Descripción y análisis empírico

Which objects are reasonable to include as knowledge questions? We provide two equivalent definitions for the set of knowledge questions. Definition 3.5 seeks a fixed point of a set equation, making three assumptions on the class of knowledge questions.

First, ‘when does agent j know’ is a reasonable knowledge question for each j ∈ J. Second, given any collection of reasonable knowledge questions, then it is reasonable to ask if agentjknows them all. Finally, the set of knowledge questions is the minimal set with these properties.

Definition 3.6 gives a recursive definition for the set of knowledge questions. New knowledge questions are formed recursively by asking when does agentj know some (finite) collection of previously formed knowledge questions. Proposition 3.7 shows that Definitions 3.5 and 3.6 are equivalent.

In set theory, sets are usually defined so that all elements of sets are themselves smaller sets. The class of sets withatomsallows for elements of sets which are non-set objects, called atoms.8 _{As the name suggests, atoms are objects which cannot be broken}

into smaller components. Definition 3.5 takes the collection of agentsJ as atoms, and builds a collection of sets around these atoms.

Definition 3.5. LetJ be a set of agents. The set of knowledge questionsQis the minimal set of ordered pairs with the first element an agentj ∈J, and the second element drawn from the collection of sets with atomsJ, such that

Q.1) For allj ∈J, then(j,∅)∈Q, and

Q.2) For allj ∈J andq1, . . . , qn ∈Q, then(j,{q1, . . . , qn})∈Q

LetFin(2Q_{)be the set of finite subsets ofQ. Then,Qis the minimal solution of the}

set equationQ=J ×Fin(2Q).

Definition 3.6. LetJ be a set of agents. Define the setQnrecursively on eachn ∈Nby

Q1 =J× {∅} Qn+1 =Qn∪ {(j,{q1, q2, . . . , qm})|j ∈J, m∈N, q1, . . . , qm ∈Qn} Let e Q= ∞ [ n=1 Qn

Proposition 3.7. LetJ be a set of agents. The set of knowledge questionsQas in Definition 3.5 is the same as the setQein Definition 3.6:

Q=Qe

8_{An example axiomatization of ZF set theory with atoms, called ZFA, is given in Chapter 4 of Jech}

There was a concern following Definition 3.5 that the set of questionsQmay not exist. Specifically, that there may not be a minimal collection of sets with property Q.1 and Q.2. Fortunately, Definition 3.6 and Proposition 3.7 guarantee that such a minimal set does indeed exist. Throughout the remainder of this work, the symbolQ

will be used to refer to the set of knowledge questions. The set of knowledge questions depends only on the finite set of agentsJ, and does not refer at all to the state spaceΩ, or the knowledge operatorsKj.

For ease of notation, the infimum symbolinfis occasionally used. For any collection of events, letinf(E1, . . . , En) =E1∩ · · · ∩En. This is the usual definition of the infimum

over2Ω_{when the set of events is endowed with the subset partial order.}9 _{Similarly, for}

collections of knowledge operators, letK = inf(K1, . . . , Kn)be defined pointwise by

KE =K1E∩ · · · ∩KnE. This is the infimum over the set of functions2Ω →2Ωwith the

partial order induced from the order on the codomain.

Each knowledge question is associated with a knowledge operator where the state- ment ‘When does agent 3 know that agent 1 knows that agents 2 and 3 know the event has occurred’, usually denoted K3(K1(K2 ∩K3)), is associated with the knowledge

questionq = (3,{(1,{(2,∅),(3,∅)})}). Definition 3.7 describes the recursive algorithm which makes this association.

Definition 3.7. Let(Ω, J,{Kj}j∈J)be a knowledge model,Qthe set of knowledge questions.

