Royal Nthrlands cadmy of rts and Scncs (KNW) Mastr Class mstrdam, Fbruary 8th, 2007 Epstmc Foundatons of Gam Thory Lctur Gacomo onanno (http://www.con.ucdavs.du/faculty/bonanno/)
QUESTION: What stratgs can b chosn by ratonal playrs who know th structur of th gam and th prfrncs of thr opponnts and who rcognz ach othr s ratonalty and knowldg? Kywords: knowldg, ratonalty, rcognton of ach othr s knowldg and ratonalty 2
Modular approach Modul : rprsntaton of blf and knowldg of an ndvdual (Hntkka, 962; Krpk, 963). Modul 2: xtnson to many ndvduals. Common blf and common knowldg ( rcognton of ach othr s blf / knowldg ) Modul 3: dfnton of ratonalty n gams (rlatonshp btwn choc and blfs) QUESTION: what ar th mplcatons of ratonalty and common blf of ratonalty n gams? 3
Modul rprsntaton of blfs and knowldg of an ndvdual α β Fnt st of stats Ω and a bnary rlaton on Ω. mans at stat α th ndvdual consdrs stat β possbl Notaton: ( ω ) = { ω Ω : ω ω } st of stats consdrd possbl at ω PROPERTIES ω, ω Ω,. ( ω) sralty 2. f ω ( ω) thn ( ω ) ( ω) transtvty 3. f ω ( ω) thn ( ω) ( ω ) ucldanns 4
Ω lf oprator on vnts: : 2 2 For E Ω, ω E f and only f ( ω) E Ω EXMPLE: p p p p α β γ δ ( α) = ( β ) = { α, β} ( γ ) = ( δ ) = { δ} Lt E = { β, δ} : th vnt that rprsnts th proposton Thn E = { γ, δ} p 5
Proprts of th blf oprator: E Ω. E E (consstncy: follows from sralty of ) 2. E E (postv ntrospcton: follows from transtvty of ) 3. E E (ngatv ntrospcton: follows from ucldannss of ) Mstakn blfs ar possbl: at γ p s fals but th ndvdual blvs p p p p p α β γ δ If E = { β, δ}, thn γ E but γ E = { γ, δ} 6
KNOWLEDGE If - n addton to th prvous proprts - th "doxastc accssblty" rflxv quvalnc rlaton s ( ω Ω, ω ( ω)) thn t s an rlaton - gvng rs to a partton of th st of stats - and th assocatd blf oprator satsfs th addtonal proprty that E Ω, E E (blfs ar corrct). In ths cas w spak of and th assocatd oprator s dnotd by K rathr than knowldg α β γ 7
Modul 2 ntractv blf and common blf St of ndvduals Ν and a bnary rlaton for vry N 2 α β γ δ p p p p Lt E = { α, β, γ} : th vnt that rprsnts th proposton p Thn K E = { α, β, γ}, K E = { α, β} 2 K K E = { α}, K K K E = 2 2 2 8
n vnt E s commonly blvd f () vrybody blvs t, (2) vrybody blvs that vrybody blvs t, (3) vrybody blvs that vrybody blvs that vrybody blvs t, tc. Dfn th vrybody blvs oprator as follows: E = E 2 E... ne Th common blf oprator * s dfnd as follows: * E = E E E... 9
Lt b th transtv closur of... Thus ω * 2 * ( ω) f and only f thr xsts a squnc ω,..., ω n Ω such that () (2) n m ω = ω ω = ω (3) for vry j =,..., m thr xsts an ndvdual N such that ω ( ω ) j+ j n : ( α) = ( β ) = { α}, ( γ ) = { γ} α β γ 2 : ( α) = { α}, ( β ) = ( γ ) = { β, γ} 2 2 2 α β γ * : ( α) = { α}, ( β ) = ( γ ) = { α, β, γ} * * * 0
