Coalitional Games Stphane Airiau and Wojtek Jamroga Stphane: ILLC @ - - PowerPoint PPT Presentation

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Coalitional Games Stphane Airiau and Wojtek Jamroga Stphane: ILLC @ - - PowerPoint PPT Presentation

Oct 07, 2022 216 likes •2.04k views

Coalitional Games Stphane Airiau and Wojtek Jamroga Stphane: ILLC @ University of Amsterdam Wojtek: ICR @ University of Luxembourg European Agent Systems Summer School Torino, Italy, September 2009 Stphane Airiau and Wojtek Jamroga

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SLIDE 1

Coalitional Games

Stéphane Airiau and Wojtek Jamroga Stéphane: ILLC @ University of Amsterdam Wojtek: ICR @ University of Luxembourg European Agent Systems Summer School Torino, Italy, September 2009

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 1/70

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SLIDE 2
  • 2. Reasoning about Coalitions

Part 2. Reasoning about Coalitions

Reasoning about Coalitions 2.1 Modal Logic 2.2 ATL 2.3 Rational Play (ATLP) 2.4 Imperfect Information 2.5 Model Checking 2.6 References

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 3/70

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SLIDE 3
  • 2. Reasoning about Coalitions

Outline In the previous chapter, we showed how coalitions can be rationally formed

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 4/70

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SLIDE 4
  • 2. Reasoning about Coalitions

Outline In the previous chapter, we showed how coalitions can be rationally formed In this chapter, we show how one can use modal logic to reason about their play and their outcome.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 4/70

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SLIDE 5
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

2.1 Modal Logic

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 5/70

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SLIDE 6
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

Why logic at all?

framework for thinking about systems, makes one realise the implicit assumptions, . . . and then we can: investigate them, accept or reject them, relax some of them and still use a part of the formal and conceptual machinery;

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 6/70

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SLIDE 7
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

Why logic at all?

framework for thinking about systems, makes one realise the implicit assumptions, . . . and then we can: investigate them, accept or reject them, relax some of them and still use a part of the formal and conceptual machinery; reasonably expressive but simpler and more rigorous than the full language of mathematics.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 6/70

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SLIDE 8
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

Why logic at all?

Verification: check specification against implementation Executable specifications Planning as model checking

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 7/70

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SLIDE 9
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

Why logic at all?

Verification: check specification against implementation Executable specifications Planning as model checking Game solving, mechanism design, and reasoning about games have natural interpretation as logical problems

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 7/70

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SLIDE 10
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

Modal logic is an extension of classical logic by new connectives

  • and ♦

♦ ♦: necessity and possibility.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 8/70

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SLIDE 11
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

Modal logic is an extension of classical logic by new connectives

  • and ♦

♦ ♦: necessity and possibility. “

  • p is true” means p is necessarily true, i.e. true in

every possible scenario, “♦ ♦ ♦p is true” means p is possibly true, i.e. true in at least

  • ne possible scenario.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 8/70

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SLIDE 12
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

Various modal logics: knowledge → epistemic logic, beliefs → doxastic logic,

  • bligations → deontic logic,

actions → dynamic logic, time → temporal logic, and combinations of the above Most famous multimodal logic: BDI logic of beliefs, desires, intentions (and time)

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 9/70

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SLIDE 13
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

Definition 2.1 (Kripke Semantics) Kripke model (possible world model): M = W, R, π, W is a set of possible worlds R ⊆ W × W is an accessibility relation π : W → P(Π) is a valuation of propositions.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 10/70

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SLIDE 14
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

Definition 2.1 (Kripke Semantics) Kripke model (possible world model): M = W, R, π, W is a set of possible worlds R ⊆ W × W is an accessibility relation π : W → P(Π) is a valuation of propositions. M, w | = ϕ iff for every w′ ∈ W with wRw′ we have that M, w′ | = ϕ.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 10/70

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SLIDE 15
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

An Example

q0 q2 q1

x=2 x=0 x=1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 11/70

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SLIDE 16
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

An Example

q0 q2 q1

x=2 x=0 x=1 s s

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 11/70

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SLIDE 17
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

An Example

q0 q2 q1

x=2 x=0 x=1 s s c c

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 11/70

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SLIDE 18
  • 2. Reasoning about Coalitions
  • 1. Modal Logic

An Example

q0 q2 q1

x=2 x=0 x=1 s s c c

x . = 1 → Ksx . = 1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 11/70

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SLIDE 19
  • 2. Reasoning about Coalitions
  • 2. ATL

2.2 ATL

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 12/70

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SLIDE 20
  • 2. Reasoning about Coalitions
  • 2. ATL

ATL: What Agents Can Achieve

ATL: Agent Temporal Logic [Alur et al. 1997] Temporal logic meets game theory Main idea: cooperation modalities

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 13/70

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SLIDE 21
  • 2. Reasoning about Coalitions
  • 2. ATL

ATL: What Agents Can Achieve

ATL: Agent Temporal Logic [Alur et al. 1997] Temporal logic meets game theory Main idea: cooperation modalities

  • A

Φ: coalition A has a collective strategy to enforce Φ

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 13/70

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SLIDE 22
  • 2. Reasoning about Coalitions
  • 2. ATL
  • jamesbond

♦win: “James Bond has an infallible plan to eventually win”

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 14/70

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SLIDE 23
  • 2. Reasoning about Coalitions
  • 2. ATL
  • jamesbond

♦win: “James Bond has an infallible plan to eventually win”

  • jamesbond, bondsgirl

fun U shot: “James Bond and his girlfriend are able to have fun until someone shoots at them”

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 14/70

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SLIDE 24
  • 2. Reasoning about Coalitions
  • 2. ATL
  • jamesbond

♦win: “James Bond has an infallible plan to eventually win”

  • jamesbond, bondsgirl

fun U shot: “James Bond and his girlfriend are able to have fun until someone shoots at them” “Vanilla” ATL: every temporal operator preceded by exactly one cooperation modality; ATL*: no syntactic restrictions;

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 14/70

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SLIDE 25
  • 2. Reasoning about Coalitions
  • 2. ATL

ATL Models: Concurrent Game Structures

Agents, actions, transitions, atomic propositions Atomic propositions + interpretation Actions are abstract

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 15/70

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SLIDE 26
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.2 (Concurrent Game Structure) A concurrent game structure is a tuple M = Agt, St, π, Act, d, o, where:

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 16/70

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SLIDE 27
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.2 (Concurrent Game Structure) A concurrent game structure is a tuple M = Agt, St, π, Act, d, o, where: Agt: a finite set of all agents

