Beyond Admissibility : Dominance between chains of strategies - - PowerPoint PPT Presentation

Beyond Admissibility : Dominance between chains of strategies

Marie van den Bogaard Joint work with N. Basset, I. Jecker, A.Pauly & J.-F. Raskin

Université Libre de Bruxelles

CSL 2018, Birmingham September 7th, 2018

MvdB (ULB) Beyond Admissibility CSL 2018 1 / 23

Beyond Admissibility : Dominance between chains of strategies

Marie van den Bogaard1 Joint work with N. Basset2, I. Jecker1, A.Pauly3 & J.-F. Raskin1

Presented at CSL 2018

1Université Libre de Bruxelles, 2Université Grenoble Alpes, 3Swansea University MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 1 / 23

Beyond Admissibility : Dominance between chains of strategies

Marie van den Bogaard Joint work with N. Basset, I. Jecker, A.Pauly & J.-F. Raskin

Université Libre de Bruxelles

CSL 2018, Birmingham September 7th, 2018

MvdB (ULB) Beyond Admissibility CSL 2018 1 / 23

Beyond Admissibility : Dominance between chains of strategies

Marie van den Bogaard1 Joint work with N. Basset2, I. Jecker1, A.Pauly3 & J.-F. Raskin1

Presented at CSL 2018

1Université Libre de Bruxelles, 2Université Grenoble Alpes, 3Swansea University MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 1 / 23

Beyond Admissibility : Dominance between chains of strategies

Marie van den Bogaard Joint work with N. Basset, I. Jecker, A.Pauly & J.-F. Raskin

Université Libre de Bruxelles

CSL 2018, Birmingham September 7th, 2018

MvdB (ULB) Beyond Admissibility CSL 2018 1 / 23

Beyond Admissibility : Dominance between chains of strategies

Marie van den Bogaard1 Joint work with N. Basset2, I. Jecker1, A.Pauly3 & J.-F. Raskin1

Presented at CSL 2018

1Université Libre de Bruxelles, 2Université Grenoble Alpes, 3Swansea University MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 1 / 23

What does it mean to act rationally in an interactive scenario?

(especially when there is no obvious optimal choice?)

Beyond Admissibility : Dominance between chains of strategies

Marie van den Bogaard Joint work with N. Basset, I. Jecker, A.Pauly & J.-F. Raskin

Université Libre de Bruxelles

CSL 2018, Birmingham September 7th, 2018

MvdB (ULB) Beyond Admissibility CSL 2018 1 / 23

Beyond Admissibility : Dominance between chains of strategies

Marie van den Bogaard1 Joint work with N. Basset2, I. Jecker1, A.Pauly3 & J.-F. Raskin1

Presented at CSL 2018

1Université Libre de Bruxelles, 2Université Grenoble Alpes, 3Swansea University MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 1 / 23

What does it mean to act rationally in an interactive scenario?

(especially when there is no obvious optimal choice?)

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Is there a strategy that lets me enter regardless of the doorman actions?

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

O H D C

NO.

Is there a strategy that lets me enter regardless of the doorman actions?

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

And now? What is the rational choice?

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

S0: do not even try

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

S1: try once, then go home if not successful

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

S1: try once, then go home if not successful S1 does better than S0 S1 dominates S0

s0 s1

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Sk: try k times, then go home if not successful

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

Sk: try k times, then go home if not successful Sk dominates Sk-1 Sk is dominated by Sk+1

sk-1 sk sk+1

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

S𝞉: never give up

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility CSL 2018 2 / 23

Let’s play. . .

