How do networks form and what is their ultimate topology? Most of the literature that addresses these questions assumes complete information: agents know in advance the value of linking to other agents, even with agents they have never met and with whom they have had no previous interaction (direct or indirect). This paper addresses the same questions under what seems to us to be the much more natural assumption of incomplete information: agents do not know in advance -but must learn -the value of linking to agents they have never met. We show that the assumption of incomplete information has profound implications for the process of network formation and the topology of networks that ultimately form. Under complete information, the networks that form and are stable typically have a star, wheel or core-periphery form, with high-value agents in the core. Under incomplete information, the presence of positive externalities (the value of indirect links) implies that a much wider collection of network topologies can emerge and be stable. Moreover, even when the topologies that emerge are the same, the locations of agents can be very different. For instance, when information is incomplete, it is possible for a hub-and-spokes network with a low-value agent in the center to form and endure permanently: an agent can achieve a central position purely as the result of chance rather than as the result of merit. Perhaps even more strikingly: when information is incomplete, a connected network could form and persist even if, when information were complete, no links would ever form, so that the final form would be a totally disconnected network. All of this can occur even in settings where agents eventually learn everything so that information, although initially incomplete, eventually becomes complete.Keywords: Network Formation, Incomplete Information, Dynamic Network Formation, Link Formation, Formation History, Externalities JEL Classification: A14, C72, D62, D83, D85 * We are grateful to William Zame, Ichiro Obara, Moritz Meyer-ter-Vehn, Sanjeev Goyal, Luca Canzian, Simpson Zhang and a number of seminar audiences for suggestions which have significantly improved the paper.

To ensure that social networks (e.g. opinion consensus, cooperative estimation, distributed learning and adaptation etc.) proliferate and efficiently operate, the participating agents need to collaborate with each other by repeatedly sharing information. However, sharing information is often costly for the agents while resulting in no direct immediate benefit for them. Hence, lacking incentives to collaborate, strategic agents who aim to maximize their own individual utilities will withhold rather than share information, leading to inefficient operation or even collapse of networks. In this paper, we develop a systematic framework for designing distributed rating protocols aimed at incentivizing the strategic agents to collaborate with each other by sharing information. The proposed incentive protocols exploit the ongoing nature of the agents' interactions to assign ratings and through them, determine future rewards and punishments: agents that have behaved as directed enjoy high ratings -and hence greater future access to the information of others; agents that have not behaved as directed enjoy low ratings -and hence less future access to the information of others. Unlike existing rating protocols, the proposed protocol operates in a distributed manner, online, and takes into consideration the underlying interconnectivity of agents as well as their heterogeneity. We prove that in many deployment scenarios the price of anarchy (PoA) obtained by adopting the proposed rating protocols is one. In settings in which the PoA is larger than one, we show that the proposed rating protocol still significantly outperforms existing incentive mechanisms such as Tit-for-Tat. Importantly, the proposed rating protocols can also operate efficiently in deployment scenarios where the strategic agents interact over time-varying network topologies where new agents join the network over time.2 Repeated information sharing, social networks, distributed networks, incentive design, distributed rating protocol, repeated games.

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