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Gossip protocols aim at arriving, by means of point-to-point or group communications, at a situation in which all the agents know each other's secrets. We consider distributed gossip protocols which are expressed by means of epistemic logic. We provide an operational semantics of such protocols and set up an appropriate framework to argue about their correctness. Then we analyze specific protocols for complete graphs and for directed rings.

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This article proposes a systematic application of recent developments in the logic of preference to a number of topics in deontic logic. The key junction is the well-known Hansson conditional for dyadic obligations. These conditionals are generalized by pairing them with reasoning about syntactic priority structures. The resulting two-level approach to obligations is tested first against standard scenarios of contrary-to-duty obligations, leading also to a generalization for the Kanger-Anderson reduction of deontic logic. Next, the priority framework is applied to model two intuitively different sorts of deontic dynamics of obligations, based on information changes and on genuine normative events. In this two-level setting, we also offer novel takes on vexed issues such as the Chisholm paradox and modelling strong permission. Finally, the priority framework is shown to provide a unifying setting for the study of operations on norms as such, in particular, adding or deleting individual norms, and even merging whole norm systems in different manners. Accordingly, deontic logic has long considered models involving betterness ordering of worlds or states, going back at least to Hansson (1969). There, statements of dyadic obligation like "it ought to be the case that φ under condition ψ" (in symbols, O(φ | ψ)) were interpreted in terms of a binary relation s t between states s, t -representing that t is at least as ideal as s -according to the following semantics: bs_bs_banner THEORIA, 2014, 80, 116-152 1 In this article only the single-agent case will be considered. Extensions to multi-agent deontic scenarios, which would not pose any technical difficulty, require indexing the ideality relations by different agents.2 Lewis (1974) is an overview of various moves in the early literature on dyadic obligation. PRIORITY STRUCTURES IN DEONTIC LOGIC3 One can also construe the order of properties differently with not-touching on top, and marrying as second best. We return to options in extracting priorities below. 118 JOHAN VAN BENTHEM ET AL.

Liquid democracy is a proxy voting method where proxies are delegable. We propose and study a game-theoretic model of liquid democracy to address the following question: when is it rational for a voter to delegate her vote? We study the existence of pure-strategy Nash equilibria in this model, and how group accuracy is affected by them. We complement these theoretical results by means of agent-based simulations to study the effects of delegations on group's accuracy on variously structured social networks. * This paper (without Appendix) appears in the proceedings of AAAI'19. We are indebted to the anonymous reviewers of IJCAI/ECAI'18 and AAAI'19 for many helpful comments on earlier versions of this paper. We are also grateful to the participants of the LAMSADE seminar at Paris Dauphine University, and the THEMA seminar at University Cergy-Pontoise where this work was presented, for many helpful comments and suggestions. Daan Bloembergen has received funding in the framework of the joint programming initiative ERA-