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 cost of each edge is shared by the users based on a cost-sharing function. One of simple cost-sharing functions is defined as the cost divided by the number of users. In fact, 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 the agents 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 do not depend on specific cost-sharing functions, and are applicable for a wide 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. In spite of the generalization, we obtain the same bounds of PoA and PoS as the cost-sharing with no overhead except PoS under sum-cost. Note that these bounds are tight. In the case of sum-cost, 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 $n$.
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