Agreement Theorems from thePerspective of
Dynamic-Epistemic Logic
Olivier Roy & Cedric Degremont
November 10, 2008
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 1
Introduction and preliminaries
Overview
1. Introduction to Agreements Theorems (AT)
2. Models of knowledge and beliefs, common priors and commonknowledge
3. Three variations on the result: static, kinematic and dynamic
Highlights:I The received view:
• ATs undermine the role of private information.
I The DEL point of view:
• ATs show the importance of higher-order information.• ATs show how “static” conditioning is different from “real”
belief dynamics.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 2
Introduction and preliminaries
Overview
1. Introduction to Agreements Theorems (AT)
2. Models of knowledge and beliefs, common priors and commonknowledge
3. Three variations on the result: static, kinematic and dynamic
Highlights:I The received view:
• ATs undermine the role of private information.
I The DEL point of view:
• ATs show the importance of higher-order information.• ATs show how “static” conditioning is different from “real”
belief dynamics.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 2
Introduction and preliminaries
Agreement Theorems in a nutshell
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 3
Introduction and preliminaries
Agreement Theorems in a nutshell
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 3
Introduction and preliminaries
Agreement Theorems in a nutshell
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 3
Introduction and preliminaries
A short result goes a long way
I Original result: [Aumann, 1976].
I No trade theorems: [Milgrom and Stokey, 1982].
I “Dynamic” versions: [Geanakoplos and Polemarchakis, 1982]
I Qualitative generalizations: [Cave, 1983], [Bacharach, 1985].
I Network structure: [Parikh and Krasucki, 1990]
I Good survey: [Bonanno and Nehring, 1997].
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 4
Introduction and preliminaries
A short result goes a long way
I Original result: [Aumann, 1976].
I No trade theorems: [Milgrom and Stokey, 1982].
I “Dynamic” versions: [Geanakoplos and Polemarchakis, 1982]
I Qualitative generalizations: [Cave, 1983], [Bacharach, 1985].
I Network structure: [Parikh and Krasucki, 1990]
I Good survey: [Bonanno and Nehring, 1997].
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 4
Introduction and preliminaries
Conclusions, the received view.
TheoremIf two people have the same priors, and their posteriors for anevent A are common knowledge, then these posterior are the same.
I How important is private information? Not quite...
I How strong is the common knowledge condition? Very...
I How plausible is the common prior assumption? Debated...
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 5
Introduction and preliminaries
Conclusions, the received view.
TheoremIf two people have the same priors, and their posteriors for anevent A are common knowledge, then these posterior are the same.
I How important is private information? Not quite...
I How strong is the common knowledge condition? Very...
I How plausible is the common prior assumption? Debated...
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 5
The point of view of Dynamic-Epistemic Logic
Conclusions, the point of view of DEL.
TheoremIf two people have the same priors, and their posteriors for anevent A are common knowledge, then these posterior are the same.
I The key is higher-order information.
I One should distinguish information kinematics vs informationdynamics, belief conditioning vs belief update.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 6
The point of view of Dynamic-Epistemic Logic
Definition (Epistemic-Doxastic Model)
An epistemic-doxastic model M is a tuple 〈W , I , {≤i ,∼i}i∈I 〉 suchthat:
I W is a finite set of states.
I I is a finite set of agents.I ≤i is a reflexive, transitive and connected plausibility ordering
on W .
• There is common priors iff ≤i = ≤j for all i and j in I .
I ∼i is an epistemic accessibility equivalence relation. We write[w ]i for {w ′ : w ∼i w ′}.
See: [Board, 2004, Baltag and Smets, 2008, van Benthem, ]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 7
The point of view of Dynamic-Epistemic Logic
Definition (Epistemic-Doxastic Model)
An epistemic-doxastic model M is a tuple 〈W , I , {≤i ,∼i}i∈I 〉 suchthat:
I W is a finite set of states.
I I is a finite set of agents.I ≤i is a reflexive, transitive and connected plausibility ordering
on W .
• There is common priors iff ≤i = ≤j for all i and j in I .
