If a person buys a single item, he can generally use that single item. If he starts reproducing that item, it's another whole ballgame.
There are caveats though. If what you are doing could fool a consumer into thinking that you are selling an item that is endorsed by the college, then you are infringing.
Let's say you buy a Mercedes Benz hood ornament. You could put that ornament on a chain around your neck, and Mercedes wouldn't care too much. You weren't selling it, and most people wouldn't think that Mercedes made it. BUT, if you put it on the front of your car, and were trying to pass off your car as a Mercedes, they would care.
Let's say you have a game that is not related to college, and you are just using the logo to decorate a game board with a number of logos, and no reasonable person would ever think that the college had endorsed the game...then you are on safer ground.
You may wish to visit with an IP attorney, in person or via phone will do. A quick explanation should make this pretty clear.
I'm not your attorney; my answer to your question includes assumptions. If you want me to be your attorney, I'm easy to find.
If the person brings you one logo to put on one "game" [I don't really know what that means] that will be used for the person's home use then, no, you are not violating any intellectual property laws -- specifically, trademark law. But if the person brings you many logos to put on many games then, yes, you would be a "contributory infringer" of the trademark rights in that logo. If putting logos on things is your business then you need to speak with your own intellectual property attorney. Good luck.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
Your question seems very strange because why would this person ask you to stick his sticker on his game? Why doesn't he just stick it on himself? It sounds like there is much more to your question than you have offered, so any advice we could provide would likely be inaccurate.
The purchase of a logo decal or logo badge or logo patch Carries with it the right to apply that logo to a single item for personal use. That does not give the right "to do whatever you want"with that logo. For example, if the"game"to which you want to apply the logo is some online app which potentially millions of people will access, that is well beyond what you are legally allowed to do. So, more facts are needed to properly advise you. Add as additional facts what you mean by"game".
So far, this is free to you. Until you pay a fee, I am not your lawyer and you are not my client, so you take any free advice at your sole risk. I am licensed in IL, MO, TX and am a Reg. Pat. Atty. so advice in any other jurisdiction is general advice and should be confirmed with an attorney licensed in that jurisdiction.