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Updating Coalition Structures
Updating Coalition Structures:
some issues and some results
Paolo Turrini
together with Jan Broersen, Rosja Mastop and John Jules Meyer
Utrecht University, The Netherlands
Logics for Dynamics of Information and PreferencesAmsterdam; June 8th, 2009
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Outline
1 Reasoning Patterns in Strategic Interaction
Strategic Reasoning
Representing Strategic Ability
Reasoning about Strategies
2 Properties
Intuitions behind the Models
The Language
A Complete Reduction
Choices as Announcements
3 Discussion
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
As you well know...
HHHHHHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
Prisoner Dilemma is an interactive situation in which the
advantage of cooperation is overruled by the individual
incentive to defect.
The rationality assumption underlying players' reasoning in a
Prisoner Dilemma warrants each player to select a move
reasoning on the opponents' possible moves.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
As you well know...
HHHHHHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
Prisoner Dilemma is an interactive situation in which the
advantage of cooperation is overruled by the individual
incentive to defect.
The rationality assumption underlying players' reasoning in a
Prisoner Dilemma warrants each player to select a move
reasoning on the opponents' possible moves.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
I play it again
HHHHHHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
If we focus on player i , we can observe this reasoning pattern:
If j plays D,
I had better play D.
If j plays C , I had better play D.
In conclusion, I had better play D.
A logic aiming at capturing strategic reasoning should make it
possible to draw this conclusion.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
I play it again
HHHHHHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
If we focus on player i , we can observe this reasoning pattern:
If j plays D, I had better play D.
If j plays C , I had better play D.
In conclusion, I had better play D.
A logic aiming at capturing strategic reasoning should make it
possible to draw this conclusion.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
I play it again
HHHHHHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
If we focus on player i , we can observe this reasoning pattern:
If j plays D, I had better play D.
If j plays C , I had better play D.
In conclusion, I had better play D.
A logic aiming at capturing strategic reasoning should make it
possible to draw this conclusion.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
I play it again
HHHHHHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
If we focus on player i , we can observe this reasoning pattern:
If j plays D, I had better play D.
If j plays C , I had better play D.
In conclusion, I had better play D.
A logic aiming at capturing strategic reasoning should make it
possible to draw this conclusion.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
I play it again
HHHHHHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
If we focus on player i , we can observe this reasoning pattern:
If j plays D, I had better play D.
If j plays C , I had better play D.
In conclusion, I had better play D.
A logic aiming at capturing strategic reasoning should make it
possible to draw this conclusion.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
Ipse dixit
Much of game theory is about the question whether
strategic equilibria exist.
But there are hardly any explicit
languages for de�ning, comparing, or combining strategies
as such the way we have them for actions and plans,
maybe the closest intuitive analogue to strategies. True,
there are many current logics for describing game
structure but these tend to have existential quanti�ers
saying that players have a strategy for achieving some
purpose, while descriptions of these strategies themselves
are not part of the logical language.
(Johan van Benthem, In Praise of Strategies, August 2007)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
Ipse dixit
Much of game theory is about the question whether
strategic equilibria exist. But there are hardly any explicit
languages for de�ning, comparing, or combining strategies
as such the way we have them for actions and plans,
maybe the closest intuitive analogue to strategies.
True,
there are many current logics for describing game
structure but these tend to have existential quanti�ers
saying that players have a strategy for achieving some
purpose, while descriptions of these strategies themselves
are not part of the logical language.
(Johan van Benthem, In Praise of Strategies, August 2007)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
Ipse dixit
Much of game theory is about the question whether
strategic equilibria exist. But there are hardly any explicit
languages for de�ning, comparing, or combining strategies
as such the way we have them for actions and plans,
maybe the closest intuitive analogue to strategies. True,
there are many current logics for describing game
structure but these tend to have existential quanti�ers
saying that players have a strategy for achieving some
purpose, while descriptions of these strategies themselves
are not part of the logical language.
(Johan van Benthem, In Praise of Strategies, August 2007)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Strategic Reasoning
Ipse dixit
Much of game theory is about the question whether
strategic equilibria exist. But there are hardly any explicit
languages for de�ning, comparing, or combining strategies
as such the way we have them for actions and plans,
maybe the closest intuitive analogue to strategies. True,
there are many current logics for describing game
structure but these tend to have existential quanti�ers
saying that players have a strategy for achieving some
purpose, while descriptions of these strategies themselves
are not part of the logical language.
