Advnd Miroonomis II Lijun Pn Ngoy Univrsiy
Dynmi Gms of Compl Informion Exnsiv-Form Rprsnion Subgm-prf Ns quilibrium
Clssifiion of Gms Si Gms Simulnous Mov Gms Gms wr plyrs oos ions simulnously. Su s prisonrs dilmm nd sl-bid uions. Plyrs mus niip srgy of ir opponn. Dynmi Gms Squnil Mov Gms Gms wr plyrs oos ions in priulr squn. Su s ss nd brgining. Plyrs mus look d in ordr o know w ion o oos now. 3
Si Gms Simulnous Mov Gms norml-form or srgi-form rprsnion: Plyrs Srgis Mum Prisonr Confss Prisonr Mum Confss - - -9 0 0-9 -6-6 Pyoffs 4
Dynmi Gms Squnil Mov Gms Exnsiv-Form Rprsnion Plyr H T Plyr Plyr H T H T - - - - 5
Exnsiv-Form Rprsnion Ss of plyrs: i= n nd psudo-plyr T ordr of movs Aion s Informion s Pyoff funions Probbiliy disribuion of psudoplyr s ion. Prfrn 6
Gm Tr An inumbn monopolis fs possibiliy of nry by llngr. T llngr my oos o nr or sy ou. If llngr nrs inumbn n oos ir o ommod or o fig. T pyoffs r ommon knowldg. Cllngr In Inumbn A F 0 0 Ou T firs numbr is pyoff of llngr. T sond numbr is pyoff of inumbn. 7
A gm r s s of nods nd s of dgs su dg onns wo nods s wo nods r sid o b djn for ny pir of nods r is uniqu p onns s wo nods Gm Tr p from x 0 o x 4 n dg onning nods x nd x 5 x 0 x x x 3 x 4 x 5 x 6 x 7 x 8 nod 8
Gm Tr A p is squn of disin nods y y y 3... y n- y n su y i nd y i+ r djn for i=... n-. W sy is p is from y o yn. W n lso us squn of dgs indud by s nods o dno p. T lng of p is numbr of dgs onind in p. Exmpl : x 0 x x 3 x 7 is p of lng 3. Exmpl : x 4 x x 0 x x 6 is p of lng 4 p from x 0 x 0 o x 4 L M x x U P U P x 3 x 4 x 5 x 6 S T x 7 x 8 9
Gm Tr Tr is spil nod x 0 lld roo of r wi is bginning of gm. T nods djn o x 0 r sussors of x 0. T sussors of x 0 r x x For ny wo djn nods nod is onnd o roo by longr p is sussor of or nod. Exmpl 3: x 7 is sussor of x 3 bus y r djn nd p from x 7 o x 0 is longr n p from x 3 o x 0 x 0 x x x 3 x 4 x 5 x 6 x 7 x 8 0
Gm Tr If nod x is sussor of nor nod y n y is lld prdssor of x. In gm r ny nod or n roo s uniqu prdssor. Any nod s no sussor is lld rminl nod wi is possibl nd of gm Exmpl 4: x4 x5 x6 x7 x8 r rminl nods x 0 x x x 3 x 4 x 5 x 6 x 7 x 8
Gm Tr Any nod or n rminl nod rprsns som plyr. For nod or n rminl nod dgs onn i wi is sussors rprsn ions vilbl o plyr rprsnd by nod Plyr H T Plyr H T Plyr H T - - - -
Gm Tr A p from roo o rminl nod rprsns ompl squn of movs wi drmins pyoff rminl nod Plyr H H Plyr T Plyr T H T - - - 3
Dynmi gms of ompl nd prf Prf informion informion All prvious movs r obsrvd bfor nx mov is osn. A plyr knows Wo s md W ois wn s s n opporuniy o mk oi 4
Dynmi gms of ompl nd imprf Imprf informion informion A plyr my no know xly Wo s md W ois wn s s n opporuniy o mk oi. Exmpl: plyr mks r oi fr plyr dos. Plyr nds o mk r dision wiou knowing w plyr s md. 5
Imprf informion: illusrion E of wo plyrs s pnny. Plyr firs ooss wr o sow Hd or Til. Tn plyr ooss o sow Hd or Til wiou knowing plyr s oi Bo plyrs know following ruls: If wo pnnis m bo ds or bo ils n plyr wins plyr s pnny. Orwis plyr wins plyr s pnny. Plyr H T Plyr H T H T - - - - 6
