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After giving it quite some thought, I have decided that I don't really care that much!
My feeling is that, if you introduce it then it should be across the board, no exceptions - everyone can see a list of who voted what.
New members should not be allowed to vote up or down at all until they have reached a certain level of time/usage of the system.
Abuse of either upvoting or downvoting should be punishable by the removal to do either, and (preferably) the removal of those votes. (this would also help prevent puppet accounts being created to upvote one's own articles).
Keep the stats of each user's voting - number of UPs vs Number of Downs and perhaps publish them, too - that will be an interesting stat! And reduce the effectiveness of a vote compared to the number of items read / the number of up or down votes.
e.g. If I read x articles and down vote them all, the 'points lost multiplier' should be reduced - so the 'grumpy old git' gets less effective over time if they don't find something to be happy about.
Flag a warning if a user consistently differs from the herd in their voting (especially down votes)
Publish the data as raw data (via an API would be lovely) and have a competition to make best use out of it.
I;m thinking of a "who hates me" app and a "Ohhhh! is he your girlfriend" app.
1) Personal (i.e. not anonymous vote) may actually be a good thing.
2) Nobody mentioned this, but a middle ground could be that you need to provide reasons for your downvote. This should be a dialog with a large listbox at the top and an auto-complete combo at the bottom, so that common down vote reasons would come up immediately(sorted by popularity, i.e. if many people put in the same reason...).
3) Anonymous voting makes the creation of claques a lot easier, running the risk of heightening the popularity of someone who does not necessarily deserves it.
The other side has merit, too: anonymizing the vote means that I feel more free to vote as I please instead of having to defend my reasons.
I am hence much more favourable to non-anonymous votes.
I'm a long-time CP member. Back when we had the 1-5 voting scheme, I voted on posts a lot. I eventually realized I was using net anonymity as an excuse for bad behavior. My New Years resolution that year was to never vote on a post again. If I like what someone says, I comment on it. If I disagree, I comment on it. No anonymity, and much less bad behavior on my part. I feel like my karma has improved somewhat.
People who want non-anonymous voting in any arena are always those who seeks to control the outcome. If you cannot handle the critics and the trolls then do not put yourself out there for them to come at you... this is not complicated stuff.