(Johan van Benthem, In Praise of Strategies, August 2007)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
Coalition Logic
φ ::= p|¬φ|φ ∨ φ|[C ]φ
The reading of the modality is:
�Coalition C can force the game to end up in a world satisfying
φ�
Marc Pauly,
A Logic for Social Software.
PhD thesis, 2001.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
E�ectivity in games
De�nition (Dynamic E�ectivity Function)
Given a �nite set of agents Agt and a set of states W , a dynamic
e�ectivity function is a function
E : W → (2Agt → 22W
).
{(4, 0)} 6∈ E (w)({i}){(4, 0)} ∈ E (w)(Agt)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
E�ectivity in games
De�nition (Dynamic E�ectivity Function)
Given a �nite set of agents Agt and a set of states W , a dynamic
e�ectivity function is a function
E : W → (2Agt → 22W
).
{(4, 0)} 6∈ E (w)({i})
{(4, 0)} ∈ E (w)(Agt)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
E�ectivity in games
De�nition (Dynamic E�ectivity Function)
Given a �nite set of agents Agt and a set of states W , a dynamic
e�ectivity function is a function
E : W → (2Agt → 22W
).
{(4, 0)} 6∈ E (w)({i}){(4, 0)} ∈ E (w)(Agt)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
E is outcome monotonic
X
Y
X ⊆ Y and X ∈ E (C ) implies Y ∈ E (C )
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
E is outcome monotonic
X
Y
X ⊆ Y and X ∈ E (C ) implies Y ∈ E (C )
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
E is outcome monotonic
X
Y
X ⊆ Y and X ∈ E (C ) implies Y ∈ E (C )
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
Coalition Logic
M = (W ,E ,V )
M,w |= [C ]φ⇔ [[φ]]M ∈ E (w)(C )
[[φ]]M = {w ∈W |M,w |= φ}
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
Coalition Logic
M = (W ,E ,V )
M,w |= [C ]φ⇔ [[φ]]M ∈ E (w)(C )
[[φ]]M = {w ∈W |M,w |= φ}
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
Coalition Logic
M = (W ,E ,V )
M,w |= [C ]φ⇔ [[φ]]M ∈ E (w)(C )
[[φ]]M = {w ∈W |M,w |= φ}
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
What we can say in Coalition Logic
HHHHHHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
[{i}]φ⇔ φ holds whatever j does
We cannot express what holds in particular given that j
defects.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Representing Strategic Ability
What we can say in Coalition Logic
HHHHHHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
[{i}]φ⇔ φ holds whatever j does
We cannot express what holds in particular given that j
defects.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
The unsung heroes of Game Theory
Therefore, I consider strategies 'the unsung heroes of
game theory' - and I want to show how the right kind of
logic can bring them to the fore.
One guide-line of
adequacy for doing so, in the fastgrowing jungle of 'game
logics', is the following: we would like to explicitly
represent the elementary reasoning about strategies
underlying many basic game-theoretic results. Or in more
general terms, we want to explicitly represent agents
reasoning about their plans.
(Johan van Benthem, In Praise of Strategies, August 2007)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
The unsung heroes of Game Theory
Therefore, I consider strategies 'the unsung heroes of
game theory' - and I want to show how the right kind of
logic can bring them to the fore. One guide-line of
adequacy for doing so, in the fastgrowing jungle of 'game
logics', is the following:
we would like to explicitly
represent the elementary reasoning about strategies
underlying many basic game-theoretic results. Or in more
general terms, we want to explicitly represent agents
reasoning about their plans.
(Johan van Benthem, In Praise of Strategies, August 2007)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
The unsung heroes of Game Theory
Therefore, I consider strategies 'the unsung heroes of
game theory' - and I want to show how the right kind of
logic can bring them to the fore. One guide-line of
adequacy for doing so, in the fastgrowing jungle of 'game
logics', is the following: we would like to explicitly
represent the elementary reasoning about strategies
underlying many basic game-theoretic results.
Or in more
general terms, we want to explicitly represent agents
reasoning about their plans.