Informion s An informion s for plyr is ollion of nods sisfying: plyr s mov vry nod in informion s nd wn ply of gm rs nod in informion s plyr wi mov dos no know wi nod in informion s s or s no bn rd. All nods in n informion s blong o sm plyr T plyr mus v sm s of fsibl ions nod in informion s. 7
Informion s: illusrion Plyr L R wo informion ss for plyr onining singl nod Plyr Plyr 3 L R 3 3 L R 3 L R L R L R L R 3 0 3 0 n informion s for plyr 3 onining r nods n informion s for plyr 3 onining singl nod 8
Informion s: illusrion All nods in n informion s blong o sm plyr Plyr C D Tis is no orr informion s Plyr Plyr 3 E F G H 3 3 0 0 3 9
Informion s: illusrion T plyr mus v sm s of fsibl ions nod in informion s. Plyr C D An informion s nno onins s wo nods Plyr Plyr E F G K H 3 0 0 3 0
Rprsn si gm s gm r: illusrion Prisonrs dilmm T firs numbr is pyoff for plyr nd sond numbr is pyoff for plyr Prisonr Mum Fink Prisonr Prisonr Mum Fink Mum Fink 4 4 5 0 0 5
Prf informion nd imprf informion A dynmi gm in wi vry informion s onins xly on nod is lld gm of prf informion. A dynmi gm in wi som informion ss onin mor n on nod is lld gm of imprf informion.
Ns quilibrium in dynmi gm W n lso us norml-form o rprsn dynmi gm T s of Ns quilibri in dynmi gm of ompl informion is s of Ns quilibri of is norml-form How o find Ns quilibri in dynmi gm of ompl informion Consru norml-form of dynmi gm of ompl informion Find Ns quilibri in norml-form 3
Enry gm Plyr s srgis In Ou Plyr s srgis In Ou Pyoffs Norml-form rprsnion Plyr In Ou Plyr In -3-3 0 Ou 0 0 0 4
Enry gm Plyr Plyr In Ou In Ou In Ou Plyr -3-3 0 0 0 0 In { In In }{ In Ou }{ Ou In }{ Ou Ou } -3-3 -3-3 0 0 Plyr Ou 0 0 0 0 0 0 5
Enry gm Tr pur srgy NE: in { Ou In } in { Ou Ou } Ou { In In } 6
Enry gm Plyr In Plyr Ou Tr pur srgy NE: in { Ou In } in { Ou Ou } Ou { In In } In Ou In Ou Plyr -3-3 0 0 0 0 In { In In }{ In Ou }{ Ou In }{ Ou Ou } -3-3 -3-3 0 0 Plyr Ou 0 0 0 0 0 0 7
Enry gm Ou { In In } is no rdibl; in { Ou Ou } is no rdibl; in { Ou In } is rdibl. ully in { Ou In } is uniqu subgm prf Ns quilibrium. 8
Rmov nonrsonbl Ns quilibrium Subgm prf Ns quilibrium is rfinmn of Ns quilibrium I n rul ou nonrsonbl Ns quilibri or nonrdibl rs W firs nd o dfin subgm 9
Subgm of dynmi gm of ompl nd prf informion A subgm of gm r bgins nonrminl nod nd inluds ll nods nd dgs following non-rminl nod Plyr H Plyr T Plyr H T H T - - - - subgm 30
Subgm: xmpl Plyr Plyr C Plyr E F Plyr G H 3 D 0 E Plyr G H 0 0 Plyr F 3 0 0 G H 0 0 3
Subgm of dynmi gm of ompl nd imprf informion bgins singlon informion s n informion s onins singl nod nd inluds ll nods nd dgs following singlon informion s nd dos no u ny informion s; is if nod of n informion s blongs o is subgm n ll nods of informion s lso blong o subgm. 3
Subgm: illusrion I E subgm 0 0 B A subgm - R D No subgm R D R D -0.5-0.5 -K -K -K -K -K -K 33
Subgm-prf Ns quilibrium A Ns quilibrium of dynmi gm is subgmprf if srgis of Ns quilibrium onsiu or indu Ns quilibrium in vry subgm of gm. Subgm-prf Ns quilibrium is Ns quilibrium. 34
