  We initiate the study of congestion games with variable demands where the
(variable) demand has to be assigned to exactly one subset of resources. The
players' incentives to use higher demands are stimulated by non-decreasing and
concave utility functions. The payoff for a player is defined as the difference
between the utility of the demand and the associated cost on the used
resources. Although this class of non-cooperative games captures many elements
of real-world applications, it has not been studied in this generality, to our
knowledge, in the past. We study the fundamental problem of the existence of
pure Nash equilibria (PNE for short) in congestion games with variable demands.
We call a set of cost functions C consistent if every congestion game with
variable demands and cost functions in C possesses a PNE. We say that C is FIP
consistent if every such game possesses the alpha-Finite Improvement Property
for every alpha>0. Our main results are structural characterizations of
consistency and FIP consistency for twice continuously differentiable cost
functions. Specifically, we show 1. C is consistent if and only if C contains
either only affine functions or only homogeneously exponential functions (c(x)
= a exp(p x)). 2. C is FIP consistent if and only if C contains only affine
functions. Our results provide a complete characterization of consistency of
cost functions revealing structural differences to congestion games with fixed
demands (weighted congestion games), where in the latter even inhomogeneously
exponential functions are FIP consistent. Finally, we study consistency and FIP
consistency of cost functions in a slightly different class of games, where
every player experiences the same cost on a resource (uniform cost model). We
give a characterization of consistency and FIP consistency showing that only
homogeneously exponential functions are consistent.
