Agreeing on Decisions: An Analysis with Counterfactuals

Moses & Nachum ([7]) identify conceptual flaws in Bacharach's generalization ([3]) of Aumann's seminal "agreeing to disagree" result ([1]). Essentially, Bacharach's framework requires agents' decision functions to be defined over events that are informationally meaningless for the agents. In this paper, we argue that the analysis of the agreement theorem should be carried out in information structures that can accommodate for counterfactual states. We therefore develop a method for constructing such "counterfactual structures" (starting from partitional structures), and prove a new agreement theorem within such structures. Furthermore, we show that our approach also resolves the conceptual flaws in the sense that, within our framework, decision functions are always only defined over events that are informationally meaningful for the agents.