With the point of view of the first (circle) player

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 2 / 23

O H D C

S𝞉: never give up S𝞉 dominates every Sk & S𝞉 is not dominated by any Sk S𝞉 is admissible

sk S𝞉

&

S𝞉 Sk

Dominance

s ª sÕ

A strategy s is dominated by a strategy sÕ (or sÕ dominates s) if : (a) for every strategy (profile) τ of the other player(s) : p(s, τ) Æ p(sÕ, τ) "s’ is always as good as s" (b) there exists a strategy (profile) τ of the other player(s) such that p(s, τ) < p(sÕ, τ) "s’ sometimes better than s"

MvdB (ULB) Beyond Admissibility CSL 2018 3 / 23

Dominance

s ª sÕ

A strategy s is dominated by a strategy sÕ (or sÕ dominates s) if : (a) for every strategy (profile) τ of the other player(s) : p(s, τ) Æ p(sÕ, τ) "s’ is always as good as s" (b) there exists a strategy (profile) τ of the other player(s) such that p(s, τ) < p(sÕ, τ) "s’ sometimes better than s"

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 3 / 23

Dominance

s ª sÕ

A strategy s is dominated by a strategy sÕ (or sÕ dominates s) if : (a) for every strategy (profile) τ of the other player(s) : p(s, τ) Æ p(sÕ, τ) "s’ is always as good as s" (b) there exists a strategy (profile) τ of the other player(s) such that p(s, τ) < p(sÕ, τ) "s’ sometimes better than s" If only (a) holds, then s ∞ sÕ : strategy sÕ weakly dominates strategy s.

MvdB (ULB) Beyond Admissibility CSL 2018 3 / 23

Dominance

s ª sÕ

A strategy s is dominated by a strategy sÕ (or sÕ dominates s) if : (a) for every strategy (profile) τ of the other player(s) : p(s, τ) Æ p(sÕ, τ) "s’ is always as good as s" (b) there exists a strategy (profile) τ of the other player(s) such that p(s, τ) < p(sÕ, τ) "s’ sometimes better than s" If only (a) holds, then s ∞ sÕ : strategy sÕ weakly dominates strategy s.

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 3 / 23

Admissibility

A strategy s is admissible if it is not dominated by any other strategy : for every sÕ, we have s ”ª sÕ.

MvdB (ULB) Beyond Admissibility CSL 2018 4 / 23

Admissibility

A strategy s is admissible if it is not dominated by any other strategy : for every sÕ, we have s ”ª sÕ.

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 4 / 23

O H D C

S𝞉 is not dominated by any Sk

S𝞉 Sk

Let’s play again . . .

MvdB (ULB) Beyond Admissibility CSL 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility CSL 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 5 / 23

O H D C

Let’s play again . . .

MvdB (ULB) Beyond Admissibility CSL 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 5 / 23

O H D C

Let’s play again . . .

MvdB (ULB) Beyond Admissibility CSL 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 5 / 23

O H D C

2 1

Let’s play again . . .

MvdB (ULB) Beyond Admissibility CSL 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 5 / 23

O H D C

2 1

What is a rational strategy?

Let’s play again . . .

MvdB (ULB) Beyond Admissibility CSL 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 5 / 23

O H D C

2 1

Sk: try k times, then go home if not successful What is a rational strategy? Sk dominates Sk-1 Sk is dominated by Sk+1

sk-1 sk sk+1

Let’s play again . . .

MvdB (ULB) Beyond Admissibility CSL 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 5 / 23

O H D C

1

S𝞉: never give up S𝞉 is not dominated by any Sk (admissible) but S𝞉 does not dominate any Sk What is a rational strategy?

sk S𝞉

&

S𝞉 Sk

Let’s play again . . .

MvdB (ULB) Beyond Admissibility CSL 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 5 / 23

O H D C

2 1

S𝞉 is not the only rational choice ! admissibility criterion is too restrictive S𝞉 and Sk incomparable w.r.t. dominance

Let’s play again . . .

MvdB (ULB) Beyond Admissibility CSL 2018 5 / 23

Let’s play again . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 5 / 23

O H D C

2 1

Now what?

