A lot of people say you should apologise - and apologise first - because you value the relationship over being right (i.e. winning the argument).
Personally, I'd say: What kind of relationship do you value?
Because personally, I don't want a relationship based around being subservient to the other person's desires, and where they get to always be right, and never take responsibility. I've had too many "friendships" that have been based around my having to apologise for every disagreement (or whatever), and the other person taking that as absolution for their part, instead of a cue to apologise in return. 😑 I'm pretty sure that's toxic...
And then waiting for them to apologise first means waiting for something that'll never happen. 😒
(And even if a lot of it is fiction, I've seen a LOT of examples where apologizing encourages abuse... not to mention is often never enough... 😢)
There's one thing I picked up from a comic: "Can we agree that we both messed up?"
There's a lot I like about that line. It requests mutual agreement, as well as mutual responsibility. It means no one person having to be the first, nor the last.
That said, I haven't had much luck, yet, with using it. 😅 I think a lot of the people I've used it with, so far, tend to see it as an accusation or attack. 😑
But that said, I think the type of relationship I value is one where it's not about winning or losing, so much as attempting to calmly and accurately assess the situation, and no one having to beat themselves up for, say, an honest mistake. One where "we both could do better" is certainly possible, if not always the conclusion. 😉