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 16/70

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SLIDE 28
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.2 (Concurrent Game Structure) A concurrent game structure is a tuple M = Agt, St, π, Act, d, o, where: Agt: a finite set of all agents St: a set of states

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 16/70

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SLIDE 29
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.2 (Concurrent Game Structure) A concurrent game structure is a tuple M = Agt, St, π, Act, d, o, where: Agt: a finite set of all agents St: a set of states π: a valuation of propositions

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 16/70

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SLIDE 30
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.2 (Concurrent Game Structure) A concurrent game structure is a tuple M = Agt, St, π, Act, d, o, where: Agt: a finite set of all agents St: a set of states π: a valuation of propositions Act: a finite set of (atomic) actions

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 16/70

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SLIDE 31
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.2 (Concurrent Game Structure) A concurrent game structure is a tuple M = Agt, St, π, Act, d, o, where: Agt: a finite set of all agents St: a set of states π: a valuation of propositions Act: a finite set of (atomic) actions d : Agt × St → P(Act) defines actions available to an agent in a state

  • : a deterministic transition function that assigns
  • utcome states q′ = o(q, α1, . . . , αk) to states and tuples
  • f actions

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 16/70

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SLIDE 32
  • 2. Reasoning about Coalitions
  • 2. ATL

Example: Robots and Carriage

1 2 1 1 2 2 1 2

pos0 pos1 pos2

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 17/70

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SLIDE 33
  • 2. Reasoning about Coalitions
  • 2. ATL

Example: Robots and Carriage

1 2 1 1 2 2 1 2

pos0 pos1 pos2

q0 q2 q1

pos0 pos1

wait,wait wait,wait wait,wait push,push push,push push,push push,wait push,wait push,wait wait,push

pos2

wait,push wait,push

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 17/70

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SLIDE 34
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.3 (Strategy) A strategy is a conditional plan.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 18/70

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SLIDE 35
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.3 (Strategy) A strategy is a conditional plan. We represent strategies by functions sa : St → Act.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 18/70

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SLIDE 36
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.3 (Strategy) A strategy is a conditional plan. We represent strategies by functions sa : St → Act. Function out(q, SA) returns the set of all paths that may result from agents A executing strategy SA from state q

  • nward.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 18/70

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SLIDE 37
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.4 (Semantics of ATL) M, q | = A Φ iff there is a collective strategy SA such that, for every path λ ∈ out(q, SA), we have M, λ | = Φ.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70

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SLIDE 38
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.4 (Semantics of ATL) M, q | = A Φ iff there is a collective strategy SA such that, for every path λ ∈ out(q, SA), we have M, λ | = Φ. M, λ | = ϕ iff M, λ[1] | = ϕ;

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70

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SLIDE 39
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.4 (Semantics of ATL) M, q | = A Φ iff there is a collective strategy SA such that, for every path λ ∈ out(q, SA), we have M, λ | = Φ. M, λ | = ϕ iff M, λ[1] | = ϕ; M, λ | = ♦ϕ iff M, λ[i] | = ϕ for some i ≥ 0;

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70

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SLIDE 40
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.4 (Semantics of ATL) M, q | = A Φ iff there is a collective strategy SA such that, for every path λ ∈ out(q, SA), we have M, λ | = Φ. M, λ | = ϕ iff M, λ[1] | = ϕ; M, λ | = ♦ϕ iff M, λ[i] | = ϕ for some i ≥ 0; M, λ | = ϕ iff M, λ[i] | = ϕ for all i ≥ 0;

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70

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SLIDE 41
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.4 (Semantics of ATL) M, q | = A Φ iff there is a collective strategy SA such that, for every path λ ∈ out(q, SA), we have M, λ | = Φ. M, λ | = ϕ iff M, λ[1] | = ϕ; M, λ | = ♦ϕ iff M, λ[i] | = ϕ for some i ≥ 0; M, λ | = ϕ iff M, λ[i] | = ϕ for all i ≥ 0; M, λ | = ϕ U ψ iff M, λ[i] | = ψ for some i ≥ 0, and M, λ[j] | = ϕ forall 0 ≤ j ≤ i.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70

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SLIDE 42
  • 2. Reasoning about Coalitions
  • 2. ATL

Definition 2.4 (Semantics of ATL) M, q | = p iff p is in π(q); M, q | = ϕ ∧ ψ iff M, q | = ϕ and M, q | = ψ; M, q | = A Φ iff there is a collective strategy SA such that, for every path λ ∈ out(q, SA), we have M, λ | = Φ. M, λ | = ϕ iff M, λ[1] | = ϕ; M, λ | = ♦ϕ iff M, λ[i] | = ϕ for some i ≥ 0; M, λ | = ϕ iff M, λ[i] | = ϕ for all i ≥ 0; M, λ | = ϕ U ψ iff M, λ[i] | = ψ for some i ≥ 0, and M, λ[j] | = ϕ forall 0 ≤ j ≤ i.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70

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SLIDE 43
  • 2. Reasoning about Coalitions
  • 2. ATL

Example: Robots and Carriage

q0 q2 q1

pos0 pos1

wait,wait wait,wait wait,wait push,push push,push push,push push,wait push,wait wait,push push,wait wait,push wait,push

pos2

pos0 → 1 ¬pos1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70

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SLIDE 44
  • 2. Reasoning about Coalitions
  • 2. ATL

Example: Robots and Carriage

q0 q2 q1

pos0 pos1

wait,wait wait,wait wait,wait push,push push,push push,push push,wait push,wait push,wait wait,push

pos2

wait,push wait,push

pos0 → 1 ¬pos1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70

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SLIDE 45
  • 2. Reasoning about Coalitions
  • 2. ATL

Example: Robots and Carriage

q0 q2 q1

pos0 pos1

wait,wait wait,wait wait,wait push,push push,push push,push push,wait push,wait wait,push push,wait wait,push wait,push

pos2

wait wait wait wait push push

pos0 → 1 ¬pos1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70

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SLIDE 46
  • 2. Reasoning about Coalitions
  • 2. ATL

Example: Robots and Carriage

q0 q2 q1

pos0 pos1

wait,wait wait wait,wait wait,wait wait push,push push,push push push,push push,wait push push,wait wait,push wait push,wait wait,push wait,push wait

pos2

pos0 → 1 ¬pos1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70

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SLIDE 47
  • 2. Reasoning about Coalitions
  • 2. ATL

Example: Robots and Carriage

q0 q2 q1

pos0 pos1

wait,wait wait wait,wait wait,wait wait push,push push,push push push,push push,wait push push,wait wait,push wait push,wait wait,push wait,push wait

pos2

pos0 → 1 ¬pos1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70

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SLIDE 48
  • 2. Reasoning about Coalitions
  • 2. ATL

