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I have some crying and frustration episodes, but the students usually bounce back. If it's really bad, I may talk to them about it in their native language (Japanese,) but not everyone has that luxury. I usually explain about winning and losing, but more so that we just play the games to have fun practicing. The real goal is not to win games or get stickers, but rather, the goal is to learn and grow.

That is usually met with the reaction you might expect. "Whatever, Mark, I want the stickers!"
But I think they get it deep down somewhere.
What I usually do is just keep playing bingo until everyone has at least made 1 bingo. I then announce to the group that the next card will be the last card. I give out small stickers for each bingo line. So, if the student has 3 bingos, they would get 3 stickers. This is a little less competitive since the players aren't really trying to beat each other. They are just trying to get the most bingo lines that they can.
They of course have lost or won in the overall, but it seems to work for me.
With older groups I make it more competitive. I tell them that when I get bingo, the game is over. So, there are times when students lose and walk away with nothing, but they are usually older and at least don't start crying.
I do the same for concentration. I award stickers for the number of pairs each student makes.
I'm a little careful about watching and making sure one student isn't losing repeatedly at the same game, over and over again. Concentration is a game that some people are just bad at. So, if I have someone like that, I'll try to avoid the game if possible.
Where are you teaching? What are your group sizes?