There was a company for a while that was trying to do that. People exploded with anger (number one, it was hard to identify what app was the cause, number two, it was as annoying as it much be for iOS users). Almost all apps removed it, the company changed strategy if I recall and Google threatened to ban their apps.

But other than one Zynga game, I never, ever get spam notifications and that Zynga game was installed for about an hour or two.