The phrase "exclusive prerogative" means having the sole right or privilege to do something. It implies that no one else has the authority or permission to do it. Full definition
With the brief and fascinating exception of the blaxploitation movies and a few other works of radical or renegade art, vengeance in the American imagination has been the virtually exclusive prerogative of white men. (nytimes.com)
As the Church teaches: «A healthcare professional's role may be to remove obstacles to conception, restore fertility, assist in fertility awareness, advise, comfort, listen, guide — but never to violate the exclusive prerogative of wives and husbands to become mother and father only through each other.» (faith.org.uk)
Choosing the leader would have been the exclusive prerogative of Labour's parliamentary elite. (leftfutures.org)