Example: Robots and Carriage

q0 q2 q1

pos0 pos1

wait,wait wait,wait wait,wait wait,wait wait,wait wait,wait push,push push,push push,push push,push push,push push,push push,wait push,wait push,wait push,wait wait,push wait,push push,wait push,wait wait,push wait,push wait,push wait,push

pos2

pos0 → 1 ¬pos1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70

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SLIDE 49
  • 2. Reasoning about Coalitions
  • 2. ATL

Example: Robots and Carriage

q0 q2 q1 q1

pos0 pos1 pos1

wait,wait wait,wait wait,wait wait,wait wait,wait wait,wait push,push push,push push,push push,push push,push push,push push,wait push,wait push,wait push,wait wait,push wait,push push,wait push,wait wait,push wait,push wait,push wait,push

pos2

pos0 → 1 ¬pos1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70

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SLIDE 50
  • 2. Reasoning about Coalitions
  • 2. ATL

Temporal operators allow a number of useful concepts to be formally specified

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 21/70

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SLIDE 51
  • 2. Reasoning about Coalitions
  • 2. ATL

Temporal operators allow a number of useful concepts to be formally specified safety properties liveness properties fairness properties

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 21/70

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SLIDE 52
  • 2. Reasoning about Coalitions
  • 2. ATL

Safety (maintenance goals): “something bad will not happen” “something good will always hold”

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 22/70

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SLIDE 53
  • 2. Reasoning about Coalitions
  • 2. ATL

Safety (maintenance goals): “something bad will not happen” “something good will always hold” Typical example: ¬bankrupt

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 22/70

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SLIDE 54
  • 2. Reasoning about Coalitions
  • 2. ATL

Safety (maintenance goals): “something bad will not happen” “something good will always hold” Typical example: ¬bankrupt Usually: ¬....

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 22/70

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SLIDE 55
  • 2. Reasoning about Coalitions
  • 2. ATL

Safety (maintenance goals): “something bad will not happen” “something good will always hold” Typical example: ¬bankrupt Usually: ¬.... In ATL:

  • s

¬crash

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 22/70

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SLIDE 56
  • 2. Reasoning about Coalitions
  • 2. ATL

Liveness (achievement goals): “something good will happen”

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 23/70

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SLIDE 57
  • 2. Reasoning about Coalitions
  • 2. ATL

Liveness (achievement goals): “something good will happen” Typical example: ♦rich Usually: ♦....

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 23/70

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SLIDE 58
  • 2. Reasoning about Coalitions
  • 2. ATL

Liveness (achievement goals): “something good will happen” Typical example: ♦rich Usually: ♦.... In ATL:

  • alice, bob

♦paperAccepted

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 23/70

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SLIDE 59
  • 2. Reasoning about Coalitions
  • 2. ATL

Fairness (service goals):

“if something is attempted/requested, then it will be successful/allocated”

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 24/70

slide-60
SLIDE 60
  • 2. Reasoning about Coalitions
  • 2. ATL

Fairness (service goals):

“if something is attempted/requested, then it will be successful/allocated” Typical examples: (attempt → ♦success) ♦attempt → ♦success

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 24/70

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SLIDE 61
  • 2. Reasoning about Coalitions
  • 2. ATL

Fairness (service goals):

“if something is attempted/requested, then it will be successful/allocated” Typical examples: (attempt → ♦success) ♦attempt → ♦success In ATL* (!):

  • prod, dlr

(carRequested → ♦carDelivered)

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 24/70

slide-62
SLIDE 62
  • 2. Reasoning about Coalitions
  • 2. ATL

Connection to Games

Concurrent game structure = generalized extensive game

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 25/70

slide-63
SLIDE 63
  • 2. Reasoning about Coalitions
  • 2. ATL

Connection to Games

Concurrent game structure = generalized extensive game

  • A

γ: A splits the agents into proponents and

  • pponents

γ defines the winning condition

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 25/70

slide-64
SLIDE 64
  • 2. Reasoning about Coalitions
  • 2. ATL

Connection to Games

Concurrent game structure = generalized extensive game

  • A

γ: A splits the agents into proponents and

  • pponents

γ defines the winning condition infinite 2-player, binary, zero-sum game

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 25/70

slide-65
SLIDE 65
  • 2. Reasoning about Coalitions
  • 2. ATL

Connection to Games

Concurrent game structure = generalized extensive game

  • A

γ: A splits the agents into proponents and

  • pponents

γ defines the winning condition infinite 2-player, binary, zero-sum game Flexible and compact specification of winning conditions

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 25/70

slide-66
SLIDE 66
  • 2. Reasoning about Coalitions
  • 2. ATL

Solving a game ≈ checking if M, q | = A γ

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 26/70

slide-67
SLIDE 67
  • 2. Reasoning about Coalitions
  • 2. ATL

Solving a game ≈ checking if M, q | = A γ But: do we really want to consider all the possible plays?

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 26/70

slide-68
SLIDE 68
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

2.3 Rational Play (ATLP)

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 27/70

slide-69
SLIDE 69
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Game-theoretical analysis of games: Solution concepts define rationality of players

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 28/70

slide-70
SLIDE 70
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Game-theoretical analysis of games: Solution concepts define rationality of players

maxmin Nash equilibrium subgame-perfect Nash undominated strategies Pareto optimality

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 28/70

slide-71
SLIDE 71
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Game-theoretical analysis of games: Solution concepts define rationality of players

maxmin Nash equilibrium subgame-perfect Nash undominated strategies Pareto optimality

Then: we assume that players play rationally ...and we ask about the outcome of the game under this assumption

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 28/70

slide-72
SLIDE 72
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Game-theoretical analysis of games: Solution concepts define rationality of players

maxmin Nash equilibrium subgame-perfect Nash undominated strategies Pareto optimality

Then: we assume that players play rationally ...and we ask about the outcome of the game under this assumption Role of rationality criteria: constrain the possible game moves to “sensible” ones

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 28/70

slide-73
SLIDE 73
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 29/70

slide-74
SLIDE 74
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

start → ¬ 1 ♦money1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 29/70

slide-75
SLIDE 75
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

start → ¬ 1 ♦money1 start → ¬ 2 ♦money2

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 29/70

slide-76
SLIDE 76
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

start → ¬ 1 ♦money1 start → ¬ 2 ♦money2

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 29/70

slide-77
SLIDE 77
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

ATL + Plausibility (ATLP)

ATL: reasoning about all possible behaviors.