I ∼i is an epistemic accessibility equivalence relation. We write[w ]i for {w ′ : w ∼i w ′}.
See: [Board, 2004, Baltag and Smets, 2008, van Benthem, ]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 7
The point of view of Dynamic-Epistemic Logic
Definition (Epistemic-Doxastic Model)
An epistemic-doxastic model M is a tuple 〈W , I , {≤i ,∼i}i∈I 〉 suchthat:
I W is a finite set of states.
I I is a finite set of agents.
I ≤i is a reflexive, transitive and connected plausibility orderingon W .
• There is common priors iff ≤i = ≤j for all i and j in I .
I ∼i is an epistemic accessibility equivalence relation. We write[w ]i for {w ′ : w ∼i w ′}.
See: [Board, 2004, Baltag and Smets, 2008, van Benthem, ]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 7
The point of view of Dynamic-Epistemic Logic
Definition (Epistemic-Doxastic Model)
An epistemic-doxastic model M is a tuple 〈W , I , {≤i ,∼i}i∈I 〉 suchthat:
I W is a finite set of states.
I I is a finite set of agents.I ≤i is a reflexive, transitive and connected plausibility ordering
on W .
• There is common priors iff ≤i = ≤j for all i and j in I .
I ∼i is an epistemic accessibility equivalence relation. We write[w ]i for {w ′ : w ∼i w ′}.
See: [Board, 2004, Baltag and Smets, 2008, van Benthem, ]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 7
The point of view of Dynamic-Epistemic Logic
Definition (Epistemic-Doxastic Model)
An epistemic-doxastic model M is a tuple 〈W , I , {≤i ,∼i}i∈I 〉 suchthat:
I W is a finite set of states.
I I is a finite set of agents.I ≤i is a reflexive, transitive and connected plausibility ordering
on W .
• There is common priors iff ≤i = ≤j for all i and j in I .
I ∼i is an epistemic accessibility equivalence relation. We write[w ]i for {w ′ : w ∼i w ′}.
See: [Board, 2004, Baltag and Smets, 2008, van Benthem, ]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 7
The point of view of Dynamic-Epistemic Logic
Key notions:
I Knowledge: w |= Kiϕ iff w ′ |= ϕ for all w ′ ∼i w .
I Everybody knows: w |= EIϕ iff w ′ |= K1ϕ ∧ ... ∧ Knϕ for1, ...n ∈ I .
I Common knowledge: w |= CKIϕ iff w ′ |= EIϕ andw ′ |= EIEIϕ and...
I Beliefs: w |= Bψi ϕ iff w ′ |= ϕ for all w ′ in max≤i ([w ]i ∩ ||ψ||).
We write Biϕ for B>i ϕ.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 8
The point of view of Dynamic-Epistemic Logic
Key notions:
I Knowledge: w |= Kiϕ iff w ′ |= ϕ for all w ′ ∼i w .
I Everybody knows: w |= EIϕ iff w ′ |= K1ϕ ∧ ... ∧ Knϕ for1, ...n ∈ I .
I Common knowledge: w |= CKIϕ iff w ′ |= EIϕ andw ′ |= EIEIϕ and...
I Beliefs: w |= Bψi ϕ iff w ′ |= ϕ for all w ′ in max≤i ([w ]i ∩ ||ψ||).
We write Biϕ for B>i ϕ.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 8
The point of view of Dynamic-Epistemic Logic
Key notions:
I Knowledge: w |= Kiϕ iff w ′ |= ϕ for all w ′ ∼i w .
I Everybody knows: w |= EIϕ iff w ′ |= K1ϕ ∧ ... ∧ Knϕ for1, ...n ∈ I .
I Common knowledge: w |= CKIϕ iff w ′ |= EIϕ andw ′ |= EIEIϕ and...
I Beliefs: w |= Bψi ϕ iff w ′ |= ϕ for all w ′ in max≤i ([w ]i ∩ ||ψ||).
We write Biϕ for B>i ϕ.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 8
The point of view of Dynamic-Epistemic Logic
Key notions:
I Knowledge: w |= Kiϕ iff w ′ |= ϕ for all w ′ ∼i w .