Enry gm Plyr In Plyr Ou Tr pur srgy NE: in { Ou In } in { Ou Ou } Ou { In In } In Ou In Ou Plyr -3-3 0 0 0 0 In { In In }{ In Ou }{ Ou In }{ OuOu } -3-3 -3-3 0 0 Plyr Ou 0 0 0 0 0 0 35
Enry gm Ou { In In } is no subgm prf; in { Ou Ou } is no subgm prf; in { Ou In } is subgm prf. ully in { Ou In } is uniqu subgm prf Ns quilibrium. 36
Find subgm prf Ns quilibri: bkwrd induion Plyr C Plyr E Plyr G H F 3 D 0 0 0 Subgm prf Ns quilibrium DG E Plyr plys D nd plys G if plyr plys E Plyr plys E if plyr plys C 37
Exisn of subgm-prf Ns quilibrium Evry fini dynmi gm of ompl nd prf informion s subgm-prf Ns quilibrium n b found by bkwrd induion. 38
Bkwrd induion: illusrion Plyr C D Plyr Plyr E F G H 3 0 0 3 Subgm-prf Ns quilibrium C EH. plyr plys C; plyr plys E if plyr plys C plys H if plyr plys D. 39
Mulipl subgm-prf Ns quilibri: illusrion Plyr C D E Plyr Plyr Plyr F G H I J K 0 0 3 Subgm-prf Ns quilibrium D FHK. plyr plys D plyr plys F if plyr plys C plys H if plyr plys D plys K if plyr plys E. 40
Mulipl subgm-prf Ns quilibri Plyr C D E Plyr Plyr Plyr F G H I J K 0 0 3 Subgm-prf Ns quilibrium E FHK. plyr plys E; plyr plys F if plyr plys C plys H if plyr plys D plys K if plyr plys E. 4
Mulipl subgm-prf Ns quilibri Plyr C D E Plyr Plyr Plyr F G H I J K 0 0 3 Subgm-prf Ns quilibrium D FIK. plyr plys D; plyr plys F if plyr plys C plys I if plyr plys D plys K if plyr plys E. 4
Squnil brgining Plyr nd r brgining ovr on dollr. T iming is s follows: A bginning of firs priod plyr proposs o k sr s of dollr lving -s o plyr. Plyr ir ps offr or rjs offr in wi s ply oninus o sond priod A bginning of sond priod plyr proposs plyr k sr s of dollr lving -s o plyr. Plyr ir ps offr or rjs offr in wi s ply oninus o ird priod A bginning of ird priod plyr rivs sr s of dollr lving -s for plyr wr 0<s <. T plyrs r impin. Ty disoun pyoff by f wr 0< < 43
Squnil brgining Priod Plyr Plyr propos n offr s -s s p -s rj Priod Plyr propos n offr s -s Plyr s -s p rj Priod 3 s -s 44
Solv squnil brgining by bkwrd induion Priod : Plyr ps s if nd only if s s. W ssum plyr will p n offr if indiffrn bwn ping nd rjing Plyr fs following wo opions: offrs s = s o plyr lving -s = -s for rslf is priod or offrs s < s o plyr plyr will rj i nd rivs -s nx priod. Is disound vlu is -s Sin -s<-s plyr sould propos n offr s * -s * wr s * = s. Plyr will p i. 45
Squnil brgining Priod Priod Plyr propos n offr s -s Plyr s p -s rj s - s Plyr propos n offr s -s Plyr s -s p rj Priod 3 s -s 46
Solv squnil brgining by bkwrd induion Priod : Plyr ps -s if nd only if -s -s *= - s or s --s * wr s * = s. Plyr fs following wo opions: offrs -s = -s *=- s o plyr lving s = --s *=-+s for rslf is priod or offrs -s < -s * o plyr plyr will rj i nd rivs s * = s nx priod. Is disound vlu is s Sin s < -+s plyr sould propos n offr s * -s * wr s * = -+s 47
Sklbrg modl of duopoly A omognous produ is produd by only wo firms: firm nd firm. T quniis r dnod by q nd q rspivly. T iming of is gm is s follows: Firm ooss quniy q 0. Firm obsrvs q nd n ooss quniy q 0. T mrk prid is PQ= Q wr is onsn numbr nd Q=q +q. T os o firm i of produing quniy q i is C i q i =q i. Pyoff funions: u q q =q q +q u q q =q q +q 48
