Why is innovation such a big deal in business but so difficult for government and the third sector to get to grips with?
It's because NGOs and the state don't face the same incentives as business does. The main priority becomes fulfilling contracts and getting more contracts - not addressing the problem in fundamentally new ways.
This isn't a problem which is easy to solve, but (you guessed it) increased transparency would certainly help. If donors were totally candid about their successes and failures then there would be more of an opportunity to piece together what works and what doesn't. Someone, somewhere, might even come up with a better way of doing things.
Another way to inject some innovation would be to flip the funding model on its head. Instead of paying development organisations for what inputs they say they will deliver (as in the normal contract model), they could be payed for the outcomes they achieve (as in the prize model). Several renegade philanthropists are now operating in this way. It makes you wonder: if the promise of a large prize can galvanize engineers into designing affordable space travel, what could it do for development?