Abstract
Given n agents each of which has a secret (a fact not known to anybody else), the classical version of the gossip problem is to achieve shared knowledge of all secrets in a minimal number of phone calls. There exist protocols achieving shared knowledge in \(2(n{-}2)\) calls: when the protocol terminates everybody knows all the secrets. We generalize that problem and focus on higher-order shared knowledge: how many calls does it take to obtain that everybody knows that everybody knows all secrets? More generally, how many calls does it take to obtain shared knowledge of order k? This requires not only the communication of secrets, but also the communication of knowledge about secrets. We give a protocol that works in \((k{+}1)(n{-}2)\) steps and prove that it is correct: it achieves shared knowledge of level k. The proof is presented in a dynamic epistemic logic that is based on the observability of propositional variables by agents.
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Acknowledgements
We would like to acknowledge several discussions about the gossip problem at the inspiring August 2015 workshop “To be announced” in Leiden, in particular with Hans van Ditmarsch, Jan van Eijck, Malvin Gattinger, Louwe Kuijer, Christian Muise, Pere Pardo, Rahim Ramezanian and Francois Schwarzentruber. We are also grateful to Davide Grossi, Emiliano Lorini and Martin Cooper.
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Herzig, A., Maffre, F. (2016). How to Share Knowledge by Gossiping. In: Rovatsos, M., Vouros, G., Julian, V. (eds) Multi-Agent Systems and Agreement Technologies. EUMAS AT 2015 2015. Lecture Notes in Computer Science(), vol 9571. Springer, Cham. https://doi.org/10.1007/978-3-319-33509-4_20
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DOI: https://doi.org/10.1007/978-3-319-33509-4_20
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