I've had this thought for a while and I wondered what people thought of it or if they have seen any statistical analysis(some searches turned up nothing) that could back it up or prove it wrong.
My thought is that saving your closer(best relief pitcher) for the end of the game doesn't really make any sense and that you should use your closer in the tightest spot not the end of the game to maximize his value.
Currently the conventional wisdom is to save your closer until you have the lead and have him pitch the 9th inning to close the game. Depending on how much work he has had or the coaches mood sometimes they will bring them in in tied games or extra innings. It seems often times that closers don't work as many innings as they should and that many games that potentially could be saved or won are lost because of this. Here are a few scenarios to illustrate my point.
You are playing St Louis and you are up 2 runs in the 7th inning. St. Louis has 1 out and a runner on 1st base, up to bat is Matt Holliday and on deck is Albert Pujols. The manager takes out the starter and brings in a reliever(because of the situation he is probably the 2nd or 3rd best reliever). Considering the situation in this game and the quality of hitters coming up I contend you should bring in your best pitcher in this situation. If we were the Yankees I would bring in Rivera right now and try to protect my lead right here. I'll take my chances with my other pitchers against the bottom of the order later. Rivera is my best pitcher I want to use him as much as I can without burning him out and I want him in the most intense situations. This game could be lost right here with a big inning and Rivera would never even see the field.
It's the 6th inning and you are down 2. The bases are loaded and 2 outs. A hit here cracks this game wide open an out keeps you right in it. I bring in my stud to get this out and then hold the next inning while I try to mount a comeback.
The downside to this theory is that at the end of the game there would be times you don't have your best pitcher around so you could potentially blow more games and depending on wether you believe only certain guys can handle the last inning pressure they could blow more saves than the "closer" saved games when he pitched earlier. I don't buy into the 9th inning is a differnt pressure theory and obviosuly I think there are more games to be won than lost by shuffling your pitcher lineup around.
So in closing I think that your best pitchers should be utilized againt the other teams best hitters and/or the most crucial situations and not always the the end of the game to best utilize the talents of your staff.