(Johan van Benthem, In Praise of Strategies, August 2007)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
The unsung heroes of Game Theory
Therefore, I consider strategies 'the unsung heroes of
game theory' - and I want to show how the right kind of
logic can bring them to the fore. One guide-line of
adequacy for doing so, in the fastgrowing jungle of 'game
logics', is the following: we would like to explicitly
represent the elementary reasoning about strategies
underlying many basic game-theoretic results. Or in more
general terms, we want to explicitly represent agents
reasoning about their plans.
(Johan van Benthem, In Praise of Strategies, August 2007)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
The unsung heroes of Game Theory
Therefore, I consider strategies 'the unsung heroes of
game theory' - and I want to show how the right kind of
logic can bring them to the fore. One guide-line of
adequacy for doing so, in the fastgrowing jungle of 'game
logics', is the following: we would like to explicitly
represent the elementary reasoning about strategies
underlying many basic game-theoretic results. Or in more
general terms, we want to explicitly represent agents
reasoning about their plans.
(Johan van Benthem, In Praise of Strategies, August 2007)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
The Subgame Operator
[C ↓ ψ]φ
�Given that coalition C chooses ψ, φ holds�
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
HHHH
HHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
The aim is to be able to express the reasoning patterns in the
Prisoner Dilemma, as well as many others, by means of the
new operator.
The idea is that a strategy 'restricts' the game and reasoning
on strategies means reasoning on restricted games.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
HHHH
HHi
jC D
C (3, 3) (0, 4)
D (4, 0) (1, 1)
The aim is to be able to express the reasoning patterns in the
Prisoner Dilemma, as well as many others, by means of the
new operator.
The idea is that a strategy 'restricts' the game and reasoning
on strategies means reasoning on restricted games.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
HHHH
HHi
jC
C (3, 3)
D (4, 0)
The aim is to be able to express the reasoning patterns in the
Prisoner Dilemma, as well as many others, by means of the
new operator.
The idea is that a strategy 'restricts' the game and reasoning
on strategies means reasoning on restricted games.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
HHHH
HHi
jD
C (0, 4)
D (1, 1)
The aim is to be able to express the reasoning patterns in the
Prisoner Dilemma, as well as many others, by means of the
new operator.
The idea is that a strategy 'restricts' the game and reasoning
on strategies means reasoning on restricted games.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
To formally capture our intuitions we need to give formal
semantics to the subgame operator:
M = (W ,E ,V )
The properties that are assumed, for all w ∈W ,
1 outcome monotonicity: if X ∈ E (w)(C ) and X ⊆ Y , then
Y ∈ E (w)(C );
2 regularity: if X ∈ E (w)(C ), then X 6∈ E (w)(C );
3 closed-worldness: E (w)(∅) = {W }
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
To formally capture our intuitions we need to give formal
semantics to the subgame operator:
M = (W ,E ,V )
The properties that are assumed, for all w ∈W ,
1 outcome monotonicity: if X ∈ E (w)(C ) and X ⊆ Y , then
Y ∈ E (w)(C );
2 regularity: if X ∈ E (w)(C ), then X 6∈ E (w)(C );
3 closed-worldness: E (w)(∅) = {W }
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
To formally capture our intuitions we need to give formal
semantics to the subgame operator:
M = (W ,E ,V )
The properties that are assumed, for all w ∈W ,
1 outcome monotonicity: if X ∈ E (w)(C ) and X ⊆ Y , then
Y ∈ E (w)(C );
2 regularity: if X ∈ E (w)(C ), then X 6∈ E (w)(C );
3 closed-worldness: E (w)(∅) = {W }
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
To formally capture our intuitions we need to give formal
semantics to the subgame operator:
M = (W ,E ,V )
