that would mean it's decided by who has the final say, Ivy or the archive? If Ivy didn't have some pull, then she wouldn't be mortal still and offering her a coin was a particularly useless plan.
I think it is a bit more complicated. According to Luccio, it is unusual for the host to inherit the Archive so early in life, poor Ivy it has been a domino effect really. First her grandmother dies in an auto accident, the Archive goes to the seventeen year old daughter, who is pregnant and in love. Because she wanted the freedom to live a "normal life," she hated her mother for dying and putting this on her and hated her unborn daughter for a future of having a "normal life," while she has a lifetime of being the Archive, so she killed herself.
So yes, when the Denarians kidnapped her, Ivy was vulnerable, or seemed that way to accepting a coin in the hopes that it might insure she had a "normal life." However except for some flashes of childlike behavior, the Archive has run the show more than Ivy has. Ivy has no clue of what a normal life is, because she never had one. So yeah, the bottom line, is the Archive was running the show and offering a coin was a pretty useless plan.