Absolutely, shutting up and multiplying is the right thing to do.
Assume: simple majority vote, 1001 voters, 1 000 000 QALY at stake, votes binomially distributed B(p=0.4), no messing with other people's votes, voting itself doesn't give you QALY.
My vote swings iff 500 <= B(1001, 0.4) < 501, with probability 5.16e-11, it is advised if takes less than 27 minutes.
Realistically, usefulness of voting is far less, due to:
actual populations are huge, and with them chance of swing-voting falls;
QALY are not quite utils (eg. other's QALY counts the same way as your own);
You will rarely see such huge rewards (if 1 QALY ~ 50 000$, our scenario gave each voter free $50 M )
So, people who 'need your vote' in real-world scenarios are either liars or just hopeless.
votes binomially distributed
So learning how 900 of the voters (other than you) voted won't shift your beliefs about how the remaining 100 voters voted in the slightest? No, I don't think so.
Note: Please see this post of mine for more on the project, my sources, and potential sources for bias.
I have written a couple of blog posts on my understanding of climate forecasting, climate change, and the Anthropogenic Global Warming (AGW) hypothesis (here and here). I also laid down the sources I was using to inform myself here.
I think one question that a number of readers may have had is: given my lack of knowledge (and unwillingness to undertake extensive study) of the subject, why am I investigating it at all, rather than relying on the expert consensus, as documented by the IPCC that, even if we're not sure is correct, is still the best bet humanity has for getting things right? I intend to elaborate on the reasons for taking a closer look at the matter, while still refraining from making the study of atmospheric science a full-time goal, in a future post.
Right now, I'm curious to hear how you formed your views on climate change. In particular, I'm interested in answers to questions such as these (not necessarily answers to all of them, or even to only these questions).