updating coalition structures: some issues and some...

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

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

Reasoning Patterns in Strategic Interaction

Strategic Reasoning

As you well know...

HHHHHHi

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.

Strategic Reasoning

As you well know...

HHHHHHi

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.

Strategic Reasoning

I play it again

HHHHHHi

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.

Strategic Reasoning

I play it again

HHHHHHi

C (3, 3) (0, 4)

D (4, 0) (1, 1)

If j plays D, I had better play D.

Strategic Reasoning

I play it again

HHHHHHi

C (3, 3) (0, 4)

D (4, 0) (1, 1)

Strategic Reasoning

I play it again

HHHHHHi

C (3, 3) (0, 4)

D (4, 0) (1, 1)

Strategic Reasoning

I play it again

HHHHHHi

C (3, 3) (0, 4)

D (4, 0) (1, 1)

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)

Strategic Reasoning

Ipse dixit

strategic equilibria exist. But there are hardly any explicit

maybe the closest intuitive analogue to strategies.

Strategic Reasoning

Ipse dixit

Strategic Reasoning

Ipse dixit

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

Marc Pauly,

A Logic for Social Software.

PhD thesis, 2001.

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)

E : W → (2Agt → 22W

{(4, 0)} 6∈ E (w)({i})

{(4, 0)} ∈ E (w)(Agt)

E : W → (2Agt → 22W

{(4, 0)} 6∈ E (w)({i}){(4, 0)} ∈ E (w)(Agt)

E is outcome monotonic

X ⊆ Y and X ∈ E (C ) implies Y ∈ E (C )

Coalition Logic

M = (W ,E ,V )

M,w |= [C ]φ⇔ [[φ]]M ∈ E (w)(C )

[[φ]]M = {w ∈W |M,w |= φ}

Coalition Logic

M = (W ,E ,V )

M,w |= [C ]φ⇔ [[φ]]M ∈ E (w)(C )

[[φ]]M = {w ∈W |M,w |= φ}

Coalition Logic

M = (W ,E ,V )

M,w |= [C ]φ⇔ [[φ]]M ∈ E (w)(C )

[[φ]]M = {w ∈W |M,w |= φ}

What we can say in Coalition Logic

HHHHHHi

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.

What we can say in Coalition Logic

HHHHHHi

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.

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.

logic can bring them to the fore. One guide-line of

logics', is the following:

we would like to explicitly

underlying many basic game-theoretic results.

Or in more

The Subgame Operator

[C ↓ ψ]φ

�Given that coalition C chooses ψ, φ holds�

Semantics

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.

Semantics

C (3, 3) (0, 4)

D (4, 0) (1, 1)

new operator.

Semantics

C (3, 3)

D (4, 0)

new operator.

Semantics

C (0, 4)

D (1, 1)

new operator.

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 }

Semantics

M = (W ,E ,V )

Y ∈ E (w)(C );

Semantics

M = (W ,E ,V )

Y ∈ E (w)(C );

Semantics

M = (W ,E ,V )

Y ∈ E (w)(C );

Semantics

M = (W ,E ,V )

Y ∈ E (w)(C );

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.

Semantics

M ↓(C ,ψM ,w).= 〈W ,E ↓(C ,ψM ,w),V 〉

Semantics

M ↓(C ,ψM ,w).= 〈W ,E ↓(C ,ψM ,w),V 〉

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}

Semantics

X u P = {ξ ∩ ψ|ξ ∈ X and ψ ∈ P}

Semantics

E ↓(C ,ψM ,w) (w)(D).= (E (w)(D) u ψM)sup for D ⊆ C and D 6= ∅

X u P = {ξ ∩ ψ|ξ ∈ X and ψ ∈ P}

Semantics

E ↓(C ,ψM ,w) (w ′)(D).= E (w)(D) for w ′ 6= w or D = ∅

X u P = {ξ ∩ ψ|ξ ∈ X and ψ ∈ P}

Properties

Restriction of opponents choices

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.

Properties

Irrelevance of hybrid coalitions

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.

Properties

once for all.

Properties

once for all.

Properties

Locality and closed-worldness

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.

Properties

Proposition

For every C , ψM ∈ E (w)(C ),w it holds that E ↓(C ,ψM ,w) is

outcome monotonic, regular and closed-world.

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.

Properties

[C ↓ ξ]p ↔ ([C ]ξ → p)

Properties

[C ↓ ξ]¬φ↔ ([C ]ξ → ¬[C ↓ ξ]φ)

Properties

[C ↓ ξ](φ ∧ ψ)↔ ([C ↓ ξ]φ ∧ [C ↓ ξ]ψ)

Properties

[C ↓ ξ][D]φ↔ [D](φ ∨ ¬ξ)( for D ⊆ C and D 6= ∅)

Properties

[C ↓ ξ][D]φ↔ A(ξ → φ)( for D ∩ C 6= ∅)

Properties

[C ↓ ξ][∅]φ↔ Aφ( for D = ∅)

Properties

[C ↓ ξ]¬A¬φ↔ [C ]φ→ ¬A¬[C ↓ ξ]φ

Properties

From ` ψ infer ` [C ↓ ξ]ψ

Properties

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.

Properties

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

Properties

relation R(a).

W ′ = φM

R ′(a) = R(a) ∩ (W × φM)

V ′(p) = V (p) ∩ φM

Properties

relation R(a).

W ′ = φM

R ′(a) = R(a) ∩ (W × φM)

V ′(p) = V (p) ∩ φM

Properties

relation R(a).

W ′ = φM

R ′(a) = R(a) ∩ (W × φM)

V ′(p) = V (p) ∩ φM

Properties

relation R(a).

W ′ = φM

R ′(a) = R(a) ∩ (W × φM)

V ′(p) = V (p) ∩ φM

Properties

relation R(a).

W ′ = φM

R ′(a) = R(a) ∩ (W × φM)

V ′(p) = V (p) ∩ φM

Properties

Model restriction 'throws worlds away'.

In fact it has a conditional reading: public announcements can

be de�ned only updating the epistemic relation.

Properties

Model restriction 'throws worlds away'.

In fact it has a conditional reading: public announcements can

be de�ned only updating the epistemic relation.

Properties

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.

Properties

[φ]p ↔ (φ→ p)

[φ]¬ψ ↔ (φ→ ¬[φ]ψ)

[φ](ξ ∧ ψ)↔ ([φ]ξ ∧ [φ]ψ)

[φ]�aψ ↔ (φ→ �a[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)

Properties

[φ]p ↔ (φ→ p)

[φ]¬ψ ↔ (φ→ ¬[φ]ψ)

[φ](ξ ∧ ψ)↔ ([φ]ξ ∧ [φ]ψ)

[φ]�aψ ↔ (φ→ �a[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)

Discussion

Who have we sung?

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.

Discussion

Who have we sung?

this issue.

Discussion

Who have we sung?

this issue.

Discussion

Who have we sung?

this issue.

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.

Discussion

Stocktaking

strategic ability.

Discussion

Stocktaking

strategic ability.

Discussion

Stocktaking

strategic ability.

Discussion

Thanks!

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