Monthly Archives: September 2007

Bounded rationality and the conjunction fallacy

There is at least one case in which an apparent instance of the conjunction fallacy is not fallacious. It happens when we consider bounded rational agents rather than the more idealized unbounded rational agents typical of Bayesianism. Bounded rationality assumes that there may be some limits on the cognitive powers of the agent and looks at a concept of rationality within these constraints. In many ways it is more relevant to human decision making than unbounded rationality, though it is much less theoretically developed. In certain cases, it is (boundedly) rational to assign a higher probability to a stronger claim than one would assign to a weaker version, if the stronger claim helps to explain why the weaker one might be true (and some other conditions are met). Consider this test question (try it for yourself, with Group 1 first):

Group 1: What is your degree of belief in the following:

a) Cats can’t see better than humans in complete darkness

Now stop and think of your degree of belief, then go on

Group 2: What is your degree of belief in the following:

b) Cats can’t see better than humans in complete darkness and no animals can see in complete darkness.

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What is Evidence?

"The sentence ‘snow is white’ is true if and only if snow is white."
        — Alfred Tarski
"To say of what is, that it is, or of what is not, that it is not, is true."
        — Aristotle, Metaphysics IV

If these two quotes don’t seem like a sufficient definition of "truth", read this.  Today I’m going to talk about "evidence".  (I also intend to discuss beliefs-of-fact, not emotions or morality, as distinguished here.)

Walking along the street, your shoelaces come untied.  Shortly thereafter, for some odd reason, you start believing your shoelaces are untied.  Light leaves the Sun and strikes your shoelaces and bounces off; some photons enter the pupils of your eyes and strike your retina; the energy of the photons triggers neural impulses; the neural impulses are transmitted to the visual-processing areas of the brain; and there the optical information is processed and reconstructed into a 3D model that is recognized as an untied shoelace.  There is a sequence of events, a chain of cause and effect, within the world and your brain, by which you end up believing what you believe.  The final outcome of the process is a state of mind which mirrors the state of your actual shoelaces.

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Radically Honest Meetings

Meetings drive … productive people especially crazy … [but] serve valuable if hidden functions. For example, meetings publicize information about status. Who speaks? Who finds it necessary to praise whom? Who displays a confident demeanor? Meetings help managers and employees figure out how to build necessary coalitions. …

Meetings also confer a sense of control. Attendees feel like insiders who have a real voice in decisions. This boosts their motivation to implement ideas discussed as a group. For this reason it is especially important to listen to the blowhards and the obstructionists, who otherwise would pursue their own agendas rather than support a common plan.  Frequent meetings help a business apply bonuses and yearly evaluations with greater precision. … meetings reaffirm the value of the individual to the company. …

That is Tyler Cowen at his best.  Now when people talk about why we have meetings, or what function meetings served, we don’t usually talk about showing and judging status and buying off blowhards.  Could we talk more honestly about the function of meetings, or would that defeat the purpose? 

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Burdensome Details

Followup toConjunction Fallacy

 "Merely corroborative detail, intended to give artistic verisimilitude to an otherwise bald and unconvincing narrative…"
            — Pooh-Bah, in Gilbert and Sullivan’s The Mikado

The conjunction fallacy is when humans rate the probability P(A&B) higher than the probability P(B), even though it is a theorem that P(A&B) <= P(B).  For example, in one experiment in 1981, 68% of the subjects ranked it more likely that "Reagan will provide federal support for unwed mothers and cut federal support to local governments" than that "Reagan will provide federal support for unwed mothers."

A long series of cleverly designed experiments, which weeded out alternative hypotheses and nailed down the standard interpretation, confirmed that conjunction fallacy occurs because we "substitute judgment of representativeness for judgment of probability".  By adding extra details, you can make an outcome seem more characteristic of the process that generates it.  You can make it sound more plausible that Reagan will support unwed mothers, by adding the claim that Reagan will also cut support to local governments.  The implausibility of one claim is compensated by the plausibility of the other; they "average out".

Which is to say:  Adding detail can make a scenario SOUND MORE PLAUSIBLE, even though the event necessarily BECOMES LESS PROBABLE.

