Again, do you realize the fact this is a 2011 thread?
Opponent: I Should Resign on move 9 a piece down. Agree?
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If I was playing someone around my rating or higher, I would probably resign if it wasn't a bullet game. If I was playing someone significantly lower rated, I might play on for a bit.
But it is a personal preference, so long as you aren't intentionally stalling, there is nothing inherently wrong with playing to mate in any position, though I do think it is good sportsmanship to resign beforehand.
The rule of thumb in online chess: never resign.
There is no 'should have' - it is your choice.
There's also no need to be polite. Because, checkmate isn't polite. It's total destruction.
Not resigning when your opponent is getting aggravated is actually a neat tactic, because it will improve their chances of messing up. As the emotional brain starts firing, the PFC gets fuzzy.
Between humans, it's more about what's more beneficial for progress and learning. It's also not about winning, a common misconception. Winning from others is simply the result of playing against standards you impose upon yourself.
If you think you're in an interesting position and have some kind of game, just play on.
This is only courtesy at GM level where players can see the entire board in their minds and 10-20 moves in.
Just consider the game to be an odds exhibition
But in this case, you didn't manage to achieve king safety. Move 18-19 was more pivotal.
If you had, you'd have had king safety. Your lines to the enemy king were declining rapidly. If you have only 1-2 left and your opponent sees them and can defend them, as was the case, it's probably not useful to play on. Then again, sometimes they quit or internet breaks, whatever, and therefore I never resign unless I'm at <5s clock time and checkmate is impending. But that's for statistical purposes only, to minimize my number of losses by checkmate.