The properties that are assumed, for all w ∈W ,
1 outcome monotonicity: if X ∈ E (w)(C ) and X ⊆ Y , then
Y ∈ E (w)(C );
2 regularity: if X ∈ E (w)(C ), then X 6∈ E (w)(C );
3 closed-worldness: E (w)(∅) = {W }
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
To formally capture our intuitions we need to give formal
semantics to the subgame operator:
M = (W ,E ,V )
The properties that are assumed, for all w ∈W ,
1 outcome monotonicity: if X ∈ E (w)(C ) and X ⊆ Y , then
Y ∈ E (w)(C );
2 regularity: if X ∈ E (w)(C ), then X 6∈ E (w)(C );
3 closed-worldness: E (w)(∅) = {W }
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
M,w |= [C ↓ ψ]φ⇔ ψM ∈ E (w)(C ) implies M ↓(C ,ψM ,w),w |= φ
The updated models are de�ned as follows:
M ↓(C ,ψM ,w).= 〈W ,E ↓(C ,ψM ,w),V 〉
The only object that actually changes is the coalitional relation.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
M,w |= [C ↓ ψ]φ⇔ ψM ∈ E (w)(C ) implies M ↓(C ,ψM ,w),w |= φ
The updated models are de�ned as follows:
M ↓(C ,ψM ,w).= 〈W ,E ↓(C ,ψM ,w),V 〉
The only object that actually changes is the coalitional relation.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
M,w |= [C ↓ ψ]φ⇔ ψM ∈ E (w)(C ) implies M ↓(C ,ψM ,w),w |= φ
The updated models are de�ned as follows:
M ↓(C ,ψM ,w).= 〈W ,E ↓(C ,ψM ,w),V 〉
The only object that actually changes is the coalitional relation.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
E ↓(C ,ψM ,w) (w)(D).= ({ψM})sup for D ∩ C 6= ∅
where for sets of sets X ,P,
(X )sup = {X ⊆W | there is Y ∈ X and Y ⊆ X ⊆W }
.
X u P = {ξ ∩ ψ|ξ ∈ X and ψ ∈ P}
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
E ↓(C ,ψM ,w) (w)(D).= ({ψM})sup for D ∩ C 6= ∅
where for sets of sets X ,P,
(X )sup = {X ⊆W | there is Y ∈ X and Y ⊆ X ⊆W }
.
X u P = {ξ ∩ ψ|ξ ∈ X and ψ ∈ P}
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
E ↓(C ,ψM ,w) (w)(D).= (E (w)(D) u ψM)sup for D ⊆ C and D 6= ∅
where for sets of sets X ,P,
(X )sup = {X ⊆W | there is Y ∈ X and Y ⊆ X ⊆W }
.
X u P = {ξ ∩ ψ|ξ ∈ X and ψ ∈ P}
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Reasoning Patterns in Strategic Interaction
Reasoning about Strategies
Semantics
E ↓(C ,ψM ,w) (w ′)(D).= E (w)(D) for w ′ 6= w or D = ∅
where for sets of sets X ,P,
(X )sup = {X ⊆W | there is Y ∈ X and Y ⊆ X ⊆W }
.
X u P = {ξ ∩ ψ|ξ ∈ X and ψ ∈ P}
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Restriction of opponents choices
E ↓(C ,ψM ,w) (w)(D).= (E (w)(D) u ψM)sup for D ⊆ C and D 6= ∅
The coalitions that have their choices updated are formed by
the opponents of C .
We can reason on what is left to one's strategic ability once
the opponents have moved.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Restriction of opponents choices
E ↓(C ,ψM ,w) (w)(D).= (E (w)(D) u ψM)sup for D ⊆ C and D 6= ∅
The coalitions that have their choices updated are formed by
the opponents of C .
We can reason on what is left to one's strategic ability once
the opponents have moved.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Restriction of opponents choices
E ↓(C ,ψM ,w) (w)(D).= (E (w)(D) u ψM)sup for D ⊆ C and D 6= ∅
The coalitions that have their choices updated are formed by
the opponents of C .
We can reason on what is left to one's strategic ability once
the opponents have moved.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Irrelevance of hybrid coalitions
E ↓(C ,ψM ,w) (w)(D).= ({ψM})sup for D ∩ C 6= ∅
After C moves, all the coalitions not made by players in C
cannot further condition the game.
The reference are strategic games: the actions are decided
once for all.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Irrelevance of hybrid coalitions
E ↓(C ,ψM ,w) (w)(D).= ({ψM})sup for D ∩ C 6= ∅
After C moves, all the coalitions not made by players in C
cannot further condition the game.
The reference are strategic games: the actions are decided
once for all.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Irrelevance of hybrid coalitions
E ↓(C ,ψM ,w) (w)(D).= ({ψM})sup for D ∩ C 6= ∅
After C moves, all the coalitions not made by players in C
cannot further condition the game.
