This section focuses on some interesting results about the model transformer [!]. More specifically, we prove that [!] realises distributed knowledge and that, in certain circumstances, it also realises distributed plausibility and hence distributed belief.
Recall that distributed knowledge is the knowledge a group of agents would have if they were to share all their hard information. Since [!] models precisely this, it should be the case that all agents have the same knowledge after deliberation and that this knowledge equals the distributed knowledge before any communication took place. This is indeed the case:
Proposition 3.5. Let N be a finite set of agents, M a multi-agent plausibility model and M! the updated model after deliberation. Then for all i∈ N :∼!
i=∼N.
Proof. By definition of∼!
i.
As a corollary, we get:
Corollary 3.6. Let N be a finite set of agents, M a multi-agent plausibility model and M! the
updated model after deliberation. Then for all i, j∈ N and allP ⊆W =W!, the following hold: 1. KiF!P =KjF!P.
2. DKFP =KiF!P.
Corollary 3.6 shows that deliberation turns distributed knowledge into actual knowledge. Therefore, we can really think of distributed knowledge as the potential knowledge of a group of agents. After sharing all their hard information, the knowledge of the agents is the same. Their plausibility relations and their beliefs, however, might still differ. The deliberation operation [!] does not, in general, realise distributed plausibility and distributed belief. However, as we anticipated in Section 2.3, it does in common prior models. First, consider the updated plausibility relations:
Proposition 3.7. LetN be a finite set of agents,M a common prior model,≤the common prior plau- sibility relation and M! the updated model after deliberation. Then for all i∈ N :≤!
i =
T
i∈N ≤i =
≤N.
Proof. Leti∈ N be arbitrary. Unfolding the definitions yields≤!
i=≤i ∩ ∼!i = (≤ ∩ ∼i)∩ Ti∈N ∼i=
≤ ∩ T
i∈N ∼i =Ti∈N(≤ ∩ ∼i) =Ti∈N ≤i =≤N.
The interpretation of this proposition is that if the differences in the plausibility of the agents are solely due to the fact that they have learned and experienced different things in their lives, i.e. solely due to differences in information, then sharing information yields agreement with respect to their plausibilities. Or more formally, [!] realises distributed plausibility in common prior models, as is reflected in the following corollary:
Corollary 3.8. Let N be a finite set of agents, M a multi-agent plausibility model and M! the updated model after deliberation. For alli, j ∈ N and all P ⊆W =W!the following hold:
1. 2Fi !P =2Fj!P. 2. D2FP =2Fi !P.
Recall that distributed belief is defined in terms of distributed knowledge and distributed plausibility. Both are realised in common prior models. Therefore, distributed belief is realised in these models as well:
Theorem 3.9. (Agreeing to Disagree) Let N be a finite set of agents,M a common prior model and M! the updated model after deliberation. Then for all i, j ∈ N and all P ⊆ W =W!, the following
hold:
1. BiF!P =BFj!P. 2. DBFP =BFi !P.
Proof. By Observation 2.10, it follows that BiF!P iff for all w0 such that w ∼!
i w0 there exists a
w00 such that w0 ≤!
i w00 and for all w000 such that w00 ≤!i w000 it holds that w000 ∈ P. According to
Propositions 3.5 and 3.7, it holds for all i ∈ N that ∼!
i = ∼N and ≤!i = ≤N. Item 2 now follows
directly from Observation 2.26. Item 1 follows from the fact that all agents have the same epistemic indistinguishability and plausibility relations.
The result of Theorem 3.9.1 is a qualitative version of Aumann’s (1976)agreeing to disagree theorem, which says that if differences in the beliefs of the agents are solely due to differences in information, then sharing of information leads them to agree on their beliefs.3 Thus, in common prior models
distributed belief can really be interpreted as form of potential group belief.
In standard common prior models, it is even more clear that distributed belief is a form of potential group belief, as one can show that the agents’ belief relations collapse after deliberation. Before proving this, we define the common belief relation and show that this relation can be thought of as the accessibility relation for distributed belief. Furthermore, we define the updated belief relations: Definition 3.10. Let N be a finite set of agents and M a standard common prior model. Let the common belief relation →N ⊆ W ×W be such that for all w, w0 ∈ W: w →N w0 iff w ∼N w0 and
w0 ∈Max(≤N).
Proposition 3.11. LetN be a finite set of agents, M a standard common prior model and P ⊆W. Then DBP = [→N]P.
Proof. Left to the reader. The proof is similar to the proof of Theorem 2.19.
Definition 3.12. Let N be a finite set of agents, M a standard common prior model and M! the updated model after deliberation. Let the updated belief relation →!
i ⊆W!×W! be such that for all
w, w0∈W!:w→! iw 0 iffw∼! i w 0 and w0 ∈Max(≤! i).
The next proposition shows that the belief relations of all the individuals after deliberation equal the distributed belief relation:
Proposition 3.13. (Agreeing to Disagree) Let N be a finite set of agents, M a standard common prior model and M! the updated model after deliberation. Then for all i ∈ N the following holds:
→!
i =→N.
Proof. Leti∈ N be arbitrary. Recall thatw→N w0iffw∼N w0andw0 ∈Max(≤N). Propositions 3.5
and 3.7 give us that∼!
i =∼N and ≤!i=≤N. By definition of →!i, it follows that →!i =→N.
It is immediately clear from this proposition that distributed belief is a form of potential group belief, because after the agents share all their information in a deliberation, they end up with the same belief relation and hence with the same beliefs. Thus, Proposition 3.13 is a qualitative verion of Aumann’s (1976) agreeing to disagree theorem in standard models.
3
D´egremont & Roy (2012) were one of the first to introduce a qualitative version of Aumann’s theorem to dynamic epistemic logic.