Well.....I'm not a Gen Con official, but, as far as I can tell, there's no strict limit by Gen Con as to age to leave your kids unattended. Here's all the information from the Gen Con FAQ about children:
Children Information
Q: Are kids welcome at the show?
Yes, Gen Con is a family friendly show and welcomes attendees of any age.
Q: How much is a child’s badge?
Children 8 and under are free when accompanied by an attending adult. Children over the age of 8 are subject to the standard badge costs. In order to prevent issues with security, it is recommended that you pick up any free badges at the Customer Service booth onsite. Free badges are only required for those children participating in events.
Q: I want to pre-register my child that is 8 and under, what do I do?
Since we cannot accurately validate the age of children over the internet we do not allow pre-registration of the free child badge. If you require a free child’s badge you will need to pick up the badge onsite at the Customer Service or Badge Registration booths.
Q: If I purchase a badge for a minor, may I accompany them for free?
No. If you wish to gain entrance into the convention center beyond any common areas you will be required to show a badge.
Q: Do you have a day care onsite?
We do not have a daycare onsite. Should you need assistance in finding care for your children while at Gen Con, we encourage you to contact the hotel where you will be staying to see what services they may offer.
Q: Do you have an area dedicated strictly to children?
Yes! Gen Con Training Grounds is an interactive program dedicated to children, ages 4-12, where they can play games, participate in art projects, etc. Children in this room MUST be accompanied by an adult.
Q: What if my child is lost?
If your child is missing please report to the Customer Service booth or any Gen Con staff member.
Soo...............it sort of depends on the kid and the event I guess. For me personally, would I have a problem leaving, say, a 9 year old in the card hall unattended at Gen Con for two hours while he played a Pokemon tournament during the day? Probably not, esp. if he had a cell phone or a family walkie talkie. Would I roll in Thursday and basically tell him he was free to run wild until Sunday afternoon? No. Would I let a 13 year old hang out in the dealer's room or the anime area for a few hours unattended? Probably, but it would depend on the kid. I think it's more a common sense rule. I find Gen Con to be pretty safe, but of course no area is completely 100% safe. I sort of think of it as a mall. If you were at a mall with your child and you'd allow your child to go to, for instance, the music store for an hour unattended, I don't see the difference between that and allowing them to go to the anime area or dealer's room for an hour unattended.
Just remember that there's no Con wide paging system, so if your child is lost, there's essentially an APB that can be put out on the radios you see some Seniors have, but there's no way to have your child paged if lost. If you are going to leave them unattended, leave them with a family walkie talkie or a cell phone, and have a clear plan of where and WHEN you are meeting; just meeting in the dealer's room is impossible; meeting at the info booth is a good idea, and Randy can entertain your child while you wait. :)
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