[PPT] - Subgame Perfect Equilibrium Quantitative Reachability Games - PowerPoint Presentation

SLIDE 1

Subgame Perfect Equilibrium Quantitative Reachability Games

Jean

Francois Raskin

Universite libre de Bruxelles

IFIP WG 2.2 meeting 2019

Vienna

Partially

based on recent

works thatappeared in

CSL '15 , Foss Acs '17, GANDALF '18 , GNCor '19

together

with Veronique Bruyeie , Thomas Brihaye , Noemi Meunier, Aline Goemine, Arno Pauly , Stephane Le Rouse, Marie Vanden Bogard .

SLIDE 2

Non zero-sum reachability games

.

8¥:::

# t)

at

a

O

8

/)

at.

o

O O

U

SLIDE 3

Non zero-sum reachability games Need for other solution concepts : NE and SPE

Player -

Ot

D Players

IN

ato

Li V J

①

⇒ no winning strategy for

Peo

/)

at.

o

O O

⇒ no winning strategy for

Pfs

U

SLIDE 4

Nash equilibrium

d.

Beef

:

Soo .rs> is

a

NE if

⑦

there is

no unilateral

✓ /)

pfikablederiakon.ie

.

Veloso

. .rs

) >Vallo

. ' .oh , too ' ↳ . v

t

Vala too , Ed > Vodka ,Ei7 , to

'

O

④

O

" !)

°

④

O

U

SLIDE 5

Nash equilibrium

I

Def

:

Soo .rs) is

a

NE if

⑦

there is

no unilateral

✓ /)

pfikablederiakon.ie

.

Val

. So . .rs

) >Vallo

. ' .oh , too ' ↳ . v

t

Val, too , Ed > Vol, Go ,Ei7 , to '

O

④

O

" !)

°

④ ①

esg

:

v. vivo

is a NE as

both

U

Players

win

SLIDE 6

Nash equilibrium

I

Def

:

Soo

. oh>

is

a

NE if

⑦

there is

no unilateral

✓ /)

pfirablederiarion.ie

.

Val

. So . . Ed >Vallo . ' .oh , too ' ↳ . v

t

Vala too , Ed > Vol, look > , to

'

④ ④

O

" !)

°

④

O

eg

: row; is a NE even if both

U

players fail to

win → no unilateral change

is profitable

.

SLIDE 7

Non credible threats

I

Def

:

Soo

, oh>

is

a

NE if

⑦

there is

no unilateral

✓ /)

pfikabkderiak.org

↳ v s

i.e

.

Valo Soo ,

> Vallo .:o) , too '

④

Val. look > uol.co.id , to

' " " / \

is

Not

SUBGAME PERFECT

④ ④

U

v, →

Ny

is not rational

= non credible threat

SLIDE 8

Subgame perfect equilibrium

I

Deff

:

Soo

, oh>

is subgame perfect if

⑦

there is

no unilateral

/ f)

profitable deviation in any

subgame

↳ v s

i. e

. for all histories h :

④ ④

O

en en

"

yo

i::÷s:÷¥:÷: :÷

.

④ ④

U

is the only sub

game perfectequilibrium

SLIDE 9

Theorems

NE

Reachability

(

Refn

%;!eggfe%

THEOREM .

NE always exist in

reachabilikggames-THEOREM.co

for

NE

is

NPcomplete

. REIT

I

musrwinrlfheir objectives

$BE

.

Reachability

true for all co - regular

/

bjectives

. THEOREM .

SPE always exist in reachabilkggames

THEOREM .

Constrained existence for

SPE

is

PSPACE - C

.

I

for

regular objectives

some complexity gaps exist . . .

