Given a state space Ω, any partition π of the state space is a knowledge partition. For ease of notation, let π(ω) be the element of the partitionπ which contains ω. A knowledge partitionπis interpreted as, ifω∗ ∈Ωis the true state of the world, then the agent believes that all statesω ∈ π(ω∗)are possible, and that all states ω /∈ π(ω∗)are
not possible. Each knowledge partitionπis associated with a knowledge operatorKπ
according to the formula
KπE ={ω∈Ω|π(ω)⊂E} (2.5)
This equation says that if partition π represents the agent’s information, then KπE
should be the set of states where every state the agent considers possible is contained in the eventE. We say an operator is partitional if there exists some partition which generates the operator using Equation 2.5.
Definition 2.18. Fix a state spaceΩ, and letKbe the associated set of knowledge operators. An operatorK ∈ Kis partitional if there exists some partitionπofΩsuch thatK =Kπ,
whereKπ is given by Equation 2.5. The set of partitional operators is denotedKP.
Remark 2.1. A partitional knowledge operator will necessarily satisfy certain regularity con- ditions. In particular, any partitional knowledge operatorK : 2Ω →2Ωwill satisfyKΩ = Ω,
K(E ∩F) = KE ∩KF, KE ⊂ E, KE ⊂ KKE and ¬KE ⊂ K(¬KE) for all events
E, F in2Ω. Moreover, any knowledge operator satisfying these properties is partitional. This
will be discussed further in Section 2.4.2.
Turning to the question of knowledge aggregators; if the knowledge operators of all agents are partitional, which knowledge aggregations will also be partitional? This question has a surprising answer. The relatively simple aggregators of somebody knows,g, and everybody knows,f, do not preserve this property of being partitional. That is, even if all agents have the very strong rationality property of a partitional information structure, nonetheless very simple statements like ‘does somebody know this’ may not have such rationality properties. Example 2.6 provides a simple model where this is the case. Proposition 2.7 shows that the more developed aggregators of common knowledge,∧, and distributed knowledge,∨, are partitional-preserving.
Definition 2.19. Fix a state spaceΩ, and letKbe the associated set of knowledge operators. An aggregatorA : Seq(K)→ K is partition-preserving if, for all sequences of partitional knowledge operators(K1, . . . , Kn)∈Seq(KP), the knowledge aggregationA(K1, . . . , Kn)is also a partitional operator.
Example 2.6. LetJ ={1,2},Ω = {a, b, c}, and
K1{a}=K1{b}=∅, K1{a, c}=K1{b, c}={c}, K1E =E otherwise
K2{b}=K2{c}=∅, K2{a, b}=K2{a, c}={a}, K2E =E otherwise
Operator K1 is a partitional operator given by the partitionπ1 = {{a, b},{c}} according to Equation 2.5. Similarly, operator K2 is a partitional operator given by the partition π2 =
{{a},{b, c}}.
The somebody knows aggregationgJ has
gJ({a, b} ∩ {b, c}) = gJ{b}=∅
gJ{a, b} ∩gJ{b, c}={a, b} ∩ {b, c}={b}
AsgJ({a, b} ∩ {b, c})=6 gJ{a, b} ∩gJ{b, c}, by Remark 2.1, the somebody knows aggregation
gJis not partitional in this case. Therefore, the somebody knows aggregatorgis not partitional
in general. Similarly, the everybody knows aggregationfJ has
fJ{a, b}={a, b} ∩ {a}={a}
fJ{a}=∅ ∩ {a}=∅
Asf{a, b} 6⊂ff{a, b}, by Remark 2.1, the everybody knows aggregationfJis not partitional
in this instance. Thus, the everybody knows aggregatorfis not partitional.
Proposition 2.7. The aggregators common knowledge and distributed knowledge are both partition-preserving aggregators.
The correspondence knowledge framework is a middle ground between the more general universe of knowledge operators and the more restrictive universe of partitions. A knowledge correspondence is a functionγ : Ω→2Ω. The usual interpretation is that,
at every stateω∈Ω, the agent considers as possible exactly those states inγ(ω), ruling out all states that do not belong toγ(ω). In other words, at stateω∈Ω, the agent thinks that an eventE is sure to happen if and only ifγ(ω)⊂E, and is sure that eventEdoes not happen if and only ifE∩γ(ω) =∅.
Each knowledge correspondence γ is associated with a knowledge operator Kγ
according to the formula
This equation says that if correspondence γ represents the agent’s information, then
KγE should be the set of states where every state the agent considers possible is con-
tained in the eventE. Definition 2.20 states that an operator is a correspondence oper- ator if there exists some correspondence which generates the operator using Equation 2.6.
Definition 2.20. Fix a state spaceΩ, and letKbe the associated set of knowledge operators. An operator K ∈ K is a correspondence operator if there exists some correspondence γ : Ω → 2Ω such that K = K
γ, where Kγ is given by Equation 2.6. The set of correspondence
operators is denotedKC.
An operatorK : 2Ω →2Ωis a correspondence operator so long as it satisfiesKΩ = Ω,
andK(E ∩F) = KE ∩KF for all eventsE, F ∈ 2Ω, as shown in Lemma 2.3. These properties are explored further in Section 2.4.2.
Lemma 2.3. Fix a state spaceΩ, and letKbe the associated set of knowledge operators. An operatorK ∈ Kis a correspondence operator if and only ifKΩ = ΩandK(E∩F) = KE∩KF for all eventsE, F ∈2Ω.
The correspondence framework is a strict generalization of the partition framework. Lemma 2.4 shows that any knowledge partition is just a knowledge correspondenceγ
that satisfies the additional properties that: P.1: ω ∈γ(ω), for everyω ∈Ω.
P.2: For everyω, ω0 ∈Ω, the setsγ(ω)andγ(ω0)either coincide or have no element in common.
Lemma 2.4. Fix a state spaceΩ, and letKbe the associated set of knowledge operators, and let
γ : Ω→2Ω be a knowledge correspondence.
Correspondenceγ satisfies properties P.1 and P.2 if and only ifKγis partitional. In general,
KC ⊂ KP.
Turning again to the question of knowledge aggregation; if the knowledge opera- tors of all agents are correspondence operators, which knowledge aggregations will also be correspondence operators? As seen in Example 2.6, the somebody knows ag- gregatorgdoes not preserve the property of being a correspondence operator. In that
example, all input operators were partitional, so by Lemma 2.4 were also correspon- dence operators. However, the output operator failed to haveK(E∩F) =KE ∩KF, and so, by Lemma 2.3, was not a correspondence operator. The remaining classical aggregators common knowledge, distributed knowledge and everybody knows all preserve the property of being a correspondence operator.
Definition 2.21. Fix a state spaceΩ, and letKbe the associated set of knowledge operators. An aggregator A : Seq(K) → K is correspondence-preserving if, for all sequences of correspondence knowledge operators (K1, . . . , Kn) ∈ Seq(K
C), the knowledge aggregation
A(K1, . . . , Kn)is also a correspondence operator.
Proposition 2.8. The aggregators common knowledge, distributed knowledge, and everybody knows, are all correspondence-preserving aggregators.