This is a really negative way of looking at it. If you have the types of players who go around hassling other players because they can, I don’t think the overlay is going to stop them from hopping into the person’s profile. If fear of bad player behavior leads to a lack of innovation or accessibility for the good players, then I think the bad players have already won 
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I personally like to thank my opponents for a game. Being kind is extremely important in a competitive atmosphere. Especially since I meet a lot of people who feel really down after a loss. Sometimes I offer compliments if I felt they played particularly great moves or had given me serious trouble during the match. I often offer follow up games or games with larger turn times, since I meet most in Blitz matches.
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I play a lot of beginners who tend to quit early. When this happens I like to message them to encourage them, sometimes offer a review, often refer them to my Newcomer’s Guide, offer them a game with slower time settings, or ask if they have any Go related questions.
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When someone leaves with a 5 minute countdown timer, I either have to wait for it to count down or I need to head to my profile to locate them. Because I play against beginners and unranked players a lot, I like to check how many games a player has under their belt (especially for those crazy unranked Go Gods that leave me trembling in the fetal position
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I play in live Tournaments and you will get these players who are in and out of the match the entire time. Most of these players are good about getting back to take their turn on time throughout the entire game. But sometimes they miss a turn or two, because they don’t get back in time. This is especially frustrating when it costs them the game and the game was really good. In these situations I would like to be able to easily PM them to notify them that I have taken my turn.