Discussion Paper No.358

Is “Unawareness Leads to Ignorance” Trivial?

Yoshihiko Tada
Chuo University
Economic Reserch Institute,
Associate Researcher
December 2021

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Note and Comment

Is “Unawareness Leads to Ignorance” Trivial?

Yoshihiko Tada1

November 16, 2021

1. Introduction
Dekel, Lipman, and Rustichini (1998, p. 166) show two properties of
unawareness: the first is that “the agent is never unaware of anything,” (let us callit Triviality), and the second is that “if the agent is unaware of anything, he knows
nothing,” (let us call it Unawareness Leads to Ignorance). They assert that “clearly,then, either property leaves us with only a trivial form of unawareness at best.” Itis certainly agreed that Triviality is a property of trivial-unawareness. However, is
it Unawareness Leads to Ignorance a property of trivial-unawareness? This article
addresses this question.

2. Preliminary
Let us first define the standard information structure 〈Ω, 𝑃〉 and the (standard)knowledge operator. We define Ω as the state space and 𝑃: Ω → 2! ∖ {∅} is the1

Graduate School of Economics, Chuo University, Japanyoshihiko.tada.4@gmail.com
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agentʼs (standard) information function. We do not assume whether 𝑃 ispartitional. Here, the knowledge operator 𝐾: 2! → 2! is defined as the following:
Given any event 𝐸 ⊆ Ω,

𝜔 ∈ 𝐾" (𝐸) if 𝑃" (𝜔) ⊆ 𝐸; and0𝜔 ∉ 𝐾" (𝐸) otherwise.

Then, by the definition, the following properties hold:

Necessitation

𝐾(Ω) = Ω, andMonotonicity

𝐸 ⊆ 𝐹 ⟹ 𝐾(𝐸) ⊆ 𝐾(𝐹).

Next, we assume three properties of the unawareness operator 𝑈: 2! → 2! asproposed by Dekel, Lipman, and Rustichini (1998) as follows:

Plausibility

𝑈(𝐸) ⊆ ¬𝐾(𝐸) ∩ ¬𝐾¬𝐾(𝐸),KU Introspection

𝐾𝑈(𝐸) = ∅, andAU Introspection

𝑈(𝐸) ⊆ 𝑈𝑈(𝐸).

Then, Dekel, Lipman, and Rustichini (1998) show the following theorem:

Theorem 1 (Dekel, Lipman, and Rustichini 1998): In a standard information

structure 〈Ω, 𝑃〉, suppose that the unawareness operator 𝑈 satisfies Plausibility,KU Introspection, and AU Introspection. Then,l(Triviality) if the knowledge operator 𝐾 satisfies Necessitation, then for any2event 𝐸 ⊆ Ω, 𝑈(𝐸) = ∅; andl(Unawareness Leads to Ignorance)2 if 𝐾 satisfies Monotonicity, then for allevents 𝐸, 𝐹 ⊆ Ω, 𝑈(𝐸) ⊆ ¬𝐾(𝐹).

Proof: By AU Introspection, 𝑈(𝐸) ⊆ 𝑈𝑈(𝐸), by Plausibility, 𝑈𝑈(𝐸) ⊆ ¬𝐾𝑈(𝐸) ∩¬𝐾¬𝐾𝑈(𝐸), and by KU Introspection, ¬𝐾𝑈(𝐸) ∩ ¬𝐾¬𝐾𝑈(𝐸) = ¬𝐾(Ω).
First, suppose that Necessitation holds. Then, because ¬𝐾(Ω) = ∅, 𝑈(𝐸) ⊆ ∅.
Second, suppose that Monotonicity holds. Then, for any 𝐹 ⊆ Ω, 𝐾(𝐹) ⊆ 𝐾(Ω),

2Galanis (2013) defines “Awareness Leads to Knowledge” as a property ofunawareness in unawareness structures proposed by Heifetz, Meier, andSchipper (2006). The property is similar to the inverse of Unawareness Leads toIgnorance. However, he first focuses only on Awareness Leads to Knowledge inunawareness structures but not in (non)standard information structures.
Unawareness structures assume a family of disjoint subjective state spaces. Somespaces might be related with and more expressive than another space. It meansthat the description of one space is greater than that of another. AwarenessLeads to Ignorance suggests that an agentʼs knowledge in a more expressive statespace is more descriptive than her knowledge in a less expressive state space. Bycontrast, the inverse of Unawareness Leads to Ignorance does not mean that in(non)standard information structures, because the state space in the information
structure does not assume another disjoint state space. Moreover, he does notconsider the property of Awareness Leads to Knowledge in standard information
structures: “...the standard model assumes an agent who is aware of everything
and knows all relevant theorems.” (Galanis 2013, P52)
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that is, ¬𝐾(Ω) ⊆ ¬𝐾(𝐹). Therefore, 𝑈(𝐸) ⊆ ¬𝐾(𝐸) for any 𝐸, 𝐹 ⊆ Ω. ∎That is, when Plausibility, KU Introspection, and AU Introspection hold,Necessitation implies Triviality, and Monotonicity implies Unawareness Leads toIgnorance. The former means that the agent cannot be aware of anything, and thelatter means that the agent cannot get any knowledge where the agent is unawareof some event. Hence, they assert that both properties are trivial, and say that “anontrivial model of unawareness requires us to abandon both necessitation andmonotonicity” (Dekel, Lipman, and Rustichini 1998, p. 166). However, the twoproperties, Necessitation and Monotonicity, must be held by a definition of theknowledge operator. Therefore, we cannot discuss nontrivial unawareness in astandard information structure.

