 99.9% right/innocent/blameless, you still have to make a choice between apologizing and not apologizing to the other person. If you refuse to apologize, things will probably get worse, because the other person thinks you're more wrong than you think you are, and they will see you not apologizing as defecting. If you apologize in a smart way, you can give an apology (which shouldn't make a difference but has the actual consequence where the other person is more probable to also apologise) without tying yourself down with too broad a commitment on your future behaviour, and without lying that you thought something was a mistake that wasn't.

(2) Using the fact that, in the limit as rationality and cooperation become arbitrarily great, apology is meaningless, as a rationalization for not apologising, when in fact you just feel embarrassed/are generally untrained and therefore not fit enough to apologise, and you're therefore avoiding the exertion of doing so.

I want to point out the difference between completely fake apologies for things one does not think were mistakes, and apologising for things that were mistakes even if the other person's mistakes were much greater. The former is less often the smart thing to do, and the latter is a lot more often than one might think. Once you get fairly strong, you can sometimes even win free points by apologising in front of a big group of people for something that everyone but the other disputant think is completely outweighed by the other disputant's actions.

E.g. 'I'm sorry I used such an abrupt tone in asking you to desist from stealing my food; it probably put you on the defensive.' If you really mean it (and you should, because you're almost certainly not a perfect communicator and there were probably things you could have done better), then often onlookers will think you're awesome and think the other person sucks for 'making you' apologise when you'd 'done nothing wrong'. Sometimes even the other disputant will be so disarmed by your unwavering 'politeness' that they will realise the ridiculousness of the situation and realise that you're being genuine and that they made a mistake, whereas when they thought you were a hostile opponent, it was much easier for them to rationalise that mistake.

Notice than in that example, your apology has not even constrained your future actions; everyone was so distracted by the ridiculousness of you apologising when you were innocent and the contrast it made between yourself and your opponent, that nobody will think to escalate against you in future the next time somebody steals your food.

That's why it's so important to know how to lose--so that you can win! Just like how the best things you could do to decrease your personal risk from fights are things like practising conflict defusion techniques, learning how to walk away from conflict, being less tempestuous, being situationally aware, or even just learning how to play dead/fake a seizure/panic attack, rather than something that just looks like winning, like practising flashy kicks.

My first reaction to this video is that it's pretty crappy. Its main message is to insist on the following sequence as everyone's moral responsibility: once you say something that someone perceives as hurtful, you're morally obliged not to "question their feelings", but to perceive that you screwed up, to feel bad, to apologize, and change your behavior so this doesn't happen again. The video repeatedly insists that your own thoughts about whether what you've done was improper are irrelevant.

Now, it should be completely obvious that this sequence is not going to be used as prescribed by anyone including the author of the video. If you were to approach him and say, "In your video, you identified my position as that of a fartbag, repeatedly, and that was very hurtful to me", he's not going to say, "Oh, thanks for telling me that, I'm sorry I screwed up, I feel really bad and will never do this again". He is, in fact, going to use his own judgement about the appropriateness of his behavior to decide whether or not he should apologize.

Why, then, is there no mention of your own moral judgement in the video, and instead, you're repeatedly encouraged not to question someone's hurt feelings, but to accept them as a proof you did something hurtful, and must feel bad and apologize?

My guess is that this is a shortcut towards setting up a double standard. Once you've gotten people to accept this sequence as their moral responsibility, some varieties of hurt feelings are going to be treated as obviously triggering the sequence, while others will just be ignored. This isn't conscious lying - the author of the video is sincere. He's going to have a blinkered view of what it means to hurt someone's feelings, and he wants you to have that same view, too. If you approach him with a hypothetical scenario in which what he's said or done was true, virtuous, and hurt the feelings of some bad person, he's not going to be able to see a problem with that, and will accuse you of sophistry.

Double standards are incredibly powerful because they allow you to be hypocritical without consciously being aware of the fact. They've replaced direct and conscious lies as the preferred method of being untruthful in arguments.

The video that is recommended at the end as the inspiration for this one provides much of the missing context; it's all about how to correctly apologize after being "called out, which in the context of this video is when you say or do something that upholds the oppression of the marginalized groups of people"; the video teaches you how to apologize and become a proper "ally", etc. So we're fully into the social-justice-warriors terminology here, which is or course very much misaligned with rationality, and the double standard described above is being built primarily with that purpose.

There are some good things about this video. It is true that many people feel that apologizing is a sign of weakness. It is also true that there should be more direct apologies and fewer non-apology "I'm sorry you feel that way" apologies. These good aspects of the video are drowned by the dogmatic, irrational claims of alleged moral responsibility to which most of the video is devoted.

Watching the video was a decent exercise in detecting some typical patterns of bad thinking.

An example, by way of "The Root of All Evil" by Richard Dawkins:

Science is about testing, comparing and corroborating this mass of evidence, and using it to update old theories of how things work. I do remember one formative influence in my undergraduate life. There was an elderly professor in my department who had been passionately keen on a particular theory for a number of years. And one day an American visiting researcher came, and he completely and utterly disproved our old man's hypothesis. The old man strode to the front, shook his hand and said, "My dear fellow, I wish to thank you. I have been wrong these 15 years." And we all clapped our hands raw. That was the scientific ideal of somebody who had a lot invested, a lifetime almost invested in a theory, and he was rejoicing that he had been shown wrong, and scientific truth had been advanced.

Relevant post from YVain's blog: http://slatestarcodex.com/2013/09/14/giving-and-accepting-apologies/