If a contest like this were to be held, there is only one path to be taken if "people choose their own games," and that is the judge the reviewee path, so that a list of objective points can be drawn up.
What this means is, in order to have people review different games, one must hold each game in equal standards to eachother, and soley judge the "performance" if you will, of the person giving the review. So for instance, if the game being reviewed is undeniably the worst game ever made, but the person reviewing it still makes the review entertaining and informative, they will have an equal advantage to someone reviewing an "awesome" game.
Granted, if these reviews are to be given over varying mediums, (video, posting, blog, etc etc.) there will be bias in judgement, simply because of the fact that two different mediums cannot be "judged" the same, because it is a completely arbitrary arguement, for example as to whether "writing skill" is regarded higher than "video editing skill." Thus, in order to have an effecient, and fair competition, one must chose a specific medium to give the review with.
tl;dr:
games choices dont matter
judge people on review skills
chose 1 specific medium to give reviews with