PROPOSITION. ω E f and only f ( ω) E. * * p p p : α β γ 2 : α β γ * : Lt E = { β, γ} : th vnt that rprsnts th proposton Thn E = { γ}, E = { β, γ}, 2 E = In fact, whl γ E = { γ}, γ E = 2 2 * p
Modul 3 Modls of gams and Ratonalty Dfnton. fnt stratgc-form gam wth ordnal payoffs s a quntupl { } { } N, S, O,, z N N = {,..., n} s a st of playrs N S s a fnt st of stratgs or chocs of playr N O s a st of outcoms s playr 's ordrng of O ( o o mans that, for playr, outcom o s at last as good as outcom o ) z : S O (whr S = S... S ) assocats an outcom wth vry stratgy profl s S n 2
Dfnton. Gvn a stratgc-form gam wth ordnal payoffs { } { } N, S, O,, z N N a rducd form of t s a trpl { } { } N, S, u N N whr u : S R s such that u ( s) u ( s ) f and only f z( s) z( s ) playr s utlty functon Playr 2 f g Playr 2 f g P l a y r C 3, 2 2, 3, 2 3, 2, 2, 2 0, 3, 4, SME S P l a y r C 9, 6 4, 9 2, 5 6, 4 3, 3 2, 5 0, 4 2, 0 8, 2 D 0, 2 0, 3, 3 D, 0 0, 8, 8 3
Dfnton. n pstmc modl of a stratgc-form gam s an ntractv blf structur togthr wth n functons σ : Ω S ( N) Intrprtaton: σ (ω) s playr s chosn stratgy at stat ω Rstrcton: f ω ( ω) thn σ ( ω ) = σ ( ω) (no playr has mstakn blfs about hr own stratgy) 4
Playr 2 EXMPLE f g P l a y r C 3, 2 2, 3, 2 3, 2, 2, 2 0, 3, 4, D 0, 2 0, 3, 3 2 α β γ δ 's stratgy: 2's stratgy: C C D f f g g t vry stat ach playr knows hs own stratgy t stat β playr plays C (h knows ths) not knowng whthr playr 2 s playng f or g and playr 2 plays f (sh knows ths) not knowng whthr playr s playng or C 5
RTIONLITY Non-probablstc (no xpctd utlty) and vry wak noton of ratonalty Dfnton. Playr s IRRTIONL at stat ω f thr s a stratgy s (of playr ) whch sh blvs to b bttr than σ (ω) (that s, f sh blvs that sh can do bttr wth anothr stratgy) Playr s RTIONL at stat ω f and only f sh s not rratonal Playr 2 f g 2 's stratgy: α β γ δ C C D P l a y r C 3, 2 2, 3, 2 3, 2, 2, 2 0, 3, 4, 2's stratgy: f f g g D 0, 2 0, 3, 3 Playr s ratonal at stat β 6
s Lt s and t b two stratgs of playr : s, t S t s ntrprtd as stratgy s s bttr for playr than stratgy t s t s tru at stat ω f u ( s, σ ( ω)) u ( t, σ ( ω)) > that s, s s bttr than t aganst σ (ω ) Playr 2 E F G profl of stratgs chosn by th playrs othr than 's stratgy: 2's stratgy: α β γ C C E F G P l a y r C 3, 2 2, 3, 2, 2, 2 0, 2 0, 3, 4, C C C C C C E 2 F F 2 G F 2 G tc. 7
{ ω σ ω σ ω } Lt s t = Ω : u ( s, ( )) > u ( t, ( )) vnt that s s bttr than t If s S, lt s { ω : σ ( ω) s} = Ω = vnt that playr chooss s Lt R b th vnt rprsntng th proposton playr s ratonal s t s R = R ( s ) t s s S t S R = R... R all playrs ar ratonal n 8
Playr 2 f g 2 's stratgy: 2's stratgy: α β γ δ C C D f f g g P l a y r C D 3, 2 2, 3, 2 0, 2 3, 2, 2, 2 0, 3 0, 3, 4,, 3 R R R R R2 KR2 R2 R2 R2 KR2 KR2 KR2 K2R K2R K2R K2R KK2R KK2R KK2R KK2R R K = { α, β, γ}, R = { α, β, γ, δ} 2 R = { α, β, γ, δ}, K R = { α, β} 2 2 K K R = { α}, K K K R = 2 2 2 K2KK2R t stat α thr s mutual knowldg of ratonalty but not common knowldg of ratonalty 9