SLIDE 10

Quantitative reachability games

↳

Minimize the member of steps

to

reach Karger

d

⑦

CEP :

⑥ ¥

,

" Constrained existence problem "

\ /

FF

: . T

is

a

NEISPE

④

. Valle) EF

412

be

B)

k

④

✓

¥2

an upper bound for each player

To

T s

U

SLIDE 11

NE - Tool: zero-sum value

WORST-CASEVAl.VE#

*

:

V - thru { to}

⑤ to

if

v belongs

to player i

/ \

p

then

Nv)

= worst-case value

5

thatPlayer i can force four

.

④ 4

Y life

④

⑦

°

to?

V O

④

to

u

T s

SLIDE 12

NE - Tool: zero-sum value

WORST-CASEVALUE-K.ir

trutta}

I

⑦

to

if

v belongs

to player i

/ \

R

then

Nv)

worst-case value

5

thatPlayer icanforafomo

.

\ /

X-CoNsiSTEN

④ 4

.

4

fish

consistentif

:

ftp.

④

⑦

Vino

U

F

¥?

'I

✓

↳ for

a

players

④

i

.

U

Tz

SLIDE 13

NE - Tool: zero-sum value

WORST

CASE

VALUE

+

*

:

V - trutta}

I

⑦

to

if

v belongs

to player i

/ \

R

then

Nv)

worst-case value

5

thatPlayer icanforafomo

.

\ /

X-CoNsiSTEN

④ 4

.

4

fish

consistentif

:

ftp.

④

⑦

Vino

U

F

¥?

'I

✓

↳ for

a

players

⑤

age

:

To

is

X

consistentand it

T s

is

a

NE

.

SLIDE 14

NE - Tool: zero-sum value

⑤ to

¥

④ 4
LEE

U

ki?

°

eye

. :

To

U

is

X

consistentand it

Tz

is

a

NE

.

SLIDE 15

Why does consistency matters ? Why does consistency matter ?

*

.

an:÷÷¥:i÷ .

Outcomes of

NE

are

X - consistent

SLIDE 16

Why does consistency matter ?

X-

al

. I

'

in

> KID D

f

Kiba

err

Ev

= determinacy

K-its Dfw. 's

Outcomes of

NE

are

X - consistent

X

consistentoutcomes

are

utcomes of NE

SLIDE 17

Why does consistency matters ? Why does consistency matter ?

*

.

al

. I

%÷j÷

.

'

÷:

> KID D

f

Kiba

err

Ev

= determinacy

K-ID Dfw. 's

Outcomes of

NE

are

X - consistent

X

consistentoutcomes

are

utcomes of NE

SLIDE 18

NE - Tool: zero-sum value

⑤ to

f is

X

consistentif

:

to ¥ ¥3 ,

Vino

\ '

r

④ 4

.

'CI

¥+2

↳ for all players

U

.-

¥2

THEOREM : f is the outcome of a

NE iff

to

!

fnt-onsisren.RU

T s

SLIDE 19

SPE: subgame perfect value

I

⑦

¥

\ /

CEP :

⑤

4)

" Constrained existence problem "

④

Ed

U

FF

: . E

is

a

SPE

¥2

. Valle) EF to

be

high

Tse

U

SLIDE 20

SPE: subgame perfect value

E

.

¥

to

. 't . . . . .

→ seq

. of values

¥

¥2

°

rope

To Ts

U

SLIDE 21

SPE: subgame perfect value

⑤ to

+a # .

to

. 't .

- - ,

→ seq

. of values

\ /

to

⑤to

° Yt ?

⑦

Xo ( v) =

if

v is

in Tourneur)

¥?

{ to

therwise

. To ° Ts

U

SLIDE 22

SPE: subgame perfect value

⑤ to

*

to

to

. 't .

- -it

→ seq

. of values

\ /

to

④to

°g¥¥Z

th

f.If

non:

e.

Towner"

¥?

Update

: To °

(

th

consistency

T a

U

Xh+z(v)=

if vet

{stfemjn.agmaseEVALihi.pl/v'ffth}

✓

↳ then

. he potentially

Owner (v

) chooses

faces

worst-case

for

succentor

among Xp consistent

paths

.

