The ticket dispensing is pretty simple really, I'd just put the tickets between two rollers and drive them with a small DC motor a few turns to eject the ticket. I can set the number of degrees to rotate with the arduino, and therefore set the number of tickets dispensed. Each winner would receive a ticket that basically allows them to challenge someone to another round of the game. Out of tickets, out of the tourney.
Hopefully, this is a "trusted environment"
. Ticket dispensers, while easy to make, aren't as easy to make robust enough to handle shenanigans. Aside from the obvious possibility that one could grab a ticket and pull out as many as they want, there's also the possibility that doing so could turn your motor into a generator and feed enough power back to the controller to cause some damage.
This is also a bad situation for a servo, depending on the size. The smaller ones have a tendency to come apart at the press fittings of the gears when forces are applied to the output spindle.
But, if it's a home setup, with folks who are well behaved, what you have will work fine.
It is indeed a trusted environment. However, I'd be amazed if anyone managed to pull tickets out without tearing them. I thought of this inevitability, and the drums I'm driving that roll out the tickets perforate the tickets length-wise. As far as the ticket count, the tickets are a standard length, so I just count the number of degrees I need to rotate the servo to dispense the ticket quantity I want. I think its something like 28* for one ticket on my setup, my notes aren't right here but it's something like that.
I guess I should have explained a little better. The spool of tickets just sits idle, and I draw the tickets between two rollers that are pressed together. I drive one of the rollers with the servo, and it *may* be a little oversized for the application. (Think small household dryer). The rollers themselves are gear mated, and have pins/sockets that interlock. As the tickets are drawn through, the pins punch holes all the way down the length of the ticket, so if anyone tries to pull too hard, the ticket tears and is invalid. The paper tickets will give up the ghost way before anyone can drive any power back through the system, besides, the gear drive is on a one way clutch, so to the would be ticket thief, best of luck getting some tickets.
If you can pull it off, I'll buy you a beer.

This whole thing is basically an excuse for me to make some neat robotic stuff. With video games.
How about a beer?
It's really an exercise in the "I think I can make that" realm.
I think you will ultimately find that the ticket thing is going to end up being an incredible amount of work and money for something you ultimately won't end up using more than a few times. I know a lot of people with a lot of games and some of them are indeed games that can dispense tickets, but I don't know of anyone actually dispensing tickets in a home environment.
Can you make a cookie dispenser for every time you exit mame? Every one is a winner? 