  • A

ϕ: agents A have some collective strategy to enforce ϕ against any response of their opponents.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 30/70

slide-78
SLIDE 78
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

ATL + Plausibility (ATLP)

ATL: reasoning about all possible behaviors.

  • A

ϕ: agents A have some collective strategy to enforce ϕ against any response of their opponents. ATLP: reasoning about plausible behaviors.

  • A

ϕ: agents A have a plausible collective strategy to enforce ϕ against any plausible response of their

  • pponents.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 30/70

slide-79
SLIDE 79
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

ATL + Plausibility (ATLP)

ATL: reasoning about all possible behaviors.

  • A

ϕ: agents A have some collective strategy to enforce ϕ against any response of their opponents. ATLP: reasoning about plausible behaviors.

  • A

ϕ: agents A have a plausible collective strategy to enforce ϕ against any plausible response of their

  • pponents.

Important The possible strategies of both A and Agt\A are restricted.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 30/70

slide-80
SLIDE 80
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

New in ATLP: (set-pl ω) : the set of plausible profiles is set/reset to the strategies described by ω. Only plausible strategy profiles are considered!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 31/70

slide-81
SLIDE 81
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

New in ATLP: (set-pl ω) : the set of plausible profiles is set/reset to the strategies described by ω. Only plausible strategy profiles are considered! Example: (set-pl greedy1) 2 ♦money2

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 31/70

slide-82
SLIDE 82
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Concurrent game structures with plausibility

M = (Agt, St, Π, π, Act, d, δ, Υ, Ω, ·)

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 32/70

slide-83
SLIDE 83
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Concurrent game structures with plausibility

M = (Agt, St, Π, π, Act, d, δ, Υ, Ω, ·) Υ ⊆ Σ: set of (plausible) strategy profiles

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 32/70

slide-84
SLIDE 84
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Concurrent game structures with plausibility

M = (Agt, St, Π, π, Act, d, δ, Υ, Ω, ·) Υ ⊆ Σ: set of (plausible) strategy profiles Ω = {ω1, ω2, . . . }: set of plausibility terms Example: ωNE may stand for all Nash equilibria

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 32/70

slide-85
SLIDE 85
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Concurrent game structures with plausibility

M = (Agt, St, Π, π, Act, d, δ, Υ, Ω, ·) Υ ⊆ Σ: set of (plausible) strategy profiles Ω = {ω1, ω2, . . . }: set of plausibility terms Example: ωNE may stand for all Nash equilibria · : St → (Ω → P(()Σ)): plausibility mapping Example: ωNE q = {(confess, confess)}

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 32/70

slide-86
SLIDE 86
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Outcome = Paths that may occur when agents A perform sA

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70

slide-87
SLIDE 87
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Outcome = Paths that may occur when agents A perform sA when only plausible strategy profiles from Υ are played

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70

slide-88
SLIDE 88
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Outcome = Paths that may occur when agents A perform sA when only plausible strategy profiles from Υ are played

  • utΥ(q, sA) =

{λ ∈ St+ | ∃t ∈ Υ(sA) ∀i ∈ N

  • λ[i + 1] = δ(λ[i], t(λ[i]))
  • }

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70

slide-89
SLIDE 89
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Outcome = Paths that may occur when agents A perform sA when only plausible strategy profiles from Υ are played

  • utΥ(q, sA) =

{λ ∈ St+ | ∃t ∈ Υ(sA) ∀i ∈ N

  • λ[i + 1] = δ(λ[i], t(λ[i]))
  • }

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70

slide-90
SLIDE 90
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Outcome = Paths that may occur when agents A perform sA when only plausible strategy profiles from Υ are played

  • utΥ(q, sA) =

{λ ∈ St+ | ∃t ∈ Υ(sA) ∀i ∈ N

  • λ[i + 1] = δ(λ[i], t(λ[i]))
  • }

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

P: the players always show same sides of their coins

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70

slide-91
SLIDE 91
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Outcome = Paths that may occur when agents A perform sA when only plausible strategy profiles from Υ are played

  • utΥ(q, sA) =

{λ ∈ St+ | ∃t ∈ Υ(sA) ∀i ∈ N

  • λ[i + 1] = δ(λ[i], t(λ[i]))
  • }

P: the players always show same sides of their coins s1: always show “heads”

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70

slide-92
SLIDE 92
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Semantics of ATLP

M, q | = A γ iff there is a strategy sA consistent with Υ such that M, λ | = γ for all λ ∈ outΥ(q, sA) M, q | = (set-pl ω)ϕ iff M ω, q | = ϕ where the new model M ω is equal to M but the new set Υω of plausible strategy profiles is set to ωq.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 34/70

slide-93
SLIDE 93
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Example: Pennies Game

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 Hh T t Ht T h Hh Ht T h T t H h Ht T h T t

M, q0 | = (set-pl ωNE) 2 ♦money2

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 35/70

slide-94
SLIDE 94
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Example: Pennies Game

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 Hh T t Ht T h Hh Ht T h T t H h Ht T h T t

M, q0 | = (set-pl ωNE) 2 ♦money2 What is a Nash equilibrium in this game? We need some kind of winning criteria!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 35/70

slide-95
SLIDE 95
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Agent 1 “wins”, if γ1 ≡ (¬start → money1) is satisfied. Agent 2 “wins”, if γ2 ≡ ♦money2 is satisfied.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70

slide-96
SLIDE 96
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Agent 1 “wins”, if γ1 ≡ (¬start → money1) is satisfied. Agent 2 “wins”, if γ2 ≡ ♦money2 is satisfied.

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70

slide-97
SLIDE 97
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Agent 1 “wins”, if γ1 ≡ (¬start → money1) is satisfied. Agent 2 “wins”, if γ2 ≡ ♦money2 is satisfied.

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

γ1\γ2 hh ht th tt HH 1, 1 0, 0 0, 1 0, 1 HT 0, 0 0, 1 0, 1 0, 1 TH 0, 1 0, 1 1, 1 0, 0 TT 0, 1 0, 1 0, 0 0, 1

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70

slide-98
SLIDE 98
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Agent 1 “wins”, if γ1 ≡ (¬start → money1) is satisfied. Agent 2 “wins”, if γ2 ≡ ♦money2 is satisfied.

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

γ1\γ2 hh ht th tt HH 1, 1 0, 0 0, 1 0, 1 HT 0, 0 0, 1 0, 1 0, 1 TH 0, 1 0, 1 1, 1 0, 0 TT 0, 1 0, 1 0, 0 0, 1 Now we have a qualitative notion of success.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70

slide-99
SLIDE 99
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Agent 1 “wins”, if γ1 ≡ (¬start → money1) is satisfied. Agent 2 “wins”, if γ2 ≡ ♦money2 is satisfied.

