Subgame Perfect Equilibrium Quantitative Reachability Games - Francois Raskin Jean Universite libre de Bruxelles WG 2.2 meeting IFIP 2019 Vienna Partially - based on recent works that appeared in ' 17 , ' 18 , ' 15 , Foss Acs GANDALF ' 19 CSL GNC or with Veronique Bruyeie together , Thomas Bri haye , Noemi Meunier , Aline Goemine , Arno Pauly Rouse , Marie Vanden Bogard , Stephane Le .
8¥ : : : Non zero-sum reachability games . o # t ) at a 8 O / ) - o at . - O O U
⇒ ⇒ 0 Player - Non zero-sum reachability games D Players Ot IN ato J no winning strategy for V Peo Li ① / ) at . - o no winning strategy for Pfs O O U Need for other solution concepts : NE and SPE
↳ Nash equilibrium NE if Beef Soo .rs > is d. a : no unilateral ⑦ there is pfikablederiakon.ie ✓ / ) ) > Vallo Veloso . oh ' ' , too . .rs . . Vala too , Ed > Vodka ,Ei7 , to " ! ) ' t v O ④ O . ° ④ O U
↳ Nash equilibrium NE if Def Soo .rs ) is I a : no unilateral ⑦ there is pfikablederiakon.ie ✓ / ) ) > Vallo Val . oh . So ' ' , too . .rs . . " ! ) Val , too , Ed > Vol , Go ,Ei7 , to ' t v O ④ O . ° ④ ① v. vivo both NE is esg as : a U Players win
↳ Nash equilibrium NE if Def Soo . oh > is I a : no unilateral ⑦ there is pfirablederiarion.ie ✓ / ) . . Ed > Vallo Val . oh . So ' ' , too . . " ! ) Vala too , Ed > Vol , look > , to ' t v O ④ ④ . ° even if both ④ O : row ; NE eg is a players fail to U win no unilateral change is profitable → .
↳ Non credible threats NE if Def , oh > Soo I is a : no unilateral ⑦ there is pfikabkderiak.org ✓ / ) > Vallo Valo Soo , . :o) ' , too i.e " / \ s v . ④ Val . look > uol.co.id , to ④ ' " is PERFECT Not SUB GAME ④ ④ U is not rational Ny v , → = non credible threat
↳ Subgame perfect equilibrium Deff is sub game perfect if Soo , oh > I : no unilateral ⑦ there is / f ) profitable deviation in any sub game . for all histories h i. e : s v O en ④ ④ en i ::÷s:÷¥:÷ : :÷ yo " . ④ ④ U is the only game perfect equilibrium sub
Theorems - Reachability Refn % ; !eggfe% ( NE NE always exist in reachabilikggames-THEOREM.co THEOREM . NP complete for NE is . I REIT musrwinrlfheir objectives - Reachability co - regular true for all $BE . / objectives SPE always exist in reachabilkggames . THEOREM . Constrained existence for SPE PSPACE - C THEOREM is . . I - regular objectives for o some complexity gaps exist . . .
↳ Quantitative reachability games - reach Karger Minimize the member of steps to d ⑦ CEP : ⑥ ¥ " Constrained existence problem " , \ / FF . T NEISPE is : a ④ . Valle ) EF k 412 B) be ④ ④ ✓ ¥2 an upper bound for each player To U T s
NE - Tool: zero-sum value WORST-CASEVAl.VE# V - thru { to } * : to player i v belongs ⑤ to if Nv ) = worst-case value then / \ p that Player i can force four 5 o . ④ 4 o Y life ④ ⑦ ° to ? V O ④ u to T s
NE - Tool: zero-sum value WORST-CASEVALUE-K.ir - trutta } to player i v belongs I if ⑦ to Nv ) - worst-case value then / \ R that Player icanforafomo 5 o . \ / X-CoNsiSTEN ④ 4 - o ftp. . fish - consistent if 4 : Vino ④ ⑦ F U ¥ ? ' I ↳ for players a ✓ o ④ U i . Tz
NE - Tool: zero-sum value - CASE WORST VALUE + V - trutta } * : to player i v belongs I if ⑦ to Nv ) - worst-case value then / \ R that Player icanforafomo 5 o . \ / X-CoNsiSTEN ④ 4 - o ftp. . fish - consistent if 4 : Vino ④ ⑦ F U ¥ ? ' I ↳ for players a ✓ o ⑤ age : X - consistent and it To is NE T s is a .
