The term comes from American sports. A team that has no hope of making the playoffs but still has to finish the season is said to be "playing out the string." The origin might be from the American frontier, where men would strike into the wilderness with a string of horses or mules, planning to ride each one until it was "played out" (too worn out to keep going), before switching to the next. If such a person was in a lost or hopeless position, he might decide to keep going until the entire string of animals was played out.
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