The reference are strategic games: the actions are decided
once for all.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Locality and closed-worldness
E ↓(C ,ψM ,w) (w ′)(D).= E (w)(D) for w ′ 6= w or D = ∅
After a coalition moves, it does not modify the choices of the
empty coalition or of the other coalitions at di�erent worlds.
The update is local.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Locality and closed-worldness
E ↓(C ,ψM ,w) (w ′)(D).= E (w)(D) for w ′ 6= w or D = ∅
After a coalition moves, it does not modify the choices of the
empty coalition or of the other coalitions at di�erent worlds.
The update is local.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Locality and closed-worldness
E ↓(C ,ψM ,w) (w ′)(D).= E (w)(D) for w ′ 6= w or D = ∅
After a coalition moves, it does not modify the choices of the
empty coalition or of the other coalitions at di�erent worlds.
The update is local.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Intuitions behind the Models
Properties
Proposition
For every C , ψM ∈ E (w)(C ),w it holds that E ↓(C ,ψM ,w) is
outcome monotonic, regular and closed-world.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
The Language
The full language
M,w |= p i� p ∈ V (w)M,w |= ¬φ i� M,w 6|= φ
M,w |= φ ∧ ψ i� M,w |= φ and M,w |= ψM,w |= [C ]φ i� φM ∈ E (w)(C )
M,w |= [C ↓ ψ]φ i� ψM ∈ E (w)(C ) implies M ↓(C ,ψM ,w),w |= φ
M,w |= Aφ i� M, v |= φ, for all v ∈W
where φM = {w ∈W |M,w |= φ} is the truth set of φ.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
A Complete Reduction
Properties
[C ↓ ξ]p ↔ ([C ]ξ → p)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
A Complete Reduction
Properties
[C ↓ ξ]¬φ↔ ([C ]ξ → ¬[C ↓ ξ]φ)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
A Complete Reduction
Properties
[C ↓ ξ](φ ∧ ψ)↔ ([C ↓ ξ]φ ∧ [C ↓ ξ]ψ)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
A Complete Reduction
Properties
[C ↓ ξ][D]φ↔ [D](φ ∨ ¬ξ)( for D ⊆ C and D 6= ∅)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
A Complete Reduction
Properties
[C ↓ ξ][D]φ↔ A(ξ → φ)( for D ∩ C 6= ∅)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
A Complete Reduction
Properties
[C ↓ ξ][∅]φ↔ Aφ( for D = ∅)
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
A Complete Reduction
Properties
[C ↓ ξ]¬A¬φ↔ [C ]φ→ ¬A¬[C ↓ ξ]φ
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
A Complete Reduction
Properties
From ` ψ infer ` [C ↓ ξ]ψ
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
A Complete Reduction
Proposition
Every formula of the abovede�ned language with the subgame
operator occurring in it can be expressed with an equivalent
formula without the subgame operator occurring in it.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Choices as announcements
Public Announcement Logic formalizes the e�ect of the
announcement of a true formula in each agent's a epistemic
relation R(a).
The operator [φ]ψ says that ψ holds after φ is announced. Its
semantics is given as follows:
M,w |= [φ]ψ ⇔ M,w |= φ implies M|φ,w |= ψ
where M|φ = (W ′,R ′(a),V ′) is de�ned as follows:
W ′ = φM
R ′(a) = R(a) ∩ (W × φM)
V ′(p) = V (p) ∩ φM
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Choices as announcements
Public Announcement Logic formalizes the e�ect of the
announcement of a true formula in each agent's a epistemic
relation R(a).
The operator [φ]ψ says that ψ holds after φ is announced. Its
semantics is given as follows:
M,w |= [φ]ψ ⇔ M,w |= φ implies M|φ,w |= ψ
where M|φ = (W ′,R ′(a),V ′) is de�ned as follows:
W ′ = φM
R ′(a) = R(a) ∩ (W × φM)
V ′(p) = V (p) ∩ φM
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Choices as announcements
Public Announcement Logic formalizes the e�ect of the
announcement of a true formula in each agent's a epistemic
relation R(a).
The operator [φ]ψ says that ψ holds after φ is announced. Its
semantics is given as follows:
M,w |= [φ]ψ ⇔ M,w |= φ implies M|φ,w |= ψ
where M|φ = (W ′,R ′(a),V ′) is de�ned as follows:
W ′ = φM
R ′(a) = R(a) ∩ (W × φM)
V ′(p) = V (p) ∩ φM
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Choices as announcements
Public Announcement Logic formalizes the e�ect of the
announcement of a true formula in each agent's a epistemic
relation R(a).
