I'm trying to get more people to rate products on a video game ratings and review website. Will allowing non registered users to upvote/downvote the reviews increase engagement and sign ups? in other words, would it be better if non-logged in users can upvote/downvote already written reviews by registered users?
You may indeed get more interactions, but conversions could easily go either way... why should I sign up if I can vote anyway?
I would be far more concerned with vote manipulation and would suggest the risks of that would outweigh any benefits.
You could allow votes and add them to a "pending" list and apply them when the user signs up (based on a cookie).
Personally I would look to make sign up following a click to vote as simple as possible and require it prior to applying the vote. Most people are happy to sign up with a Facebook or Google Id. If you don't ask for any extra fields or access your conversion rate should be high.
This brings me to my final point. You could try to track conversion rate based on different approaches and keep the method that works best. You also need to consider how to encourage shares... again integration with social networks can help there too.