This appears specifically designed to not permit people to take vacations as they rightfully should. I will say this: whoever created this design is smart, but heartless, cruel, and inhumane. The employees who find themselves battling against each other as a result are definitely not the assholes here.

If there's a way that you can switch jobs to a better organization where such a culture is not promoted, that may ultimately improve your qualify of life. You can even use a random "vacation day" here and there to interview - use your company's policies against it, and in your favor.