Contests have a very different appeal to them compared to other question types. Also, there are usually no quality concerns for answers to contests.
Do you think voting on contest *answers* should be different to voting on other objects. If so, which voting mechanics would you apply to contest answers?
I would not want to change the voting system for this purpose. The arguments in favor of "one vote per user" apply pretty much everywhere. Competitions are not really about the outcome only, but also about the elegance of the code and so on. If we decide that more seasoned users have more say in voting on ordinary answers, we do so probably because we think their opinion should have more weight because they have more experience, can better gauge what a good answer is, and can tell which codes are elegant and efficient. However, if this is the case, we can make the same argument for the competitions.
Personally I do not care too much about one way or another, nor about the votes in general. However, I do care about overall consistency. And to me it seems odd to apply different rules in different situations. It is also complicated and may lead to confusion.
I'd say that for contest answers each user should be able to award one star to each answer, not the usual reputation-dependent number of stars.
Contests for us are generally of creative nature and the technical expertise of users shouldn't affect their voting power on creative content.