1. I beat him at pingpong.

2. We defeated them at basketball.
3. He beat the big man at wrestling.
4. I defeated him at bowling.
5. We defeated the opponent team at basketball.
6. They beat us at soccer.
7. I was beaten at chess by him.
8. I was defeated at tennis by her.
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Are they all grammatical? Do we have to use 'at' alone before game names?

Sometimes "in" can be used instead of "at" but using "at" makes the competition clearer.

In 5, I'd say "the opposing team" -- but all the rest are fine.

Thank you for your help.

What about the following?

1-1. I beat him at a pingpong match.
1-2. I beat him in a ping-pong match.

2-1. We defeated them at a basketball match.

2-2. We defeated them in a basketball match.

(Which ones are correct? Are they all grammatical?)

1-1 should be "I beat him at ping-pong."

1-2 is fine.

2-1 should be "We defeated them at basketball."
2-2 is fine.

Do you see the difference?

Yes, all the sentences you provided are grammatically correct. When talking about winning or losing a game or a sport, it is common to use the preposition "at" before the name of the game.

For example, you can say "I beat him at ping pong" or "We defeated them at basketball" to describe winning, and "He beat the big man at wrestling" or "I was defeated at tennis by her" to describe losing.

Using "at" before the game name is the general pattern, but it's not the only possible construction. In some cases, you can also use other prepositions like "in" or "on" depending on the context. For instance, you could say "We defeated the opponent team in basketball" or "They beat us in soccer."

Overall, using "at" before the game names is a common and straightforward way to describe winning or losing a game or sport.