Congestion Game with Varying Price
I think the answer is yes, there is a Nash equilibrium, and it can be attained just by letting each person in turn change his selection to the best possible alternative. 

For more information, read this paper:

Weighted congestion games with separable preferences

by: Igal Milchtaich

http://www.citeulike.org/user/erelsegal-halevi/article/12205199