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Become intimidated or frightened. For example, The polls don't look too good for our candidate and he's running scared, or The shrinking market has many businesses running scared. [Mid-1900s]
American Heritage Idioms
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Phrase(s): run scared
to behave as if one were going to fail. • The mayor was running scared, but won anyway. • When we lost that big contract, everyone in the office was running scared. We thought we’d be fired.
McGrawhill's American Idioms And Phrasal Verbs