Forgive me if this comes across as patronizing, but I think you might avoid this in the future by doing the following:
* Charge daily or weekly, not hourly.
* Don't anchor your results to "free" (this includes not getting paid unless specific results happen).
* If a customer has been repeatedly burned by other consultants and believes this problem could be solved in an hour, consider either 1) avoiding those customers or 2) working on an agreeable scope that precludes an absurd amount of time.
I'm not saying that will fix all your problems, but if it helps, I've never encountered this issue with this formula.
This can work for the provider of a service, in certain circumstances.
But my opinion is that it is always better to charge a price that reflects the value delivered, no matter how long it takes to deliver that value.
There is a (possibly apocryphal) anecdote I read once in which a consultant was brought in to service a machine with many parts. He replaced one part and invoiced (say) $10,000 for replacing a part worth $1. When asked why his bill was so high, he said: "I charge $1 for the replacement part. The other $9,999 is for knowing which part to replace."
As knowledge workers, those who are skilled must remember that "hours spent sitting in a seat" is not a valid measure of their productivity and should not permit clueless managers/clients from judging them in that way.
That's good advice, but I don't think getting paid to produce a specific result is anchoring to "free". I guess it depends on how precise the scope is. If it sounds vague and up to the client's discretion, that they just want free work on spec, then run away. But if it's easy to define and measure, then it's a useful way to signal competence and avoid anchoring to a rate instead of the business need.
Anyway, I'm happy to say that I avoided this customer (and others that care excessively about timesheets) and I have no regrets.
Anchoring is a different effect than you might have thought. In breakthrough research, it was shown that when people were asked to write down completely random numbers (such as the last four digits of their social security number), before taking a guess at the value of a home (say; might have been something else), it had an absurdly strong effect on the value of their guesses. (People with coincidentally high last-four-digits wrote much, much higher ones than people with coincidentally low last-four-digits, even though it's obvious they wouldn't have even thought twice about it. If I asked you to think about the last four digits of your social security number, what that happened to be would impact any other valuation you were making immediately after.
The mere mention of a number - any number - "anchors" psychologically in an absurdly strong way.
So in that sense, the statement: "I offered a fixed-price 'no win, no fee' rate" includes the literal mention (like the social security ending digits 0000) of "no fee" -- so it doesn't matter what the sentence is. The sentence could read "Obviously I wouldn't be able to do it for no fee" and it would anchor to $0, whereas, if you said, "I obviously wouldn't charge a hundred billion dollars" it will anchor it to a hundred billion dollars.
These effects are incredibly bizarre. I'm not an expert salesperson but I do know about them.
Is a good introductory article. Daniel Kahneman is the name most associated with this research, or at least reporting it, and he won a Nobel prize* in Economics in 2002 for his work.
* (the Nobel Memorial Prize in Economic Sciences by the Swedish National Bank)
FWIW the phrasing I use with clients is "fixed-price for successful completion of the following objectives" so it's a bit of a moot point.
I do think you're overstating the research though. Many studies of unconscious priming haven't been replicated, and psychology as a whole is very subject to publication bias. [1]
Anchoring is a real effect, and I'm a fan of Kahneman (especially prospect theory), but the evidence about "incidental" influences is very weak.
(the research I read about was way stronger than just statistically significant, or something like it. I would be pretty shocked if it failed to be replicated. It's a very strong effect.)
The phrasing you say you actually use isn't an anchor. But you can still try different ways of phrasing it!
Extremely subtle differences in phrasing can have huge effects on people's reactions. If people are not reacting reasonably, try a slightly different phrasing, as well as different offers :)
* Charge daily or weekly, not hourly.
* Don't anchor your results to "free" (this includes not getting paid unless specific results happen).
* If a customer has been repeatedly burned by other consultants and believes this problem could be solved in an hour, consider either 1) avoiding those customers or 2) working on an agreeable scope that precludes an absurd amount of time.
I'm not saying that will fix all your problems, but if it helps, I've never encountered this issue with this formula.