Some formalities

Model : G = ÈP, G, (pi)i∈PÍ multiplayer turn-based games on finite graphs game graph : G = (V = ‡i∈PVi, E) Player i strategies : Σi = {s : V ∗Vi æ V } (from histories to vertices) payoff functions : pi : V ω æ R (from outcomes to reals) Key points : focus on one player point of view no "adversarial opponent" hypothesis :

no assumptions about the other player(s) objectives / preferences

MvdB (ULB) Beyond Admissibility CSL 2018 6 / 23

Some formalities

Model : G = ÈP, G, (pi)i∈PÍ multiplayer turn-based games on finite graphs game graph : G = (V = ‡i∈PVi, E) Player i strategies : Σi = {s : V ∗Vi æ V } (from histories to vertices) payoff functions : pi : V ω æ R (from outcomes to reals) Key points : focus on one player point of view no "adversarial opponent" hypothesis :

no assumptions about the other player(s) objectives / preferences

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 6 / 23

Boolean case

Admissibility is a good criterion of rationality in the boolean case : always exist for ω-regular winning objectives admissible strategies coincide with winning strategies (when these exist) every strategy is either : admissible or dominated by an admissible strategy

MvdB (ULB) Beyond Admissibility CSL 2018 7 / 23

Boolean case

Admissibility is a good criterion of rationality in the boolean case : always exist for ω-regular winning objectives admissible strategies coincide with winning strategies (when these exist) every strategy is either : admissible or dominated by an admissible strategy

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 7 / 23

Fundamental property!

Boolean case

Admissibility is a good criterion of rationality in the boolean case : always exist for ω-regular winning objectives admissible strategies coincide with winning strategies (when these exist) every strategy is either : admissible or dominated by an admissible strategy Bonus : iteration, synthesis . . . [Berwanger ’07, Faella ’09, Raskin et al.+ ]

MvdB (ULB) Beyond Admissibility CSL 2018 7 / 23

Boolean case

Admissibility is a good criterion of rationality in the boolean case : always exist for ω-regular winning objectives admissible strategies coincide with winning strategies (when these exist) every strategy is either : admissible or dominated by an admissible strategy Bonus : iteration, synthesis . . . [Berwanger ’07, Faella ’09, Raskin et al.+ ]

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 7 / 23

Fundamental property!

Quantitative case

MvdB (ULB) Beyond Admissibility CSL 2018 8 / 23

Quantitative case

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 8 / 23

O H D C

Quantitative case

MvdB (ULB) Beyond Admissibility CSL 2018 8 / 23

Quantitative case

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 8 / 23

O H D C

s0

Quantitative case

MvdB (ULB) Beyond Admissibility CSL 2018 8 / 23

Quantitative case

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 8 / 23

O H D C

s0 s1

Quantitative case

MvdB (ULB) Beyond Admissibility CSL 2018 8 / 23

Quantitative case

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 8 / 23

O H D C

s0 s1 s2

Quantitative case

MvdB (ULB) Beyond Admissibility CSL 2018 8 / 23

Quantitative case

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 8 / 23

O H D C

s0 s1 s2 s3

Quantitative case

MvdB (ULB) Beyond Admissibility CSL 2018 8 / 23

Quantitative case

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 8 / 23

O H D C

s0 s1 s2 s3 sk

Quantitative case

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 8 / 23

O H D C

s0 s1 s2 s3 sk s𝞉

Quantitative case

As soon as there are 3+ payoffs : Admissible strategies represent rational choices but . . . they do not always exist, even when they do, they do not cover all rational behaviours, no guarantee to satisfy the fundamental property : dominated strategies not dominated by an admissible strategy exist [Brenguier, Perez, Raskin, Sankur FSTTCS’16]

MvdB (ULB) Beyond Admissibility CSL 2018 9 / 23

Quantitative case

As soon as there are 3+ payoffs : Admissible strategies represent rational choices but . . . they do not always exist, even when they do, they do not cover all rational behaviours, no guarantee to satisfy the fundamental property : dominated strategies not dominated by an admissible strategy exist [Brenguier, Perez, Raskin, Sankur FSTTCS’16]

