NASH EQUILIBRIUM AND ROBUST STABILITY IN DYNAMIC GAMES: A SMALL-GAIN PERSPECTIVE

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1 NASH EUILIBIUM AND OBUS SABILIY IN DYNAMIC GAMES: A SMALL-GAIN PESPECIE Iao arafyll * Zog-Pg Jag ** ad George Aaaou *** * Depare of Evroeal Egeerg eccal Uvery of Cree 73 Caa Greece eal: karafyl@eveg.uc.gr ** Depare of Elecrcal ad Copuer Egeerg Polyecc Iue of Ne York Uvery S Meroec Ceer Brookly NY U.S.A. eal: zag@corol.poly.edu *** Hellec Poak Facal Servce -6 Peazoglou r. 75 Ae Greece eal: g.aaaou@ak.gr Arac paper develop a ovel eodology o udy rou aly propere of Na eulru po dyac gae. Sall-ga ecue oder aeacal corol eory are ued for e fr e o derve codo guaraeeg uuee ad gloal aypoc aly of Na eulru po for ecooc odel decred y fucoal dfferece euao. Specfcao o a Couro olgopoly gae uded deal o deorae e poer of e propoed eodology. eyord: Dyac gae Couro olgopoly Na eulru ou aly Sall ga.. Iroduco Dyacal gae-eorecal odel ave ere uceray ay apec. e uceray relaed rogly o a uer of ope ueo c cao e aered a pror:. Sould e odel e forulaed couou e or dcree e? e aer o e aove ueo crucal: odel dcree e ll e decred y dfferece euao ee [35633 le odel couou e are geerally decred y dffereal euao or ou delay; ee [43. e aer o e aove ueo a gfca coeuece: e percepo of e for eac player a dyac gae-eorecal odel affec er eavor.. Wa are e ecao rule a a player a for e oer player? Aga e aer o e aove ueo crucal: e eavor of a player ll eavly rely o e ecao for e aco of e re player. ere a large ecooc leraure o e effec of ecao rule e.g. aïve ackard-lookg raoal ecao ee [35 ad referece ere. Moreover f ecao rule are ug delay er e e coeuece o aly ca e pora ee [4. 3. Wa are e value of e varou coa volved a dyac gae-eorecal odel? I ay dyac gae e rae of cage of e aco of oe player aued o e proporoal o eer e devao of e aco fro e e reply ee for eaple [6 or e grade of e payoff fuco ee for eaple [63. e value of e proporoaly coa cao e ko a pror.

2 erefore e aer o pora ueo uc a e eece of a Na eulru po e uuee of a Na eulru po ad aly propere are uually relaed o e pecfc aupo ade order o cope e uceray. Coeuely e follog ueo are: Ca e erac rou forao fro a ucera olear ecooc odel c ll old o aer a e uceray? e pree ork aer affravely. I oe cae e ca eve o e eece of a Na eulru po uuee ad gloal aypoc aly propere for all pole ucerae. I order o e ale o do e propoe e follog eodology: Fr e forulae our odel couou e y ea of Fucoal Dfferece Euao ee [ By dog o e cover a fe-deoal prole o a fe-deoal prole c ee o e a clear dadvaage a fr g. Hoever ay e ca oa all feaure of couou e ad dcree e odel. Ideed e ll o a ay odel appeared e leraure ca e codered a pecal cae of our propoed odel. Secod e do o aue a pecfc ecao rule: ead e ll oly aue a e ecao coe e ory of e gae coe ackard lookg ecao; ee Defo. elo. I order o e ale o erac pora forao fro e ucera odel e ue advaced aly eod. Ideed y applyg all-ga aaly ee [789 e ca guaraee a e Na eulru po uue ad gloally aypocally ale ee eore 3. ad eore 4. elo. o our koledge e fr e a uc reul are preeed for dyacal gae-eorecal olear odel. e oly oer ork c e ave foud ad ca addre uc ueo [: our reul geeralze e reul [. Moreover e reul of [ are appled a dcree-e fraeork ad cao e ued for e aaly of odel couou e. A a yproduc of our ork e ll alo gve codo for uuee of a fed po ee Corollary 3. ad Corollary 3.3 elo c ca e ued couco clacal fed-po eore ad are dffere fro oer uuee codo e leraure ee [8. I ould e oced a e aly/uuee codo oaed y e propoed eodology ll e ore deadg codo a e oe c ca e oaed fro e udy of a pecfc odel pecfc ecao rule pecfc value for e coa volved e odel ad a pecfc percepo of e. Hoever eced ce e aly/uuee codo oaed y e propoed eodology are uffce codo for gloal aypoc aly for a ucera odel c coa ay oer odel a pecal cae. o ed aly aaly y ea of olear all-ga eore ulzed. Sall-Ga reul ave ee ued freuely aly ude ee [ ad are aed o varao of e Ipu-o- Sae Saly propery roduced y E. D. Soag [6 ad e Ipu-o-Oupu Saly propery ee [4678. e rucure of e paper a follo: Seco e apply e aove decred eodology o e Couro dyac olgopoly prole. ere a va leraure o ell-uded prole ee for ace [ For pecfc prole e decre deal our propoed eodology ad e o o e ca oa reul o e aly propere of e Couro eulru c do o deped o e for of e uceray. e preeao of e pecal cae of e Couro gae efore e geeral cae a preferred for uoral purpoe: all ue arg e geeral cae are pree e Couro gae. I Seco 3 e proceed o e ore geeral cae of dyac raegc gae ad Seco 4 e dcu e prole of accoodag e raoal ecao. Our cocludg reark are gve Seco 5. Fally e Apped e gve e proof of cera reul of ork. Noao rougou paper e adop e follog oao: For a vecor e deoe y uual Eucldea or. deoe e e of o-egave real uer. For every [ deoe e eger par of.e. e large eger eg le a or eual o. We ay a a o-decreag couou fuco γ : of cla N f γ. Le : I [ a I ad up <. We deoe y up. I [ a a Le U e a cloed cove e. By PrU e deoe e proeco of o U.

