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 how well they know that, etc.).
But this sort of problem seems to me to depend on very fragile details—perfect reasoning, common knowledge, absolutely predictable behaviour, and so forth. Even a small deviation from those details can make the catastrophe (mass exodus or execution, or mass movement to index funds) not happen.
… But it’s “can make”, not “will definitely make”. For instance, let’s modify the blue-eyes problem slightly by allowing for a little uncertainty: everyone starts off with Pr(I have blue eyes) = 1⁄2; if that probability reaches p then they leave with probability q (both these probabilities are very close to 1). Leave everything else the same. What then? Well, it turns out that here the catastrophe still happens, with probability very close to 1, for any n; the uncertainty doesn’t expand exponentially and break the inferences.
There’s another difficulty in matching up the cuckolded wives (hmm, I think “cuckolded” is a gender-specific term; is there a word for a female victim of marital infidelity?) with the underperforming investors: the very first parallel in your list states, in effect, that investors are extremely irrational when contemplating their own performance, which is hard to square with making them all perfect logicians who make all available inferences.
[EDITED to add: it turns out that the wife of an unfaithful husband is a “cuckquean”; the word is obsolete and has many variant spellings. And there’s an old joke: the word for a man whose wife is unfaithful is “cuckold”; the word for a woman whose husband is unfaithful is “wife”.]
Good points, all. You’re right that I didn’t get the 100 wives problem phrased completely; it would need to stipulate that the killing happens at e.g. the next day’s noon, just as the blue eye problem has departures at discrete intervals. Then each wife expects that some wife will kill at the first chance after the announcement, and when none do, they can all conclude that their initial assumption of husband faithfulness is false.
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 how well they know that, etc.).
But this sort of problem seems to me to depend on very fragile details—perfect reasoning, common knowledge, absolutely predictable behaviour, and so forth. Even a small deviation from those details can make the catastrophe (mass exodus or execution, or mass movement to index funds) not happen.
… But it’s “can make”, not “will definitely make”. For instance, let’s modify the blue-eyes problem slightly by allowing for a little uncertainty: everyone starts off with Pr(I have blue eyes) = 1⁄2; if that probability reaches p then they leave with probability q (both these probabilities are very close to 1). Leave everything else the same. What then? Well, it turns out that here the catastrophe still happens, with probability very close to 1, for any n; the uncertainty doesn’t expand exponentially and break the inferences.
There’s another difficulty in matching up the cuckolded wives (hmm, I think “cuckolded” is a gender-specific term; is there a word for a female victim of marital infidelity?) with the underperforming investors: the very first parallel in your list states, in effect, that investors are extremely irrational when contemplating their own performance, which is hard to square with making them all perfect logicians who make all available inferences.
[EDITED to add: it turns out that the wife of an unfaithful husband is a “cuckquean”; the word is obsolete and has many variant spellings. And there’s an old joke: the word for a man whose wife is unfaithful is “cuckold”; the word for a woman whose husband is unfaithful is “wife”.]
Good points, all. You’re right that I didn’t get the 100 wives problem phrased completely; it would need to stipulate that the killing happens at e.g. the next day’s noon, just as the blue eye problem has departures at discrete intervals. Then each wife expects that some wife will kill at the first chance after the announcement, and when none do, they can all conclude that their initial assumption of husband faithfulness is false.
Very nice!
Typo near the end: “Queen announces presence of one faithful man”: faithful → unfaithful.