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 9 / 23 s0 s1 s2 s3 s𝞉 sk

Quantitative case

New approach : shift from singleton strategy analysis to consider families of strategies Idea : cover rational behaviours dismissed by admissibility criterion

MvdB (ULB) Beyond Admissibility CSL 2018 10 / 23

Quantitative case

New approach : shift from singleton strategy analysis to consider families of strategies Idea : cover rational behaviours dismissed by admissibility criterion

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 10 / 23 s0 s1 s2 s3 s𝞉 sk

Quantitative case

New approach : shift from singleton strategy analysis to consider families of strategies Idea : cover rational behaviours dismissed by admissibility criterion

MvdB (ULB) Beyond Admissibility CSL 2018 10 / 23

Quantitative case

New approach : shift from singleton strategy analysis to consider families of strategies Idea : cover rational behaviours dismissed by admissibility criterion

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 10 / 23 s0 s1 s2 s3 s𝞉 sk s0 s1 s2 s3 s𝞉 sk

Chains of strategies (sequences of strategies ordered by dominance)

Chains of strategies in (Σi, ∞)

A chain of strategies (sα)α<β is a sequence of strategies, indexed by an

• rdinal β > 0, that respects the dominance quasiorder :

for every α, αÕ < β such that α < αÕ, we have sα ∞ sαÕ. Increasing chain : (sα)α<β such that sα ª sαÕ for every α < αÕ.

MvdB (ULB) Beyond Admissibility CSL 2018 11 / 23

Chains of strategies in (Σi, ∞)

A chain of strategies (sα)α<β is a sequence of strategies, indexed by an

• rdinal β > 0, that respects the dominance quasiorder :

for every α, αÕ < β such that α < αÕ, we have sα ∞ sαÕ. Increasing chain : (sα)α<β such that sα ª sαÕ for every α < αÕ.

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 11 / 23 s0 s1 s2 s3 sk

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility CSL 2018 12 / 23

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 12 / 23

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility CSL 2018 12 / 23

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 12 / 23 s’0 s’1 s’2 s’3 s'k s0 s1 s2 s3 sk

(s’) (s)

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility CSL 2018 12 / 23

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 12 / 23 s’0 s’1 s’2 s’3 s'k s0 s1 s2 s3 sk

(s’) (s)

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility CSL 2018 12 / 23

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 12 / 23 s’0 s’1 s’2 s’3 s'k s0 s1 s2 s3 sk

(s’) (s)

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility CSL 2018 12 / 23

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 12 / 23 s’0 s’1 s’2 s’3 s'k s0 s1 s2 s3 sk

(s’) (s)

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility CSL 2018 12 / 23

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 12 / 23 s’0 s’1 s’2 s’3 s'k s0 s1 s2 s3 sk

(s’) (s)

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

A chain (sα)α<β is maximal if for every chain (sÕ

αÕ)αÕ<βÕ, we have

(sα)α<β ı (sÕ

αÕ)αÕ<βÕ ∆ (sÕ αÕ)αÕ<βÕ ı (sα)α<β

.

MvdB (ULB) Beyond Admissibility CSL 2018 12 / 23

Dominance between chains :

A chain (sα)α<β is weakly dominated by a chain (sÕ

αÕ)αÕ<βÕ if :

for every α < β, there exists αÕ < βÕ such that sα ∞ sαÕ. (sα)α<β ı (sÕ

αÕ)αÕ<βÕ

A chain (sα)α<β is maximal if for every chain (sÕ

αÕ)αÕ<βÕ, we have

(sα)α<β ı (sÕ

αÕ)αÕ<βÕ ∆ (sÕ αÕ)αÕ<βÕ ı (sα)α<β

.

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 12 / 23 s’0 s’1 s’2 s’3 s'k s0 s1 s2 s3 sk

(s’) (s)

Chains of chains !

Considering (IC(Σi), ı) : increasing chains of strategies and the quasi-order ı, we can build chains of chains of strategies : (s2k)k<ω ı (s2k+1)k<ω ı (sk)k<ω

MvdB (ULB) Beyond Admissibility CSL 2018 13 / 23

Chains of chains !

