I will echo what a lot of other people have said that doing a second look probably best serves an applicant because it gives them a more comprehensive view of the program. However, I feel that people are understating the potential value of a second look from a matching perspective. I know we would all like to think of ourselves as really special and that we all stand out in the crowd, but most programs are probably interviewing at least 10 people per spot and it is easy to get lost in the mix since the vast majority of applicants are outstanding. Doing a second look without question will make you more memorable and if you come off well it at the very least has the potential of improving your chances of matching at a program. Obviously, it might not make a difference at every program and it probably is a reasonable generalization that it will matter more to smaller programs like my own. However, I think it is naive to think that if you spent a few days working with a chairman at most programs that they would not have the ability to affect your position in a rank list in a positive or negative way. Plus, doing a second look at a program shows you are interested and even though that probably should not matter, I think it is similar to the rest of life--people inevitably tend to "like" people better when they seem interested in who they are.
In regards to playing the game, I figure it is like football and you have to fight not just for every yard but every inch because when clock ticks down to zero, it can mean the difference between a win and a loss. In the match, getting bumbped up a programs rank list even a few spots could mean the difference between matching at that program or maybe not matching anywhere at alll. You never really know until you are holding the match letter in your hand and even then you still will never truly know how close you may have been to not matching.
For what it is worth, I personally was a middle of the road applicant and did a second look at a competitive program. I doubt I was ranked very highly after the interview day, but before the match I received favorable phone calls from residents and the chairman. I am 100% confident that would not have happened without a second look.
I definitely think doing a second is far from required. But if you really liked a program, I think there is more than one good reason to do it.