From Sega Retro
Atari Games (or Atari Games Corporation) was a company formed in 1984, after the original Atari Inc. split into two halves (the other half being Atari Corporation, sold by Warner Communications to Jack Tramiel). Atari Games started life off as the arcade division of Atari Inc., producing a number of top arcade games during the 1980s. One of the conditions of the split requires Atari Games to always include "Games" in its title and logos - Atari Corporation did not have that restriction.
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Unlike Atari Corporation, Atari Games was comprised of many former Atari Inc. imployees, and as such continued operating in a similar manner to Atari Inc. In 1985 Namco bought a controlling interest in the company, though in the following year these shares were bought back by Atari Games employees.
Atari Games' home console division was Tengen, and was the company engaged in a legal battle with Nintendo over the NES release of Tetris. In 1993, Atari Games was bought again by Time Warner Interactive though kept its identiy until 1996 when it was sold to WMS Industries. At the time, its cousin, Atari Corporation was also being purchased and re-organised by numerous companies, so WMS Industries brought together Atari Games and the remnants of Tengen and named the studio Midway Games West to avoid further confusion.
In 2003 Midway Games declared bankruptcy, and so Atari Games does not exist in any form. Much of is assets are now owned by Atari Corporation (now confusingly known as a different "Atari Inc.").
Note: In most cases, Atari Games built the arcade versions, Tengen/Time Warner Interactive created the Sega ports.