I looked for an answer to this but couldn't find one. Perhaps that's because I'm misunderstanding something about the solo game. (In which case, I may move this to the Rules forum instead of the Strategy forum.)
In a solo game, I am always making the decision of which die to place on which Initiative card, so I'll place the die I want on the Initiative card I want to choose.
If I choose Initiative 1: 0 gold and no Market card is trashed before I purchase.
If I choose Initiative 2: +1 gold and a Market card might be trashed (50/50 chance).
If I choose Initiative 3: 0 gold and a Market card will be trashed.
Initiative 3 gets me no benefit, but it has a 33% chance of trashing the card I want to purchase. Whereas Initiative 2 gets me one gold at a 16% chance of trashing the card I want. And Initiative 1 guarantees that I can purchase the card I want (but without gaining any gold). Why would I ever choose Initiative 3?
What they said!