Throw up the sponge. Give up; confess oneself beaten. The metaphor is from boxing matches.
“We must stand up to our fight now, or throw up the sponge. There's
no two ways about the
“We hear that the followers of the Arab chief have thrown up the sponge.” —Newspaper paragraph, April 2nd, 1888.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894