Think tanks very often claim they are "fighting the war of ideas" - but indeed it is difficult to assess success and casualties in this kind of "war". All non-profits have problems in finding suitable metrics for their achievements - but think tanks particularly so. After all, if your goal is to feed the poor, you can assess your impact by counting how many you are feeding, in proportion to the area in which you operate. But what kind of assessment can you make, if your aim is to change the climate of opinion?
Well, first of all this kind of success can only be seen in the long run: but research centres need to finance themselves in the short run, and produce results that can please their supporters.
Measuring productivity is relatively easy. Most institutes keep track - among others - of their media hits: op-eds their researchers publish, articles that discuss their papers, TV interviews and the like. This is all nice, but I fear it is not very meaningful. It is one thing to react to new proposals and laws, another - quite different - is to be the agenda setter. If your researchers get often invited to radio and TV shows, they are certainly talented enough to entertain and speak out in the political debate - but this does not necessarily mean they will influence and change the way people think, in the long term (though if they go to TV they are more likely to do it, than if they live in a monastery).
James McGann, a political scientist now with the University of Pennsylvania, has long been trying to provide think tanks with an interesting tool. His "Global Go To Think Tank" is based on extensive polling of the community of peers, and other opinion formers. The ranking aims to reflect think tanks' reputations, in the eyes of those most interested in this trade.
I think McGann does a great work, keeping track of more than 6,000 think tanks all over the world. The definition of think tank he uses is necessarily a rather loose one, which produces some paradoxes (he numbers some 80 think tanks in Italy, for example, but they vary greatly for impact, size, research capabilities, et cetera). The nomination and selection process relied on as many as over 700 experts, plus think tankers, NGO people, journalists and (rather few) donors as inputs. No work of this kind can be perfect - but it makes for an interesting read. By the way the top world wide think tank in 2013 is the Brookings Institution. The first free market oriented ones are the Heritage Foundation (17), the Cato Institute (18), the Fraser Institute (22), the American Enterprise Institute (24) and the Indian Center for Civil Society (50). If you never heard of CCS, check it out. Their amazing campaign for school choice in India is particularly noteworthy.