I would probably use the word diffident when someone is acquiescing to another's authority. The word connotes not just "simple" shyness, but also a lack of self-confidence and assertiveness.
Definition: (http://dictionary.reference.com/browse/diffident)
diffident
adjective
1. lacking confidence in one's own ability, worth, or fitness; timid; shy.
Another option would be timid. It's a fairly common word and I don't think I need to define it here.
To more directly put across the point of submission to authority, you can consider submissive.
Definition: (http://dictionary.reference.com/browse/submissive)
submissive
adjective
1. inclined or ready to submit or yield to the authority of another; unresistingly or humbly obedient:
submissive servants.
2. marked by or indicating submission or an instance of yielding to the authority of another: a submissive reply.
Obedient is another (more positive) option. Again, a word in common usage, and I won't be defining it here (it's easy to look up).
The most negative options, generally used critically, are pushover and doormat. They are idiomatic nouns that describe a person who seriously lacks self-assertiveness and may be defined as follows:
Doormat (http://dictionary.reference.com/browse/doormat):
doormat
- Slang One who submits meekly to domination or mistreatment by others.
Pushover (http://www.thefreedictionary.com/pushover)
push·o·ver
- One that is easily defeated or taken advantage of.
In the same vein as the above two, there is another idiom, which I believe is peculiar to the US: milquetoast.
Milquetoast (http://dictionary.reference.com/browse/milquetoast)
milquetoast
noun, (sometimes initial capital letter)
- a very timid, unassertive, spineless person, especially one who is easily dominated or intimidated:
a milquetoast who's afraid to ask for a raise.
And, as included in the definition of that word spineless is another negative adjective for someone lacking self-assertiveness. Again, it's a highly critical word. Given that the spine supports your body in an upright posture, I believe you should be able to see how the metaphorical meaning comes about.
Best Answer
I've often heard "lies like a rug"
From McGraw-Hill Dictionary of American Idioms and Phrasal Verbs: