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I will disagree with you Cavalier: although the second game will of course be different, I don't think it would be quite as different as a game against an entirely new opponent.
Also, I'm not sure why I'm supposed to give up 1...e6 and play something I don't know just because it's "courteous" to avoid playing the same opening twice apparently, even if it's from a different side.
I agree that a new person is a completely different player, but I'd bet that the second game with the first player will be just as varied. Due to memorized openings, it is possible for you and the new opponent to end up with the same 20 moves as the first game, making the first half of the game will feel the same. I think the chances of the other player liking the same opening and variation as the first is just as slim as the first player letting the game go into the same opeing a second time.
In the previous example, not playing 1... e6 isn't a courteous act, unless it's to one's self in the hopes of variety and growing in a different position. In the game with the new opponent you have a 50% chance of playing the same color again. You made a point to suggest that you don't play openings you're not familiar with, but in the above example it suggests you played 1. e4. That means your chances of playing 1. e4 in the next game are high, unless you don't do it out of some courtesy.
The common first moves in response for 1. e4 are e5, e6, c5, and d5, so there is still a chance that in your game with the new player you end up with 1. e4 e6. The chances that both players will steer you into a game you're not comfortable with are probably equal. Neither one of them want to play into your demands, and both want to win. Neither knows nor cares what systems you like or dislike.
The main thing I wonder is what great insights will you have learnt about the first player in one game that will let you predict the outcome of the second?
No particularly great insight, but it may not feel quite as fresh.
Most of the time I do not accept a rematch because there was something in the game I feel needs study, and I would rather check it out while all the moves and variations considered are still fresh on my mind. It is rarely ever by any fault of the person I was playing. Occasionally there is someone who irks me in some way and I decline to play them again though.
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