A network in a society consists of individuals with utility depending on their reputation

In recent studies on network formation, it is assumed that a player directly receives a utility from the network, namely she/he receives a utility from any other players by forming links with them. Jackson and Wolinsky gave a simple model leading the empty, the complete, or the star network as the stable networks. The star network consists of one central player and other peripheral players who form links only with the central player. Under a different condition of link formation and deletion, Hummon showed that the ring network can be also stable. In the real world, however, we often observe a general structure of social networks rather than simple ones such as the empty, the complete, the star and the ring networks. In a certain social network such as an environmental conservation group network, however, it is natural that a player receives a utility not from a network but from the corresponding public goods. In such a group, forming links means that a player joins in conservation activities of public goods such as the global environment, and it improves the level of the public goods. In this paper, we treat a mathematical model which is taking into account the effect of individuals' reputation in a society with interest in public goods, and we show that a general structure of networks in which there exist multiple disjoint components connecting some star networks is also stable.