A Forward-Looking Nash Game and Its Application to Achieving Pareto-Efficient Optimization

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1 Applied Mhemics Pulished Online Decemer 013 (hp://wwwscirporg/journl/m) hp://ddoiorg/10436/m A Forwrd-Looing Nsh Gme nd Is Applicion o Achieving Preo-Efficien Opimizion Jie Ren 1 Ki-Ki Wong Jinjun Hou 1 1 School of Elecronics nd Informion Engineering eijing Jioong Universiy eijing Chin Deprmen of Elecronic nd Elecricl Engineering Universiy College London Torringon Plce London UK Emil: jeeren@livecn i-iwong@uclcu houjj@jueducn Received Sepemer ; revised Ocoer ; cceped Ocoer Copyrigh 013 Jie Ren e l This is n open ccess ricle disriued under he Creive Commons Ariuion License which permis unresriced use disriuion nd reproducion in ny medium provided he originl wor is properly cied ASTRACT Recognizing he fc h plyer s cogniion plys defining role in he resuling equilirium of gme of compeiion his pper provides he foundion for Nsh gme wih forwrd-looing plyers y presening forml definiion of he Nsh gme wih considerion of he plyers elief We use simple wo-firm model o demonsre is fundmenl difference from he sndrd Nsh nd Scelerg gmes Then we show h he plyers elief funcions cn e regrded s he opimizion prmeers for direcing he gme owrds much more desirle equilirium Keywords: elief; Cogniion; Ierive Algorihm; Nsh Equilirium; Preo-Opimliy; Scelerg 1 Inroducion Gme heory hs een very well recognized s rnch of pplied mhemicl ools es for nlyzing he phenomenon of selfish compeiion which rises in numerous rel-life pplicions rnging from economics o socil sciences nd even o engineering prolems Mny opimizion prolems cn e viewed s compeiion prolem If here re shred resources hen inherenly here will e compeiion in he llocion of such resources An individul who pricipes in he resource llocion lwys wishes o mimize is pyoff Neverheless he plyer s reurn no only depends on is own sregy u is lso dependen on compeiors responses As resul idelly plyer should choose or opimize is sregy sed on no only is immedie reurn u lso he possile oucomes of how ohers migh respond o is sregy Apprenly plyer s cogniion (ie is elief on how he environmen s whole would rec o ny of is cion) will e pivol in he opimizion of is sregy for mimizing is pyoff nd in defining he equilirium of he gme In he lierure here re numer of gme-heoreic models nd hey differ in heir ssumpions on he plyers cogniion Due o his fundmenl difference hese models represen differen compeiion environmens herey resuling in very differen oucomes which he sedy se re referred o s equiliri When he compeiion process reches n equilirium y definiion no plyer would hve incenive o chnge heir sregies furher o devie from he equilirium which is herefore usully considered s sisfcory oucome o ll plyers The mos populr model hs o e he Nsh gme [1] in which plyer hs he elief h oher plyers sregies re fied nd will no chnge regrdless of wh i does A Nsh gme is sed on such ssumpion on he plyers cogniion I is noed h he gme is sill very well defined lhough he plyer s elief is olly inccure On he oher hnd in Scelerg gme [] here is super plyer commonly nown s leder who nows ll he informion ou is compeiors nown s followers In his cse he leder is considered o hve perfec cogniion nd herefore cn oin he mos rewrding sregy In mos prcicl compeiion siuions he efficiency of Nsh gme resuling in he mos celered Nsh equilirium is ofen poor (which will e reveled y wo-firm emple ler in his pper) The min reson is h he ssumpion for he Nsh gme is oo much simplified nd i fils o ccoun for he inercion eween

2 1610 J REN ET AL he plyers In conrs he ssumpion for he Scelerg model is oo sringen nd i is ofen oo difficul o e qulified o e leder in mos prcicl scenrios Neverheless he Scelerg equilirium is highly eneficil o he leder The limiion is h if here were wo or more leders ll of which possess perfec cogniion nd wish o e he igges winner his would led o rgedy [3] In his pper we recognize he impornce of plyers cogniion in gme nd focus on how he ssumpion of he plyers cogniion ffecs he equilirium sed on he nlysis of Nsh nd Scelerg equiliri we provide new definiion for he Nsh gme wih consider ion of he plyers elief As useful yproduc he new definiion fcilies he inerpreion of he plyers elief s he opimizion prmeers which if opimized properly cn direc he gme owrds Preo-efficien equilirium Preliminries 1 The Two-Firm Model For illusrive purpose in his pper we use simple wo-firm model in economics s n emple In his simple model here re wo firms Firm A nd Firm They mnufcure n idenicl produc nd is uni cos is he sme for oh firms In ddiion he profi per uni produc diminishes s he numer of producs ville in he mre increses In priculr le 0 denoe he numer of producs mnufcured y Firm A nd 0 e h mnufcured y Firm The uni cos is denoed y c nd he price per produc which is se he sme y oh Firm A nd Firm is ssumed o e for some consn >0 Therefore he profi funcions for Firm A nd Firm re respecively given y f c f c In vrious gme-heoreic models he ojecive would e for Firm A nd Firm o ierively opimize heir respecive sregies nd for mimizing heir profis (1) in compeiive fshion efore we emine differen equiliri of his wo-firm gme we find i useful o firs presen he generl noions of gme For gme wih K >1 plyers we denoe he sregy profile for plyer s nd use 1 K (1) () o represen he sregy profile for ll he plyers Moreover we use o denoe specific choice of sregy from ll he plyers where is he sregy doped y plyer nd (3) K denoes he doped sregy y ll he plyers ecep plyer Liewise he rewrd funcion for plyer is denoed y f If he gme converges o n equilirium hen we will use he superscrip o highligh h he corresponding prmeers re he equilirium For insnce we hve nd f he equilirium Nsh Compeiion Given he compeiion of K plyers if some poin no one cn gin ny furher y deviing from is presen sregy hen he sregy of ll he plyers is sid o hve reched o n equilirium [4] Mhemiclly we hve nd : f f (4) Now proceed o derive he Nsh equilirium for he wo-firm emple According o (4) Firm A solves m m f c (5) 0 0 As such i cn e esily shown h df c c (6) d Similrly cn e derived resuling he se of simulneous equions c (7) c As resul he gme governed y (7) will rech he Nsh equilirium c c 3 3 (8) which leds o he profis f c c f f (9) 9 9 I is worh poining ou here h in Nsh equilirium when ing d f in he opimizion for plyer A d plyer A elieves h ny given ime insnce is d opiml nd fied or 0 This is oviously d inccure A he Nsh equilirium in priculr from d (17) we cully hve 05 0 This shows d

3 J REN ET AL 1611 h Nsh plyer s cogniion nd he reliy hve con- siderle difference nd he profi funcion f is in fc only he profi in he idel siuion u does no represen he cul profi funcion due o inercion from plyer 3 Scelerg Compeiion A plyer cn enefi more from he res of he plyers in gme if his plyer hs perfec cogniion ou how ohers would rec o is sregy This is sudied formlly y he Scelerg equilirium In he generl seing wih K plyers if plyer is he leder i should now perfecly he response funcion nd herefore is le o oin he mos effecive sregy such h : f f (10) For oher plyers s hey do no hve ny informion ou ny of he oher plyers sregies hey cn only ssume h oher plyers sregies re fied nd will c lie Nsh plyer see (4) As resul we hve nd : f f (11) In Scelerg gme here is very sric order of how he plyers ply he gme [15] In priculr leder needs o firs give ou is sregy nd les oher plyers compee o rech Nsh equilirium gins his sregy efore i revises is sregy for noher round of compeiion mong he res of he plyers Achieving he Scelerg equilirium will hus require wo-level gme The meri of Scelerg equilirium is h leder hs n solue dvnge over oher plyers u he drwc is h nowing he funcion wou ld e oo difficul o chieve in prcice if no impossile A sndrd pproch would require he le der o ry ehusively ll possile o idenify he es sregy Reclling from he wo-firm emple if we le Firm A e he leder nd Firm e he follower hen ccording o (11) plyer s cogniion is h plyer A s sregy is opiml nd fied nd plyer herefore ims o solve m m f c (1) 0 0 which y seing df 0 d gives c (13) which is he ec response funcion of plyer wih respec o ny cion According o (10) nd using (13) plyer A finds is es sregy y c m f m c (14) 0 0 As resul he es sregy for plyer A cn e nlyiclly oined s c (15) nd he corresponding sregy for plyer is c 4 Hence he Scelerg equilirium we hve which leds o he profis f c c (16) 4 c c (17) 8 16 Noe h in order o chieve he Nsh equilirium erlier here is no specific order of oining nd in solving he simulneous Equions (7) nd he equilirium cn e chieved y free compeiion This is however no rue for he Scelerg equilirium where he leder s sregy mus e oined firs nd he follower(s) respond As he numer of plyers increses he compleiy of he Scelerg gme will increse considerly nd he simpliciy of he Nsh gme will previl In ddiion gme wih ll leders degeneres o Nsh gme In he wo-firm emple ecuse Firm A nows precisely he sregy doped y Firm (13) i is le o chieve higher profi hn wh is chieved y he Nsh equilirium However he ey quesions re: If (13) is no nown y Firm A or Firm A only nows pril informion ou (13) eg d d or only d d would i sill help Firm A o oin eer or even Scelerg sregy? If he nswer is yes his would men h he level of cogniion for leder could e significnly reduced Also wh if wo plyers hve pril informion ou ech oher s sregies? Wh will hppen? This hs moived us o invesige he Nsh gme wih simulneous forwrd-looing plyers in which ech plyer opimizes is sregy sed on is own elief 3 Forwrd-Looing Compeiion Afer he discussion of Nsh nd Scelerg equiliri ove i is cler h plyers cogniion is ey o defining he resuling equilirium of gme A he sme ime we

4 161 J REN ET AL undersnd he euy of he Nsh gme where plyers cn compee freely (wihou following specific order) o rech he equilirium sed on hese wo poins we presen he Nsh gme wih forwrd-looing plyers To fcilie our nlysis we find i useful o hve he following definiions Environmenl funcion In compeiion process he rewrd for plyer depends no only is own sregy u lso ohers sregies ny given ime insn We use he environmenl funcion r o qunify he influence of oher plyers sregies ( ime insn ) ono plyer s rewrd Oviously oher plyers sregies cn lwys e reed s some form of response o given plyer s sregy ime insn elief funcion A plyer s undersnding on is environmenl funcion reflecs is cogniion ou he compeiion in he gme We ssume h plyer possesses he nowledge of elief funcion which is denoed s r where denoes he sregies from ll he plyers ime insn ecep plyer nd clerly r r The ler relionship furher suggess o formule he elief response using some form of Tylor series epnsion For emple we my wrie d r r r (18) where is regrded s he inerd ference derivive (o e discussed in Secion 33) The elief funcion r is plyer s cogniion on wh he environmenl funcion r would e given sregy nd he presen se If r r hen plyer s cogniion will e perfec Oherwise i is only predicio n Prediced rewrd The prediced rewrd funcion denoed y f r indices he moun of rewrd plyer elieves o chieve y he sregy nd oher plyers sregies ime insn r sed on he elief funcion 31 A Moiving Emple In Secion 3 if plyer A is Scelerg leder hen i hs he environmenl funcion (or response) r nd nows perfecly he sregy of plyer (13) Therefore i hs he perfec elief funcion r r 05 c (19) If plyer A s cogniion is reduced; for insnce nows ny given ime insn d d 05 in (13) nd d d cn oserve he environmenl funcion presen ime r r plyer A cn formule is elief funcion using firs-order Tylor series s ie 05 r (0) nd is prediced rewrd funcion cn e epressed s 05 c 05 f r r c (1) From he side of plyer if i hs s low cogniion s Nsh plyer hen i will hve he elief funcion r Hence plyer s rewrd funcion will e c f r r c As resul plyer A s sregy cn e opimized y 0 m f r m 05 c05 0 () (3) y seing d f 0 his suggess n upding process d 05 c (4) 1 On he oher hnd for plyer i ims o solve m m f r c (5) 0 0 f y seing d 0 plyer updes is sregy y d c 1 (6) Applying he wo upding rules (4) nd (6) re- pe edly s we ge c c 4 (7) which ends up he sme sregy in he Scelerg equilirium ( 16) u vi compleely differen process The sriing resul here is h he Scelerg equilirium cn now e chieved y free compeiion eween he plyers wihou following specific order of how he gme should e plyed nd h he so-clled leder ie plyer A here does no need perfec cogniion of he environmenl funcion (13) u good elief (0) 3 Definiion of he New Equilirium Moived y he poenil of eing forwrd-looing we here presen he Nsh equilirium wih forwrd-looing

5 J REN ET AL 1613 plyers (ie wih some cogniion in he form of elief funcions) Mhemiclly i is wrien s nd S : f r f r (8) where r is he elief funcion reflecing plyer s cogniion iliy nd f is he prediced rewrd funcion for plyer Noe h (8) cn e rewrien s nd : S f f r ecuse ccording o (18) (9) r r nd s such he equilirium we hve f r f r f (30) In his model he elief funcion r cn e chosen rirrily nd i only serves o indice plyer s undersnding ou he compeiion environmen In fc (9) emrces he convenionl Nsh equilirium (4) in which plyers hve he elief funcion r r which effecively res he environmen plyer oserves ny presen ime insn s fied nd consn nd ignores he susequen chnges in oher plyers sregies provoed y plyer s new sregy We refer o he equilirium of Nsh gme wih elief funcions s elief-direced Nsh equilirium (NE) 33 From Inerference Derivive o Preo-Opimliy In he proposed model of he Nsh gme wih forwrdlooing plyers every plyer is free o choose is own elief funcion Alogeher he cominion of he plyers elief funcions defines he resuling equilirium of he plyers compeiion nd hs numerous possiiliies To emine his we consider in he wo-firm emple h r (31) r where λ re regrded s he inerference derivives which cn e inerpreed s he re of chnge of he environmenl funcion wih respec o one s sregy Also (31) cn e viewed s firs-order Tylor series pproimion for he response Wih (31) we cn epress he prediced rewrd funcions for he plyers s f r c f r c (3) For convenience we refer o his wo-firm emple s NE gme y NE(λ) In his gme plyer A ims o 0 0 m f r m c which cn e solved y seing df d c 1 1 (33) 0 This hen gives (34) Similrly we cn esily derive he corresponding resul for opimizing As (fer sufficien numer of ierions) he equilirium we hve c c (35) The ove sregies will led o he profis (or rewrds) c 1 1 f 1 (36) c 1 1 f 1 Very ineresingly we cn oserve h he equilirium vries ccording o he elief prmeers of he plyers λ which offers n opporuniy o opimize he equilirium In order o illusre his we le nd consider h is n opimizion prmeer of he gme Then cn e opimized y c m The ove opimizion mimizes oh he sum-profi nd he individul profis of he plyers nd herefore is Preo-opiml Noe h 3 is no permied s his would led o unounded soluions o nd I cn e esily shown h he opiml vlue of is 1 which gives nd f c c 4 4 (37) 1 (38) 1 c c 8 8 (39)

6 1614 J REN ET AL We summrize our resuls nd include some ineresing scenrios in Tle 1 Noe h in his wo-firm emple he Scelerg equilirium is no Preoopiml I cn e seen y compring i wih NE(1) h if he leder is eing less ggressive is profi is no reduced u he profi for he follower cn e furher incresed Anoher oservion is h chieving Preo-opimum does no require h he plyers cogniion e ccure In priculr for he cse of Preo-opimum ie 1 we see from (34) h he equilirium we hve dr d d 3 (40) d However inriguingly he plyer s cogniion is very differen nd for plyer A we hve dr d 1 3 (41) This revels h he elief funcion plyer hs is no required o reflec he rue reliy u cn sill chieve he es oucome from he compeiion of he plyers This hs chnged our undersnding on how plyer s cogniion influences he equilirium of gme A euiful world my e n oucome from people s mises On he conrry if 1 hen i cn e shown h Tle 1 Sregies nd profis for vrious equiliri Model Sregy Profi/Rewrd f d dr dr dr 1 nd 1 (4) d d r d d nd he elief funcions re perfec in erms of represening he reliy Unforunely he profis for oh plyers he equilirium will e 0 showing h if oh plyers re perfecly smr he oucome could e rgedy In his le we lso included he cse wih 3 which ineresingly resuls in he sme profis s he Nsh equilirium u wih hlf producion To show how he elief (or he inerference derivive ) ffecs he sregy nd he rewrd f f f we provide he resuls in Figure 1 ssuming h c Eisence Since he irh of gme heory he ques for he eisence of n equilirium hs een he focl poin in his re In [6] Nsh proved he eisence of he widely nown Nsh equilirium we now ody Ler numerous reserchers presened furher heorems nd proofs ou he eis- ence of he gme-heoreic equiliri [78] sed on wh is nown in he lierure we provide he sufficien condiion for he eisence of he generl NE gme Corollry 1 Consider K -plyer gme wih he sregy spces for 1 K which re no empy compc nd conve suses of n Eucliden spce If he prediced rewrd funcion f r is coninuous nd qusi-concve in S for ny fied hen here eiss n equilirium Nsh Equilirium NE λ wih λ 00 NE λ wih λ 33 c c 3 3 c c 6 6 c c Scelerg Equilirium NE λ wih λ 050 NE λ wih λ 11 NE λ wih λ 1 1 c c 4 c c 4 4 # c c c c 8 16 c c 8 8 I hs he sme profis s Nsh equilirium u hlf producion I chieves he Scelerg equilirium u is no Preo-opiml # This is he Preo-opiml equilirium 00 Figure 1 The sregy nd rewrd vrious equiliri gins he elief λ ssuming c 10 The solid line refers o he resuls for he sregy while he dsh line shows he rewrds

7 J REN ET AL 1615 Proof The NE gme (wih forwrd-looing plyers) presened in Secion 3 follows he sme spiri s ypicl Nsh gme where ny plyer cs owrds n equilirium ssuming h oher plyers sregies re fied Their only difference lies in heir undersnding ou he pyoff funcions due o differen level of cogniion o he environmen Consequenly he sme resul regrding eisence of Nsh equilirium [7] cn e direcly pplied o NE y simply replcing he py off funcion y he prediced rewrd funcion which complees he proof 4 Conclusion This pper showed he impornce of plyers cogniion o he equilirium of he gme nd presened he Nsh equilirium of forwrd-looing plyers Using wo-firm emple we demonsred how Nsh gme wih elief (referred o s NE) cn e mde o chieve he Nsh nd Scelerg equiliri On he oher hnd his pper hs lso illusred he poenil of using he elief funcion s n opimizion prmeer which mes possile he gme converging o Preo-opiml equilirium However his is new regime in gme heory nd fuure wor is required o eer undersnding of elief-direced gmes REFERENCES [1] M J Osorne nd A Ruinsein A Course in Gme Theory MIT Press Cmridge 1994 [] G H Von Scelerg Mre Srucure nd Equilirium Springer-verlg erlin Heidelerg 011 hp://ddoiorg/101007/ [3] E Rsmusen Gme nd Informion sil cwell Ld Oford 1989 [4] J Nsh Non-Cooperive Gmes The Annls of Mhemics Vol 54 No 1951 pp hp://ddoiorg/10307/ [5] D Fudenerg nd J Tirole Gme Theory MIT Press Cmridge 1991 [6] J Nsh Equilirium Pins in n-person Gmes Proceedings of he Nionl Acdemy of Science Vol 36 No pp hp://ddoiorg/101073/pns36148 [7] D A Dereu Socil Equilirium Eisence Proceed- ings of he Nionl Acdemy of Science Vol 38 No pp hp://ddoiorg/ /pns [8] R Myerson Gme Theory: Anlysis of Conflic Hrvrd Universiy Cmridge 1991