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

γ1\γ2 hh ht th tt HH 1, 1 0, 0 0, 1 0, 1 HT 0, 0 0, 1 0, 1 0, 1 TH 0, 1 0, 1 1, 1 0, 0 TT 0, 1 0, 1 0, 0 0, 1 Now we have a qualitative notion of success. M, q0 | = (set-pl ωNE) 2 (¬start → money1) where ωNE q0 = “all profiles belonging to grey cells”.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70

slide-100
SLIDE 100
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

How to obtain plausibility terms?

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 37/70

slide-101
SLIDE 101
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

How to obtain plausibility terms?

Idea Formulae that describe plausible strategies! (set-pl σ.θ)ϕ: “suppose that θ characterizes rational strategy profiles, then ϕ holds”.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 37/70

slide-102
SLIDE 102
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

How to obtain plausibility terms?

Idea Formulae that describe plausible strategies! (set-pl σ.θ)ϕ: “suppose that θ characterizes rational strategy profiles, then ϕ holds”. Sometimes quantifiers are needed... E.g.: (set-pl σ. ∀σ′ dominates(σ, σ′))

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 37/70

slide-103
SLIDE 103
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Characterization of Nash Equilibrium

σa is a’s best response to σ (wrt γ): BR

γ a(σ) ≡ (set-pl σ[Agt\{a}])

  • a

γa → (set-pl σ) ∅ γa

  • Stéphane Airiau and Wojtek Jamroga · Coalitional Games

EASSS’09 @ Torino 38/70

slide-104
SLIDE 104
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Characterization of Nash Equilibrium

σa is a’s best response to σ (wrt γ): BR

γ a(σ) ≡ (set-pl σ[Agt\{a}])

  • a

γa → (set-pl σ) ∅ γa

  • σ is a Nash equilibrium:

NE

γ(σ) ≡

  • a∈Agt

BR

γ a(σ)

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 38/70

slide-105
SLIDE 105
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Example: Pennies Game revisited

γ1 ≡ (¬start → money1); γ2 ≡ ♦money2.

q0

start

q1

money1

q2

money2

q3 money1

money2

q4 q5

money2 H h T t H t T h Hh H t T h T t Hh H t T h T t

γ1\γ2 hh ht th tt HH 1, 1 0, 0 0, 1 0, 1 HT 0, 0 0, 1 0, 1 0, 1 TH 0, 1 0, 1 1, 1 0, 0 TT 0, 1 0, 1 0, 0 0, 1 M1, q0 | = (set-pl σ.NEγ1,γ2(σ)) 2 (¬start → money1) ...where NEγ1,γ2(σ) is defined as on the last slide.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 39/70

slide-106
SLIDE 106
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Characterizations of Other Solution Concepts

σ is a subgame perfect Nash equilibrium: SPN

γ(σ) ≡

∅ NE

γ(σ)

σ is Pareto optimal: PO

γ(σ)

≡ ∀σ′

  • a∈Agt

((set-pl σ′) ∅ γa → (set-pl σ) ∅ γa) ∨

  • a∈Agt

((set-pl σ) ∅ γa ∧ ¬(set-pl σ′) ∅ γa

  • .

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 40/70

slide-107
SLIDE 107
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

σ is undominated: UNDOM

γ(σ)

≡ ∀σ1∀σ2∃σ3

  • (set-pl σ{a}

1

, σAgt\{a}

2

) ∅ γa → (set-pl σ{a}, σAgt\{a}

2

) ∅ γa

  • ∨
  • (set-pl σ{a}, σAgt\{a}

3

) ∅ γa ∧ ¬(set-pl σ{a}

1

, σAgt\{a}

3

) ∅ γa

  • .

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 41/70

slide-108
SLIDE 108
  • 2. Reasoning about Coalitions
  • 3. Rational Play (ATLP)

Theorem 2.5 The characterizations coincide with game-theoretical solution concepts in the class of game trees.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 42/70

slide-109
SLIDE 109
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

2.4 Imperfect Information

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 43/70

slide-110
SLIDE 110
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

How can we reason about extensive games with imperfect information?

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 44/70

slide-111
SLIDE 111
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

How can we reason about extensive games with imperfect information? Let’s put ATL and epistemic logic in one box.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 44/70

slide-112
SLIDE 112
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

How can we reason about extensive games with imperfect information? Let’s put ATL and epistemic logic in one box. Problems!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 44/70

slide-113
SLIDE 113
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

q0 q1 q9 q8 q7 q1

1

q1

2

q1

3

q1

4

q1

5

q1

6

q1

7

q1

8

win win start win win win win

q1 q2 q3 q4 q5 q6

( , )

  • -

( Q , K ) (A,K) (A,Q) (K,A) (K,Q) (Q,A) (Q,K) ( A , K ) ( K , Q ) ( A , Q ) ( Q , A ) ( K , A ) ( A , Q ) ( K , Q ) ( K , A ) ( Q , A ) ( A , K ) ( Q , K )

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 45/70

slide-114
SLIDE 114
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

q0 q1 q9 q8 q7 q1

1

q1

2

q1

3

q1

4

q1

5

q1

6

q1

7

q1

8

win win start win win win win

q1 q2 q3 q4 q5 q6

( , )

  • -

( Q , K ) (A,K) (A,Q) (K,A) (K,Q) (Q,A) (Q,K) ( A , K ) ( K , Q ) ( A , Q ) ( Q , A ) ( K , A ) ( A , Q ) ( K , Q ) ( K , A ) ( Q , A ) ( A , K ) ( Q , K )

keep keep keep keep keep keep trade trade trade trade trade trade

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 45/70

slide-115
SLIDE 115
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

q0 q1 q9 q8 q7 q1

1

q1

2

q1

3

q1

4

q1

5

q1

6

q1

7

q1

8

win win start win win win win

q1 q2 q3 q4 q5 q6

( , )

  • -

( Q , K ) (A,K) (A,Q) (K,A) (K,Q) (Q,A) (Q,K) ( A , K ) ( K , Q ) ( A , Q ) ( Q , A ) ( K , A ) ( A , Q ) ( K , Q ) ( K , A ) ( Q , A ) ( A , K ) ( Q , K )

keep keep keep keep keep keep trade trade trade trade trade trade

start → a ♦win

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 45/70

slide-116
SLIDE 116
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

q0 q1 q9 q8 q7 q1

1

q1

2

q1

3

q1

4

q1

5

q1

6

q1

7

q1

8

win win start win win win win

q1 q2 q3 q4 q5 q6

( , )