NE - Tool: zero-sum value ⑤ to ¥ o ④ 4 LEE o U ki ? ° eye . : X U - consistent and it is To NE is a . Tz
Why does consistency matters ? Why does consistency matter ? * . an :÷÷¥:i÷ . Outcomes of X - consistent NE are
Why does consistency matter ? X - . I al ' in > KID D f err Kiba = determinacy Ev K - its Dfw . 's Outcomes of outcomes of NE - consistent outcomes X X - consistent NE are are
Why does consistency matter ? * Why does consistency matters ? . . I al ÷ : %÷j÷ ' > KID D f err Kiba . = determinacy Ev K - ID Dfw . 's Outcomes of outcomes of NE - consistent outcomes X X - consistent NE are are
NE - Tool: zero-sum value ⑤ to X - consistent if to ¥ ¥3 f is : Vino o , \ ' r ④ 4 ' CI . ¥+2 ↳ for all players o U ¥2 . - fnt-onsisren.RU THEOREM : f is the outcome of NE iff a ! to T s
SPE: subgame perfect value I ⑦ ¥ \ / CEP : " Constrained existence problem ⑤ " 4) Ed ④ FF SPE . E U ¥2 is : a . Valle ) EF high be to U Tse
SPE: subgame perfect value E . 't . . of values to . → seq ¥ . . . . ¥ ¥2 ° rope To U Ts
SPE: subgame perfect value ⑤ to . 't . . of values to → seq + a # - - - , . \ / to ⑤ to ° Yt ? o if in Tourneur ) ⑦ Xo ( v ) { to is v = ¥ ? otherwise . ° To U Ts
SPE: subgame perfect value ⑤ to * - - - it . 't . . of values to → seq to o \ / to ④ to °g¥¥Z th Towner " - f. If non : ¥ ? e. Update - consistency : th ( ° To Xh+z(v)= { stfemjn.agmaseEVALihi.pl/v'ffth } if vet 0 U T a - . he potentially ✓ ↳ then ) chooses Owner ( v faces worst-case among Xp consistent for succentor paths .
⑤ to + a # o Efi \ 1/3 MIN §µ * 4 * ¥3 y . o o o . . . DDD U HEE Yeezy Max ma , Halil ° To T s
⑤ to l l o . to o Efi \ / MIN 3 a K€ ④ 4 000 4 ? Yeezy Max ma , { ' ° To Tz
⑤ to + a ¥ ¥ o , Efi \ / MIN 3 #\ +3 * ¥2 + . o o o . . 000 U top . Yeezy Max . " E ° to T s
⑤ to 4x¥ ¥ o , EP " \ / MIN 3 ① 5¥ ④ 3 a 000 ¥ ? Yet Max ma , sialic ° To T s
¥5 " ¥ ° Efi \ / MIN 3 TIX ¥3 * ¥2 . o o o . . 000 U top . Yeezy Max ma , E ° To Tse
fined pinheaded ⑤ 5 " ¥ ° Efi \ / MIN 3 IT ¥3 * ¥2 . o o o . . 000 U of ? Yeezy Max ma , { ' ° To T s
- reached fixed point ⑤ 5 4¥ o ' ' ' ④ 3 o 41.2 ④ ④ • to ? V O is the ¥ - consistent outcome ④ u to T s
- reached fixed point ⑤ 5 it ! ¥ o , ' ' ' ④ 3 o 41.2 ④ ④ U ¥2 ' ttteoremfztheoukmener.ofaspeeffJ.sn ° u to T s
- reached fixed point Termination ⑤ 5 → No { to : V ] X 4¥ ¥ s X if o , the V : the Icu ) k \ / s ④ 3 ¥2 ↳ well ° order queasy U ¥ ? - monotone Update f : Better complexity ° ) through ( Pspace U to ( exponential ) Ts on values bounds . ( Nlt 3) ( III + D) O ( / v1
Theorems - Quantitative Reachability NE NE always exist in reachability games THEOREM . Constrained existence for NP complete THEOREM NE is . . Quantitative Reachability SPE - SPE always exist in reachability games THEOREM . Constrained existence for SPE PSPACE - C THEOREM is . . → exponentially large on extended graph → fined point is compared ( v , Pfs Players that have already seen their objective ( no nationality assumed ) ( Nlt 3) ( t ITI +2 ) ) O ( / v1 → bounds on values :
Open questions we have no examples with on values ? ① Better bounds . If I c x Nl →
Open questions we have no examples with on values ? ① Better bounds . If I c x Nl → the problem FPT in the number of players ? ② Is
Open questions we have no examples with on values ? ① Better bounds . If I ex Nl → the problem FPT in the number of players ? ② Is - payoff . ? What about mean ③ extends readily to NE - value approach , not to SPE
Open questions we have no examples with on values ? ① Better bounds . If I ex Nl → the problem FPT in the number of players ? ② Is - payoff . ? What about mean ③ extends readily to NE - value approach , not to SPE may not exist - SPE : a ) s kid A kid 0=-0 O_0 - t
Open questions we have no examples with on values ? ① Better bounds . If I ex Nl → the problem FPT in the number of players ? ② Is - payoff . ? What about mean ③ extends readily to NE - value approach , not to SPE may not exist - SPE : s j s kid A kid A 0=-0 O_0 - i N E a-gtfs.ua/-I f E - ← → - I - problem is open .
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