# MoL-2010-14: Agreeing to Disagree in Probabilistic Dynamic Epistemic Logic

MoL-2010-14: Demey, Lorenz (2010) Agreeing to Disagree in Probabilistic Dynamic Epistemic Logic. [Report]

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Aumann's agreeing to disagree theorem is a central theorem of game theory. This result says that if two agents have a common prior, then they cannot agree (have common knowledge of their posteriors) to disagree (while these posteriors are not identical). This thesis looks at the agreeing to disagree theorem from the perspective of probabilistic dynamic epistemic logic. The first goal of the thesis is to establish a new connection between game theory and epistemic logic. We prove (local model-based versions and global frame-based versions of) several semantic agreement theorems, and show that these are natural formalizations of Aumann's original result. We also provide axiomatizations of (dynamic) agreement logics, in which the first of these agreement theorems can be derived syntactically. The second goal is the further technical development of probabilistic dynamic epistemic logic. We mention three examples. First, to model the experiment dynamics, we enrich the probabilistic Kripke models with experiment relations', thus establishing a link with the dynamic epistemic logic of questions. Second, to model the communication dynamics, we introduce the notion of a dialogue about a proposition \varphi', which is a particular sequence of public announcements; we show that this sequence always has a fixed point, and that at this fixed point the agents' probabilities for \varphi have become common knowledge. Thirdly, to make sure that both types of dynamics are well-defined, we introduce the constraint that \mu_i(w)(w)>0 for all states w in any Kripke model, and discuss the technical and methodological consequences of this constraint. The third goal is to use the technical results for the purpose of clarifying some conceptual issues surrounding the agreement theorem. In particular, we discuss the role of common knowledge (which we claim to be smaller than often thought), and the importance of explicitly representing the experimentation and communication dynamics, which is central in the intuitive motivation behind Aumann's result. Recently D\'{e}gremont and Roy have formalized Aumann's agreement theorem in the context of epistemic plausibility models. Our fourth and final goal is to provide a detailed comparison between their approach and the one developed in this thesis, focusing on the representation of the agents' soft information (quantitatively versus qualitatively).