What is the expression "You beat me"?
About 10 years ago, I heard the expression "You beat me" used in non-literal way; if my perception was correct, they should mean "Your argument or action are too good/outstanding/weird for me to say something further. For instance, I'm having an argument with a friend in order to justify the point or an issue I advocate...All of a sudden, he brings up a so convincing point against my argument that I become "speechless". On this kind of occasion, may I say "hey~ you beat me. You're right in this."?

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Answer

Yes, in that way it means, "you've won (the argument)"  or "I have no good response to that."

The feeling is one of friendly surrender and the tone is casual, so I wouldn't ever put it in a larger sentence.  "You beat me" stands alone just fine.  Smile and shrug while you say it. :)

A related phrase meaning "I don't know" is "beats me".

"Do you know where Peachey got to?"  "Beats me."

Answer

Yes. You "beat" someone in a competition. Like a football match or a race.

Other words to describe speechlessness would be
dumbstruck /dumbfound /dumb fold(ed) 
Or you would "have no point".