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Confidence helps you stay focused and believe that you can find solutions, whereas lack of confidence can lead to panicking and making rash decisions.
It probably varies from one person to another... for me I don't find it's that important during the game. I don't really let my emotions effect me much in game, it's just a logic problem that needs to be solved. But some other people do seem to feel confidence matters.... I do think confidence outside of the game is very important for driving you to put the time in to succeed though. If you don't believe you can succeed you probably won't try as hard.
I strongly disagree with #1. If you have no confidence, your elo will drop all the way to 100, so technically confidence = 3400 rating.
Well, confidence is great, but just make sure you don't have too much. I got way overconfident today and lost to somebody about 300 below me . So while I'd say confidence might get you those 150 rating points, it might then push you to thing that you can't lose to lesser opponents and then you won't be paying them enough due respect.