A suggestion: since you have Joy's number, perhaps consider giving the Lottery Winner the option to "summon" Joy to the island for poker, chats, or to have some "quality time" at the cost of Company Favor? Say perhaps 50 favor for one day or 500 favor to have available at will?
We are rats in Joy's lab and she is yanking our chain not the reverse.
She is willing to make sure her little maze runs well (visits) and get some sex out of it too (highly sexed and the MC is exceptional by all accounts) but the power and choice nonetheless rests with her and only her, we can solicit her presence not command it.
And HI favor is made by her, it holds no value for her beyond as a tool.
Even without those things existing, she would, by her nature and persona never been available at will to anyone.
A basic premise of HI is that while the Main Character seem to be seducing the girls and thus "in control", this is not the case, it is an illusion.
As the story and girls hint at from time to time, there are three plays occurring on the Island,
1. Joy's reasons and experiments (which is the foundation of the HI game).
2. The girls competing for fun against each other, the MC being both an sex and affection interest for them yes, but he also a proxy prize, a tool, used to measure progress and standing in
their internal competition
3. And last, and least important, the MC's game (which is our part of it), which first takes on any meaning as we progress their personal quests and the girls start to view him with
real affection or
actual interest.