Considering (IC(Σi), ı) : increasing chains of strategies and the quasi-order ı, we can build chains of chains of strategies : (s2k)k<ω ı (s2k+1)k<ω ı (sk)k<ω

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 13 / 23

O H D C

Chains of chains !

Considering (IC(Σi), ı) : increasing chains of strategies and the quasi-order ı, we can build chains of chains of strategies : (s2k)k<ω ı (s2k+1)k<ω ı (sk)k<ω

MvdB (ULB) Beyond Admissibility CSL 2018 13 / 23

Chains of chains !

Considering (IC(Σi), ı) : increasing chains of strategies and the quasi-order ı, we can build chains of chains of strategies : (s2k)k<ω ı (s2k+1)k<ω ı (sk)k<ω

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 13 / 23

O H D C

Chains of chains !

Considering (IC(Σi), ı) : increasing chains of strategies and the quasi-order ı, we can build chains of chains of strategies : (s2k)k<ω ı (s2k+1)k<ω ı (sk)k<ω

MvdB (ULB) Beyond Admissibility CSL 2018 13 / 23

Chains of chains !

Considering (IC(Σi), ı) : increasing chains of strategies and the quasi-order ı, we can build chains of chains of strategies : (s2k)k<ω ı (s2k+1)k<ω ı (sk)k<ω

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 13 / 23

O H D C

A sufficient criterion. . .

. . .to recover a similar fundamental property as in the boolean case :

Theorem

If the chains of chains of strategies have at most a countable number of elements (chains of strategies), then every chain of strategies is either maximal or dominated by a maximal chain of strategies.

MvdB (ULB) Beyond Admissibility CSL 2018 14 / 23

A sufficient criterion. . .

. . .to recover a similar fundamental property as in the boolean case :

Theorem

If the chains of chains of strategies have at most a countable number of elements (chains of strategies), then every chain of strategies is either maximal or dominated by a maximal chain of strategies.

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 14 / 23

A sufficient criterion. . .

. . .to recover a similar fundamental property as in the boolean case :

Theorem

If the chains of chains of strategies have at most a countable number of elements (chains of strategies), then every chain of strategies is either maximal or dominated by a maximal chain of strategies. Some proof ingredients : (i) every increasing chain has countable length (ii) every increasing chain of increasing chains has an upper bound (iii) Zorn’s Lemma !

MvdB (ULB) Beyond Admissibility CSL 2018 14 / 23

A sufficient criterion. . .

. . .to recover a similar fundamental property as in the boolean case :

Theorem

If the chains of chains of strategies have at most a countable number of elements (chains of strategies), then every chain of strategies is either maximal or dominated by a maximal chain of strategies. Some proof ingredients : (i) every increasing chain has countable length (ii) every increasing chain of increasing chains has an upper bound (iii) Zorn’s Lemma !

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 14 / 23

A sufficient criterion. . .

(i) every increasing chain has countable length : as each strategy s is an increasing chain : (sk)k<1

MvdB (ULB) Beyond Admissibility CSL 2018 15 / 23

A sufficient criterion. . .

(i) every increasing chain has countable length : as each strategy s is an increasing chain : (sk)k<1

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 15 / 23

A sufficient criterion. . .

(i) every increasing chain has countable length : as each strategy s is an increasing chain : (sk)k<1 (ii) every increasing chain of increasing chains has an upper bound :

MvdB (ULB) Beyond Admissibility CSL 2018 15 / 23

A sufficient criterion. . .

(i) every increasing chain has countable length : as each strategy s is an increasing chain : (sk)k<1 (ii) every increasing chain of increasing chains has an upper bound :

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 15 / 23

𝞉 𝞉

A sufficient criterion. . .

(i) every increasing chain has countable length : as each strategy s is an increasing chain : (sk)k<1 (ii) every increasing chain of increasing chains has an upper bound :

MvdB (ULB) Beyond Admissibility CSL 2018 15 / 23

A sufficient criterion. . .

