Well, sure. If the problem is important enough to be worth a prize, it will have already attracted the attention of the people in the field. The problems that are left after the people in the field have tried to solve it are almost by definition the ones that require an outsider's perspective.
Prize winners don't gain only the money, but the status too, and this future gain affects all the way down. In some fields like mathematics, a prize is synonymous of real advance, with is not so clear in others, like philosophy.
Let's say I was King of Saudi Arabia and I wanted to offer a prize to be given to the first organization to resurrect a frozen adult dog. Because I'm King of Saudi Arabia, I have great personal wealth as well as the resources of the rest of the Saudi government to draw upon. How big would the prize have to be in order to make venture capitalists (or other profit-seeking entities) interested in funding cryonics research?
There are (at least) two kinds of things that are difficult to do:
1) Things we know how to do, but lack the resources to implement 2) Things we don't know a way of doing.
For example, if I had several billion dollars, I could probably build a duplicate of the Great Pyramid of Egypt. (Modern skyscrapers cost similar amounts to build.) The Apollo program cost about $170 billion in today's dollars. A manned mission to Mars, including a return trip, would probably be doable if some organization was willing to spend a few hundred billion dollars to do it. (A few hundred billion dollars spent on improving developing world health could also have a huge impact.) These are all difficult things to do, but prizes don't seem to be a particularly good way of getting them done - if you have the money to spend on them, you can often spend the money on them directly rather than on prizes, and if you don't have the money to fund them directly, you won't be able to offer a big enough prize to get someone else to do it.
On the other hand, prizes can certainly be a way of attracting attention to a problem that's being neglected and that is capable of succumbing to innovation in a way that stacking stone blocks isn't. There also seem to be problems that don't seem like they would benefit from having bigger prizes for solving them; would offering a billion dollar prize for a proof of the Riemann Hypothesis get us a proof any faster than the current million dollar prize would? (The person who proved the Poincare conjecture didn't even accept the million dollars!)
There is also the issue of extrinsic incentives vs intrinsic motivation, however.
Relevant keywords (google scholar): extrinsic motivation intrinsic motivation creativity
"... in general, the perception of an activity as a means to some extrinsic goal can undermine intrinsic motivation and, as demonstrated here, creativity."
I am skeptical about the applicability of those sorts of psych experiments to general life. Sure, it takes someone two minutes instead of one to solve a puzzle when they're stressed out by it. But the sort of problems these prizes are used for are typically problems that take much more than two minutes to think about. If prizes buy you the attention of people, then they can be very worthwhile.
It seems to me that the creativity stopping effect is greater when the money is too high. Then the attention is focused on getting the money rather than the problem itself. For example offering me $1,000,000 would paralyze me, and I would also think that too many people will jump at the opportunity, so I don't have a chance.
If the amount offered is less than the work is worth (less that the effort spent, multiplied by the chance of victory), it should be good with regard to the intrinsic motivation. The money will help to overcome akrasia, but seeing that rationally it is not enough, will just strengthen the intinsic motivation by the power of rationalization.
Also, offering money is a signal: "we need the solution, and we will look at all proposals carefully". This may be important for someone who is not a recognized expert in the field -- it says that so far the experts did not provide the solution, and that their contribution will not be ignored just because it comes from outside of the field. This increases the percieved probability of success.
The (rather flimsy) Harvard Business School paper suggests that prizes are not a very good deal (financially at least) for the solvers; chances-of-winning x prize-money / hours-spent does not look good.
Competing for a big prize against many qualified others reminds me too much of the dollar auction.
The attraction of prizes that is relevant here is mostly to the prize awarders, not the prize recipients.
Surely the costs/benefits to everybody, including third-parties, counts. Surely the real issue is the ultimate economic efficiency of these prizes as a way to allocate our collective resources toward achieving the most collective benefit from solved problems.
What counts depends on your perspective. Awarders and recipients have their own opinions on the issue.
That partially answers my question: in order to be a good (financial) deal for the solvers, prizes generally have to be a lot bigger than they are now.
To took at a totally different area, see the Google Lunar X Prize, from their FAQ:
How much will teams spend to win the Google Lunar X PRIZE?
Three centuries of history have shown that teams competing to win incentive prizes often spend more than the prize value itself. Teams in competitions such as the Orteig Prize, the Ansari X PRIZE, and the DARPA Grand Challenges have spent as much as 5 times the prize purse value to fund their entries; and expenditures of 2.5 times the prize purse value by individual teams are relatively common.
We expect that teams pursuing the Google Lunar X PRIZE will follow these historical trends. A broad range of team expenditures—from as low as $15 million to as high as $100 million—are expected. Past prizes have shown the best funded teams do not necessarily win the prize, however. We look forward to learning from our teams as they pioneer new methods to raise money and to trim costs for a lunar mission.
Obviously, you will gain many benefits (financial and non-financial) beyond the prize if you win, but I think it is safe to say that these prizes by themselves are about inspiring people, not about (direct) financial gain.
Financial gain as a result of prizes often goes to the awarders of the prize - rather than the prize recipients.
Prizes leave a bit of a bad taste in my mouth, frankly.
They are only so efficient because you're having many people spend their resources on a job that you're only going to pay a few of them for doing. It makes good sense for the person paying prize money, but that's part of the reason I tend to avoid competitions where prizes are offered: it feels like it will be a waste of my time. This is apparently an easy way to exploit online artists, too: ask several of them to do your design work for you, and pay only the one you like best. You get better designs, and most of them get a day of unpaid work.
On the other hand, these competitions are fairly small-scale, you're paying everyone who has an exercise good enough to test, and it's likely not a waste of mental effort even if I don't come up with anything spectacular. So I don't have a particular ethical problem with the SoTW prizes.
Unfortunately after a brief glance at InnoCentive I didn't see any straightforward algorithm design challenges.
Several commenters on SoTW: Be Specific suggest that prizes may not be a good idea because they might be counterproductive. As a counterpoint, Tony Barrett of GCRi pointed me to this piece at NYT blogs: