Some people have been discussing this paper on apologizing, which notes that there is some (limited) evidence that apologies don't tend to result in less punishment, and may in fact lead people to want to increase the punishment. This is actually consistent with a number of other things we know; there is a lots of evidence that in one way or another suggests something like this. If your goal is to avoid punishment, or to reduce it, apologizing is not a way to do that. What I've found surprising is that so many people find this surprising whenever a new bit of evidence shows this to be the case.
Apologizing always has the following logical structure, which can be done in lots of different ways but are all conditions for it to be a genuine apology:
(1) You admit that you have done something that deserves punishment.
(2) You recognize that the person to whom you are apologizing has the right or authority to punish you.
(3) You 'turn yourself in', so to speak, to be punished.
That's it. It's why you should never, ever apologize if you are not genuinely willing to be punished; apologizing is giving people permission to punish you. It always has been. It's also why a certain kind of toxic personality is always trying to force people to apologize: once you've apologized to them, you've given them power. It's the venomous dialectic of our day, in fact, that people who want power over you are always trying to get you to apologize for something; they will even make something up for you to apologize about, if that's what it takes.
So why, then, do people so often assume that apologizing will naturally lead to lessening of punishment? In reality, you would obviously expect that refusing to apologize would tend to deter people from bringing the full force of punishment, by making clear that if they want to punish you, they have to fight you, while you'd expect apologies, which make clear that you are open to being punished and will not fully resist, to lead to more and harsher punishment. And this is exactly we find in many cases. So what's the reason people are astonished that permitting people to punish you often leads to more and harsher punishment?
Of course, it depends on the audience. People are sometimes mind-boggled by the willingness some have shown in the past for forgiving televangelists for their indiscretions when they apologize. But they know their audience, a broadly evangelical group who have been raised to believe that repentance is very, very important, that it takes some courage to do and therefore can in that sense be admirable, and that mercy to those who repent is a duty. In such a context, apologies are brave expressions of repentance which require that one act mercifully; in that context, apologies are obviously to be encouraged and so obviously you want people to be punished more harshly if they don't apologize than they do. Things can get a little more complicated when we insist on sincerity of repentance, but in fact the kind of people who believe all these other things are also inclined to think that we have a duty to give people the benefit of the doubt. Outside that context, people will take apologies for what they are: confessions of wrongdoing in which the wrongdoer agrees to be punished, and thus either not affecting the punishment or giving more reason to punish.
If you want a society in which apologies are taken seriously rather than being ignored or being treated as reasons for intensifying punishment, you have to build it. People have to be raised to stay their hand in the face of apology. I suppose the fact that so many people are surprised by the ineffectiveness of apology to reduce punishment is a sign that many still are. But you have only to look around to see that this seems to be a dwindling minority. In such a society, the rules are: Don't apologize unless you are genuinely sorry; never apologize to someone you don't want to have power over you; do not tolerate people demanding apologies for minor or made-up reasons. It's the only way to survive.