I Everybody knows: w |= EIϕ iff w ′ |= K1ϕ ∧ ... ∧ Knϕ for1, ...n ∈ I .
I Common knowledge: w |= CKIϕ iff w ′ |= EIϕ andw ′ |= EIEIϕ and...
I Beliefs: w |= Bψi ϕ iff w ′ |= ϕ for all w ′ in max≤i ([w ]i ∩ ||ψ||).
We write Biϕ for B>i ϕ.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 8
The point of view of Dynamic-Epistemic Logic
Theorem (Static agreement)
For any epistemic-doxastic model M with common priors, for all wwe have that
w 6|= CKI (Bi (E ) ∧ ¬Bj(E ))
where E ⊆ W.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 9
The point of view of Dynamic-Epistemic Logic
The key property: Sure-thing principle
If, first, you would believe, conditional on the fact that itis cloudy, that it will rain and,
second, you would believe,conditional on the fact that it is not cloudy, that it willrain, then you unconditionally believe that it will rain.
[Savage, 1954, Bacharach, 1985]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 10
The point of view of Dynamic-Epistemic Logic
The key property: Sure-thing principle
If, first, you would believe, conditional on the fact that itis cloudy, that it will rain and, second, you would believe,conditional on the fact that it is not cloudy, that it willrain, then
you unconditionally believe that it will rain.
[Savage, 1954, Bacharach, 1985]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 10
The point of view of Dynamic-Epistemic Logic
The key property: Sure-thing principle
If, first, you would believe, conditional on the fact that itis cloudy, that it will rain and, second, you would believe,conditional on the fact that it is not cloudy, that it willrain, then you unconditionally believe that it will rain.
[Savage, 1954, Bacharach, 1985]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 10
The point of view of Dynamic-Epistemic Logic
The key property: Sure-thing principle
W
W
Agent 1’s IP
Agent 2’s IP
P1
Q1 Q2 Q3
P2
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 11
The point of view of Dynamic-Epistemic Logic
The key property: Sure-thing principle
W
W
Agent 1’s IP
Agent 2’s IP
not B1(E/P1)
B2(E/Q1) B2(E/Q2) B3(E/Q3)
not B1(E/P2)
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 11
The point of view of Dynamic-Epistemic Logic
The key property: Sure-thing principle
W
W
B2(E)
The CK cell
By the Sure-Thing Principle
The CK cell
not B1(E)
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 11
The point of view of Dynamic-Epistemic Logic
The key property: Sure-thing principle
W
not B1(E)
W
B2(E)
Hence no common priors
=
The CK cell
The CK cell
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 11
The point of view of Dynamic-Epistemic Logic
Lesson:
I They might not have the same (first-order) information, but...
what is common knowledge is precisely where their(higher-order) information coincide.
The cornerstone is higher-order information.
Question:I How can CK obtain with respect to each others’ beliefs?
• “Dialogues” or repeated announcements.[Geanakoplos and Polemarchakis, 1982, Bacharach, 1985]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 12
The point of view of Dynamic-Epistemic Logic
Lesson:
I They might not have the same (first-order) information, but...what is common knowledge is precisely where their(higher-order) information coincide.
The cornerstone is higher-order information.
Question:I How can CK obtain with respect to each others’ beliefs?
• “Dialogues” or repeated announcements.[Geanakoplos and Polemarchakis, 1982, Bacharach, 1985]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 12
The point of view of Dynamic-Epistemic Logic
Lesson:
I They might not have the same (first-order) information, but...what is common knowledge is precisely where their(higher-order) information coincide.
The cornerstone is higher-order information.
Question:I How can CK obtain with respect to each others’ beliefs?
• “Dialogues” or repeated announcements.[Geanakoplos and Polemarchakis, 1982, Bacharach, 1985]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 12
The point of view of Dynamic-Epistemic Logic
Lesson:
I They might not have the same (first-order) information, but...what is common knowledge is precisely where their(higher-order) information coincide.
The cornerstone is higher-order information.
