“Psychophysics,” despite the name, is the respectable field that links physical effects to sensory effects. If you dump acoustic energy into air—make noise—then how loud does that sound to a person, as a function of acoustic energy? How much more acoustic energy do you have to pump into the air, before the noise sounds twice as loud to a human listener? It’s not twice as much; more like eight times as much.

Acoustic energy and photons are straightforward to measure. When you want to find out how loud an acoustic stimulus sounds, how bright a light source appears, you usually ask the listener or watcher. This can be done using a bounded scale from “very quiet” to “very loud,” or “very dim” to “very bright.” You can also use an unbounded scale, whose zero is “not audible at all” or “not visible at all,” but which increases from there without limit. When you use an unbounded scale, the observer is typically presented with a constant stimulus, the modulus, which is given a fixed rating. For example, a sound that is assigned a loudness of 10. Then the observer can indicate a sound twice as loud as the modulus by writing 20.

And this has proven to be a fairly reliable technique. But what happens if you give subjects an unbounded scale, but no modulus? Zero to infinity, with no reference point for a fixed value? Then they make up their own modulus, of course. The ratios between stimuli will continue to correlate reliably between subjects. Subject A says that sound X has a loudness of 10 and sound Y has a loudness of 15. If subject B says that sound X has a loudness of 100, then it’s a good guess that subject B will assign loudness in the vicinity of 150 to sound Y. But if you don’t know what subject C is using as their modulus—their scaling factor—then there’s no way to guess what subject C will say for sound X. It could be 1. It could be 1,000.

For a subject rating a single sound, on an unbounded scale, without a fixed standard of comparison, nearly all the variance is due to the arbitrary choice of modulus, rather than the sound itself.

“Hm,” you think to yourself, “this sounds an awful lot like juries deliberating on punitive damages. No wonder there’s so much variance!” An interesting analogy, but how would you go about demonstrating it experimentally?

Kahneman et al. presented 867 jury-eligible subjects with descriptions of legal cases (e.g., a child whose clothes ...

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