3. Unawareness Leads to Ignorance without Triviality
This section addresses the question “Is Unawareness Leads to Ignorance trivial?”
In some cases, the answer is no. Let us consider an unemployed person, Alice, who
cannot access information about the unemployment insurance program. Then,
she may not know that she can receive a jobless insurance, or how to apply for thejobless insurance. In other words, she cannot know everything about theunemployed insurance program.
We mathematically model this situation. Let Ω = {𝜔# , 𝜔$ }, and let us interpret
𝜔# as “Alice can receive jobless insurance,” and 𝜔$ as “Alice cannot receive
jobless insurance.” Each event is interpreted as follows.

4𝐸# = {𝜔# }:Alice can receive jobless insurance

𝐸$ = {𝜔$ }:Alice cannot receive jobless insuranceΩ = {𝜔# , 𝜔$ }:Aliceʼs country has an unemployed insurance program.

Here, let 𝐾: 2{&! ,&" } → 2{&! ,&"} be Aliceʼs knowledge operator, and for any event𝐸 ⊆ Ω , 𝐾(𝐸) is interpreted as “Alice knows the event 𝐸 .” Let 𝑈: 2{&! ,&"} →2{&! ,&" } be Aliceʼs unawareness operator, and for any event 𝐸 ⊆ Ω , 𝑈(𝐸) isinterpreted as “Alice is unaware of the event 𝐸.” Suppose that for any event 𝐸 ⊆Ω, 𝐾(𝐸) = ∅, that is, she does not know everything. Then, 𝐾 does not satisfy
Necessitation, but it satisfies Monotonicity. Moreover, 𝑈 obviously satisfies
Plausibility, KU Introspection, and AU Introspection because 𝑈(𝐸) = Ω for any
𝐸. Then, it represents that if she is unaware of something about the unemployed
insurance program, she does not know everything about it.
This example represents Unawareness Leads to Ignorance. In this, the property
is interpreted as Alice is unaware of the true event related to the decision making,therefore she cannot be aware that she faces a decision. In other words, Alicecannot be aware that she faces the decision to apply for jobless insurance. A similar
situation often occurs in the real world. Hence, in this case, Unawareness Leads
to Ignorance is not a property of trivial unawareness.
However, this case has a crucial problem. In the standard information structure,both Necessitation and Monotonicity must hold. Hence, we cannot discuss
Monotonicity without Necessitation. In other words, we cannot consider
Unawareness Leads to Ignorance without Triviality. Can we not explore whether
nontrivial Unawareness Leads to Ignorance? To answer this question, we consider5

a nonstandard information structure in the next section.

4. Information structure without Necessitation
To consider Unawareness Leads to Ignorance without Triviality, let us firstdefine a nonstandard information function. The standard information function 𝑃
does not imply that the information set is empty. Every information set is anonempty set. In contrast, we allow some information sets to be empty. Let 𝑃L: Ω →2! be the agentʼs nonstandard information function. We do not assume whether
it is partitional or not. Some state 𝜔 ∈ Ω may satisfy 𝑃L (𝜔) = ∅. It is interpreted
as “the agent cannot access information (set) about 𝜔.” Let us reconsider Aliceʼsexample: suppose that the true state is 𝜔# , and 𝑃L (𝜔# ) = ∅. Then, the information
set is interpreted as “Alice cannot access information about her applying forjobless insurance.” Let 〈Ω, 𝑃L〉 be a nonstandard information structure.
M : 2! → 2! in 〈Ω, 𝑃L〉.
Next, let us define the nonstandard knowledge operator 𝐾
Given any event 𝐸, 𝐾(𝐸) is defined as follows:

M (𝐸 ) if 𝑃L (𝜔) ⊆ 𝐸 and 𝑃L(𝜔) ≠ ∅; and𝜔∈𝐾0M (𝐸 ) otherwise.
𝜔∉𝐾At 𝜔, the agent knows 𝐸 if not only the information set given 𝜔 is a subset of𝐸, but also the information set is not empty. That is, to know 𝐸, she must accessM are generalizations of 𝑃 andinformation about 𝜔. It is obvious that 𝑃L and 𝐾𝐾.
Here, we show the following properties.

