What do you call a system in which the payer is not the decision maker? Is there a name for a system in which the person/entity who makes the buying decision is not the same as the person/entity who pays?

Specific examples of this might be situations which arise in insurance but you could also say a date in which a male is paying for a female (or vice-versa) might qualify.

EDIT:
My interest in the problem arose a couple of years ago when I came down with shingles.  My doctor prescribed an anti-viral and I noted that my insurance covered almost the entire prescription cost.  I think I paid about \$10 but the full price was listed somewhere as around $250.  I then pondered the idea of whether or not I would have bought the prescription if I were faced with the entire \$250.  I was honestly pretty torn, leaning towards probably not.  But this overall experience made me ponder how the demand curve is augmented by the situation.

My interest here is primarily in the demand side effects of this kind of a situation.  My example of the date was an attempt to broaden the scope beyond purely an insurance based question.
This is the principal-agent problem or agency problem.  It's one of the most studied problems in economics over the last 40 years.  The Wikipedia entry offers a thorough summary.  One topic of research is mechanism design, which is how to design institutions to avoid or minimize agency problems.