For anyq ∈Q1, whereq= (j,∅), the knowledge operatorKq : 2Ω →2Ω is defined as

Kq =Kj

For anyq ∈Q\Q1, whereq = (j,{q1, q2, . . . , qn}), the knowledge operatorKq : 2Ω →2Ω is

recursively defined as

Kq=Kj(inf(Kq1, . . . Kqn)) (3.3)

For a fixed knowledge questionq ∈Q, Definition 3.7 suggestsqcan be interpreted as a function which takes as input the|J|knowledge operatorsK1, . . . , K|J|, and outputs

a new knowledge operatorKq. Viewed in this way, each knowledge questionqis the

restriction of a knowledge aggregator, as in Chapter 2, to a fixed input length.

An event is defined to be common knowledge, under Behavioral Common Knowl- edge, if all knowledge questions are known.

9_{The subset partial order is the ordering whereX}

Definition 3.8. Let(Ω, J,{Kj}j∈J)be a knowledge model, andQthe associated set of knowl-

edge questions. Behavioral Common KnowledgeCB : 2Ω →2Ωis

CB = inf

q∈QKq (3.4)

Hence, for all eventsE ∈2Ω

CBE =

\

q∈Q

KqE

The construction of Behavioral Common Knowledge admits a recursive set-theoretic characterization. This representation will be useful for proving many of the prop- erties of the Behavioral Common Knowledge operator. Definition 3.9 describes the representation, and Proposition 3.8 shows it to be equivalent to Behavioral Common Knowledge.

Definition 3.9. Let(Ω, J,{Kj}j∈J)be a knowledge model. For each eventE ∈ 2Ω, define a

sequence of collections of events,(Si[E])i∈Naccording to: S1[E] ={KjF |j ∈J, F =E}, and

Sn[E] ={KjF |j ∈J, F ∈Sn−1[E]}

∪ {F1∩F2 |F1, F2 ∈Sn−1[E]}, for all n > 1 (3.5)

That is,S1[E]is the collection of events of the formKjEfor somej; and givenSn−1[E], a

new event inSn[E]is constructed either by(i)prependingKj to some set inSn−1[E], or(ii)

taking the intersection of two events inSn−1[E].10

DefineCeby: e CE = ∞ \ n=1 \ F∈Sn[E] F (3.6)

This sequence of collections of events(Si[E])i∈_Nis a non-decreasing, or telescoping,

series, as shown in Lemma 3.3. This fact, combined with the finiteness of2Ω, is espe- cially useful when showing thatCe : 2Ω → 2Ω is the same as the Behavioral Common

KnowledgeCB, as in Proposition 3.8.

Lemma 3.3. Let(Ω, J,{Kj}j∈J)be a knowledge model. Then, the sequence(Sk[E])k∈_Ngiven

by Equation 3.5 is a non-decreasing sequence. That is,Sm[E]⊂Sm+1[E]for all eventsE ∈2Ω,

andm ∈_N.

10_{Taking intersections of two events is sufficient for the construction. Allowing arbitrary finite inter-}

Proposition 3.8. Let(Ω, J,{Kj}j∈J)be a knowledge model. LetCB : 2Ω →2Ω be Behavioral

Common Knowledge given by Equation 3.4, and Ce : 2Ω → 2Ω be defined by Equation 3.6.

Then,CB =Ce.

As a robustness check, Proposition 3.9 shows that if the set of knowledge questions were made any smaller, this would lead to a different definition of common knowledge. If we remove questions from the question setQ, then the Behavioral Common Knowl- edge operator becomes weakly more inclusive. Proposition 3.9 shows that if even a single knowledge question is removed, there are examples of knowledge models with a strict change in the resulting operator.

Proposition 3.9. Let J be a set of agents, Qthe associated set of knowledge questions, and

b

Q₍Q. Then, there exists a knowledge model(Ω, J,{Kj}j∈J)such thatCB 6= inf_q∈QbKq.

One consequence of Proposition 3.9 is that, absent additional information about the state spaceΩ, it is not sufficient to define the set of knowledge questionsQas the union

SN

n=1Qnfor some very largeN. It is necessary to allow arbitrarily ‘long’ questions in

the set of knowledge questions in order to have the definition of common knowledge given here. This mirrors the result of Rubinstein [1989] on almost common knowledge.