I think its still possible to have a scenario like this. Lets say each trader would buy or sell a certain amount when the price is below/above what they think it to be, but the transition being very steep instead of instant. Then you could still have long price intervalls where the amounts bought and sold remain constant, and then every point in there could be the market price.

I’m not sure if this is significant. I see no reason to set the traders up this way other than the result in the particular scenario that kicked this off, and adding traders who don’t follow this pattern breaks it. Still, its a bit worrying that trading strategies seem to matter in addition to beliefs, because what do they represent? A traders initial wealth is supposed to be our confidence in its heuristics—but if a trader is mathematical heuristics and trading strategy packaged, then what does confidence in the trading strategy mean epistemically? Two things to think about:

Is it possible to consistently define the set of traders with the same beliefs as trader X?

It seems that logical induction is using a trick, where it avoids inconsistent discrete traders, but includes an infinite sequence of continuous traders with ever steeper transitions to get some of the effects. This could lead to unexpected differences between behaviour “at all finite steps” vs “at the limit”. What can we say about logical induction if trading strategies need to be lipschitz-continuous with a shared upper limit on the lipschitz constant?

The continuity property is really important.

I think its still possible to have a scenario like this. Lets say each trader would buy or sell a certain amount when the price is below/above what they think it to be, but the transition being very steep instead of instant. Then you could still have long price intervalls where the amounts bought and sold remain constant, and then every point in there could be the market price.

I’m not sure if this is significant. I see no reason to set the traders up this way other than the result in the particular scenario that kicked this off, and adding traders who don’t follow this pattern breaks it. Still, its a bit worrying that trading strategies seem to matter in addition to beliefs, because what do they represent? A traders initial wealth is supposed to be our confidence in its heuristics—but if a trader is mathematical heuristics and trading strategy packaged, then what does confidence in the trading strategy mean epistemically? Two things to think about:

Is it possible to consistently define the set of traders with the same beliefs as trader X?

It seems that logical induction is using a trick, where it avoids inconsistent discrete traders, but includes an infinite sequence of continuous traders with ever steeper transitions to get some of the effects. This could lead to unexpected differences between behaviour “at all finite steps” vs “at the limit”. What can we say about logical induction if trading strategies need to be lipschitz-continuous with a shared upper limit on the lipschitz constant?