What about bidding ? Where players can control the narrative by winning a blind bid. I think there are a bunch of games that use that with tokens. The only thing would be divorcing that from numbers, swapping 'x tokens' for abstract representations, allowing them to chose what they bid... Rats, I've come back to rock paper scissors, back to the drawing board.
Edit: rereading the thread, it seems there have already been suggestions similar to this.