Sklbrg modl of duopoly Find subgm-prf Ns quilibrium by bkwrd induion W firs solv firm s problm for ny q 0 o g firm s bs rspons o q. T is w firs solv ll subgms bginning firm. Tn w solv firm s problm. T is solv subgm bginning firm 49
Sklbrg modl of duopoly Solv firm s problm for ny q 0 o g firm s bs rspons o q. Mx u q q =q q +q subj o 0 q + FOC: q q = 0 Firm s bs rspons R q = q / if q = 0 if q > 50
Sklbrg modl of duopoly Solv firm s problm. No firm n lso solv firm s problm. T is firm knows firm s bs rspons o ny q. Hn firm s problm is Mx u q R q =q q +R q subj o 0 q + T is Mx u q R q =q q / subj o 0 q + FOC: q / = 0 q = / 5
Sklbrg modl of duopoly Subgm-prf Ns quilibrium / R q wr R q = q / if q = 0 if q > T is firm ooss quniy / firm ooss quniy R q if firm ooss quniy q. T bkwrd induion ouom is / /4. Firm ooss quniy / firm ooss quniy /4. 5
Sklbrg modl of duopoly Firm produs q = / nd is profi q q + q = /8 Firm produs q = /4 nd is profi q q + q = /6 T ggrg quniy is 3 /4. 53
Courno modl of duopoly Firm produs q = /3 nd is profi q q + q = /9 Firm produs q = /3 nd is profi q q + q = /9 T ggrg quniy is /3. 54
Monopoly Suppos only on firm monopoly produs produ. T monopoly solvs following problm o drmin quniy q m. Mx q m q m subj o 0 q m + FOC: q m = 0 q m = / Monopoly produs q m = / nd is profi q m q m = /4 55
Squnil-mov Brrnd modl of duopoly diffrnid produs Two firms: firm nd firm. E firm ooss pri for is produ. T pris r dnod by p nd p rspivly. T iming of is gm s follows. Firm ooss pri p 0. Firm obsrvs p nd n ooss pri p 0. T quniy onsumrs dmnd from firm : q p p = p + bp. T quniy onsumrs dmnd from firm : q p p = p + bp. T os o firm i of produing quniy q i is C i q i =q i. 56
Squnil-mov Brrnd modl of duopoly diffrnid produs Solv firm s problm for ny p 0 o g firm s bs rspons o p. Mx u p p = p + bp p subj o 0 p + FOC: + p + bp = 0 p = + + bp / Firm s bs rspons R p = + + bp / 57
Squnil-mov Brrnd modl of duopoly diffrnid produs Solv firm s problm. No firm n lso solv firm s problm. Firm knows firm s bs rspons o p. Hn firm s problm is Mx u p R p = p + br p p subj o 0 p + T is Mx u p R p = p + b + + bp / p subj o 0 p + FOC: p + b + + bp /+ +b / p = 0 p = ++b+b b // b 58
Squnil-mov Brrnd modl of duopoly diffrnid produs Subgm-prf Ns quilibrium ++b+b b // b R p wr R p = + + bp / Firm ooss pri ++b+b b // b firm ooss pri R p if firm ooss pri p. 59
Exmpl: muully ssurd dsruion Two suprpowrs nd v nggd in provoiv inidn. T iming is s follows. T gm srs wi suprpowr s oi ir ignor inidn I rsuling in pyoffs 0 0 or o sl siuion E. Following slion by suprpowr suprpowr n bk down B using i o los f nd rsul in pyoffs - or i n oos o prod o n omi onfronion siuion A. Upon is oi wo suprpowrs ply following simulnous mov gm. Ty n ir rr R or oos o doomsdy D in wi world is dsroyd. If bo oos o rr n y suffr smll loss nd pyoffs r -0.5-0.5. If ir ooss doomsdy n world is dsroyd nd pyoffs r -K -K wr K is vry lrg numbr. 60
