Agreeing to Disagree in Probabilistic Dynamic Epistemic Logic
Lorenz Demey
Abstract:
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).
Keywords: agreement theorem, probabilistic dynamic epistemic logic,
common knowledge, common prior.