Good game is an option to consider. Strikingly strong play is good as well. The ones related to being the master of a certain piece.
Which trophy do you send in this situation?

Good suggestion about being the master of a certain piece - I'll try that. Saying "Good Game" may sound a little sarcastic.

Depending on the game, the trophy, "Don't Be Too Critical" might be appropriate. In similar situations, I think it is more about the personalized message you send with the trophy than the trophy itself. I try to make my messages concise, yet personal.

"I'm Sorry To Hear Of Your Loss" is the best option xD. I literally laughed. I can't wait to use this on someone I don't like too much

Good Sportsmanship is always a good one to send (especially if it were a Daily game when they're down yet haven't given up!)

Good Sportsmanship is always a good one to send (especially if it were a Daily game when they're down yet haven't given up!)
Agreed.

I just want to send good trophies if they win or if they lose, the current ones are a bit cheesy.
There are less cheesy ones in another section. Go into the game archive and click the game you are relating the trophy towards, then click game "info" and there is a place that says "send award." This gives more options that are more specific to the game itself.
Please give me some advice here.
You play someone who has recently joined and you win easily but they were a good person. You would like to send them a trophy to thank them for the game and encourage them to stay. They may be a kid. You think they may like trophies. You don't want to be their friend.
With all that information, which are the best trophies to send? I don't believe that there are any.