How would you design a currency so that its value fluctuates as little as possible, without centralized control of the money supply?
Bitcoin's value fluctuates wildly. I think this is because Bitcoin really functions as a commodity, rather than a currency: people are just using it as a convenient vehicle for financial speculation. Almost nobody actually buys or sells things with Bitcoins, aside from other digital currencies, because their value is so unstable. If a dollar buys 1 loaf of bread today, but there's a non-negligible chance it will buy 1,000 loaves tomorrow, who would spend the dollar?
I think what you'd want is a currency that has a built-in negative feedback mechanism -- as its value increases, the money supply also increases. In fact, Bitcoin has this feature, to some extent -- Bitcoins are mined by computers, and running these computers costs electricity. As the value of Bitcoins rise, people are willing to pour more and more electricity into creating new Bitcoins. But, empirically, it turns out that this by itself isn't nearly strong enough to counteract the positive feedback loop of financial speculators in lemming mode.
1. Can you create a stable currency simply using negative feedback? (If so, what is the nature of the feedback -- proportional? power-law? exponential? time-lagged?)
2. Once you've figured out the answer to (1), here's a harder question: how do you get people to use your new currency? Presumably the answer to (1) will involve the de novo creation of new money somehow. So, who gets the new money? Is it disbursed to people proportional to the amount of the currency they already have? Does everyone get an equal share? Is it a lottery somehow?
As an aside, I think the fact that Bitcoins are "mined" via a somewhat complicated procedure helped get them established in the first place, because people felt invested in them -- they mined them with their computers, so they felt like, "I WORKED for these coins, they're mine and I want their value to go up!"