If so, then, hypothetically speaking, we might find futurists spinning unconscionably plausible and detailed future histories, or find people swallowing huge packages of unsupported claims bundled with a few strong-sounding assertions at the center.

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Your Future Has Detail

Our visions of future events don’t include a lot of detail.  In the Sept. 7 Science, Gilbert and Wilson discuss the many resulting biases: 

We feel better when we imagine going to the theater than to the dentist, but we feel better imagining either event on a sunny day than on a rainy day, or when we are well rather than ill. … When people who have missed trains in the past are asked to imagine missing a train in the future, they tend to remember their worst train-missing experience rather than their typical train-missing experience. … which leads them to overestimate how painful the next train-missing experience will be.

Similarly, when people experience an unpleasant episode that ends in brief relief – for example, submerging their arms for 90 s in a bath of ice water that is slightly warmed in the final 30 s – they tend to remember the closing moments of the experience rather than the most typical moments … which leads them to underestimate how painful the recurrence will be. It seems that everyone remembers their best day, their worst day, and their yesterday. Because unusual events and recent events are so memorable, people tend to use them when constructing simulations of future events. … Because simulations omit inessential features, people tend to predict that good events will be better and bad events will be worse than they actually turn out to be …

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Conjunction Controversy (Or, How They Nail It Down)

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Followup toConjunction Fallacy

When a single experiment seems to show that subjects are guilty of some horrifying sinful bias – such as thinking that the proposition “Bill is an accountant who plays jazz” has a higher probability than “Bill is an accountant” – people may try to dismiss (not defy) the experimental data.  Most commonly, by questioning whether the subjects interpreted the experimental instructions in some unexpected fashion – perhaps they misunderstood what you meant by “more probable”.

Experiments are not beyond questioning; on the other hand, there should always exist some mountain of evidence which suffices to convince you.  It’s not impossible for researchers to make mistakes.  It’s also not impossible for experimental subjects to be really genuinely and truly biased.  It happens.  On both sides, it happens.  We’re all only human here.

If you think to extend a hand of charity toward experimental subjects, casting them in a better light, you should also consider thinking charitably of scientists.  They’re not stupid, you know.  If you can see an alternative interpretation, they can see it too.  This is especially important to keep in mind when you read about a bias and one or two illustrative experiments in a blog post.  Yes, if the few experiments you saw were all the evidence, then indeed you might wonder.  But you might also wonder if you’re seeing all the evidence that supports the standard interpretation.  Especially if the experiments have dates on them like “1982” and are prefaced with adjectives like “famous” or “classic”.

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Why Teen Paternalism?

Though in centuries past 15-19 year olds were treated as adults, today we often paternalistically restrict their behavior because of "immature" brains.   An OpEd in Monday’s New York Times says 35-54 year olds actually behave worse:

A spate of news reports have breathlessly announced that science can explain why adults have such trouble dealing with teenagers: adolescents possess "immature," "undeveloped" brains that drive them to risky, obnoxious, parent-vexing behaviors. … But the handful of experts and officials making these claims are themselves guilty of reckless overstatement. More responsible brain researchers … caution that scientists are just beginning to identify how systems in the brain work. …

Our most reliable measures show Americans ages 35 to 54 are suffering ballooning crises: … 46,925 fatal accidents and suicides in 2004, leaving today’s middle-agers 30 percent more at risk for such deaths than people aged 15 to 19 … 21 million binge drinkers (those downing five or more drinks on one occasion in the previous month), double the number among teenagers and college students combined …

Overdose rates for heroin, cocaine, pharmaceuticals and drugs mixed with alcohol far higher than among teenagers. … More than half of all new H.I.V./AIDS diagnoses in 2005 were given to middle-aged Americans, … It’s true that 30 years ago, the riskiest age group for violent death was 15 to 24. But those same boomers continue to suffer high rates of addiction and other ills throughout middle age, while later generations of teenagers are better behaved. Today, the age group most at risk for violent death is 40 to 49, including illegal-drug death rates five times higher than for teenagers.

