I'll start by defining terms. First is "pace", well defined in articles like this one. For our purposes, we'll borrow that article's assertions: "a scene has its own pace", that pace can be slow or fast, that pace ought to vary, and that slow scenes are more common and more helpful to story than fast scenes.
"The challenge economy" is a way to answer the question of "how do we keep the scene going at the pace it ought to have?" The usual answer to this question is "present challenges for PCs to overcome". In other RPGs, this would be the frequency of dice rolls to determine something. For example, in D&D, a PC is confronted with a challenge in every combat round: what to do, and whether it will succeed. In PBTA games, a player rolls when many moves are triggered.
Games can introduce new challenges through fiction, or mechanically through failed (or complicated) dice rolls. Since any game can do the former, we'll concentrate on the latter.
Finally, for the challenge economy to work, a couple things must be true.
- Effortless success is rare enough to keep challenges coming at a good pace
- New challenges introduced by meeting current challenges ought not overwhelm the game
A 2d6+0 roll gives a 10+ 16% of the time. With +1, it's 27% of the time. And with +2, it's 41% of the time. That leaves an even split of 41% for a soft vs. hard move at +0, a 27% hard move or 44% soft move at +1, and a 16% hard move or 41% soft move at +2.
How many rolls is typical? I'll use a recent Dungeon World one-shot as a model. It's not rigorous, but eh. I rolled 14 times in 2 hours, and the other player rolled 8 times. Taking the difference gives us about 11 rolls per person, in a typical session.
A starting flip-a-card character has 3 cards they can use to generate, effectively, 10+ results. After that, they must either start taking conditions, trigger flip-side effects on their character cards, or take advantage of challenge cards through clever fiction. If they must meet 11 challenges a session, and succeed at 3 of those without complication, that's 27%, or effectively a +1 skill level in a PBTA game. If they flip 4 cards back (36%), take 2 condition cards (18%), and leverage 2 challenge cards (18%), you get a roughly similar balance between soft and hard moves. To be clear, I'd consider condition cards akin to a "hard move", since you can't just narrate your way out of them in the same scene.
If anyone feels like checking my math, let me know.