3 e or of a ored lear pace ll e deoed y deoe e ored lear pace of ouded fuco. If. More pecfcally e pree ork ll :[ or up for gve :[ a ere a a ouded appg e [ a defed y { : } a uually ye delay ee [9. For a vecor... S S e ll ue e oao ee [ for < < ad for.e. e vecor of order afer deleg e - copoe S of e vecor... S. S. Dyac Couro Olgopoly We coder e cae of Couro olgopoly ere player produce uae of a gle oogeeou produc. e payoff fuco for eac player reed y: π p c.... ere c... are coa [... e uay of e coody produced y e - player e au level of produco of e produc for e - player ad p e prce of e coody. Aug a lear dead fuco: p a. ere a are coa afyg a oe of e player: ad e oa e e reply appg for eac.. a c f : a....3 We defe: f F : f 3 S : [ [ [.4... S.5 a c a.6 a c a

4 ad e oce a e e S a defed y.4 copac ad cove ad a e ap F : S S a defed y.6 couou. Coeuely Brouer fed po eore guaraee e eece of a lea oe Na eulru a c S f a for.... Ne e aue a e dyac of e gae are decred couou e a follo: every player for a ecao for e eavor of all oer player a eac e : e ecao of e - player for e produco level of e - player a e ll e deoed y [... every player deere er produco level a a cove coao of a pa produco level ad e e reply repoe aed o e ecao for e eavor of all oer player a eac e.e. θ { a{ }} θ a a c....7 ere θ : [ : [ r... are geeral uko fuco < < r are coa geeral uko. e reader ould oce a.7 a odel a evolve couou e.e.. If e ecao rule... ad e fuco θ : [ : [ r... ere ko e ould ave a accurae decrpo of e dyac of e Couro olgopoly gae. Hoever e ll o aue eac koledge of e ecao rule u a pecfc coecy codo. Fr e gve e defo for a Coe Backard-lookg ecao repec o e Na eulru po S. Defo.: A ecao rule ere... called a Coe Backard-lookg ecao repec o e Na eulru po S f ere e coa < r uc a: up r [ r for all.8 I oer ord e coecy codo.8 recogze a o logcal for - player o ec a e produco level of e - aufacurer ll devae fro eulru level ore a e ge devao e a ereced e pa. Ne e pree oe eaple of Coe Backard-lookg ecao rule: a l l a l ere a [ l r l for all ad l.... I dcree-e odel e cae l [ l l l a l l l e uual ackard-lookg ecao c gve l k l l l for [ k k. 4

5 r a d a ere < r < a [ for all :[ r a Leegue egrale fuco for alo all [ r ad d. Of coure cae reured a u e Leegue egrale ad eeally ouded. r We oce e follog pora fac for coe ackard-lookg ecao: FAC I: ere... a Coe Backard-lookg ecao repec o e Na eulru po S f ad oly f ere e coa < r ad a fuco d : [ uc a: a d.9 [ r Proof of Fac I: Aue fr a ere... a Coe Backard-lookg ecao repec o e Na eulru po S.e. a.8 old. We dgu e follog cae. Cae : If e.8 ple a.9 old. [ r. I cae e defe d ad eualy Cae : If e.8 ple a eualy.9 old. [ r. I cae e defe d ad Cae 3: If ad e eualy.9 old d [ r g. Ieualy.8 ple a d [ r. [ r d [. Cae 4: If ad e eualy.8 ple a eualy.9 old for arrary. I cae O e oer ad f.9 old e [ for all. Moreover e reader ca verfy a eualy.8 old. e proof coplee. For e dyacal ye.7 e ake e follog aupo: H: All ecao rule... are Coe Backard-lookg ecao repec o e Na eulru po S. e prevou fac o a ypoe H euvale o e eece of coa d : [... uc a e follog euale old for all... : < r ad fuco 5

6 θ θ { a{ }} a a c a d [ r. I geeral e fuco θ : [ : [ r d : [... a ell a e coa < < r are uko. erefore e dyacal ye. a ucera dyacal ye decred y Fucoal Dfferece Euao FDE ee [ I order o udy e eavor of e oluo of. e defe e deole devao varale... ad e oa fro. for... : θ { L a{ L }} θ L a L M L g { a{ L d }} [ r. ere L [ c afy M a c g for... are coa a M g L for all.... L eark ad Eaple aou ye..: a e reader ould oce a ye. a fe-deoal dyacal ye ae pace eg e ored lear pace of ouded fuco :[ or up. Ideed y ug e eod of ep gve a al codo d : [ 6 ad fuco θ : [ : [ r... e oe ca prcple deere fro. e oluo... for r... for all [. e e ca deere fro. e oluo... for rr. Coug ay e ca deere fro. e oluo... for kr k r ere k a pove eger. e oluo deed ouded ad e for all ce. guaraee a [ L L for all.... e ae of ye. ll e deoed y { : } a uually ye delay ee [9 ad e copoe of e ae y { : } for.... e reader ould alo oce a a eulru po for ye.. Ideed for every fuco θ : [ : [ r d : [. eulru po correpod o e Na eulru po S defed y for.... c All dcree-e odel of e for: a c k θ k k θ k a... ple for all e devao varale ave ee k....

7 k a k l k k l a k.3 l ere k are o-egave eger a k [... θ k [... [ l k l k for all k ad l are cluded e ucera odel l. ad euvale reo. e ee a for every odel of e for..3 oe ca gve fuco θ : [ : [ r d : [... uc a e oluo of. cocde e oluo oaed y e dcree-e odel..3. d All couou-e odel of e for: a c a....4 ere are coa ad... are Coe Backard-lookg ecao repec o e Na eulru po S are cluded e ucera odel.. Ideed for r e oluo of.4 ple e follog egral euao: a c r r a d... r Fro e aove reo uder e aupo a e appg... are couou e ca coclude a for all r ad... ere e [ r... uc a r r r a g a c g... e reader ay verfy a for Coe Backard-lookg ecao repec o e Na eulru po S e aove odel ca e decred y e ucera odel. θ r r... ad : a r <.... e crucal ueo a ca e poed e ueo of rou aypoc aly of e Na eulru S for ye. or euvalely e ueo of rou aypoc aly for ye.. e reader ca oa rgorou defo for rou gloal aypoc aly [5-9. e follog eore e a reul of e pree eco ad o a for cera value of e paraeer volved e Na eulru S rouly gloally aypocally ale for ye. e ee a for every al codo ad for every e of fuco θ : [ : [ r d : [... e oluo... of ye. afe l. eore.: old for eac ouly Gloally Aypocally Sale for ye. f e follog e of codo p... : for all {... } k f k. p <.5 p 7

8 I oer ord f codo.5 old e e Na eulru po S rouly Gloally Aypocally Sale repec o all pole Coe Backard-lookg ecao rule repec o e Na eulru po S.... I ould e oced a codo.5 are ore deadg euale a oer aly codo e leraure. Hoever eced ce codo.5 are uffce codo for gloal aypoc aly for e ucera odel. c coa ay odel uded e leraure a pecal cae. Codo.5 are ered a all-ga codo Maeacal Corol eory ee [789. For codo.5 are euvale o e eualy: < For 3 codo.5 are euvale o e follog four euale: 4 < < 4 3 < 8 3 < For 4 codo.5 are euvale o e follog eleve euale: 9 < 9 3 < 9 4 < 9 3 < 9 4 < < 7 3 < 7 4 < 7 34 < 7 34 < 8 34 < e proof of eore. rele eavly o rece reul o dyacal ye ee [9 ad ecue developed for delay ye ee [89 ad provded a e Apped. A ereg corollary for e Couro olgopoly gae gve e. Corollary.3: If codo.5 old for eac p... ad for all {... } k f k e e Na eulru po S uue for e gae decred y.3.4 ad.5. e reader ould oce a Brouer fed po eore guaraee e eece of e Na eulru S u doe o guaraee uuee. Proof of Corollary.3: Suppoe a ere e S a c a for... ad. ple a y : L a L M L g L y for.... Ug e prevou euale e reader ca verfy a e oluo of. al codo Py ere Py : { y : } y y... y correpodg o e coa pu θ f y d : g y f y f y <... afe Py for all : [ r... are rreleva ad coeuely e cao ave l. pole accordg o eore.. e proof coplee. 8

9 3. Eeo o e Geeral Cae of Dyac Gae e approac decred e prevou eco for e Couro olgopoly gae ca e eeded o ay raegc gae. k Coder a raegc gae player ad S... eg e aco pace for eac oe of e player. We aue a e e reply appg for eac oe of e player a fuco f : S... S S... S S for < < 3 ad f : S... S S f : S... S S afyg e follog euale: π < π f for all S f ere π e payoff fuco of e - player. We aue e eece of a Na eulru S for e gae ere S : S S e oucoe pace for e gae.e. ere e... S uc a f e eece of a Na eulru ca e guaraeed y Brouer fed po eore e all aco pace S k... are copac ad cove ad e all e e reply appg f : S... S S... S S for < < 3 ad f : S... S S f : S... S S are couou appg. k Ne e aue a S... are cloed cove e ad a e dyac of e gae are decred couou e a follo: every player for a ecao for e eavor of all oer player a eac e : e ecao of e - player for e produco level of e - player a e ll e deoed y S... every player deere er aco a a cove coao of a pa aco ad e e reply repoe aed o e ecao for e eavor of all oer player a eac e.e. θ Pr S θ Pr θ Pr S S θ f θ f θ f ere θ : [ : [ r... are geeral uko fuco < < r are coa geeral uko all ecao rule S... are Coe Backard-lookg ecao repec o e Na eulru po S.e. ere e coa < r uc a: up r [ r for all 3.4 We oce e follog fac for coe ackard-lookg ecao: 9

10 k FAC II: Suppoe a S a cloed cove e.... a Coe Backardlookg ecao repec o e Na eulru po S f ad oly f ere e coa < r ad a fuco : { : } k d d d uc a: PrS d [ r 3.5 Proof of Fac II: Ideed ug e fac a Pr Pr y y for every cloed cove e oe ca verfy a for every : { : } k d d d U U y ere U a e fuco defed y 3.5 afe 3.4 ad S for all. Hece a Coe Backard-lookg ecao repec o e Na eulru po S. O e oer ad f S a Coe Backard-lookg ecao repec o e Na eulru po S afyg 3.4 for all e e fuco defed y: d f [ r [ r k afe d { d : d } f d [ r. Moreover 3.5 old for all. e proof coplee. Fac II o a f all aco pace < r < ad fuco θ : [ : [ r... uc a: θ Pr S θ Pr S I geeral e coa S k... are cloed cove e e ere e coa θ f PrS d : { : } k d d d θ f Pr d...pr d k { d : d } S [ r [ r...prs d [ r S [ r 3.6 < r < ad e fuco θ : [ : [ r d :... are uko. erefore e dyacal ye 3.6 a ucera dyacal ye decred y Fucoal Dfferece Euao FDE ee [ I order o udy e eavor of e oluo of 3.6 e defe e devao varale... ad e oa fro 3.6: θ θ θ θ Pr S f PrS d... PrS d [ r Pr S f PrS d... PrS d. [ r [ r 3.7 [ r

11 Fally e aue a ere e fuco γ N for all S : f a... uc a e follog euale old γ Ug aga e fac a Pr U PrU y y for every y ere U a cloed cove e ad euale 3.8 e oa fro 3.7 for all ad : a a γ [ r [ r eark ad Eaple aou ye : a e reader ould oce a ye 3.7 a fe-deoal dyacal ye ae pace eg N e ored lear pace of ouded fuco :[ ere N k... k or up. Ideed y ug e eod of ep gve a al codo ad fuco : { : } k d d d θ : [ : [ r... e oe ca prcple deere fro 3.8 e oluo... for r... N for all [. e e ca deere fro 3.8 e oluo... for rr. N Coug ay e ca deere fro 3.8 e oluo... for kr k r ere k a pove eger. e ae of ye 3.8 ll e deoed y { : } a uually ye delay ee [9 ad e copoe of e ae y { : } for.... e oluo e for all ad afe { : } for all. o ee oce a 3.9 ple e eece of a fuco G N uc a: [ r [ r up G 3. [ Wou lo of geeraly e ay aue a G for all. Ieualy 3. ple a G for all [ r [ [ ad G [ r r [ 3. Workg ay ad ug duco e ay eal a for every pove eger k old a k k G [ [ ere G : G... G. erefore 3. ple a k e ere [ / r deoe e eger par of / r. for all [ kr 3. [ / r [ / r G G for all 3.3 [ [ e reader ould alo oce a a eulru po for ye 3.7. Ideed for ay fuco θ : [ : [ r : { : } k d d d... ple for all. eulru po correpod o e Na eulru po S og a e devao varale ave ee defed y for....

12 c All dcree-e odel of e for: k θ k k θ k f k θ k k θ k f k θ k k θ k f k... k... k... k k k k... k 3.4 k gve y.3 ere k are o-egave eger a k [... θ k [... [ l k l k for all k ad l are l cluded e ucera odel 3.6 ad euvale reo 3.7 e ee a for every odel of e for oe ca gve fuco θ : [ : [ r : { : } k d d d... uc a e oluo of 3.6 cocde e oluo oaed y e dcree-e odel d Slarly a o prevou eco f S [ for all... ad f all ecao rule... are Coe Backard-lookg ecao repec o e Na eulru po S ad all appg... are couou e all couou-e odel of e for: f... f... ere are coa are cluded e ucera odel 3.6. e e reader ould oce a o couy aupo ade for e e reply appg of e player f : S... S S... S S for < < 3 ad f : S... S S f : S... S S. k Moreover e ave o aued a e aco pace S... are copac e: e ply reure a e aco pace are cloed cove e. Hoever e ave aued e eece of a Na eulru po S ad e eece of fuco γ N... afyg 3.8. e crucal ueo a ca e poed e ueo of rou aypoc aly of e Na eulru S for ye 3.6 or euvalely e ueo of rou aypoc aly for ye 3.7. e follog eore e a reul of eco ad o a rou gloal aly ca e deered y e fuco γ N... afyg 3.8. eore 3.: ouly Gloally Aypocally Sale for ye 3.7 f ere e ω uc a e follog e of codo old for eac p... : γ γ... < γ 3 p for all {... } k f k ere γ : ω γ ω. 3.5

13 I oer ord f codo 3.5 old e e Na eulru po S rouly Gloally Aypocally Sale repec o all pole Coe Backard-lookg ecao rule repec o e Na eulru po S.... Codo 3.5 are ered a cyclc all-ga codo Maeacal Corol eory ee [789. For codo 3.5 are euvale o e euale: γ < ad < γ γ γ For 3 codo 3.5 are euvale o e follog elve euale for all : γ < γ < γ γ γ < γ < 3 γ 3 3 γ 3 γ < γ < 3 γ 3 3 γ 3 γ γ < γ γ < γ γ < 3 γ 3 3 γ γ γ γ < γ γ < γ γ < 3 γ γ 3 γ 3 e reader ould oce a ay of e aove euale are euvale. For eaple for e eualy γ < for all ple e eualy γ < for all. Slarly for e cae 3 e γ γ follog fve euale γ γ < γ 3 γ 3 < γ 3 γ 3 < γ 3 γ 3 < γ γ < for all ply all elve euale c re codo 3.5 cae. 3 γ 3 γ I ould e oced a for e Couro olgopoly gae uded e prevou eco e e reply appg f... are defed y.3. Coeuely ug e covey of e e S [... e oa e follog euale for... : f a e aove euale ply a euale 3.8 old : γ ere :. eore 3. ad e aove defo guaraee rou gloal aypoc aly of e Na eulru provded a ere e ω uc a e follog e of codo old for eac p... :... p p ω p < for all {... } k f k. Codo.5 are eceary ad uffce codo for e eece of a uffcely all coa ω afyg e aove euale for eac p... ad for all {... } k f k. erefore e coclude a eore. a pecal cae of eore 3.. A ore careful aaly lar o e aove aaly reveal a e Na eulru for e Couro olgopoly gae decred Seco ll e aypocally ale provded a ere e e of pove real uer A { a }... a a for all... ad... uc a e follog e of codo old for eac p... : a... a... a < 3 p p 3

14 for all {... } k f k. e aove codo are le rercve a codo.5; deed codo.5 are pled y e aove codo for e pecal cae a for all.... For eaple for 3 e aove all-ga codo are euvale o e eece of ε ε ε uc a: ε ε < ε ε ε < ε ε 3 3 < ε ε ε < ε ε 3 3 < 3 For e aove euale e ave ued a ε 3 3 a 3 ε ε a ε. By elecg ε ε ε e oa euale.6. 3 a a 3 ε 3 ε 3 a ad I ould e epazed a e paraeer r c are volved e defo of e Coe Backard-lookg ecao Defo. play o role e all-ga codo. Coeuely e allga codo ca elp u o decde eer e Na eulru po rouly ale ou ay koledge of e ecao rule. e all-ga codo 3.5 dead koledge of e Na eulru po S ad e e reply appg f : S... S S... S S for < < 3 ad f : S... S S f : S... S S for c euale 3.8 old. e proof of eore 3. rele eavly o rece reul o dyacal ye ee [9 ad ecue developed for e-delay ye ee [89 ad provded e Apped. A ereg corollary gve e. Corollary 3.: If ere e ω uc a codo 3.5 old for eac p... ad for all {... } k f k e e Na eulru po S uue. e proof of Corollary 3. eacly e ae e proof of Corollary.3: e o a e eece of a addoal Na eulru S coradc rou gloal aypoc aly of for ye 3.7. Deal are lef o e reader. Ug Corollary 3. e ay oa codo for uuee for a fed po. Ideed e ave: k Corollary 3.3: Le S... e cloed cove e ad le fuco : S... S S... S f S for < < 3 ad f : S... S S f : S... S S for c ere e... S ere S : S S afyg 3.3. Furerore uppoe a ere e fuco γ N... uc a euale 3.8 old for all S ad a ere e ω uc a codo 3.5 old for eac p... ad for all {... } k f k. e... S e uue fed po of e appg S F : f... f S. e reader ould oce a Corollary 3.3 doe o guaraee e eece of a fed po for e appg S F : f... f S. Corollary 3.3 ca e ued couco clacal fed-po k eore e.g. Brouer fed po eore e all aco pace S... are copac ad cove ad e e appg S F : f... f S couou order o guaraee uuee of e fed po. 4

15 4. eark o e aoal Epecao Cae I clear fro Defo. a raoal ecao are o ecearly coe ackardlookg ecao repec o e Na eulru po S. erefore e cae of raoal ecao o ecearly covered y e reul of e prevou eco. po ovae e follog defo for e raegc gae decred e prevou eco. Defo 4.: A ecao rule... called a aoal-coe Backard-lookg ecao repec o e Na eulru po S f ere e a coa < uc a: up [ for all 4. Clearly raoal ecao are aoal-coe Backard-lookg ecao repec o e Na eulru po S. Moreover a Coe Backard-lookg ecao e ee of Defo. a aoal-coe Backard-lookg ecao repec o e Na eulru po S. Ca e coder ye 3.3 ere all ecao rule S... are aoal-coe Backard-lookg ecao repec o e Na eulru po S? e key aeacal prole a are cae eer e ca oa a ell-defed dyacal ye: eark a prevou eco doe o apply. Hoever e ca eed e aaly of e prevou eco uder e follog ypoe: H ere e de e J l {... } l... J J for l k ad {... } a: l k J l l... uc All player po S. J are ug Coe Backard-lookg ecao repec o e Na eulru Moreover for every k... e follog aee old: All player J k are ug aoal-coe Backard-lookg ecao S l k ad Coe Backard-lookg ecao S f oere. f J l Ideed f ypoe H old e ye 3.3 reed devao varale e follog euao: θ θ θ θ : : PrS f Pr d... Pr d PrS f Pr d... Pr d S S [ [ r ere θ : [ : [ r f J k ad J l l k f J ad J l k k S l S : { : } k d d d y 4. 5

16 Le u la e y ye 4. a fe-deoal dyacal ye ae pace eg e N ored lear pace of ouded fuco :[ ere N k... k or up. Ideed y ug e eod of ep gve a al codo ad fuco θ : [ : [ r : { : } k d d d... e oe ca prcple deere fro 4. e oluo... for r... for all [ ug e follog procedure: Sep : Fr deere e oluo for r ad for all player J o are ug Coe Backardlookg ecao repec o e Na eulru po S. I cae e are a poo o deere e copoe of e oluo for J ad r y ea of 4.. Furerore cae eualy 3.9 old for r J ad coeuely ere e fuco G N uc a: Sep : [ r up G for J 4.3 [ Ne deere e oluo for r ad for all player J o are ug aoal-coe Backard-lookg ecao S J ad Coe Backard-lookg ecao f S f oere. I cae 3.8 ple a e follog eualy old for all r : a γ a γ [ 4.4 r [ J Hoever e copoe of e oluo for J ad r ave ee deered y Sep. erefore e are a poo o deere e copoe of e oluo for J ad r y ea of 4.. Ug 4.3 ad 4.4 e oa e eece of a fuco G N uc a: Sep k 3 k : [ r J [ up G for J J 4.5 We deere e oluo for r ad for all player J k o are ug aoal-coe Backard-lookg ecao S J l k ad Coe Backard-lookg f l ecao S f oere. I cae 3.8 ple a e follog eualy old for all r : a γ a γ [ 4.6 r [ Jl Jl l k l k Hoever e copoe of e oluo for J l l k ad r ave ee deered y prevou ep. erefore e are a poo o deere e copoe of e oluo for J k ad r y ea of 4.. Moreover y vrue of prevou ep ere e ere e fuco G k N uc a: [ r k up G for J l 4.7 [ l k... 6

17 Ug 4.6 ad 4.7 e oa e eece of a fuco G N uc a: [ r k k up G for J l 4.8 [ l k... Afer e copleo of e ep e ave deered all copoe of e oluo for... ad r. Moreover e ave alo coruced a fuco G N uc a: [ r up G 4.9 Wou lo of geeraly e ay aue a G for all. Moreover y ug 4.9 e ay coclude eacly a e prevou eco a eae ad 3.3 old. [ e proof of e follog eore eacly e ae e proof of e eore 3. ad erefore oed. eore 4.: ouly Gloally Aypocally Sale for ye 4. uder ypoe H f ere e ω uc a e e of codo 3.5 old for eac p... ad for all {... } k f k ere γ : ω γ ω. I ould e epazed a e paraeer r c are volved e defo of e Coe Backard-lookg ecao Defo. play o role e all-ga codo. Moreover e uer of e de e J l {... } volved ypoe H or e parcular eer of eac de e play aoluely o role e all-ga codo 3.5. Furerore all ee paraeer are alloed o cage e: ere o eed o aue a ee paraeer rea coa. Coeuely e all-ga codo ca elp u o decde eer e Na eulru po rouly ale ou ay koledge of e ecao rule. Aga e all-ga codo 3.5 dead koledge of e Na eulru po S ad e e reply appg f : S... S S... S S for < < 3 ad f : S... S S f : S... S S for c euale 3.8 old. Fally ould e oed a for a pecfc raegc gae eve le deadg ypoee a ypoe H ca e ued order o guaraee a ye 3.3 gve a fe-deoal dyacal ye ae pace N eg e ored lear pace of ouded fuco :[ ere N k... k or up. ca e doe y log pecal propere of e e reply appg f : S... S S... S S for < < 3 ad f : S... S S f : S... S S e.g. f oe of e fuco are depede of cera argue. 5. Cocluo I ork advaced aly eod ave ee ued order o provde uffce codo called cyclc all ga codo c guaraee rou gloal aypoc aly of e Na eulru dyac gae. e oaed reul are poerful ecaue ey ca e appled o ucera odel for c e player for coe ecao aed o e ory of e gae. I addo y forulag dyac gae-eorecal odel y ea of Fucoal Dfferece Euao pole o oa all feaure of couou-e ad dcree-e odel. A Couro olgopoly gae a ee ued order o llurae e eorecal reul. Fuure reearc ca addre e ecooc eag of all-ga codo o oer gae ued ecooc reearc e.g. e udy of e aly propere of e Walraa eulru of a arac ecooy. A ep oard reearc dreco e fac a e reul preeed ork ca e drecly eeded o e cae ere e e reply appg are e-valued ap ead of fuco.e. f S. Hoever cae euale 3.8 u e odfed e follog ay: 7