(i) every increasing chain has countable length : as each strategy s is an increasing chain : (sk)k<1 (ii) every increasing chain of increasing chains has an upper bound :

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 15 / 23

𝞉 𝞉

A sufficient criterion. . .

(iii) Zorn’s Lemma : Let (X, Ù) be a (quasi-)ordered set. If every chain in (X, Ù) has an upper bound, then every element of X is below a maximal element.

MvdB (ULB) Beyond Admissibility CSL 2018 16 / 23

A sufficient criterion. . .

(iii) Zorn’s Lemma : Let (X, Ù) be a (quasi-)ordered set. If every chain in (X, Ù) has an upper bound, then every element of X is below a maximal element.

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 16 / 23

A sufficient criterion. . .

(iii) Zorn’s Lemma : Let (X, Ù) be a (quasi-)ordered set. If every chain in (X, Ù) has an upper bound, then every element of X is below a maximal element. Apply to (IC(Σi), ı)

MvdB (ULB) Beyond Admissibility CSL 2018 16 / 23

A sufficient criterion. . .

(iii) Zorn’s Lemma : Let (X, Ù) be a (quasi-)ordered set. If every chain in (X, Ù) has an upper bound, then every element of X is below a maximal element. Apply to (IC(Σi), ı)

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 16 / 23

A sufficient criterion. . .

(iii) Zorn’s Lemma : Let (X, Ù) be a (quasi-)ordered set. If every chain in (X, Ù) has an upper bound, then every element of X is below a maximal element. Apply to (IC(Σi), ı)

Theorem

If the chains of chains of strategies have at most a countable number of elements (chains of strategies), then every chain of strategies is either maximal or dominated by a maximal chain of strategies.

MvdB (ULB) Beyond Admissibility CSL 2018 16 / 23

A sufficient criterion. . .

(iii) Zorn’s Lemma : Let (X, Ù) be a (quasi-)ordered set. If every chain in (X, Ù) has an upper bound, then every element of X is below a maximal element. Apply to (IC(Σi), ı)

Theorem

If the chains of chains of strategies have at most a countable number of elements (chains of strategies), then every chain of strategies is either maximal or dominated by a maximal chain of strategies.

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 16 / 23

A "trick" ?

MvdB (ULB) Beyond Admissibility CSL 2018 17 / 23

A "trick" ?

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 17 / 23

A "trick" ?

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 17 / 23

O H D C Q

Length 𝞉1 !

(fn)0

s1 s2 s3 sk

(fn)1 (fn)2 (fn)3 (fn)𝞉

A "trick" ?

A way to ensure chains of chains with countable numbers of elements : Restrict to a countable set of strategies from the start !

MvdB (ULB) Beyond Admissibility CSL 2018 17 / 23

A "trick" ?

A way to ensure chains of chains with countable numbers of elements : Restrict to a countable set of strategies from the start !

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 17 / 23

A "trick" ?

A way to ensure chains of chains with countable numbers of elements : Restrict to a countable set of strategies from the start ! Sufficient to cover finite-memory strategies

MvdB (ULB) Beyond Admissibility CSL 2018 17 / 23

A "trick" ?

A way to ensure chains of chains with countable numbers of elements : Restrict to a countable set of strategies from the start ! Sufficient to cover finite-memory strategies

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 17 / 23

A "trick" ?

A way to ensure chains of chains with countable numbers of elements : Restrict to a countable set of strategies from the start ! Sufficient to cover finite-memory strategies Mealy automaton :

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 17 / 23

O H D C

Parameterized automata to handle chains of strategies

Parameterized automaton : Mealy automaton with a single counter in counter-access states : transition depends on the counter-value being > 0 or = 0.

