There probably is no fixed point, however: imagine a game with two players, the hunter and the prey where the turn consists of choosing whether to go place A or to place B. If both players end up in the same place, the hunter wins, otherwise the prey wins. Keep revealing to them the other player's decision, and they will keep adjusting their decisions without ever reaching any fixed point because logical negation doesn't have a fixed point. So I would wager this result generalizes to almost any antagonistic game.
> where a new forecast doesn't change anybody's reaction any more
That's the original proposition I was working with. Obviously, if the amount of published forecasts finite, then the amount of decision changes in response to such publications will be finite too.
Now that I think about it, a forecast that causes nobody to change any plans is pretty useless and is effectively indistinguishable from the trivial "empty" forecast which got to be the unique fixed point of this whole forecasting exercise.
Yeah, the forecast itself has to factor the changing of plans, e.g. don't pick a solution that won't get picked. If there is low trust placed onto the forecasting system, then no viable solutions is the only answer. There is a scale of influence to it as well, if the forecaster is giving a forecast to someone who can only purchase $1000 worth of shares, that won't influence the economy in any meaningful way. Someone with a $1B dollars who can actually present adversarial strategies to other players in the market? A lot less opportunities can be presented (assuming other players also have a forecaster), and a lot more trust in the forecaster would be required. Equilibrium only exists if everyone wins, and moreover, everyone is satisfied with the victory that they have been given.
It's all based on trust though, if one forecaster suspects that another forecaster has been hacked to optimize profits for that individual player, the equilibrium is broken.
Even with everyone in the world having an all powerful genie, the greed of a few but powerful can ruin the entire thing.
