Let's say I have a website where people can rate movies on a scale of 0-10. We say people vote honestly when the rating they give to a movie is what they actually think. People vote dishonestly when they try to get the website's rating to be closer to their own rating: for example, if a movie's current average rating is 7.2, but I believe it deserves an 8, I will rate it a 10 to bump the average higher. (Everyone who votes dishonestly always gives a 0 or a 10.)
It's clear that if everyone votes honestly, the website's rating will reflect the true average viewer rating of the movie. My question is: if everyone votes dishonestly, will the website's rating still reflect the true average viewer rating? In other words, does it matter?
Hypothesis: It doesn't matter. Presumably the order of who votes when affects things (when everyone votes dishonestly), but when the number of voters $N$ gets large enough, it should converge.
Follow up questions:
If it does make a difference, why?
If it doesn't make a difference, how large of an $N$ do we need before we start to see convergence with the true average? Is that only true if the true ratings follow a certain distribution (uniform vs. normal vs. two-humped, etc.)?
What if it's actually a mix: some people vote honestly, and some vote dishonestly?