Lt S = S... S S... S st of stratgy profls of all playrs xcpt + n Dfnton. Lt s, t S. W say that t s strctly domnatd by s f u ( t, s ) < u ( s, s ) for all s S ITERTED DELETION OF STRICTLY DOMINTED STRTEGIES Playr 2 f g Playr 2 f g P l a y r C 3, 2 2, 3, 2 3, 2, 2, 2 0, 3, 4, C 3, 2 2, 3, 2 3, 2, 2, 2 0, 3, 4, D 0, 2 0, 3, 3 (by C) (by ) f 3, 2 2, 3 (by ) 3, 2 2, 3 f 3, 2, 2 (by ) 3, 2 2, 3 3, 2, 2 C, 2, 2 (by ) 20
Lt G b a stratgc-form gam wth ordnal payoffs and G b th gam obtand aftr applyng th procdur of Itratd Dlton of Strctly Domnatd Stratgs. Lt S dnot th stratgy profls of gam G Gvn a modl of G, lt S dnot th vnt { ω Ω : σ ( ω) S } P l a y r G C D Playr 2 f g 3, 2 3, 0, 2, 3 2, 2 3,, 2, 2 4, 0, 2 0, 3, 3 G 3, 2 2 's stratgy: S = {(, )} α β γ δ C C S = {δ} 2's stratgy: f f 2
PROPOSITION. R S * If at a stat t s commonly blvd that all playrs ar ratonal, thn th stratgy profl chosn at that stat blongs to th gam obtand aftr applyng th tratd dlton of strctly domnatd stratgs. Playr 2 f g 2 's stratgy: 2's stratgy: α β γ δ C C D f f g g P l a y r C D 3, 2 2, 3, 2 0, 2 3, 2, 2, 2 0, 3 0, 3, 4,, 3 R R R R R2 KR2 R2 R2 R2 KR2 KR2 K R K2R K2R K2R K2R KK2R KK2R KK2R KK2R t stat α thr cannot b 2 common knowldg of ratonalty snc σ ( α) (, ) K2KK2R 22
Evry normal oprator satsfs th proprty that f E F thn E F. s a normal oprator. Thus from * * t follows that R S. * * * * * * y transtvty of w hav that E Thus E * * * It follows that * for vry vnt E. R R. * R * S R S 2 's stratgy: 2's stratgy: Sam as: P l a y r C D 2, 3 α β γ δ 2 3, 2, 2 0, 2 Playr 2 f 3, 2, 2, 2 0, 3 g 0, 3, 4,, 3 's stratgy: 2's stratgy: 23
REMRK. In gnral t s not tru that S * R Playr 2 f g P l a y r C 3, 2 3, 0, 2, 3 2, 2 3,, 2, 2 4, 2 's stratgy: α β γ δ C C D 0, 2 0, 3, 3 2's stratgy: f f S = {δ} K * R = R K 2 = { α, δ}, R = { α, β, γ, δ} 2 R = 24
PROPOSITION 2. Fx a stratgc-form gam wth ordnal payoffs G and lt s S. Thn thr xsts an pstmc modl of G and a stat ω such that σ ( ω) = s and ω R. * EXMPLE P l Playr 2 f 3, 2 3, 3 2, 3 4, 2 2 α β γ δ In ths gam vry stratgy profl survvs tratv dlton * α β γ δ σ σ 2 f f In ths modl R = R = Ω and vry stratgy profl occurs at som stat * 25
REMRK. Gvn th abov noton of ratonalty, thr s no dffrnc btwn common blf of ratonalty and common knowldg of ratonalty. Th prvous two propostons can b rstatd n trms of knowldg and common knowldg. PROPOSITION. K * R S PROPOSITION 2. Fx a stratgc-form gam wth ordnal payoffs G and lt s S. Thn thr xsts an pstmc modl of G and a stat ω such that σ ( ω) = s and ω K R. * 26
