Copying moves from other games in progress is perfectly legal, even copying from your opponent's games. Duplicate games happen often between in main-line openings at OTB tournaments where the players can easily observe each other's games in progress. There is no rule prohibiting players from looking at or getting ideas from the live tournament games visible within the playing hall. If someone claims otherwise, please point to your source.
But there is a simple method to prevent this strategy in timed games, especially with time controls that set a fixed maximum time per move. Let's say Alice and Bob are playing 2 simultaneous games, each player having white in one game, and black in the other. Bob wants to use the copy-cat strategy, so he waits for Alice to play her white move
Game 1: Alice 1.e4
Bob immediately plays the same move in his game as white
Game 2: Bob 1.e4
So far, so good. But Bob's clock in his game as black (Game 1) has been ticking longer than Alice's clock in her game as black (Game 2), so if Alice just waits long enough, Bob must make his first move as black before Alice must make her first move as black, otherwise Bob loses on time. Once Bob makes his first move as black,
Game 1: Alice 1.e4, Bob 1...e5
Alice quickly makes a different move in her game as black (having previously decided on at least 2 decent candidate moves while waiting for Bob to move).
Game 2: Bob 1.e4, Alice 1...c5
Each player risks losing on time by waiting until the last few seconds, but the point is that the copy-cat strategy can be prevented. If anyone wants to challenge my anti-copy-cat strategy, just challenge me to 2 unrated games.
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Thriller : That doesn't work either. He simply does the same. Whatever time you take , he will do too . You are basically playing 1 game , against yourself. It is a symmetrical situation , unless there is a difference in vacation time.
The problem is : You want him to move first in his White-game , where he won't be able to copy your move ( because you haven't moved in the other game yet). It is only the clock that can force him to do his move.
This is not totally true:
Think about it - Let's say you have already played 1.e4 e5 2.Nf3 Nc6 Bb5, and he has done the same. In one of the two games, it's his move, and since you played Bb5 before he did, the game where it's his move there is a SHORTER amount of time left on the clock. It may only be 4 minutes, but it's shorter.
Think about it, if I made the move 3.Bb5 at 10:34 today, and he "copied" me at 10:38 today, I have until 10:38 tomorrow, he has until 10:34 tomorrow. What if I wait until 10:36 to make my move, where I have 2 minutes left on the clock?
He better have made his move already. If he played 3...a6, I play 3...Nf6. If he didn't play 3...a6, I play 3...a6.
If he let his clock run out, I play whatever I feel like at that time!
This is what I'm saying.