This post was done by voice dictation. There may be some funny grammar when the computer does not understand what I said.
You don't have to be a lawyer to understand the basic ideas about copyright laws. The copyright holder does not want you to use this copyrighted material in such a way that it would deride him some of the benefit he could have by selling or renting his copyrighted material. You can compare it to a book. When you buy the book, you have the right to read it. After you read it, you can let somebody else read it. And that sometime in the future you could sell the book. But in no case can you make a duplicate of the book and sell it. That is just wrong and it violates the copyright law.
Applying that to computer software, it is against the law to make a copy of your software and let somebody else use it because that would deprive the copyright holder the game he would get if somebody would buy their own copy. Of course, you could sell your software to somebody else. But while somebody else has your software, you can't use it. It's like you can't read the book if it's in somebody else's house.
It sounds like you might have got confused with the public performance clauses found in some copyright statements. That usually applies to music and plays that are given in public in order to make a profit or to promote some kind of advantage or benefit for the performers. But that's a different issue. I don't think there's any copyright clauses in software that have anything to do with public performance.
Here's the crux of the matter. Are you guys making extra copies of the piece of software so that you can play together at a LAN party? If so, you guys are breaking the law about copyrights. You can let other people use your programs, but it's against the law for you to use the program while somebody else is using it. In other words, only one person should have the software in their position. The fact that it's so easy to copy does not change the copyright law. For example with a fax scanner you could easily make copies of several pages in a book. You can break the law by doing that if you copy too many pages.
Years ago I bought manuals for repair of floppy drives they came from the manufacturer and I included it in the kit that I sold to help people prepare their floppy drives. Well, one day my supplier ran out of the the manuals, so he sent me copies of the manuals which I then passed on to my customers. One day I got a call from some legal people at the company that makes the floppy drives and had to explain to them how come I was distributing copies of their manual. So I had broken the law by distributing manuals that been copied by somebody else. They excepted my explanation and after that I could no longer include manuals in the kit that I was selling.
Now that story may not directly relate to your problem, but it illustrates the idea of what it means to have a copyright. The copyright helps protect the owner of the copyright so that he can make a reasonable profit on the sale and distribution of his products. If you duplicate his product and use it for some advantage or benefit, then you are breaking the copyright law.
This sent five explanation should be enough for you to understand that you cannot see making copies of the game software and distributing it to everybody at the party. Everybody has to buy their own copy of the game if they wish to play together at the same time. Now some cases the game itself may have a provision that allows you to play the game with other people at the same time. You have to read the instruction manual and see if it really allows you to do that.
The simple answer to your question is this: it's not about where you have your LAN party. It's about two you get some benefit from the software that would be a violation of the copyright law. Making copies of software and enjoying it is a violation of law. It's that simple.