STRONGER NOTION OF RTIONLITY Stll non-probablstc (no xpctd utlty) Dfnton. Playr s IRRTIONL at stat ω f thr s a stratgy s whch sh blvs to b at last as good as σ (ω) and sh consdrs t possbl that s s bttr than σ (ω) Playr s RTIONL at stat ω f and only f sh s not rratonal 2 's stratgy: α β γ C C P l a y r 3, 2 2, 3 Playr 2 C 4, 2, 2 2, f 3,, 2 g 0, 3, 2's stratgy: f f g Playr s rratonal at stat β: s at last as good as C at both β and γ and t s bttr than C at γ R = { α}, R = 2 27
Playr s IRRTIONL at stat ω f thr s a stratgy s whch sh blvs to b at last as good as σ (ω) and sh consdrs t possbl that s s bttr than σ (ω) s t s t s R ( s ) t s t s R = s S t S R = R... R all playrs ar ratonal n 28
Dfnton. { } { } Gvn a gam G = N, S, O,, z, a subst of stratgy profls X S and a stratgy profl x X, w say that x s nfror (thus s N N rlatv to X f thr xst a playr and a stratgy s S of playr nd not blong to th projcton of X onto S ) such that:. z( s, x ) z( x, x ) and 2. for all s S, f ( x, s ) X thn z( s, s ) z( x, s ). Itratd Dlton of Infror Profls : m that ar nfror rlatv to. for m N dfn m 0 T S rcursvly as follows: T = S and, for m, m m m m m T = T \ I, whr I T s th st of stratgy profls T m N m Lt T = T. 29
Playr Playr 2 Playr 2 d f d f 2, 0, 2, 2, 0, 2, Playr, 0, 0,, 0, C, 4, 3 0, 3 C, 4, 3 T 0 T Playr Playr 2 Playr 2 d f d f 2, 0, 2, 2, 0, 2, Playr C C, 4 T = T 3 T 2 0 0 T = S = d f d f C d C C f I = C f {(, ),(, ),(, ),(, ),(, ),(, ),(, ),(, ),(, )}, {(, ),(, )} (th lmnaton of (, ) s don through playr 2 and stratgy f, whl th lmnaton of ( C, f ) s don through playr and stratgy ); T = d f d f C d C I = d f C d f {(, ),(, ),(, ),(, ),(, ),(, ),(, )}, ¹ {(, ),(, ),(, )} (th lmnaton of (, ) and (, ) s don through playr and stratgy, whl th lmnaton of ( C, ) s don through playr 2 and stratgy d); T ² = {(, d),(, ),(, f ),( C, d)}, I ² = {( C, d)} (th lmnaton of ( C, d) s don through playr and stratgy ); T ³ = {(, d),(, ),(, f )}, I ³ = ; thus T = T ³. 30
PROPOSITION 3. R T K* If at a stat t s commonly known that all playrs ar ratonal, thn th stratgy profl chosn at that stat blongs to th gam obtand aftr applyng th tratd dlton of Infror stratgy profls. PROPOSITION 4. Fx a stratgc-form gam wth ordnal payoffs G and lt s T. Thn thr xsts an pstmc modl of G and a stat ω such that σ ( ω) = s and ω K R. * 3
NOT TRUE f w rplac common knowldg wth common blf Playr Playr 2 c d,, 0, 0, : 2 : α α β β R = { α, β}, R = { α, β} 2 Thr s common blf of ratonalty at vry stat and yt at stat α th stratgy profl playd s (,d) whch s nfror * : σ : σ 2 : d c T S = = {(,c),(,c)} {(,c),(,d),(,c),(,d)} 32
PROILISTIC ELIEFS Dfnton. aysan fram s an ntractv blf fram togthr wth a collcton { p } of probablty, ω N, ω Ω dstrbutons on Ω such that () f ω ( ω) thn p =, ω, ω, ω (2) p ( ω ) > 0 f and only f ω ( ω) (th support of p concds wth ( ω)), ω p : /2 /2 α β γ : /3 2/3 2 33
Dfnton. stratgc-form gam wth von Numann-Morgnstrn payoffs s a quntupl N = {,..., n} s a st of playrs { } { } N, S, O, U, z N S s th st of stratgs of playr N O whr s a st of U : O R s playr 's von Numann-Morgnstrn utlty functon outcoms z : S O (whr S = S... S ) assocats an outcom wth vry stratgy profl n s S N { } { π } Its rducd form s a trpl N, S, whr π ( s ) U ( z ( s )). = N N 34
n pstmc modl of a stratgc-form gam s a aysan fram togthr wth n functons σ : Ω S ( N) such that f ω ( ω) thn σ ( ω ) = σ ( ω) Strongr dfnton of Ratonalty than th prvous ons Playr s RTIONL at stat α f hr choc at α maxmzs hr xpctd payoff, gvn hr blfs at α: for all t S ( ) p π ( t ) σ ( α), σ ( ω) ( ω), σ ( ω) ( ω), α, α ω ( α ) ω ( α ) π p 35
α β γ δ ε Playr 2 d : d d 2/3 /3 /2 /2 R = d { δ, ε} P l a y r C 3, 0, 0 0, 3 0, 2, 2 3, 2 2 4 Playr s not ratonal at α bcaus hr xpctd payoff s + 2 = 3 3 3 2 whl f sh had chosn stratgy hr payoff would hav bn 3+ 0 = 2 3 3 3 On th othr hand, Playr s ratonal at δ bcaus hr xpctd payoff s 3+ 0 = 2 2 2 3 and f sh had chosn stratgy hr payoff would hav bn + 2 = 2 2 2 3 and f sh had chosn stratgy C hr payoff would hav bn 0 + 3 = 2 2 2 36
What ar th mplcatons of Common lf of ths strongr noton of ratonalty? Dfnton. mxd stratgy of playr s a probablty dstrbuton ovr S Th st of mxd stratgs of playr s dnotd by (S ) Lt t S and ν ( S ). W say that t s strctly domnatd by ν f, for vry s S, π ( t, s ) < ν ( s ) π ( s, s ) d Playr 2 s S P l a y r C 3, 0 0, 0 0, 3 0, 2, 2 3, 2 In ths gam stratgy of playr s strctly domnatd by th mxd stratgy 5 6 6 C 37
Playr 2 f g Playr 2 ITERTIVE DELETION OF PURE STRTEGIES THT RE STRICTLY P l a y r C D 3, 0, 0 0,, 0, 2, 0, 0 4, 2, 2 0, 3, 0 3, 2 (a) Th gam G s strctly domnatd by (/2, /2 D) P l a y r C D f g 3, 0, 0 0, 0, 0 4, 2, 2 0, 3, 0 3, 2 (b) Th gam G Now f s strctly domnatd by g DOMINTED Y (POSSILY Playr 2 g Playr 2 MIXED) STRTEGIES P l a y r C D 3, 0 0, 0 0, 3 0, 2, 2 3, 2 Playr g 3, 0 0, D 0, 3 3, 2 (c) 2 Th gam G Now C s strctly domnatd by (/6, 5/6 D) (d) 3 Th gam G = G No stratgy s strctly domnatd 38
Lt G b a stratgc-form gam wth von Numann-Morgnstrn payoffs and G b th gam obtand aftr applyng th procdur of Itratd Dlton of Pur Stratgs that ar Strctly Domnatd by Possbly Mxd Stratgs. Lt S dnot th pur-stratgy profls of gam G m { Ω Sm} Gvn a modl of G, lt S b th vnt ω : σ ( ω) m PROPOSITION 5. * R S m PROPOSITION 6. Fx a stratgc-form gam wth von Numann-Morgnstrn payoffs G and lt s S. Thn thr xsts a aysan modl of G and a stat ω such that σ ( ω) = s and ω R. m * 39
Gvn ths strongr noton of ratonalty, thr s a dffrnc btwn common blf of ratonalty and common knowldg of ratonalty. Th mplcatons of common knowldg of ratonalty ar strongr. Wth knowldg, a playr s blfs ar always corrct and ar blvd to b corrct by vry othr playr. Thus thr s corrctnss and common blf of corrctnss of vrybody s blfs. 40
