Gradual information flow makes intuitive sense to me even for fully rational investors who are aware that they are at an informational disadvantage.

The investors who are at an informational disadvantage are more likely to be long term holders of the securities that they purchase than market participants as a whole. Because of that, it's rational for them to model many of the short term events which cause major (but transitory) market swings as if they were noise. This means that their process for updating their assumptions has to heavily discount the information content of short term market movements because they don't have the understanding of short term movements necessary to recognize their causes (and therefore whether they reflect long term changes in fundamentals).

The main testable prediction from the way that I'm looking at this is that investors who know they are at an informational disadvantage (at least in their understanding of short term movements) should trade less often -- either by owning index funds or by trading only on long term fundamental changes where they don't consider themselves at an informational disadvantage.

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One reason (I'd say) for irrationality in markets is that the information flow also includes what other players in the market are doing. Investors are concerned that others have more information than they do; hence, when the market goes down (up), the investor infers that others have more information, and then sells (buys) accordingly. Also, investors suffer mightily from recency bias, weighting recent information more heavily than what is already known.

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