Question:I How can CK obtain with respect to each others’ beliefs?
• “Dialogues” or repeated announcements.[Geanakoplos and Polemarchakis, 1982, Bacharach, 1985]
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 12
The point of view of Dynamic-Epistemic Logic
Towards kinematic agreement
DefinitionA kinematic dialogue about A is an epistemic-doxastic modelM = 〈W , I , {≤i , {∼n,i}n∈N}i∈I 〉 with
, for all i ∈ I , the sequenceof epistemic accessibility relation {∼n,i}n∈N is inductively definedas follows (for 2 agents).
I ∼0,i is a given epistemic accessibility relation.
I for all w ∈ W :
[w ]n+1,i = [w ]n,i ∩
{Bn(A) if w |= Bn,j(A)
¬Bn,j(A) otherwise.
with Bn,j(A) = {w ′ : max≤j [w′]n,j ⊆ A}.
Intuition: Bn+1,iϕ⇔ BBn,jϕn,i ϕ.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 13
The point of view of Dynamic-Epistemic Logic
Towards kinematic agreement
DefinitionA kinematic dialogue about A is an epistemic-doxastic modelM = 〈W , I , {≤i , {∼n,i}n∈N}i∈I 〉 with , for all i ∈ I , the sequenceof epistemic accessibility relation {∼n,i}n∈N is inductively definedas follows (for 2 agents).
I ∼0,i is a given epistemic accessibility relation.
I for all w ∈ W :
[w ]n+1,i = [w ]n,i ∩
{Bn(A) if w |= Bn,j(A)
¬Bn,j(A) otherwise.
with Bn,j(A) = {w ′ : max≤j [w′]n,j ⊆ A}.
Intuition: Bn+1,iϕ⇔ BBn,jϕn,i ϕ.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 13
The point of view of Dynamic-Epistemic Logic
Towards kinematic agreement
DefinitionA kinematic dialogue about A is an epistemic-doxastic modelM = 〈W , I , {≤i , {∼n,i}n∈N}i∈I 〉 with , for all i ∈ I , the sequenceof epistemic accessibility relation {∼n,i}n∈N is inductively definedas follows (for 2 agents).
I ∼0,i is a given epistemic accessibility relation.
I for all w ∈ W :
[w ]n+1,i = [w ]n,i ∩
{Bn(A) if w |= Bn,j(A)
¬Bn,j(A) otherwise.
with Bn,j(A) = {w ′ : max≤j [w′]n,j ⊆ A}.
Intuition: Bn+1,iϕ⇔ BBn,jϕn,i ϕ.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 13
The point of view of Dynamic-Epistemic Logic
Towards kinematic agreement
DefinitionA kinematic dialogue about A is an epistemic-doxastic modelM = 〈W , I , {≤i , {∼n,i}n∈N}i∈I 〉 with , for all i ∈ I , the sequenceof epistemic accessibility relation {∼n,i}n∈N is inductively definedas follows (for 2 agents).
I ∼0,i is a given epistemic accessibility relation.
I for all w ∈ W :
[w ]n+1,i = [w ]n,i ∩
{Bn(A) if w |= Bn,j(A)
¬Bn,j(A) otherwise.
with Bn,j(A) = {w ′ : max≤j [w′]n,j ⊆ A}.
Intuition: Bn+1,iϕ⇔ BBn,jϕn,i ϕ.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 13
The point of view of Dynamic-Epistemic Logic
Lemma (Fixed-point)
Every kinematic dialogue about A has a fixed-point, i.e. there is an∗ such that
[w ]n∗,i = [w ]n∗+1,i
for all w and i.
Lemma (Common knowledge)
The posteriors beliefs at the fixed-point of a kinematic dialogue arecommon knowledge.
Theorem (Kinematic agreement)
For any kinematic dialogue about A, if there is common priors thenat the fixed-point either all agents believe that A or they all don’tbelieve that A.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 14
The point of view of Dynamic-Epistemic Logic
Lemma (Fixed-point)
Every kinematic dialogue about A has a fixed-point, i.e. there is an∗ such that
[w ]n∗,i = [w ]n∗+1,i
for all w and i.