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 18 / 23

O H D C

Parameterized automata to handle chains of strategies

A chain is uniform if it is realized by a parameterized automaton

MvdB (ULB) Beyond Admissibility CSL 2018 19 / 23

Parameterized automata to handle chains of strategies

A chain is uniform if it is realized by a parameterized automaton

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 19 / 23

O H D C

s0 s1 s2 s3 sk

Generalised safety/reachability games

Games equipped with a set of leaves such that ending in leaf ¸n yields payoff n (∈ Z), while avoiding them yields payoff 0.

MvdB (ULB) Beyond Admissibility CSL 2018 20 / 23

Generalised safety/reachability games

Games equipped with a set of leaves such that ending in leaf ¸n yields payoff n (∈ Z), while avoiding them yields payoff 0.

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 20 / 23

O H D C

Leaf l1 Leaf l2

{ {

Parameterized automata to handle chains of strategies

In generalised safety/reachability games, considering finite-memory strategies : every dominated f.-m. strategy is dominated by an admissible f.-m. strategy or by a maximal uniform chain given a parameterized automaton, it is decidable whether it realizes an (increasing) chain dominance between two strategies is decidable dominance between two uniform chains is decidable

MvdB (ULB) Beyond Admissibility CSL 2018 21 / 23

Parameterized automata to handle chains of strategies

In generalised safety/reachability games, considering finite-memory strategies : every dominated f.-m. strategy is dominated by an admissible f.-m. strategy or by a maximal uniform chain given a parameterized automaton, it is decidable whether it realizes an (increasing) chain dominance between two strategies is decidable dominance between two uniform chains is decidable

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 21 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . .

MvdB (ULB) Beyond Admissibility CSL 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis :

MvdB (ULB) Beyond Admissibility CSL 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis :

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour

MvdB (ULB) Beyond Admissibility CSL 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . .

MvdB (ULB) Beyond Admissibility CSL 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . .

MvdB (ULB) Beyond Admissibility CSL 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . .

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

(s) is maximal

• r dominated

by (s’) maximal

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . . . . .when games satisfy a sufficient criterion

MvdB (ULB) Beyond Admissibility CSL 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . . . . .when games satisfy a sufficient criterion

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . . . . .when games satisfy a sufficient criterion

MvdB (ULB) Beyond Admissibility CSL 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . . . . .when games satisfy a sufficient criterion

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

Chains of chains have countably many elements

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . . . . .when games satisfy a sufficient criterion In practice : stick to finite-memory strategies

MvdB (ULB) Beyond Admissibility CSL 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . . . . .when games satisfy a sufficient criterion In practice : stick to finite-memory strategies

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . . . . .when games satisfy a sufficient criterion In practice : stick to finite-memory strategies Ò → Parameterized automata to handle chains

MvdB (ULB) Beyond Admissibility CSL 2018 22 / 23

To sum-up

Admissibility works well as a rationality measure in the boolean case. . . . . .but fails in the quantitative one Ò → Departure from the singleton strategy analysis : Chains of strategies to represent "types" of behaviour Possible to recover the fundamental property . . . . . .when games satisfy a sufficient criterion In practice : stick to finite-memory strategies Ò → Parameterized automata to handle chains

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 22 / 23

Perspectives

Algorithm to decide if an uniform chain is maximal ?

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 23 / 23

Perspectives

Algorithm to decide if an uniform chain is maximal ? Pursue parameterized game model investigation

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 23 / 23

Perspectives

Algorithm to decide if an uniform chain is maximal ? Pursue parameterized game model investigation Compare chain analysis approach with other rationality criteria (regret ?)

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 23 / 23

Perspectives

Algorithm to decide if an uniform chain is maximal ? Pursue parameterized game model investigation Compare chain analysis approach with other rationality criteria (regret ?) Scratch on the surface of quantitative games :

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 23 / 23

Perspectives

Algorithm to decide if an uniform chain is maximal ? Pursue parameterized game model investigation Compare chain analysis approach with other rationality criteria (regret ?) Scratch on the surface of quantitative games : new approach to tend towards other rationality criteria ?

MvdB (ULB) Beyond Admissibility GT ALGA, October 16th 2018 23 / 23