Is there a "percentage fallacy"?

The "percentage fallacy" is a real thing, and it does get covered in standard JDM/behavioral economics. See the classic Kahneman & Tversky jacket-calculator study, for instance. K&T also cite a study which shows that if you compare the prices of a consumer good at various stores, the standard deviation in price is roughly proportional to the price of the good.

I don't think it has a handy name, but the money version ($20 out of $2500 or $40) usually gets covered under mental accounting, as reflecting the shape of the value function (diminishing marginal sensitivity). It also gets connected more broadly to Weber's Law in psychophysics, which basically says that people's perceptions of a continuous quantity (including weight, brightness, etc.) follow a log scale - e.g., if someone is holding a weight and you add more weight to it, the smallest change that they'll notice is proportional to the total amount of weight that they're holding.

There has been some research applying this to temporal discounting, including a recent paper by Zauberman & colleagues (pdf).

Zauberman, G. B., Kim, K., Malkoc, S., & Bettman, J. R. (2009). Discounting time and time discounting: Subjective time perception and intertemporal preferences. Journal of Marketing Research, 46, 543–556.

It seems to me it's more important to save a larger percent on the $40 items than a much smaller percent on $2500 items since in the long run you'll buy many more of the first type of items than the second. So it looks like a useful heuristic.

This has been discussed before.

My first thought is that you buy $40 items much more often than $2500 items; saving $20 on each $40 dollar purchase will save more money than the same savings on $2500 purchases. Especially given costs of driving around town and negotiating in high value purchases. I'll agree that if you include these hidden costs and convert to a utility calculation people are probably not perfectly efficient, but doing the calculations themselves can be expensive.

One way in which I've observed some very smart and numerate people falling for this fallacy is when they run out of cash in bars and are forced to take money out of those rip-off ATMs that charge $3 or so per transaction. People will often take a larger amount of money than necessary, rationalizing that the rip-off isn't that bad if it's only a small percentage of the amount (I've heard this "reasoning" expressed loudly several times). This despite the fact that tomorrow they'll walk by their own bank's ATM from which they can take money without any fees, so there's absolutely no benefit from taking more money than necessary from the expensive one.

I am myself not immune to this feeling, even though I'm perfectly aware it's completely irrational. I would feel awful if I paid $3 to take out a twenty and then took $100 without a fee next day, but paying $3 to take $120 feels much less bad.

Thinking about some of the examples people have posted, I think people might be making decisions based on an assumption of "what if I did this many times?". For example, if you spent all your money on blenders, you'd waste half of it if you didn't go out of your way to get the cheaper ones, whereas if you spent all your money on laptops, you'd only waste a small percentage. Likewise with the ripoff ATMs - if you withdrew all your money from them, you'd be much better off withdrawing as much as possible each time.

There was an article in Scientific American a few years ago about the Traveler's Dilemma and how human beings make more money than the Nash Equilibrium tells them to. Edit: Wikipedia summary

It occurred to me that the percentage fallacy might explain why people give high numbers in this version -- the Nash equilibrium is pocket change compared to the max payoff. The same is true for the reward for undercutting; you might not be so motivated to low-ball if your reward for doing so is 2% of the max payoff.

It would be interesting to see an experiment where the payoff for giving the low estimate varied. If you were playing the game with a $10 bonus for lowballing, would you give the Nash equilibrium of $10? Or would you maybe go for the $40s or $50s hoping the other person would go even higher? My guess would be that as the reward for undercutting as a percentage of the max reward increases people get more and more vicious, and at some percentage people will default to the Nash equilibrium.

Random thoughts:

If you make a bad investment and lose 50% of your net worth, you now need to increase your net worth by 100%, not 50%, in order to get back to where you started from.

Expressing probabilities as percents gets a bit weird, because subtraction doesn't really work like it should. The difference between a 1% chance and a 2% chance is bigger than the difference between a 2% chance and a 3% chance. (1% is 1 in 100, 2% is 1 in 50, and 3% is about 1 in 33.)

On the waiting thing, my intuitions say to take the $120 tomorrow, so I might be unrepresentative - but couldn't this be explained just as well in terms of whether you have to wait at all, not necessarily just the specific amount of a day? Would you get similar results if you offered $100 tomorrow or $120 the day after? That way you have to wait either way.