  • -

( Q , K ) (A,K) (A,Q) (K,A) (K,Q) (Q,A) (Q,K) ( A , K ) ( K , Q ) ( A , Q ) ( Q , A ) ( K , A ) ( A , Q ) ( K , Q ) ( K , A ) ( Q , A ) ( A , K ) ( Q , K )

keep keep keep keep keep keep trade trade trade trade trade trade

start → a ♦win start → Ka a ♦win

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 45/70

slide-117
SLIDE 117
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

q0 q1 q9 q8 q7 q1

1

q1

2

q1

3

q1

4

q1

5

q1

6

q1

7

q1

8

win win start win win win win

q1 q2 q3 q4 q5 q6

( , )

  • -

( Q , K ) (A,K) (A,Q) (K,A) (K,Q) (Q,A) (Q,K) ( A , K ) ( K , Q ) ( A , Q ) ( Q , A ) ( K , A ) ( A , Q ) ( K , Q ) ( K , A ) ( Q , A ) ( A , K ) ( Q , K )

keep keep keep keep keep keep trade trade trade trade trade trade

start → a ♦win start → Ka a ♦win Does it make sense?

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 45/70

slide-118
SLIDE 118
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Problem: Strategic and epistemic abilities are not independent!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 46/70

slide-119
SLIDE 119
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Problem: Strategic and epistemic abilities are not independent!

  • A

Φ = A can enforce Φ

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 46/70

slide-120
SLIDE 120
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Problem: Strategic and epistemic abilities are not independent!

  • A

Φ = A can enforce Φ It should at least mean that A are able to identify and execute the right strategy!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 46/70

slide-121
SLIDE 121
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Problem: Strategic and epistemic abilities are not independent!

  • A

Φ = A can enforce Φ It should at least mean that A are able to identify and execute the right strategy! Executable strategies = uniform strategies

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 46/70

slide-122
SLIDE 122
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Definition 2.6 (Uniform strategy) Strategy sa is uniform iff it specifies the same choices for indistinguishable situations: (no recall:) if q ∼a q′ then sa(q) = sa(q′) (perfect recall:) if λ ≈a λ′ then ⇒ sa(λ) = sa(λ), where λ ≈a λ′ iff λ[i] ∼a λ′[i] for every i.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 47/70

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SLIDE 123
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Definition 2.6 (Uniform strategy) Strategy sa is uniform iff it specifies the same choices for indistinguishable situations: (no recall:) if q ∼a q′ then sa(q) = sa(q′) (perfect recall:) if λ ≈a λ′ then ⇒ sa(λ) = sa(λ), where λ ≈a λ′ iff λ[i] ∼a λ′[i] for every i. A collective strategy is uniform iff it consists only of uniform individual strategies.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 47/70

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SLIDE 124
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Note: Having a successful strategy does not imply knowing that we have it!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 48/70

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SLIDE 125
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Note: Having a successful strategy does not imply knowing that we have it! Knowing that a successful strategy exists does not imply knowing the strategy itself!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 48/70

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SLIDE 126
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Levels of Strategic Ability

From now on, we restrict our discussion to uniform memoryless strategies.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 49/70

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SLIDE 127
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Levels of Strategic Ability

From now on, we restrict our discussion to uniform memoryless strategies. Our cases for A Φ under incomplete information:

2 There is σ such that, for every execution of σ, Φ holds

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 49/70

slide-128
SLIDE 128
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Levels of Strategic Ability

From now on, we restrict our discussion to uniform memoryless strategies. Our cases for A Φ under incomplete information:

2 There is σ such that, for every execution of σ, Φ holds 3 A know that there is σ such that, for every execution of

σ, Φ holds

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 49/70

slide-129
SLIDE 129
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Levels of Strategic Ability

From now on, we restrict our discussion to uniform memoryless strategies. Our cases for A Φ under incomplete information:

2 There is σ such that, for every execution of σ, Φ holds 3 A know that there is σ such that, for every execution of

σ, Φ holds

4 There is σ such that A know that, for every execution

  • f σ, Φ holds

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 49/70

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SLIDE 130
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Case [4]: knowing how to play

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 50/70

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SLIDE 131
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Case [4]: knowing how to play Single agent case: we take into account the paths starting from indistinguishable states (i.e.,

  • q′∈img(q,∼a) out(q, sA))

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 50/70

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SLIDE 132
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Case [4]: knowing how to play Single agent case: we take into account the paths starting from indistinguishable states (i.e.,

  • q′∈img(q,∼a) out(q, sA))

What about coalitions? Question: in what sense should they know the strategy? Common knowledge (CA), mutual knowledge (KA), distributed knowledge (DA)?

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 50/70

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SLIDE 133
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Given strategy σ, agents A can have: Common knowledge that σ is a winning strategy. This requires the least amount of additional communication (agents from A may agree upon a total order over their collective strategies at the beginning of the game and that they will always choose the maximal winning strategy with respect to this order)

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 51/70

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SLIDE 134
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Given strategy σ, agents A can have: Common knowledge that σ is a winning strategy. This requires the least amount of additional communication (agents from A may agree upon a total order over their collective strategies at the beginning of the game and that they will always choose the maximal winning strategy with respect to this order) Mutual knowledge that σ is a winning strategy: everybody in A knows that σ is winning

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 51/70

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SLIDE 135
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Distributed knowledge that σ is a winning strategy: if the agents share their knowledge at the current state, they can identify the strategy as winning

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 52/70

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SLIDE 136
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Distributed knowledge that σ is a winning strategy: if the agents share their knowledge at the current state, they can identify the strategy as winning “The leader”: the strategy can be identified by agent a ∈ A

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 52/70

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SLIDE 137
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Distributed knowledge that σ is a winning strategy: if the agents share their knowledge at the current state, they can identify the strategy as winning “The leader”: the strategy can be identified by agent a ∈ A “Headquarters’ committee”: the strategy can be identified by subgroup A′ ⊆ A

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 52/70

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SLIDE 138
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Distributed knowledge that σ is a winning strategy: if the agents share their knowledge at the current state, they can identify the strategy as winning “The leader”: the strategy can be identified by agent a ∈ A “Headquarters’ committee”: the strategy can be identified by subgroup A′ ⊆ A “Consulting company”: the strategy can be identified by some other group B

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 52/70

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SLIDE 139
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Many subtle cases...

