Psychology of Meeting Dates

I'm a believer ... Meeting Wizard is a great, free program. I've only used it three times so far but saves tremendous time when trying to find a date for a meeting that can work for multiple people.

There's one feature of the program that I think has a psychology to it: Whether or not to allow the ones invited to participate in the meeting to know who else is invited and displaying to all what dates/times each can do.

* If I show all the names, and later replacement name(s) needed, then everyone would know who was asked last;
* If everyone can see who can do what date(s), then the fact the first person rules out dates may make others rule out dates they really could do, so the first one to respond may have a real advantage;
* If I don't show the names and dates then someone might not realize that there is a date that lots of others can do, and they may be more open to including a specific date as an option than they would otherwise.

So far, I'm not showing the list of names and displaying what days everyone can do. If you've used this or a similar program, what's your meeting dates philosophy: display or in the dark?