The operator [φ]ψ says that ψ holds after φ is announced. Its
semantics is given as follows:
M,w |= [φ]ψ ⇔ M,w |= φ implies M|φ,w |= ψ
where M|φ = (W ′,R ′(a),V ′) is de�ned as follows:
W ′ = φM
R ′(a) = R(a) ∩ (W × φM)
V ′(p) = V (p) ∩ φM
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Choices as announcements
Public Announcement Logic formalizes the e�ect of the
announcement of a true formula in each agent's a epistemic
relation R(a).
The operator [φ]ψ says that ψ holds after φ is announced. Its
semantics is given as follows:
M,w |= [φ]ψ ⇔ M,w |= φ implies M|φ,w |= ψ
where M|φ = (W ′,R ′(a),V ′) is de�ned as follows:
W ′ = φM
R ′(a) = R(a) ∩ (W × φM)
V ′(p) = V (p) ∩ φM
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Choices as announcements
Public Announcement Logic formalizes the e�ect of the
announcement of a true formula in each agent's a epistemic
relation R(a).
The operator [φ]ψ says that ψ holds after φ is announced. Its
semantics is given as follows:
M,w |= [φ]ψ ⇔ M,w |= φ implies M|φ,w |= ψ
where M|φ = (W ′,R ′(a),V ′) is de�ned as follows:
W ′ = φM
R ′(a) = R(a) ∩ (W × φM)
V ′(p) = V (p) ∩ φM
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Choices as Announcements
Model restriction 'throws worlds away'.
In fact it has a conditional reading: public announcements can
be de�ned only updating the epistemic relation.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Choices as Announcements
Model restriction 'throws worlds away'.
In fact it has a conditional reading: public announcements can
be de�ned only updating the epistemic relation.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Public Announcement axioms
[φ]p ↔ (φ→ p)
[φ]¬ψ ↔ (φ→ ¬[φ]ψ)
[φ](ξ ∧ ψ)↔ ([φ]ξ ∧ [φ]ψ)
[φ]�aψ ↔ (φ→ �a[φ]ψ)
From ` ψ infer ` [φ]ψ
[C ↓ ξ]p ↔ ([C ]ξ → p)
[C ↓ ξ]¬φ↔ ([C ]ξ → ¬[C ↓ ξ]φ)
[C ↓ ξ](φ∧ψ)↔ ([C ↓ ξ]φ∧[C ↓ ξ]ψ)
[C ↓ ξ][D]φ↔ [D](φ ∨ ¬ξ)( for D ⊆C and D 6= ∅)[C ↓ ξ][D]φ↔ A(ξ →φ)( for D ∩ C 6= ∅)[C ↓ ξ][∅]φ↔ Aφ( for D = ∅)From ` ψ infer ` [C ↓ ξ]ψ
The structure of the two axiom systems is very similar in the
atomic and boolean case, but very di�erent in the modal case.
The di�erence lies on the way the relation is updated.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Public Announcement axioms
[φ]p ↔ (φ→ p)
[φ]¬ψ ↔ (φ→ ¬[φ]ψ)
[φ](ξ ∧ ψ)↔ ([φ]ξ ∧ [φ]ψ)
[φ]�aψ ↔ (φ→ �a[φ]ψ)
From ` ψ infer ` [φ]ψ
[C ↓ ξ]p ↔ ([C ]ξ → p)
[C ↓ ξ]¬φ↔ ([C ]ξ → ¬[C ↓ ξ]φ)
[C ↓ ξ](φ∧ψ)↔ ([C ↓ ξ]φ∧[C ↓ ξ]ψ)
[C ↓ ξ][D]φ↔ [D](φ ∨ ¬ξ)( for D ⊆C and D 6= ∅)[C ↓ ξ][D]φ↔ A(ξ →φ)( for D ∩ C 6= ∅)[C ↓ ξ][∅]φ↔ Aφ( for D = ∅)From ` ψ infer ` [C ↓ ξ]ψ
The structure of the two axiom systems is very similar in the
atomic and boolean case, but very di�erent in the modal case.