Strangely, the experts never mention even more damning new "discoveries" about the middle-aged brain, like the 2004 study of scans by Harvard researchers revealing declines in key memory and learning genes that become significant by age 40.

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Conjunction Fallacy

The following experiment has been slightly modified for ease of blogging.  You are given the following written description, which is assumed true:

Bill is 34 years old.  He is intelligent, but unimaginative, compulsive, and generally lifeless.  In school, he was strong in mathematics but weak in social studies and humanities.

No complaints about the description, please, this experiment was done in 1974.  Anyway, we are interested in the probability of the following propositions, which may or may not be true, and are not mutually exclusive or exhaustive:

A:  Bill is an accountant.
B:  Bill is a physician who plays poker for a hobby.
C:  Bill plays jazz for a hobby.
D:  Bill is an architect.
E:  Bill is an accountant who plays jazz for a hobby.
F:  Bill climbs mountains for a hobby.

Take a moment before continuing to rank these six propositions by probability, starting with the most probable propositions and ending with the least probable propositions.  Again, the starting description of Bill is assumed true, but the six propositions may be true or untrue (they are not additional evidence) and they are not assumed mutually exclusive or exhaustive.

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Beware Monkey Traps

It is said you can trap a monkey by putting a nut through a small hole in a gourd.  The monkey reaches in and grabs the nut, but then his fist won’t fit back through the hole.  Greedy monkeys will literally let themselves be caught rather than let go of the nut.  So far, no commenter on my essay seems willing to let go of the nut of effective medicine, held in the gourd of the second half of medical spending. 

As an analogy, imagine you ran a software company, whose many offices had different wage levels and work cultures, with average work hours ranging from seven to fourteen per day.  Surprised to see these offices were equally productive, you randomly changed wages, inducing changes in work hours.  You again found offices that worked more did not produce more; after seven hours people got tired and added as many bugs as they fixed.  If instead of just cutting wages to get only seven hours of work, you just told everyone "watch out for bugs," you would be in a monkey trap, refusing to let go of the nut of productive work in the gourd of extra work hours.

So begins my first reply at CATO Unbound.   I go on to argue that it is a monkey trap that keeps health policy experts from endorsing my proposal to "Cut Medicine In Half."  You might think that humans wouldn’t fall for such a simple trap, but consider our military policy of "Leave No Man Behind":

Depicted in the film Black Hawk Down, this mission resulted in the deaths of 18 soldiers. In fact, the strategy of enemy Somali militiamen focused on the American policy of not leaving any soldier behind; they knew that if they managed to shoot down a helicopter, the Americans would move in to defend the helicopter’s crew.

Since I suggest "showing that you care" signals explain our inclination toward excess medicine, and solidarity signals are said to explain "leave no man behind," perhaps monkeys are inclined to never let go of food as a signal to would-be thieves. 

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Outputs Require Inputs

It is a simple point: mechanisms give outputs from inputs.  With more inputs, we expect more outputs.  So when comparing mechanisms, correct for input variation. 

For example, over $100 million was spent trying to win the $10 million Ansari prize, as competitors also wanted credibility in the near-Earth space market.  So now the Google moon X-Prize offers $30 million, seemingly far too little for such an effort, as there is no moon market to win.  I worry that when the prize is not won, people will take this as a failure of the prize mechanism, rather than as a failure of the prize amount offered.

Also, every week I see another startup whose business model is to sell info from play money "competitive forecasting" (like prediction markets).  (E.g., see yesterday’s New York Times article where I’m quoted).  Professionals who would otherwise charge for their insight will supposedly instead tell all for the "community" of a few token prizes, chat rooms, comment sections, leader boards, and social networking.  "Crowd-sourcing" software experts have assured them this, and a marketing budget, is all it takes to make a volunteer community they can sell.  (Curiously, these software experts have not suggested replacing themselves with free open source volunteers.)

I worry that when these businesses fail, people will take this as a failure of mechanisms like prediction markets, rather than as a failure to get people to work for free.  Prizes are a promising way to induce research or development, and prediction markets are a promising way to gain information, even when you must on average pay contributors market wages for their time and efforts. 

Added: InTrade now lets you bet on whether the Google Moon prize will be won.   

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