The Blue Eyes Puzzle (solution) depicts a paradox: people engage in coordinated action despite having no new information, when "I know you know he knows" reaches a critical mass. Apparently the formal system invented to address this is called Common Knowledge.
I wonder if any serious investor could actually explain what new information "the market" has which could explain why DJIA should be worth 11% less than it was 2 weeks ago.
The typical, compelling, explanation for this sort of thing is herd behavior. In the absence of new information, the market is modeled as a random walk, and when the amplitude of its swing happens to get high enough, people see a trend, anticipate it continuing, and thereby create the trend and cause a massive swing.
I wonder if you could instead model stock market swings, or other seemingly unmotivated coordinated activity, as common knowledge reaching critical mass. Say new information was injected into the market two weeks ago, and it took that long to reach a blue eyes catastrophe.
I have no evidence for this other than random pattern matching.
There is a similar common-knowledge-based argument regarding the difficulty in beating the market, based on an isomorphism to a similar logic problem.
This is the logic problem:
"There is a village of 100 couples, where each wife has the unusual ability to know when any man except her husband has been unfaithful, and she knows all other wives have that ability, but they cannot directly communicate. Each woman must also publicly kill her husband [EDIT: at noon the next day] if she can deduce he has been unfaithful. All women are also perfect logicians, capable of deducing everything that can be."
"It also happens to be the case, that every man has been unfaithful. One day, the queen, whom everyone trusts, announces that at least one man has been unfaithful. Then, all women kill their husbands. Why?"
The answer basically involves each wife doing a proof by contradiction by assuming her husband has been faithful, while leads her to believe that there exists one woman who believes all other wives believe their husbands faithful, which must be false per the queen's announcement [EDIT: once they notice no executions next noon], rendering the initial assumption of husband faithfulness false.
So, the connection to bias in investment:
"Each woman is capable of seeing other husbands' infidelity, but not her own." --> "Each active investor is capable of seeing other active investors' likelihood of underperforming the market, but not their own."
"If a woman can deduce her husband unfaithful, she kills him." --> "If an investor can deduce they are incapable of beating the market, they switch to index funds."
"Every man is unfaithful." --> "All investors are incapable of beating the market."
"Queen announces presence of one unfaithful man." --> "Market statistics announce the presence of some underperforming investors."
"Each woman does proof by contradiction that her husband has been unfaithful. --> "Each investor does proof by contradiction that they will underperform if they actively manage rather than use an index fund." (???)
The unfaithful-husbands problem isn't merely similar to the blue-eyes problem; it's exactly isomorphic. ("I have blue eyes" = "My husband is unfaithful".) In particular, that proof by contradiction involves the same sort of recursive unwinding as in the blue-eyes problem. The only difference is that the blue-eyes problem is synchronous and the unfaithful-husbands problem is asynchronous (which, actually, makes it not work without some further hypothesis about how quickly the women make their deductions, and how well they know that, and ... (read more)