  The cost-sharing connection game is a variant of routing games on a network.
In this model, given a directed graph with edge costs and edge capacities, each
agent wants to construct a path from a source to a sink with low cost. The
users share the cost of each edge based on a cost-sharing function. One of the
simple cost-sharing functions is defined as the cost divided by the number of
users. Most of the previous papers about cost-sharing connection games
addressed this cost-sharing function. It models an ideal setting where no
overhead arises when people share things, though it might be quite rare in real
life; it is more realistic to consider the setting that the cost paid by an
agent is the original cost per the number of agents using the edge plus the
overhead. In this paper, we model the more realistic scenario of cost-sharing
connection games by generalizing cost-sharing functions. The arguments on the
model are based on not concrete cost-sharing functions but cost-sharing
functions under a reasonable scheme; they are applicable for a broad class of
cost-sharing functions satisfying the following natural properties: they are
(1) non-increasing, (2) lower bounded by the original cost per the number of
the agents, and (3) upper bounded by the original cost, which enables to
represent various scenarios of cost-sharing. We investigate the Price of
Anarchy (PoA) and the Price of Stability (PoS) under sum-cost and max-cost
criteria with the generalized cost-sharing function. Despite the
generalization, we obtain the same tight bounds of PoA and PoS as the
cost-sharing with no overhead except PoS under sum-cost. Moreover, for the
sum-cost case, the lower bound on PoS increases from $\log n$ to $n+1/n-1$ by
the generalization, which is also almost tight because the upper bound is $n$.
