Aumann famously showed that two Bayesian agents with common priors cannot “agree to disagree” about a posterior that is common knowledge. One might wonder, does this generalize to decision functions other than Bayesianism? In the early 80s, Cave (1983) and Bacharach (1985) did precisely that, stating that likemindedness (we take the same decision if we have the same information) and a sure thing principle that only implicitly used the knowledge operator. This recent paper in GEB by Dov Samet shows that the sure thing principle they use is problematic, and rederives conditions necessary for agreement.
The problem essentially is this. Give me two agents with two information partitions. I want to say that A is more knowledgeable than B if A’s knowledge is given by a set E and B’s knowledge by a set E+F, where + here represents the union operator. The problem is that this is impossible with the standard partitional formulation of knowledge that philosophers and economists use. If two agents do not have exactly the same information, then each one knows something the other does not. This is true even if one has an information partition that is a strict refinement of the other. Why? Let A know event G at state w. Let B not know event G. The knowledge operator itself also defines events, and by a property of knowledge, A does not know that A does not know G at w, while B knows that he does not know G.
The intuition from Cave and Bacharach can still work, though. Let [j>=i] be all the states where no matter what event E occurs, j knows E whenever i knows it. Assume that if i knows j is at least as knowledgeable at he is in state w, then i takes the same decision as j. Finally assume that if we add a third agent who knows less than i or j at w, then all three agents make the same decision. When these assumptions hold, agents cannot agree to disagree.
Samet quotes a story from Aumann that sums up how the theory works. Alice and Bob are detectives. Bob collects data until 5 with his partner Alice. Alice then stays at work until late at night collecting more information. Both were trained in the same academy, and therefore make the same decisions if they have the same information. Intuitively, if Bob knows that Alice has every bit of information he has plus more, then he should just make the same decision in the end as Alice about the guilt of the suspect. The conditions in the prior paragraph capture this intuition.
http://www.tau.ac.il/~samet/papers/generalized-agreement-theorem.pdf (Final GEB version)
You should say “Assume that if i knows j is at least as knowledgeable at he is in state w, *and he knows what j will decide* then i takes the same decision as j.” Similarly for the ignorant dummy agent…he makes the same decision as others *if* he knows what they will decide.
A good exercise is to create an example showing necessity of the stronger condition saying you can create a dummy agent (such example strangely missing from the paper; the referee was softer than I would have been!)