The public confession of wrongdoing and apology has long been an element of punishment, going back to back at least to the Inquisition. It doesn't matter whether the apology is sincere (Galileo muttering "it still moves" while recanting his views on Copernican astronomy.) What matters is that the person apologizing is demonstrating submission to the one who demands the apology. In some respects an apology understood by all to be insincere is an even better demonstration of power and subordination.

So the purpose of making your kids apologize is to understand the balance of power? And that if they are powerful, they do not need to apologize? What about when the wrong is between two equals?

If the goal is to have the child learn how to apologize genuinely and from the heart (which is when it is meaningful in social situations), these would be terrible lessons to internalize. When I apologize to my wife about screwing something up, it is not because she is more powerful or that I am her subordinate. It is because I know that I screwed up, and want to acknowledge this beyond just making the corrective action. Plus it helps the relationship to move along as equals.

The purpose of making your kids apologize is to have them compensate the person they have injured by abasing themselves. That's what you do when you apologize to your wife. Whatever may be the long term dynamic, in the short term "you were right and I was wrong" is an admission of inferiority.

By that reasoning, a parent should almost never apologize to a child, since it reverses the power dynamic. The point of an apology should really be to acknowledge having wronged someone, not necessarily to abase themselves. Parents should apologize at times to kids, in part to model the behavior, and in part to provide object lessons later about how it feels to be wronged and how it feels to receive the apology/acknowledgement.

The relationship of apologies to power is an excellent point. Reading the article, I kept thinking of the times my parents had forced me to apologise for something that even they agreed wasn't my fault, just so that it would smooth things over with the other child (typically the progeny of friends of theirs -- within the family I had the power to appeal and argue my side of the story). Did this affect me as an adult? I can think of plenty of times where I've copped to taking the blame for something I was blameless for just to get a fight to stop, so I'd have to say "yes".

Then there's the other adults I've met who seem to think saying "sorry" is a magic word that leaves them free to make the exact same transgression next time they feel like it, and who are bewildered when people get fed up with their behaviour -- they apologised! Doesn't that mean they're supposed to be forgiven, enjoy a clean slate?

Finally, there are the adults (and I include myself here as well) who cannot accept apologies, and assume they are always insincere. We were forced to pretend to be gracious while someone else pretended to be apologetic, or vice versa, too many times as children. Despite having difficulty believing, I find myself wishing I could believe -- that I could simply accept what the person was saying and move on from there. Instead, I watch the person's behaviour carefully and decide for myself if they've really mended their ways or not -- which must be uncomfortable for them if they catch me at it.

Not a very happy state to find forgiveness in.

I disagree with this analysis that an apology should be a demonstration of power. I see it rather as a gesture of commonality, an acknowledgement that the person wronged is of equal moral standing as the person apologizing and has rights just the same.

Being made to apologize "just to smooth things over" when one is blameless, now that seems like a gesture of submission, and it's not surprising that it would be resented.