The previous page is Sacchan doing both sides to give an example of how the game is played. (This is one of the cases where looking at a later page, then going back to the earlier page can give a false impression of who was speaking.)
The reason past tense is used for a game that has not started yet, I don’t know for certain. But here is my speculation:
The line is essentially, “The one who became tired first loses”.
However, Japanese tense doesn’t work quite the same as it does in English. Rather than thinking of た as being “past tense”, it’s more “completed action”. Thus, when the action of becoming tired has completed, that person loses.