Suppose I spin a Wheel of Fortune device as you watch, and it comes up pointing to 65. Then I ask: Do you think the percentage of countries in the United Nations that are in Africa is above or below this number? What do you think is the percentage of UN countries that are in Africa? Take a moment to consider these two questions yourself, if you like, and please don’t Google.
Also, try to guess, within five seconds, the value of the following arithmetical expression. Five seconds. Ready? Set . . . Go!
1 × 2 × 3 × 4 × 5 × 6 × 7 × 8
Tversky and Kahneman recorded the estimates of subjects who saw the Wheel of Fortune showing various numbers.1 The median estimate of subjects who saw the wheel show 65 was 45%; the median estimate of subjects who saw 10 was 25%.
The current theory for this and similar experiments is that subjects take the initial, uninformative number as their starting point or anchor; and then they adjust upward or downward from their starting estimate until they reach an answer that “sounds plausible”; and then they stop adjusting. This typically results in under-adjustment from the anchor—more distant numbers could also be “plausible,” but one stops at the first satisfying-sounding answer.
Similarly, students shown “1 × 2 × 3 × 4 × 5 × 6 × 7 × 8” made a median estimate of 512, while students shown “8 × 7 × 6 × 5 × 4 × 3 × 2 × 1” made a median estimate of 2,250. The motivating hypothesis was that students would try to multiply (or guess-combine) the first few factors of the product, then adjust upward. In both cases the adjustments were insufficient, relative to the true value of 40,320; but the first set of guesses were much more insufficient because they started from a lower anchor.
Tversky and Kahneman report that offering payoffs for accuracy did not reduce the anchoring effect.
Strack and Mussweiler asked for the year Einstein first visited the United States.2 Completely implausible anchors, such as 1215 or 1992, produced anchoring effects just as large as more plausible anchors such as 1905 or 1939.
There are obvious applications in, say, salary negotiations, or buying a car. I won’t suggest that you exploit it, but watch out for exploiters.
And watch yourself thinking, and try to notice when you are adjusting a figure in search of an estimate.
Debiasing manipulations for anchoring have generally proved not very effective. I would suggest these two: First, if the initial guess sounds implausible, try to throw it away entirely and come up with a new estimate, rather than sliding from the anchor. But this in itself may not be sufficient—subjects instructed to avoid anchoring still seem to do so.3 So, second, even if you are trying the first method, try also to think of an anchor in the opposite direction—an anchor that is clearly too small or too large, instead of too large or too small—and dwell on it briefly.
1Amos Tversky and Daniel Kahneman, “Judgment Under Uncertainty: Heuristics and Biases,” Science 185, no. 4157 (1974): 1124–1131.
2Fritz Strack and Thomas Mussweiler, “Explaining the Enigmatic Anchoring Effect: Mechanisms of Selective Accessibility,” Journal of Personality and Social Psychology 73, no. 3 (1997): 437–446.
3George A. Quattrone et al., “Explorations in Anchoring: The Effects of Prior Range, Anchor Extremity, and Suggestive Hints” (Unpublished manuscript, Stanford University, 1981).
Let's make the debiasing technique more rigorous.
How much more unlikely is it that I will throw 15 consecutive snake-eyes, than that I will throw 11 consecutive snake eyes?
I should allocate about -170 dB of belief to the likelihood of throwing 11 snake-eyes, and about -232 dB to the likelihood I will throw 15 snake-eyes. The ~60 dB difference indicates the latter event is 6 orders of magnitude more unlikely.
What does it mean if someone thinks the difference is smaller?
If 6 orders of magnitude of improbability are glossed over, that means the person does not comprehend it in gut terms.
To what other event might I allocate -60 dB to? How about flipping a coin 20 times and getting all Heads?
Now we're getting somewhere. Let us ask ourselves a series of restricted Aumann Questions (on various statements in general knowledge) and calculate our joint belief. The difference between the belief we allocated, and the belief we ought to have allocated, is a measure of our flattened sense of improbability. We can take this into account, and adjust our anchors accordingly. We can, in effect, see how finely-tuned is our sense of improbability.
i.e. Suppose I take a restricted Aumann test of 40 questions regarding various general facts. I assign a joint probability of -150 dB to the survey. If I were better calibrated, my priors ought to have increased this to -100. I now know I must be aware a possible 50 dB gap between my beliefs and reality, I ought to be wary of any parochial adjustment. How wary? I should attach very little confidence to any adjustment under one order of magnitude...
I ran that one in my head and thought, "that's got to be about a million times less likely." And indeed it was, 6 orders of magnitude. To some extent, I may just have gotten lucky... but I think that lurking on Less Wrong for the last couple years may have made me appreciate probabilities at a more intuitive level.
So does this mean Less Wrong actually works?