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player_score>=computer_score makes the computer win even when they score the same. A comparison should be > instead of >= (same for the next clause).

The game, as coded, may terminate early (before reaching 5 rounds). For me, it contradicts the requirements.

User input shall be validated (is it really a number?) somehow.

I don't know if you covered functions already. If so, factor out all the copy-pastes into functions.