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If people trust human forecasters over machine forecasters, but the machine forecasts are better, just use the machine secretly, and take all the credit.
Indeed. That is precisely what the so-called "closet index funds" are doing. They are said to be actively managed funds, but are in reality so-called index trackers, which just are tracking the stock market index.
The reason the managers of the fund are using index-tracking algorithms rather than human experts is, however, not so much that the former are better (as I understand they are roughly on par) but that they are much cheaper. People think that the extra costs that active management brings with it are worth it, however, since they erroneously believe that human experts can consistently beat the index.
Maybe human experts tend to track the index anyway?