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 53/70

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SLIDE 140
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Many subtle cases... Solution: constructive knowledge operators

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 53/70

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SLIDE 141
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Constructive Strategic Logic (CSL)

  • A

Φ: A have a uniform memoryless strategy to enforce Φ

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 54/70

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SLIDE 142
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Constructive Strategic Logic (CSL)

  • A

Φ: A have a uniform memoryless strategy to enforce Φ Ka a Φ: a has a strategy to enforce Φ, and knows that he has one For groups of agents: CA, EA, DA, ...

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 54/70

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SLIDE 143
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Constructive Strategic Logic (CSL)

  • A

Φ: A have a uniform memoryless strategy to enforce Φ Ka a Φ: a has a strategy to enforce Φ, and knows that he has one For groups of agents: CA, EA, DA, ... Ka a Φ: a has a strategy to enforce Φ, and knows that this is a winning strategy For groups of agents: CA, EA, DA, ...

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 54/70

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SLIDE 144
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Non-standard semantics: Formulae are evaluated in sets of states M, Q | = A Φ: A have a single strategy to enforce Φ from all states in Q

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 55/70

slide-145
SLIDE 145
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Non-standard semantics: Formulae are evaluated in sets of states M, Q | = A Φ: A have a single strategy to enforce Φ from all states in Q Additionally:

  • ut(Q, SA) =

q∈Q out(q, SA)

img(Q, R) =

q∈Q img(q, R)

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 55/70

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SLIDE 146
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Non-standard semantics: Formulae are evaluated in sets of states M, Q | = A Φ: A have a single strategy to enforce Φ from all states in Q Additionally:

  • ut(Q, SA) =

q∈Q out(q, SA)

img(Q, R) =

q∈Q img(q, R)

M, q | = ϕ iff M, {q} | = ϕ

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 55/70

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SLIDE 147
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Definition 2.7 (Semantics of CSL) M, Q | = p iff p ∈ π(q) for every q ∈ Q;

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 56/70

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SLIDE 148
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Definition 2.7 (Semantics of CSL) M, Q | = p iff p ∈ π(q) for every q ∈ Q; M, Q | = ¬ϕ iff not M, Q | = ϕ;

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 56/70

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SLIDE 149
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Definition 2.7 (Semantics of CSL) M, Q | = p iff p ∈ π(q) for every q ∈ Q; M, Q | = ¬ϕ iff not M, Q | = ϕ; M, Q | = ϕ ∧ ψ iff M, Q | = ϕ and M, Q | = ψ;

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 56/70

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SLIDE 150
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Definition 2.7 (Semantics of CSL) M, Q | = p iff p ∈ π(q) for every q ∈ Q; M, Q | = ¬ϕ iff not M, Q | = ϕ; M, Q | = ϕ ∧ ψ iff M, Q | = ϕ and M, Q | = ψ; M, Q | = A γ iff there exists SA such that, for every λ ∈ out(Q, SA), we have that M, λ[1] | = ϕ;

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 56/70

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SLIDE 151
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

M, Q | = KAϕ iff M, q | = ϕ for every q ∈ img(Q, ∼K

A) (where

K = C, E, D);

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 57/70

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SLIDE 152
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

M, Q | = KAϕ iff M, q | = ϕ for every q ∈ img(Q, ∼K

A) (where

K = C, E, D); M, Q | = ˆ KAϕ iff M, img(Q, ∼K

A) |

= ϕ (where ˆ K = C, E, D and K = C, E, D, respectively).

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 57/70

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SLIDE 153
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Example: Simple Market

q0 q1

bad-market loss success 2 2 1 c c c c c

wait subproduction

  • wn-production
  • wn-production
  • wn-production

s u b p r

  • d

u c t i

  • n

s u b p r

  • d

u c t i

  • n

q2 ql qs

  • ligopoly

s&m

wait wait

@ q1 :

¬Kc c ♦success

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 58/70

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SLIDE 154
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Example: Simple Market

q0 q1

bad-market loss success 2 2 1 c c c c c

wait subproduction

  • wn-production
  • wn-production
  • wn-production

s u b p r

  • d

u c t i

  • n

s u b p r

  • d

u c t i

  • n

q2 ql qs

  • ligopoly

s&m

wait wait

@ q1 :

¬Kc c ♦success ¬E{1,2} c ♦success ¬K1 c ♦success ¬K2 c ♦success

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 58/70

slide-155
SLIDE 155
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Example: Simple Market

q0 q1

bad-market loss success 2 2 1 c c c c c

wait subproduction

  • wn-production
  • wn-production
  • wn-production

s u b p r

  • d

u c t i

  • n

s u b p r

  • d

u c t i

  • n

q2 ql qs

  • ligopoly

s&m

wait wait

@ q1 :

¬Kc c ♦success ¬E{1,2} c ♦success ¬K1 c ♦success ¬K2 c ♦success ¬D{1,2} c ♦success

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 58/70

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SLIDE 156
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Theorem 2.8 (Expressivity) CSL is strictly more expressive than most previous proposals.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 59/70

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SLIDE 157
  • 2. Reasoning about Coalitions
  • 4. Imperfect Information

Theorem 2.8 (Expressivity) CSL is strictly more expressive than most previous proposals. Theorem 2.9 (Verification complexity) The complexity of model checking CSL is minimal.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 59/70

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SLIDE 158
  • 2. Reasoning about Coalitions
  • 5. Model Checking

2.5 Model Checking

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 60/70

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SLIDE 159
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Model Checking Formulae of CTL and ATL

Model checking: Does ϕ hold in model M and state q?

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 61/70

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SLIDE 160
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Model Checking Formulae of CTL and ATL

Model checking: Does ϕ hold in model M and state q? Natural for verification of existing systems; also during design (“prototyping”) Can be used for automated planning

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 61/70

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SLIDE 161
  • 2. Reasoning about Coalitions
  • 5. Model Checking

function plan(ϕ). Returns a subset of St for which formula ϕ holds, together with a (conditional) plan to achieve ϕ. The plan is sought within the context of concurrent game structure S = Agt, St, Π, π, o. case ϕ ∈ Π : return {q, − | ϕ ∈ π(q)} case ϕ = ¬ψ : P1 := plan(ψ); return {q, − | q / ∈ states(P1)} case ϕ = ψ1 ∨ ψ2 : P1 := plan(ψ1); P2 := plan(ψ2); return {q, − | q ∈ states(P1) ∪ states(P2)} case ϕ = A ψ : return pre(A, states(plan(ψ))) case ϕ = A ψ : P1 := plan(true); P2 := plan(ψ); Q3 := states(P2); while states(P1) ⊆ states(P2) do P1 := P2|states(P1); P2 := pre(A, states(P1))|Q3 od; return P2|states(P1) case ϕ = A ψ1 U ψ2 : P1 := ∅; Q3 := states(plan(ψ1)); P2 := plan(true)|states(plan(ψ2)); while states(P2) ⊆ states(P1) do P1 := P1 ⊕ P2; P2 := pre(A, states(P1))|Q3 od; return P1 end case

