Lexicographic Agreeing to Disagree and Perfect Equilibrium

Aumann's seminal agreement theorem deals with the impossibility for agents to acknowledge their distinct posterior beliefs. It has also led to important economic implications, for example, in the context of speculation and trade. The reasoning of the agents gains in depth when also admitting merely-foreseeable contingencies. We model these contingencies by means of lexicographic probability systems and explore agreeing to disagree in this framework. A weak agreement theorem in the sense of identical posteriors only at the first lexicographic level obtains. Somewhat surprisingly however, a possibility result does emerge for the deeper contingencies. Accordingly, agents can agree to lexicographically disagree on their posteriors beyond the first lexicographic level. Intensifying the agents' like-mindedness beyond the common prior assumption, a strong agreement theorem with equal posteriors at every lexicographic level emerges. Subsequently, we turn to games and explore the classical solution concept of perfect equilibrium in terms of interactive reasoning. We provide the first epistemic foundation for perfect equilibrium with more than two players. Our lexicographic agreement theorems turn out to be pivotal in deriving this result