Lemma (Common knowledge)
The posteriors beliefs at the fixed-point of a kinematic dialogue arecommon knowledge.
Theorem (Kinematic agreement)
For any kinematic dialogue about A, if there is common priors thenat the fixed-point either all agents believe that A or they all don’tbelieve that A.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 14
The point of view of Dynamic-Epistemic Logic
Lemma (Fixed-point)
Every kinematic dialogue about A has a fixed-point, i.e. there is an∗ such that
[w ]n∗,i = [w ]n∗+1,i
for all w and i.
Lemma (Common knowledge)
The posteriors beliefs at the fixed-point of a kinematic dialogue arecommon knowledge.
Theorem (Kinematic agreement)
For any kinematic dialogue about A, if there is common priors thenat the fixed-point either all agents believe that A or they all don’tbelieve that A.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 14
The point of view of Dynamic-Epistemic Logic
Lesson:
I Common knowledge arise from “dialogues”...
Warnings from DEL:
I This is only a kinematic (i.e. conditioning) dialogue: the truthof A is fixed during the process.
I In general, this is not the case. A might be about the agents’information.
I Another way to look at it:
kinematic agreement = “virtual” agreement
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 15
The point of view of Dynamic-Epistemic Logic
Lesson:
I Common knowledge arise from “dialogues”...
Warnings from DEL:
I This is only a kinematic (i.e. conditioning) dialogue: the truthof A is fixed during the process.
I In general, this is not the case. A might be about the agents’information.
I Another way to look at it:
kinematic agreement = “virtual” agreement
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 15
The point of view of Dynamic-Epistemic Logic
Lesson:
I Common knowledge arise from “dialogues”...
Warnings from DEL:
I This is only a kinematic (i.e. conditioning) dialogue: the truthof A is fixed during the process.
I In general, this is not the case. A might be about the agents’information.
I Another way to look at it:
kinematic agreement = “virtual” agreement
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 15
The point of view of Dynamic-Epistemic Logic
Lesson:
I Common knowledge arise from “dialogues”...
Warnings from DEL:
I This is only a kinematic (i.e. conditioning) dialogue: the truthof A is fixed during the process.
I In general, this is not the case. A might be about the agents’information.
I Another way to look at it:
kinematic agreement = “virtual” agreement
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 15
The point of view of Dynamic-Epistemic Logic
Towards dynamic agreement
DefinitionA dynamic dialogue about A is sequence of epistemic-doxasticpointed models {(Mn,w)}n∈N such that:
I M0 is a given epistemic-doxastic model.I Mn+1 = 〈Wn+1, I ,≤n+1,i ,∼n+1,i 〉 with
• Wn+1 = {w ′ ∈ Wn : w ′ |= Bn(An)} with:I An = {w ′ ∈ Wn : w ′ |= A}I Bn(An) is Bn,i (An)∧Bn,j(An) if w |= Bn,i (An)∧Bn,j(An), etc...
• ≤n+1,i ∼n+1,i are the restrictions of ≤n,i and ∼n,i to Wn+1.
Intuition: Bn+1,iϕ⇔ [Bnϕ!]Bn,iϕ
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 16
The point of view of Dynamic-Epistemic Logic
Lemma (Fixed-point)
Every dynamic dialogue about A has a fixed-point, i.e. there is an∗ such that:
Mn∗ = Mn∗+1
Lemma (Common knowledge)
The posteriors beliefs at the fixed-point of a dynamic dialogue arecommon knowledge.
Theorem (Dynamic agreement)
For any dynamic dialogue about A, if there is common priors thenat the fixed-point n∗ either all agents believe that An∗ or they alldon’t believe that An∗ .
But...
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 17
The point of view of Dynamic-Epistemic Logic
Lemma (Fixed-point)
Every dynamic dialogue about A has a fixed-point, i.e. there is an∗ such that:
Mn∗ = Mn∗+1
Lemma (Common knowledge)
The posteriors beliefs at the fixed-point of a dynamic dialogue arecommon knowledge.
