What was the first commercial game sold for use on IBM/MS/PC DOS after the IBM August 12, 1981 release.
The first game sold for use on the IBM PC was Microsoft Adventure, which was available on the day the IBM PC was released (it was part of the launch, along with VisiCalc, Easywriter etc.). It was developed by Gordon Letwin (later of OS/2 fame) in 1979, based on the Colossal Cave mainframe game. It didn’t run on DOS though, it was a “booter” — you booted the computer with the game floppy, and it took care of everything, without a separate operating system. You can see it in action on the Internet Archive. (Booters were quite common in those days, on the PC as well as on other micros; see Mobygames’ list of PC booters for other examples.)
Based on Mobygames’ list of DOS games released in 1981, the likeliest candidate for first commercial DOS game is IBM’s Arithmetic Games Set 1. Its contents suggest it was in development by August 1981, and probably complete not long after; the latest files on the original disk are dated September 23, 1981. Strictly speaking it’s a BASIC game, so it could have run using only the PC’s built-in ROM BASIC, without DOS; but as sold it used DOS. It’s also available to play on the Internet Archive. It’s not obvious whether any other games were released commercially earlier than that — some games were published in magazines and later published as commercial games. (Games published in magazines are arguably commercial games too, but it’s not quite the same.)