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 62/70

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SLIDE 162
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Complexity od Model Checking ATL

Theorem (Alur, Kupferman & Henzinger 1998) ATL model checking is P-complete, and can be done in time linear in the size of the model and the length of the formula.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 63/70

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SLIDE 163
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Complexity od Model Checking ATL

Theorem (Alur, Kupferman & Henzinger 1998) ATL model checking is P-complete, and can be done in time linear in the size of the model and the length of the formula. So, let’s model-check!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 63/70

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SLIDE 164
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Complexity od Model Checking ATL

Theorem (Alur, Kupferman & Henzinger 1998) ATL model checking is P-complete, and can be done in time linear in the size of the model and the length of the formula. So, let’s model-check! Not as easy as it seems.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 63/70

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SLIDE 165
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Nice results: model checking ATL is tractable.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 64/70

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SLIDE 166
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Nice results: model checking ATL is tractable. But: the result is relative to the size of the model and the formula

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 64/70

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SLIDE 167
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Nice results: model checking ATL is tractable. But: the result is relative to the size of the model and the formula Well known catch: size of models is exponential wrt a higher-level description

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 64/70

slide-168
SLIDE 168
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Nice results: model checking ATL is tractable. But: the result is relative to the size of the model and the formula Well known catch: size of models is exponential wrt a higher-level description Another problem: transitions are labeled So: the number of transitions can be exponential in the number of agents.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 64/70

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SLIDE 169
  • 2. Reasoning about Coalitions
  • 5. Model Checking

3 agents/attributes, 12 states, 216 transitions

nofuel roL caR fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK nofuel fuelOK

1 5 6 2 3 4 8 7 9 10 12 11

roL roP roL roL roL roL roP roP roP roP roP caL caL caL caL caR caR caR caP caP caP caP

< > load ,nop ,fuel

1 2

< > unload ,unload ,fuel

1 2

< > nop ,nop ,nop

1 2 3

< > load ,unload ,nop

1 2 3

< > nop ,unload ,load

1 2 3

< > unload ,unload ,nop

1 2 3

< > unload ,nop ,nop

1 2 3

< > unload ,nop ,fuel

1 2

< > load ,unload ,fuel

1 2

< > nop ,nop ,fuel

1 2

< > nop ,unload ,fuel

1 2

< > nop ,nop ,load

1 2 3

< > load ,nop ,load

1 2 3

< > load ,unload ,load

1 2 3

< > load ,nop ,nop

1 2 3

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 65/70

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SLIDE 170
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Model Checking Temporal & Strategic Logics

m, l n, k, l nlocal, k, l CTL ATL CSL

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 66/70

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SLIDE 171
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Model Checking Temporal & Strategic Logics

m, l n, k, l nlocal, k, l CTL P [1] P [1] PSPACE [2] ATL CSL

[1] Clarke, Emerson & Sistla (1986). [2] Kupferman, Vardi & Wolper (2000).

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 66/70

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SLIDE 172
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Model Checking Temporal & Strategic Logics

m, l n, k, l nlocal, k, l CTL P [1] P [1] PSPACE [2] ATL P [3] ∆P

3 [5,6]

EXPTIME [8,9] CSL

[1] Clarke, Emerson & Sistla (1986). [2] Kupferman, Vardi & Wolper (2000). [3] Alur, Henzinger & Kupferman (2002). [5] Jamroga & Dix (2005). [6] Laroussinie, Markey & Oreiby (2006). [8] Hoek, Lomuscio & Wooldridge (2006).

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 66/70

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SLIDE 173
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Model Checking Temporal & Strategic Logics

m, l n, k, l nlocal, k, l CTL P [1] P [1] PSPACE [2] ATL P [3] ∆P

3 [5,6]

EXPTIME [8,9] CSL ∆P

2 [4,7]

∆P

3 [7]

PSPACE [9]

[1] Clarke, Emerson & Sistla (1986). [2] Kupferman, Vardi & Wolper (2000). [3] Alur, Henzinger & Kupferman (2002). [4] Schobbens (2004). [5] Jamroga & Dix (2005). [6] Laroussinie, Markey & Oreiby (2006). [7] Jamroga & Dix (2007). [8] Hoek, Lomuscio & Wooldridge (2006). [9] Jamroga & Ågotnes (2007).

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 66/70

slide-174
SLIDE 174
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Main message: Complexity is very sensitive to the context!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 67/70

slide-175
SLIDE 175
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Main message: Complexity is very sensitive to the context! In particular, the way we define the input, and measure its size, is crucial.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 67/70

slide-176
SLIDE 176
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Even if model checking appears very easy, it can be very hard.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 68/70

slide-177
SLIDE 177
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Even if model checking appears very easy, it can be very hard. Still, people do automatic model checking!

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 68/70

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SLIDE 178
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Even if model checking appears very easy, it can be very hard. Still, people do automatic model checking! LTL: SPIN

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 68/70

slide-179
SLIDE 179
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Even if model checking appears very easy, it can be very hard. Still, people do automatic model checking! LTL: SPIN CTL/ATL: MOCHA, MCMAS, VeriCS

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 68/70

slide-180
SLIDE 180
  • 2. Reasoning about Coalitions
  • 5. Model Checking

Even if model checking appears very easy, it can be very hard. Still, people do automatic model checking! LTL: SPIN CTL/ATL: MOCHA, MCMAS, VeriCS Even if model checking is theoretically hard, it can be feasible in practice.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 68/70

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SLIDE 181
  • 2. Reasoning about Coalitions
  • 6. References

2.6 References

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 69/70

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SLIDE 182
  • 2. Reasoning about Coalitions
  • 6. References

[Alur et al. 2002] R. Alur, T. A. Henzinger, and

  • O. Kupferman.

Alternating-time Temporal Logic. Journal of the ACM, 49:672–713, 2002. [Emerson 1990] E. A. Emerson. Temporal and modal logic. Handbook of Theoretical Computer Science, volume B, 995–1072. Elsevier, 1990. [Fisher 2006] Fisher, M.. Temporal Logics. Kluwer, 2006. [Jamroga and Ågotnes 2007] W. Jamroga and T. Ågotnes. Constructive knowledge: What agents can achieve under incomplete information. Journal of Applied Non-Classical Logics, 17(4):423–475, 2007.

Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 70/70