Exmpl: muully ssurd dsruion I E 0 0 B A - R D R D R D -0.5-0.5 -K -K -K -K -K -K 6
Find subgm prf Ns quilibri: bkwrd induion I E subgm 0 0 B A subgm Sring wi os smlls subgms Tn mov bkwrd unil roo is rd - R D On subgmprf Ns quilibrium IR AR R D -0.5-0.5 -K -K R D -K -K -K -K 6
Find subgm prf Ns quilibri: bkwrd induion I E subgm 0 0 B A subgm Sring wi os smlls subgms Tn mov bkwrd unil roo is rd - R D Anor subgmprf Ns quilibrium ED BD R D -0.5-0.5 -K -K R D -K -K -K -K 63
Bnk runs Two invsors nd v dposid D wi bnk. T bnk s invsd s dposis in long-rm proj. If bnk liquids is invsmn bfor proj murs ol of r n b rovrd wr D > r > D/. If bnk s invsmn murs proj will py ou ol of R wr R>D. Two ds wi invsors n mk widrwls from bnk. 64
Bnk runs: iming of gm T iming of is gm is s follows D bfor bnk s invsmn murs Two invsors ply simulnous mov gm If bo mk widrwls n rivs r nd gm nds If only on mks widrwl n s rivs D or rivs r-d nd gm nds If nir mks widrwl n proj murs nd gm oninus o D. D fr bnk s invsmn murs Two invsors ply simulnous mov gm If bo mk widrwls n rivs R nd gm nds If only on mks widrwl n s rivs R-D or rivs D nd gm nds If nir mks widrwl n bnk rurns R o invsor nd gm nds. 65
Bnk runs: gm r W NW W: widrw NW: no widrw W NW W NW D r r D r D r D D subgm W NW D On subgm-prf Ns quilibrium NW W NW W W NW W NW R R R D D D R D R R 66
Bnk runs: gm r W NW W: widrw NW: no widrw W NW W NW D r r D r D r D D subgm W NW D On subgm-prf Ns quilibrium W W W W W NW W NW R R R D D D R D R R 67
Triffs nd imprf inrnionl ompiion Two idnil ounris nd simulnously oos ir riff rs dnod rspivly. Firm from ounry nd firm from ounry produ omognous produ for bo om onsumpion nd xpor. Afr obsrving riff rs osn by wo ounris firm nd simulnously ooss quniis for om onsumpion nd for xpor dnod by nd rspivly. Mrk pri in wo ounris P i Q i = Q i for i=. Q = + Q = +. Bo firms v onsn mrginl os. E firm pys riff on xpor o or ounry. 68
69 Triffs nd imprf inrnionl ompiion Firm 's pyoff is is profi: ] [ ] [ Firm 's pyoff is is profi: ] [ ] [
70 Triffs nd imprf inrnionl ompiion Counry 's pyoff is is ol wlfr: sum of onsumrs' surplus njoyd by onsumrs of ounry firm 's profi nd riff rvnu Q W wr Q. Counry 's pyoff is is ol wlfr: sum of onsumrs' surplus njoyd by onsumrs of ounry firm 's profi nd riff rvnu Q W wr Q.
7 Bkwrd induion: subgm bwn wo firms Hr w will find Ns quilibrium of subgm bwn wo firms for ny givn pir of. Firm mximizs ] [ ] [ FOC: 0 0 Firm mximizs ] [ ] [ FOC: 0 0
7 Bkwrd induion: subgm bwn wo firms Hr w will find Ns quilibrium of subgm bwn wo firms for ny givn pir of. Givn Ns quilibrium * * * * of subgm sould sisfy s quions. Solving s quions givs us 3 3 3 3 * * * *
73 Bkwrd induion: wol gm Bo ounris know wo firms' bs rspons for ny pir Counry mximizs Q Q W Plugging w w go ino ounry 's objiv funion 3 3 3 3 3 3 3 3 3 3 8 FOC: 3 By symmry w lso g 3
74 Triffs nd imprf inrnionl ompiion T subgm-prf Ns quilibrium 3 3 3 3 3 3 * * T subgm-prf ouom 9 9 4 9 9 4 3 3 * * * * * *