Dfnton. Gvn a stratgc-form gam wth von Numann-Morgnstrn payoffs G, a pur-stratgy profl x X S s nfror rlatv to X f thr xsts a playr and a (possbly mxd) stratgy ν of playr (whos support can b any subst of S, not ncssarly th projcton of X onto S ) such that: π ( x ) < (), x π ( s, x ) ν ( s ) ( ν ylds a hghr xpctd payoff than x aganst x ) s S π ( x, s ) (2) for all s S such that ( x, s ) X, π ( s, s ) ν ( s ) s S Playr 2 D E F Playr 2, 0 2, 2 0, 2 2, 2, 2 5, C 2, 0, 0, 5 Hr (C,F) s nfror rlatv to S (for playr, wakly domnats C and s strctly bttr than C aganst F) and (,D) s nfror rlatv to S (for playr 2, E wakly domnats D and s strctly bttr than D aganst ) 4
ITERTED DELETION OF INFERIOR PURE STRTEGY PROFILES Playr 2 D E F Playr 2, 0 2, 2 0, 2 2, 2, 2 5, C 2, 0, 0, 5 (a) S 0 s = S, D 0 s = {(, D), (C, F)} Playr 2 D E F Playr 2, 2 0, 2 2, 2, 2 C 2, 0 (c) S 2 s = {(, E), (, F), (,D), (, E), (C, D) }, D 2 s = {(, E)}. Playr 2 D E F Playr 2, 2 0, 2 2, 2, 2 5, C 2, 0, 0 (b) S s = {(, E), (, F), (,D), (, E), (, F), (C, D), (C, E)} D s = {(C, E), (, F)} Playr 2 D E F Playr 2, 2 0, 2 2, 2 C 2, 0 (d) S 3 s = S s = {(, E), (, F), (,D), (C, D) }, D 3 s =. 42
Lt G b a stratgc-form gam wth von Numann-Morgnstrn payoffs and G b th gam obtand aftr applyng th procdur of Itratd Dlton of Infror Pur-Stratgy Profls. Lt S dnot th pur-stratgy profls of gam G s { Ω Ss } Gvn a modl of G, lt S b th vnt ω : σ ( ω) s PROPOSITION 7. K* R S s PROPOSITION 8. Fx a stratgc-form gam wth von Numann-Morgnstrn payoffs G and lt s S. Thn thr xsts a aysan modl of G and a stat ω such that σ ( ω) = s and ω K R. s * 43
Playr 2 D E F Playr 2, 0 2, 2 0, 2 2, 2, 2 5, C 2, 0, 0, 5 In ths gam m whl S = {(, E),(, F),(, D),( C, D)} s S = S = S Thus vry stratgy profl s compatbl wth common blf of ratonalty whl only (,E), (,F), (,D) and (C,D) ar compatbl wth common knowldg of ratonalty 44
CREDITS Th lnk btwn th tratd dlton of strctly domnatd stratgs and th nformal noton of common blf of ratonalty was frst shown by rnhm (984) and Parc (984) Th frst xplct pstmc charactrzaton was provdd by Tan and Wrlang (998) usng a unvrsal typ spac. Th stat spac formulaton usd n Propostons 5 and 6 s du to Stalnakr (994), but t was mplct n randnburgr and Dkl (987). Propostons 7 and 8 ar du to Stalnakr (994) (wth a corrcton gvn n onanno and Nhrng, 996b). To my knowldg, Propostons, 2, 3 and 4 hav not bn xplctly statd bfor. Rfrncs and furthr dtals can b found n attgall, Prpaolo and onanno Gacomo, Rcnt rsults on blf, knowldg and th pstmc foundatons of gam thory, Rsarch n Economcs, 53 (2), Jun 999, pp. 49-225. For a syntactc vrson of Propostons, 2, 3 and 4 s Gacomo onanno, syntactc approach to ratonalty n gams, Workng Papr, Unvrsty of Calforna, Davs (http://www.con.ucdavs.du/faculty/bonanno/pdf/cr.pdf) 45