From what I remember, the Emperor hated that Olivia’s so called father and the Duchess had a perfect relationship in society. So to tear them up, he purposefully drugged Olivia’s mother and father and had them sleep together to tarnish his reputation. Her mother was presumably blackmailed or coerced and she was a victim herself.
But from what I’m guessing the reason for the engagement stems from having the Crown Prince married to a woman from a high ranking family which the Madeline’s are in society. If not for Olivia, then the engagement would be pushed onto her younger sister but everyone knows the Prince has a lover of his own.
Meanwhile the Emperor wanted his daughter to marry the Grand Duke and rule there.
umm okay a lot of people are saying stuffs regarding the background of the characters. let me add some that is not too confusing
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so the king wants Olivia to marry crown prince coz her father is the head of the Royalist faction, to strengthen their connection. it doesn't really have to be olivia but they need a woman from his family, that's the reason she got adopted (he doesn't want his real daughter to suffer in politics)
the queen and princess want Olivia to be married to crown prince coz they need a slave to do their work, Olivia is the one doing the full work of Princess while the Princess take all the credits, (aside from doing the princess' job, she's also doing her job as the prince's fiance)
that's why it's gonna be fun to see their work tumbling down since they can't do it properly w/out MC

So what's the deal with the royalty here? Why are they adamant to keep the engagement with her?