18 p a γ for all p f ad e aove e of euale drecly ple a e Na eulru po afe f { } for all.... W odfcao eore 3. Corollary 3. Corollary 3.3 ad eore 4. old cae a ell. Fuure reearc ca alo addre e ue of udyg dyac gae-eorecal odel y ea of rece reul yrd ye eory ee [45 or e ue of alzao of Na eulra for dyac gaeeorecal odel y ea of olear feedack la ug recely propoed eodologe ee for eaple [3. Ackoledge e ork of Z.-P. Jag a ee uppored par y NSF gra ECS-9376 ad DMS eferece [ Agza H. N. G. I. Bc ad M. opel Mulaly a Dyac Couro Gae ree Olgopol Maeac ad Copuer Sulao [ Aaaou G. I. arafyll ad S. oo Prce Salzao Ug Buffer Sock Joural of Ecooc Dyac ad Corol [3 Bc G. ad M. opel Eulru eleco a olear duopoly gae adapve ecao Joural of Ecooc Beavor & Orgazao [4 Carella C. ad F. Szdarovzky Dyac olgopole ou full forao ad cououly drued e lag Joural of Ecooc Beavor & Orgazao [5 Droe E. C. Hoe ad J. ura Edogeou flucuao uder evoluoary preure Couro copeo Gae ad Ecooc Beavor [6 Forgo F. J. Szep ad F. Szdarovzky Iroduco o e eory of Gae. Cocep Meod Applcao luer Acadec Puler 999. [7 Grue L. Ipu-o-Sae Dyacal Saly ad Lyapuov Fuco Caracerzao IEEE raaco o Auoac Corol [8 Guo. ad P. Nu Soe eore o Eece ad Uuee of Fed Po for Decreag Operaor Copuer ad Maeac Applcao [9 Hale J.. ad S. M.. Luel Iroduco o Fucoal Dffereal Euao Sprger-erlag Ne York 993. [ Hoe C. H. O e coecy of ackard-lookg ecao: e cae of e coe Joural of Ecooc Beavor & Orgazao [ Huag W. Iforao lag ad dyac aly Joural of Maeacal Ecooc [ Io H. ad Z.-P. Jag Nolear Sall-Ga Codo Coverg ISS Sye: Necey ad Suffcecy fro a Lyapuov Perpecve Proceedg of e 44 IEEE Coferece o Deco ad Corol ad Europea Corol Coferece 5 Sevlle Spa [3 Io H. ad Z.-P. Jag Neceary ad Suffce Sall-Ga Codo for Iegral Ipu-o-Sae Sale Sye: A Lyapuov Perpecve IEEE raaco o Auoac Corol [4 Jag Z.P. A. eel ad L. Praly Sall-Ga eore for ISS Sye ad Applcao Maeac of Corol Sgal ad Sye [5 arafyll I. A Sye-eorec Fraeork for a Wde Cla of Sye I: Applcao o Nuercal Aaly Joural of Maeacal Aaly ad Applcao [6 arafyll I. A Sye-eorec Fraeork for a Wde Cla of Sye II: Ipu-o-Oupu Saly Joural of Maeacal Aaly ad Applcao [7 arafyll I. ad Z.-P. Jag A Sall-Ga eore for a Wde Cla of Feedack Sye Corol Applcao SIAM Joural o Corol ad Opzao [8 arafyll I. P. Pepe ad Z.-P. Jag Saly eul for Sye Decred y Coupled earded Fucoal Dffereal Euao ad Fucoal Dfferece Euao Nolear Aaly eory Meod ad Applcao [9 arafyll I. ad Z.-P. Jag A ecor Sall-Ga eore for Geeral Nolear Corol Sye Proceedg of e 48 IEEE Coferece o Deco ad Corol 9 Saga Ca Alo avalale p://arv.org/a/ [ L G. ad Y. Hog Delay duced ocllao predaor-prey ye Beddgo-DeAgel fucoal repoe Appled Maeac & Copuao

19 9 [ Lueerger D. G. Coplee Saly of Nocooperave Gae Joural of Opzao eory ad Applcao [ Mazec F. ad P.-A. Bla Backeppg Deg for e-delay Nolear Sye IEEE raaco o Auoac Corol [3 Mazec F. M. Maloff ad Z. L Furer eul o Ipu-o-Sae Saly for Nolear Sye Delayed Feedack Auoaca [4 Nec D. ad A.. eel Ipu oupu aly propere of eorked corol ye IEEE raaco o Auoac Corol [5 Nec D. ad A.. eel Ipu o ae aly of eorked corol ye Auoaca [6 Soag E.D. Soo Salzao Iple Copre Facorzao IEEE raaco o Auoac Corol [7 Soag E.D. ad Y. Wag Noo of Ipu o Oupu Saly Sye ad Corol Leer [8 Soag E.D. ad Y. Wag Lyapuov Caracerzao of Ipu-o-Oupu Saly SIAM Joural o Corol ad Opzao [9 Pacuk A. ad. Puu Saly a No-Auooou Ierave Sye: A Applcao o Olgopoly Copuer ad Maeac Applcao [3 Ok E. A. eal Aaly Ecooc Applcao Prceo Uvery Pre 7. [3 Okuguc. ad. Yaazak Gloal aly of uue Na eulru Couro olgopoly ad reeekg gae Joural of Ecooc Dyac ad Corol [3 Pepe P. e Lapuov ecod eod for couou e dfferece euao Ieraoal Joural of ou ad Nolear Corol [33 Szdarovzky F. ad W. L A Noe o e Saly of a Couro-Na Eulru: e Mulproduc Cae Adapve Epecao Joural of Maeacal Ecooc Apped Proof of eore.: Noce a. ple a e follog euao old for all : [ [ [ [ Pr Pr Pr r d c a θ θ... A. Ug e fac a y y U U Pr Pr for every y ere U a cloed cove e couco c a Pr [ ad [ θ < e oa fro A. for... : c a c a Pr Pr [ [ θ θ θ θ θ θ A. Cog A..8 ad ug defo g... e coclude a for every ad for every e of fuco [ : θ [ : r

