In the sense I'm using it here, forgiveness is an arbitrary ritual whereby the 'I'm angry at you about this' tag is switched to 'Nah, 's cool' for the forgiver. For the forgived, it's a way of saying that they're 'allowed' to stop feeling guilty about their past action because the forgiver has 'granted them' forgiveness.
Forgiveness can be useful if you don't have the self-awareness to know when you've fully learned your lesson from a mistake, and someone attuned to it (the forgiver) is better positioned to discern when you have. It can also serve a social function as a 'lowering of weapons', or as a way of saying 'I am now over my emotional disgust and am ready to engage again', or so forth.
Insomuch as forgiveness is an approximation to these various component propositions and is coloured by magical thinking (e.g. intuitively thinking that it makes sense to have an epiphenomenal 'mad at you' tag that should determine your disposition towards someone), advanced cooperating rationalists would not use forgiveness, because it encourages magical thinking about the various components, and this magical thinking is susceptible to lost purposes, e.g. turning into a game of 'winning forgiveness' that is divorced from the actual purpose of doing better in future or understanding mistakes better.
'Forgive and forget' is even worse; advanced cooperating rationalists would not permit each other to forget misdeeds, including their own, because that would be throwing away evidence. Of course, 'forget' here does not literally mean forget; misdeeds might be brought up again if the same mistake is made in future. But this is still a binary thing of 'Allowed to bring this up as evidence'/'Not allowed to' which is a crude approximation to the continuous and constant nature of past misdeeds as evidence about a person.
I don't remember ever seeing a forgiveness ritual take place on LW, but I do know that I've seen lots of cases in an exchange where someone explained their own misdeed and its cause to prove they could avoid it in future (and sometimes committed to avoiding it in future), and that was good enough for all involved.
I probably phrased this a bit strongly in the first place, since I could see e.g. Bay Area instrumental x-rationality pros using forgiveness rituals as an informal time-saving shorthand for the underlying rigorous game-theoretical/Bayesian concepts. But I suspect they would be less susceptible to losing sight of that underlying core (e.g. less susceptible to 'win forgiveness' games). This would be 'post-rigorous forgiveness', but I'd remain suspicious of pre-rigorous forgiveness.
Now, it is said we all here pride ourselves on our intelligence, rationality, and moral sense. It is also said, however, that we are a fiercely independent bunch, and that we can let this pride of ours get the better of us. There have also been comments that the live communities that appear at meetups provide much more positive interactions than what goes on on this site's discussions; this might merit further investigation.
My point is; we've done a lot of research on how to do proper ethical and metaethical calculations, and on how to achieve self-empowerment and deal with our own akrasia, which is awesome. We've also done some work on matters of gender equality, which is very positive as well. But I haven't seen us do anything about the basic details of human interaction, what one would call "politeness" and "basic human decency". And I think it might be useful if we started tackling these, for our own sakes, that of those who surround us, and that of easing our mission along, which is, as I understand it so far, to save the world (from existential risk (at the hands of (unfriendly and self-modifying) artificial intelligence))).
What inspired me to propose this post was a video I just saw from Hank Green of the famed and fabled vlogbrothers. I hold these two individuals in very high esteem, and I would expect many here to share my feelings about them, on account of their values and sensibilities largely overlapping with ours; namely the sense that intelligence, knowledge and curiosity are awesome, and that intellectuals ought to use their power to help improve themselves and the world around them.
Here it is; I hope you enjoy it