The di�erence lies on the way the relation is updated.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Properties
Choices as Announcements
Public Announcement axioms
[φ]p ↔ (φ→ p)
[φ]¬ψ ↔ (φ→ ¬[φ]ψ)
[φ](ξ ∧ ψ)↔ ([φ]ξ ∧ [φ]ψ)
[φ]�aψ ↔ (φ→ �a[φ]ψ)
From ` ψ infer ` [φ]ψ
[C ↓ ξ]p ↔ ([C ]ξ → p)
[C ↓ ξ]¬φ↔ ([C ]ξ → ¬[C ↓ ξ]φ)
[C ↓ ξ](φ∧ψ)↔ ([C ↓ ξ]φ∧[C ↓ ξ]ψ)
[C ↓ ξ][D]φ↔ [D](φ ∨ ¬ξ)( for D ⊆C and D 6= ∅)[C ↓ ξ][D]φ↔ A(ξ →φ)( for D ∩ C 6= ∅)[C ↓ ξ][∅]φ↔ Aφ( for D = ∅)From ` ψ infer ` [C ↓ ξ]ψ
The structure of the two axiom systems is very similar in the
atomic and boolean case, but very di�erent in the modal case.
The di�erence lies on the way the relation is updated.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Who have we sung?
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Who have we sung?
Coalition Logic is extremely abstract: [C ]φ does not mean
that C can select all φ worlds.
But it does not even mean that they can't.
[C ↓ φ] inherits this ambiguity.
Updating with the nonmonotonic core (those formulas whose
subsets are not present in the e�ectivity function) may solve
this issue.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Who have we sung?
Coalition Logic is extremely abstract: [C ]φ does not mean
that C can select all φ worlds.
But it does not even mean that they can't.
[C ↓ φ] inherits this ambiguity.
Updating with the nonmonotonic core (those formulas whose
subsets are not present in the e�ectivity function) may solve
this issue.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Who have we sung?
Coalition Logic is extremely abstract: [C ]φ does not mean
that C can select all φ worlds.
But it does not even mean that they can't.
[C ↓ φ] inherits this ambiguity.
Updating with the nonmonotonic core (those formulas whose
subsets are not present in the e�ectivity function) may solve
this issue.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Who have we sung?
Coalition Logic is extremely abstract: [C ]φ does not mean
that C can select all φ worlds.
But it does not even mean that they can't.
[C ↓ φ] inherits this ambiguity.
Updating with the nonmonotonic core (those formulas whose
subsets are not present in the e�ectivity function) may solve
this issue.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Stocktaking
The subgame operator extends the update paradigm of
Dynamic Epistemic Logic to account for the dynamics of
strategic ability.
The framework explicitly expresses how a coalitional move
modi�es the ability of all the players involved in the
interaction, providing a useful framework for capturing
coalitional reasoning in strategic interaction.
The results are limited to Coalition Logic.
The results are limited to updating with a very abstract
representation of strategic ability.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Stocktaking
The subgame operator extends the update paradigm of
Dynamic Epistemic Logic to account for the dynamics of
strategic ability.
The framework explicitly expresses how a coalitional move
modi�es the ability of all the players involved in the
interaction, providing a useful framework for capturing
coalitional reasoning in strategic interaction.
The results are limited to Coalition Logic.
The results are limited to updating with a very abstract
representation of strategic ability.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Stocktaking
The subgame operator extends the update paradigm of
Dynamic Epistemic Logic to account for the dynamics of
strategic ability.
The framework explicitly expresses how a coalitional move
modi�es the ability of all the players involved in the
interaction, providing a useful framework for capturing
coalitional reasoning in strategic interaction.
The results are limited to Coalition Logic.
The results are limited to updating with a very abstract
representation of strategic ability.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Stocktaking
The subgame operator extends the update paradigm of
Dynamic Epistemic Logic to account for the dynamics of
strategic ability.
The framework explicitly expresses how a coalitional move
modi�es the ability of all the players involved in the
interaction, providing a useful framework for capturing
coalitional reasoning in strategic interaction.
The results are limited to Coalition Logic.
The results are limited to updating with a very abstract
representation of strategic ability.
Turrini2009 Updating Coalition Structures
Updating Coalition Structures
Discussion
Thanks!
Thanks!
Turrini2009 Updating Coalition Structures