The game is called "Killers". At least, that's what we called it in college. University College Dublin has a bunch of clubs and societies, like any other college, and one is GameSoc. The Games Society. This was the mid 1990s, I presume it may have taken time to filter across the Atlantic. Also, paintball guns weren't really a thing in a country suffering from an insurgency (they are categorised as firearms in Irish law), so we used water pistols which also are less damaging. But these are quibbles. You can use paint, nerf guns, stickers, whatever you like.

The rules were first published by Steve Jackson Games in 1981. and can be considered the first LARP (Live Action Role Playing Game)

The concept, however, is as you describe. As we ran it, in order to play, you had to submit a photograph of yourself and your official class schedule. Three 'rings' of hits were in play at the same time, so really you submitted three photos and three copies of your schedule. At game start, you were handed three 'targets', one from each ring. There was really only one rule: "Kill" your target by squirting them (any form of wetness, however, was acceptable, to include pushing them in the lake, but only do that to friends), then take their target as your next in that 'ring'. Of course, your target was also armed and could shoot in self defense. A successful defensive shooting would result in a 24-hour immunity. Eventually you will kill all other persons in the 'ring', be the last man standing, and win the round.

There were no time/place restrictions in the variant we played. If you happened to run into the person on the bus, or knew where they lived, or randomly encountered them downtown, it was valid. It tended to make friendships a bit tense, you never knew if your buddy had you on his list when you met up for a drink. However, for players you did not know (14,000 people in college is a lot, by Irish standards at least), your best bet was to use the provided class schedule. As a result, the scene of a class lecture being interrupted by a running 'gunfight' was not unheard of, though it was usually polite to wait until the end of the class.

The game was not particularly approved of by our college authorities. Too much chance of someone slipping on a wet floor. Meh. Collateral damage is a thing in real life as well....

You'd be amazed how much extra time you get before being recognised if you normally wear a baseball cap, but don't wear it on the hunt...

Anyway, a quick bit of digging indicates that the game is still played. And, sadly, U-Texas Dallas has prohibited it. Must remind my kid not to go there for college.