Unsurprisingly related to: Ugh fields.
If I had to choose a single piece of evidence off of which to argue that the rationality assumption of neoclassical economics is totally, irretrievably incorrect, it's this article about financial incentives and medication compliance. In short, offering people small cash incentives vastly improves their adherence to life-saving medical regimens. That's right. For a significant number of people, a small chance at winning $10-100 can be the difference between whether or not they stick to a regimen that has a very good chance of saving their life. This technique has even shown promise in getting drug addicts and psychiatric patients to adhere to their regimens, for as little as a $20 gift certificate. This problem, in the aggregate, is estimated to cost about 5% of total health care spending -$100 billion - and that may not properly account for the utility lost by those who are harmed beyond repair. To claim that people are making a reasoned decision between the payoffs of taking and not-taking their medication, and that they be persuaded to change their behaviour by a payoff of about $900 a year (or less), is to crush reality into a theory that cannot hold it. This is doubly true when you consider that some of these people were fairly affluent.
A likely explanation of this detrimental irrationality is something close to an Ugh field. It must be miserable having a life-threatening illness. Being reminded of it by taking a pill every single day (or more frequently) is not pleasant. Then there's the question of whether you already took the pill. Because if you take it twice in one day, you'll end up in the hospital. And Heaven forfend your treatment involves needles. Thus, people avoid taking their medicine because the process becomes so unpleasant, even though they know they really should be taking it.
As this experiment shows, this serious problem has a simple and elegant solution: make taking their medicine fun. As one person in the article describes it, using a low-reward lottery made taking his meds "like a game;" he couldn't wait to check the dispenser to see if he'd won (and take his meds again). Instead of thinking about how they have some terrible condition, they get excited thinking about how they could be winning money. The Ugh field has been demolished, with the once-feared procedure now associated with a tried-and-true intermittent reward system. It also wouldn't surprise me the least if people who are unlikely to adhere to a medical regimen are the kind of people who really enjoy playing the lottery.
This also explains why rewarding success may be more useful than punishing failure in the long run: if a kid does his homework because otherwise he doesn't get dessert, it's labor. If he gets some reward for getting it done, it becomes a positive. The problem is that if she knows what the reward is, she may anchor on already having the reward, turning it back into negative reinforcement - if you promise your kid a trip to Disneyland if they get above a 3.5, and they get a 3.3, they feel like they actually lost something. The use of a gambling mechanism may be key for this. If your reward is a chance at a real reward, you don't anchor as already having the reward, but the reward still excites you.
I believe that the fact that such a significant problem can be overcome with such a trivial solution has tremendous implications, the enumeration of all of which would make for a very unwieldy post. A particularly noteworthy issue is the difficulty of applying such a technique to one's own actions, a problem which I believe has a fairly large number of workable solutions. That's what comments, and, potentially, follow-up posts are for.