Calendar/Appointment Book for each user

I run an online gaming site through wordpress hosted on my own server. What I have been looking, unsuccessfully for, is either a calendar or an appointment book type of plugin that each user can list their availability on.

In essence, the site is a central hub for people of the community to come to and see who can play when for pick up games or league games. This way if someone wants to know when they can play that specific person they can go to their calendar/schedule and see when their availability is.

A second step, not as important as the first, would be, for the 2nd person to see the first is available and then be able to book an appointment with them during that available time.

I’ve google searched and searched directly on here. I am ok with a free plugin or one I have to pay for as long as its what I need.