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Theorem 2: In a nonstandard information structure 〈Ω, 𝑃L〉 possessing a

M, 𝐾M satisfies the following properties:
nonstandard knowledge operator 𝐾l

M (Ω) = Ω.
(Necessitation) for any 𝜔 ∈ Ω, 𝑃L(𝜔) ≠ ∅ if and only if 𝐾l

M (𝐸) ⊆ 𝐾
M (𝐹)(Monotonicity) 𝐸 ⊆ 𝐹 ⟹ 𝐾

M is equivalent to the standard
Proof: Suppose for any 𝜔 ∈ Ω, 𝑃L(𝜔) ≠ ∅. Then, 𝐾
knowledge operator 𝐾 . Hence, Necessitation holds. Next, suppose that there
M (𝐸) for any 𝐸 ⊆ Ω. That is, 𝜔 ∉exists 𝜔 ∈ Ω satisfying 𝑃L(𝜔) = ∅, then 𝜔 ∉ 𝐾M(Ω). Hence, 𝐾
M (Ω) ⊊ Ω.
𝐾M (𝐸). Then, by definition of 𝐾
M , 𝑃L (𝜔) ⊆Next, suppose that 𝐸 ⊆ 𝐹, given 𝜔 ∈ 𝐾
M (𝐹) , that is,
𝐸 and 𝑃L(𝜔) ≠ ∅ . Because 𝐸 ⊆ 𝐹 , 𝑃L(𝜔) ⊆ 𝐸 ⊆ 𝐹 . Hence, 𝜔 ∈ 𝐾
M(𝐸) ⊆ 𝐾
M (𝐹). ∎𝐾Interestingly, the nonstandard knowledge operatorM𝐾

may not satisfy

Necessitation. If there exists some information set that is empty, thenNecessitation does not hold, and vice versa. By contrast, Monotonicity holds.
M satisfying
Hence, it is obvious that given the unawareness operator based on 𝐾
Plausibility, KU Introspection, and AU Introspection, Unawareness Leads toIgnorance holds, whereas Triviality holds if and only if there is empty noinformation set.

Corollary 1:

In a nonstandard information structure 〈Ω, 𝑃L〉 possessing theM , let the nonstandard unawareness operator
nonstandard knowledge operator 𝐾
M : 2! → 2! satisfy Plausibility (𝑈M (𝐸) ⊆ ¬𝐾
M (𝐸) ∩ ¬𝐾M ¬𝐾M (𝐸)), KU Introspection
𝑈7

M𝑈M(𝐸) = ∅ ), and AU Introspection ( 𝑈M(𝐸) ⊆ 𝑈
M𝑈
M(𝐸) ). Then, 𝑈
M satisfies the(𝐾following:lM(𝐸) = ∅.
(Triviality) If 𝑃L (𝜔) ≠ ∅, then 𝑈lM(𝐸) ⊆ ¬𝐾
M (𝐹).
(Unawareness Leads to Ignorance) For any 𝐸, 𝐹 ⊆ Ω, 𝑈

M and 𝑈
M are based on theLet us reconsider Unawareness Leads to Ignorance. 𝐾
nonstandard information function, 𝑃L. It allows some information set to be empty,and given such an information set, the agent cannot know the relevant event. Here,M (𝐸), that is, 𝜔 ∈ ¬𝐾M (𝐸).
given 𝜔 ∈ Ω, suppose 𝑃L (𝜔) = ∅. Then, for any 𝐸, 𝜔 ∉ 𝐾This means that at 𝜔, if the agent cannot receive relevant information (set), thenM (𝐸), then 𝜔 ∈ ¬𝐾
M(𝐹) = ¬𝐾
M ¬𝐾
M (𝐸) isshe cannot know every event. Let 𝐹 = ¬𝐾
obvious. This means that, at 𝜔, the agent cannot know the ignorance. Therefore,M(𝐸) for any 𝐸. From the above, 𝜔 ∈ 𝑈M(𝐸) means that at 𝜔, because she𝜔∈𝑈cannot receive any relevant information (set), she cannot perceive or understandthat she faces some (nonstandard) information structure. Hence, she cannot
perceive every event in the nonstandard information structure. Then,
Unawareness Leads to Ignorance means that the agent who cannot perceive her(nonstandard) information structure cannot know everything.

5. Conclusion
This note characterizes Unawareness Leads to Ignorance by allowing an empty
information set. Dekel, Lipman, and Rustichini (1998) suggest that UnawarenessLeads to Ignorance is trivial in a standard information structure. That is true.
However, in a nonstandard information structure that allows the information set
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to be empty, Unawareness Leads to Ignorance means that because the agent
cannot perceive her information structure, the agent cannot know everything in
the information structure. Hence, the answer to the question, “Is UnawarenessLeads to Ignorance trivial?” is no in the case where some information set is empty.

Reference
Dekel, Eddie, Barton L. Lipman and Aldo Rustichini (1998), “Standard State-Space
Models Preclude Unawareness”, Econometrica, 66: 159-173.
Galanis, Spyros (2013), “Unawareness of Theorems”, Economic Theory, 52: 41-73.
Heifetz, Aviad, Martin Meier and Burkhard C. Schipper (2006), “InteractiveUnawareness”, Journal of Economic Theory, 130: 78-94.

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