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I really don't think The Joker qualifies in any version, as regardless of his backstory, he is always portrayed as proud of and revelling in his own evilness.
What's the difference between this and Woobie, Destroyer of Worlds?
Motive. The short, not totally complete answer is: Woobie is a sympathetic character doing evil things, a Tragic Villain is a character doing evil things for sympathetic reasons. Similar, and they can definitely overlap, but still different tropes.
Example moved here because Examples Are Not Arguable. Can someone look at this to see whether it is an example, and rewrite it to cut down the Word Cruft, if it is?
This seems like a real misuse of the word "tragic". The whole point of tragedy is that a character's downfall is their own fault — their mistakes or innate flaws set events in motion which ruin them. Most classic villains are tragic figures, because they set the plot in motion out of their own greed, ambition or hubris, and they're punished for it. For the same reason most films with a Villain Protagonist are tragedies.
The description of this trope suggests it's more to do with how sympathetic their motives are.
No, sympathetic motives make more for a Well-Intentioned Extremist. It's tragic because they're evil and there's nothing they can do about it. It may even be their own fault. The tragedy is that they are fully aware, not Obliviously Evil, but instead of wearing their villainy like a badge of honor, they utterly hate themselves (and probably the world they live in) because of it.
Michael Dawson IS NOT a Villain. If he is a villain, so are Jack Shephard, James Ford, Kate Austen, John Locke, Mr. Eko, Ana-Lucia Cortez, Sayid Jarrah, Shannon Rutherford, Desmond Hume and many others.
Why on earth did you label Michael as a villain, when other "LOST" characters who were survivors on the island, had committed several crimes themselves. Your label smacks of hypocrisy.
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How well does it match the trope?