Well, Detroit didn't appear in any game, and Chicago (or rather its surroundings) was in Fallout Tactics, not in FO1/2. Chicago is actually being run by the Midwestern Brotherhood of Steel:
http://fallout.wikia.com/wiki/Midwestern_Brotherhood_of_Steel
But there is one city run by the mafia in Fallout 2:
http://fallout.wikia.com/wiki/New_Reno
And I do have a question..while the 40's, 50's, etc style of things in Fallout is one of it's hallmarks, does anybody else find it cloyingly out of place given what YEAR the blast is said to have occurred? yet another reason I said the next fallout game should get some consideration as a prequel...I'd LOVE to see the What If story behind some of the details of the game atmosphere....
Fallout is set in an alternate timeline that diverged from ours in the 1950s. In the 21st century, it looked like the future as imagined in the 1950s pulp science fiction. E.g. instead of transistors you have huge vacuum tube-based computers with artificial intelligence and monochromatic terminals, robotic servants, a pseudo-utopian society based on 1950s values, black and white TV, music from the 1950s considered all-time classics, and most modern musical genres never appearing, etc,. Weapon development was also different from ours and thus plasma and laser weaponry was introduced, 10mm became the most common ammo type, etc. The modern high-tech weapons were never created - instead, weapons based on how people in the 1950s thought 21st century weaponry would look like are used.
For more information, see:
http://fallout.wikia.com/wiki/Fallout_world
http://fallout.wikia.com/wiki/Divergence
http://fallout.wikia.com/wiki/Timeline