I had an idea about that. Publish the algorithm by which you come up with each game's multiplier. Presumably it will be seeded somehow. Pick a random seed that you will use for game 1 billion. sha256 hash it to get the seed you will use for the game before that, and so on. Each games seed is the sha256 hash of the game that comes after it. So you have a billion-long chain of sha256 hashes. Before each game you publish the hash of the current game's seed, which allows everyone to verify all the previous games but none of the future games. Now you can't be accused of rigging the outcomes.
Hmm. I just tried it, and was able to create a 0.2% undetectable house advantage for me, for the first year of operation. The problem is quite simple, I don't (and can't!) publish the starting seed -- which means I can cherry-pick the best one for me. Certain sequences are going to be more or less favorable to me. When I tried generating thousands of starting seeds -- and picking the one the lead the the most favorable opening year for me -- I was able to pretty quickly find one that would give me a 0.2% advantage. If I spent a bit more time, I'm sure I could do even better.
Given this, I'm not sure if I should bother continuing down this route?