20 d : [... e oluo of. al codo correpodg o pu θ : [ : [ r d : [... afe: [ r θ θ g for all A.3 [ r We oce a ye. a auooou ucera dyacal ye e ee decred [567. Ne e o a a rou eulru po for ye. e ee decred [567.e. for every ε ere e δ : δ ε uc a f δ e for every e of fuco θ : [ : [ r d : [... e oluo of. al codo correpodg o pu θ : [ : [ r d : [... afe ε for all [. o ee oce a A.3 ple e eece of a coa G uc [ a: up [ r G A.4 [ r [ Wou lo of geeraly e ay aue a G. Ieualy A.4 ple a G for all [ r A.5 [ [ Workg ay ad ug duco e ay eal a for every pove eger k old a erefore A.6 ple a k G [ [ for all [ kr A.6 [ / r [ / r G G for all A.7 [ [ ere [ r / deoe e eger par of / r. Coeuely A.7 ple a for every ε ere e : [ / r δ δ ε ε G uc a f δ e for every e of fuco θ : [ : [ r d : [... e oluo of. al codo correpodg o pu θ : [ : [ r d : [... afe ε for all [. erefore a rou eulru po for ye. e ee [ decred [567. e reader ould oce a eualy A.3 ple e follog eualy for all... ad : a g r for all [ [ r A.8 Le ad coder e faly of fucoal :... defed y: up A.9 Le r ad e arrary. Defo A.9 ad eualy A.8 ply a:

21 up up a up up a up up [ [ g r r Ug defo A.9 ad e fac a g... ad up up [ r r r e oa fro e aove eualy: up a up a up up a up up up up a r r Coeuely for every... r ad old a: up a up a A. Ug duco ad A. e ca o a for every... r ad for every o-egave eger k old a: up a up a k k k k A. erefore A. ple a for every... ad e follog eualy old: up a up a A. Ne e aue a l <. e reader ould oce a defo A.8 ple a up for l < ad coeuely:

22 [ up A.3 ere :. I follo fro A. A.3 ad defo A.9 c ple for : a a e follog euale old for every... ad < l : a L up a up A.4 4 L a 4 up A.5 ere L : ad : a. I follo fro A.3 A.4 A.5 ad eore 3. [9 a... ouly Gloally Aypocally Sale for ye. provded a e follog e of codo old for eac p... : for all {... } k f k ad p p p < A.6 p < A.7 Noce a f codo.5 old for eac p... ad for all {... } k f k e e codo A.6 A.7 old for eac p... ad for all {... } k f k for uffcely all ad for uffcely cloe o. e proof coplee. Proof of eore 3.: We fr oce a ye 3.7 a auooou dyacal ye e ee decred [567. Ne e o a a rou eulru po for ye 3.7 e ee decred [567.e. for every ε ere e δ : δ ε uc a f δ e for every e of fuco θ : [ : [ r : { : } k d d d... e oluo of 3.7 al codo correpodg o pu θ : [ : [ r k { d : d } d :... afe ε for all [. Wou lo of [ geeraly e ay aue a e fuco G N volved 3.3 a rcly creag fuco. Defe G [ / r κ : for every c a rcly creag couou fuco κ ad l κ recall a G for all ad defe κ : o e e vere fuco of κ o. Ideed 3.3 ple a for every ε ere e δ : δ ε κ ε uc a f δ e for every e of fuco θ : [ : [ r : { : } k d d d... e oluo of 3.7 al codo correpodg o pu θ : [ : [ r : { : } k d d d... afe ε for all [ [. erefore a rou eulru po for ye 3.7 e ee decred [567. Le ad coder e faly of fucoal :... defed y:

23 3 up A.8 Ug defo A.8 ad 3.9 e oa for every... ad : up a up a a up up a up a up up up a a up up a up up a up up [ [ γ γ γ γ Coeuely for every... ad old a: up a up a γ A.9 Ug duco ad A.9 e ca o a for every... ad for every o-egave eger k old a: up a up a k k k k γ A. erefore A. ple a for every... ad e follog eualy old: up a up a γ A. e reader ould oce a defo A.8 ple a up for l ad coeuely: [ up A. Wou lo of geeraly e ay aue. Defe : ω ω ad le e coa afy e euale: l l ω ad < ω ω l ere ω e coa volved e ypoee of e eore. Noce a e ypoe < ad prevou defo ply a <

24 ω ad ω. I follo fro A. A. ad defo A.8 c ple a e follog euale old for every... ad : a L B up a ω γ ω up A.3 L a aω γ ω up B up A.4 ere L : ad B : <. I follo fro A. A.3 A.4 ad eore 3. [9 a ouly Gloally Aypocally Sale for ye 3.7 provded a e e of codo 3.5 old for eac p... ad for all {... } k f k. e proof coplee. 4