SLIDE 23

⑤ to

+a #

\ 1/3

Efi

MIN

y

* 4

§µ

* ¥3

.

. . o

U

HEE

. DDD

To

°

Halil

Yeezy

Max ma,

T s

SLIDE 24

⑤ to

l l

.

to

\ /

3

Efi

MIN

④ 4

a K€

4?

000

To

°

{'

Yeezy

Max ma,

Tz

SLIDE 25

⑤ to

+a¥ ¥

,

\ /

3

Efi

MIN

+

+3

#\

* ¥2

.

. . o

U

top

.

000

to

°

E

Yeezy

Max

."

T s

SLIDE 26

⑤ to

4x¥ ¥

,

\ /

3

EP

"

MIN

④

3

①

a

5¥

¥?

000

To

°

sialic

Yet

Max ma,

T s

SLIDE 27

¥5

"¥

°

\ /

3

Efi

MIN

¥3

TIX

* ¥2

.

. . o

U

top

.

000

To

°

E

Yeezy

Max ma,

Tse

SLIDE 28

fined pinheaded

⑤ 5

"¥

°

\ /

3

Efi

MIN

¥3

IT

* ¥2

.

. . o

U

f?

000

To

°

{'

Yeezy

Max ma,

T s

SLIDE 29

fixed point

reached

⑤ 5

4¥

' '

'

④

3

41.2

④

to?

V O

④

is the ¥- consistentoutcome

to

u

T s

SLIDE 30

fixed point

reached

⑤ 5

it! ¥ ,

' '

'

④

3

41.2

④

U

¥2

'

ttteoremfztheoukmener.ofaspeeffJ.sn

°

to

u

T s

SLIDE 31

Termination

fixed point

reached

⑤ 5

X

: V → No { to

]

4¥ ¥

,

\ /

s

k s X if

the V : the Icu)

④

3 °

¥2

↳ well

queasy

rder

U

¥?

Update

:

f

monotone

° to

U

Better complexity

(Pspace

) through

Ts

bounds

n values

(exponential)

.

O ( / v1

( Nlt 3) ( III +D)

SLIDE 32

Theorems

NE

Quantitative Reachability

THEOREM .

NE always exist in reachability games

THEOREM .

Constrained existence for

NE

is

NPcomplete

.

SPE

Quantitative Reachability

THEOREM .

SPE always exist in reachability games

THEOREM .

Constrained existence for

SPE

is

PSPACE - C

.

→ exponentially large

→ fined point is compared

n extended graph

( v ,Pfs Players thathave already

seen their

bjective

( no nationality assumed) → bounds

n values

:

O ( / v1

( Nlt 3) ( t ITI +2))

SLIDE 33

Open questions

① Better bounds

n values ?

→ we have no examples with c x Nl . If I

SLIDE 34

Open questions

① Better bounds

n values ?

→ we have no examples with c x Nl . If I

②

Is

the problem FPT in the number of players ?

SLIDE 35

Open questions

① Better bounds

n values ?

→ we have no examples with

ex Nl

. If I

②

Is

the problem FPT in the number of players ?

③

Whatabout mean

payoff

. ?

value approach

extends readily to NE

, notto SPE

SLIDE 36

Open questions

① Better bounds

n values ?

→ we have no examples with

ex Nl

. If I

②

Is

the problem FPT in the number of players ?

③

Whatabout mean

payoff

. ?

value approach

extends readily to NE

, notto SPE

SPE

may not exist

:

A kid

a) s

kid

0=-0

O_0

t

SLIDE 37

Open questions

① Better bounds

n values ?

→ we have no examples with

ex Nl

. If I

②

Is

the problem FPT in the number of players ?

③

Whatabout mean

payoff

. ?

value approach

extends readily to NE

, notto SPE

SPE

may not exist

:

A kid

sj s

A

kid

0=-0

O_0

i

N E

E

f

←

→

a-gtfs.ua/-I

I

problem is open

.