They played a game of chicken to see who would back down first.
Они играли в блеф, чтобы увидеть, кто сдастся первым.
Definitions
A game of chicken is a situation in which two parties engage in a conflict or competition where each side threatens to take a dangerous or extreme action unless the other backs down, with the risk that if neither yields, both will suffer negative consequences. The term originates from a daredevil driving game where two drivers head towards each other and the first to swerve is considered the 'chicken.'