Theorem (Dynamic agreement)
For any dynamic dialogue about A, if there is common priors thenat the fixed-point n∗ either all agents believe that An∗ or they alldon’t believe that An∗ .
But...
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 17
The point of view of Dynamic-Epistemic Logic
Kinematic agreements can be different from dynamic agreements.
W
2p not p
1 1
w1 w2
I Let A = p ∧ ¬B2p
I At the fixed points for the kinematic and the dynamicdialogue about A, we have that [w1]n∗,1 = [w1]n∗,2 = {w1}
I Kinematic beliefs: w |= Bn∗,i (p ∧ ¬B0,2p), for i = 1, 2 .
I Dynamic beliefs: w |= ¬Bn∗,i (p ∧ ¬Bn∗,2p), for i = 1, 2.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 18
The point of view of Dynamic-Epistemic Logic
Kinematic agreements can be different from dynamic agreements.
W
2p not p
1 1
w1 w2
I Let A = p ∧ ¬B2p
I At the fixed points for the kinematic and the dynamicdialogue about A, we have that [w1]n∗,1 = [w1]n∗,2 = {w1}
I Kinematic beliefs: w |= Bn∗,i (p ∧ ¬B0,2p), for i = 1, 2 .
I Dynamic beliefs: w |= ¬Bn∗,i (p ∧ ¬Bn∗,2p), for i = 1, 2.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 18
The point of view of Dynamic-Epistemic Logic
Kinematic agreements can be different from dynamic agreements.
W
2p not p
1 1
w1 w2
I Let A = p ∧ ¬B2p
I At the fixed points for the kinematic and the dynamicdialogue about A, we have that [w1]n∗,1 = [w1]n∗,2 = {w1}
I Kinematic beliefs: w |= Bn∗,i (p ∧ ¬B0,2p), for i = 1, 2 .
I Dynamic beliefs: w |= ¬Bn∗,i (p ∧ ¬Bn∗,2p), for i = 1, 2.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 18
The point of view of Dynamic-Epistemic Logic
Kinematic agreements can be different from dynamic agreements.
W
2p not p
1 1
w1 w2
I Let A = p ∧ ¬B2p
I At the fixed points for the kinematic and the dynamicdialogue about A, we have that [w1]n∗,1 = [w1]n∗,2 = {w1}
I Kinematic beliefs: w |= Bn∗,i (p ∧ ¬B0,2p), for i = 1, 2 .
I Dynamic beliefs: w |= ¬Bn∗,i (p ∧ ¬Bn∗,2p), for i = 1, 2.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 18
The point of view of Dynamic-Epistemic Logic
Kinematic agreements can be different from dynamic agreements.
W
2p not p
1 1
w1 w2
I Let A = p ∧ ¬B2p
I At the fixed points for the kinematic and the dynamicdialogue about A, we have that [w1]n∗,1 = [w1]n∗,2 = {w1}
I Kinematic beliefs: w |= Bn∗,i (p ∧ ¬B0,2p), for i = 1, 2 .
I Dynamic beliefs: w |= ¬Bn∗,i (p ∧ ¬Bn∗,2p), for i = 1, 2.
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 18
Conclusion
I Agreements theorems:
• Undermine the role of private information?
• A better way to look at it (“DEL methodology”):I Rest on higher-order information and its role in interactive
reasoning.I Highlight the difference between belief kinematics and belief
dynamics.
I (Hopefully not so distant) future work :
• General (countable) case?• Announcements of reasons and not only of opinions?• Relaxing the common prior assumption? Agreements on
everything?
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 19
Conclusion
I Agreements theorems:
• Undermine the role of private information?• A better way to look at it (“DEL methodology”):
I Rest on higher-order information and its role in interactivereasoning.
I Highlight the difference between belief kinematics and beliefdynamics.
I (Hopefully not so distant) future work :
• General (countable) case?• Announcements of reasons and not only of opinions?• Relaxing the common prior assumption? Agreements on
everything?
Olivier Roy & Cedric Degremont: Agreement Theorems & Dynamic-Epistemic Logic, 19
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