We are concerned with a model of network formation games, called local connection games, in which networks are formed based on players’ strategies. Each player may decide to build links to other players by paying a certain cost fixed in advance, and a strategy of each player is a selection of links to be built. Each player determines and/or changes her or his strategy depending on cost for building links and cost of contacting all other players on the entire network, which influence the structure of the entire network. One of the main interests on the study of local connection games is to characterize all the stable strategy profiles.<br />
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In this paper, we analyze the influences on network structure of stable strategy profiles caused by the cost for building links. For the unit cost case, we provide, in terms of network structure, a necessary and sufficient condition for strategy profiles to be stable. Moreover, we investigate the relationship between integral cost cases and non-integral cost cases as well.