Game theoy he study of multeson decsons Fou tyes of games Statc games of comlete nfomaton ynamc games of comlete nfomaton Statc games of ncomlete nfomaton ynamc games of ncomlete nfomaton Statc v. dynamc Smultaneously v. sequentally Comlete v ncomlete nfomaton Playes ayoffs ae ublc nown o vate nfomaton Concet of Game Equlbum Nash equlbum (NE Statc games of comlete nfomaton Subgame efect Nash equlbum (SPNE ynamc games of comlete nfomaton ayesan Nash equlbum (NE Statc games of ncomlete nfomaton Pefect ayesan equlbum (PE ynamc games of ncomlete nfomaton Lectue Notes II- Statc Games of Comlete Infomaton Nomal fom game he sone s dlemma efnton and devaton of Nash equlbum Counot and etand models of duooly Pue and mxed stateges Statc Games of Comlete Infomaton Fst the layes smultaneously choose actons; then the layes eceve ayoffs that deend on the combnaton of actons ust chosen he laye s ayoff functon s common nowledge among all the layes Nomal- Fom Reesentaton he nomal-fom eesentaton of a games secfes ( the layes n the game ( the stateges avalable to each laye (3 the ayoff eceved by each laye fo each combnaton of stateges that could be chosen by the layes Game defnton enotaton n laye game Stategy sace S : the set of stateges avalable to laye s S : s s a membe of the set of stateges S Playe s ayoff functon u (s,.,s n : the ayoff to laye f layes choose stateges (s,.,s n efnton he nomal-fom eesentaton of an n-laye game secfes the layes stategy saces S,.,S n and the layoff functons u,.,u n. We denote game by G={S,.,S n ; u,.,u n }
Examle: he Psone s lemma Psone Examle: he Psone s lemma um (slent Fn (confess Psone Psone um Fn -,- -9,0 0,-9 -,- Stategy sets S =S ={um, Fn } Payoff functons u (um, um= -, u (um, Fn= -9, u (Fn, um=0, u (Fn, Fn= - u (um, um= -, u (um, Fn=0, u (Fn, um= -9, u (Fn, Fn= - Psone um Fn um (slent -,- (R,R 0,-9 (,S Fn (confess -9,0 (S, -,- (P,P (temtaton>r (ewad>p (unshment>s (suces (Fn, Fn would be the outcome Stctly omnated Stateges efnton In the nomal-fom game G={S,.,S n ; u,.,u n }, let s and s be feasble stateges fo laye. Stateges s s stctly domnated by stategy s f fo each feasble combnaton of the othe lays stateges, s ayoff fom layng s s stctly less than s ayoff fom ayng s : u (s,.,s -, s,s +,.,s n < u (s,.,s -, s,s +,.,s n fo each (s,.,s -,s +,.,s n that can be constucted fom the othe layes stategy saces S,.,S -,.,S n Iteated Elmnaton of omnated Stateges,0, 0, 0,3 0,,0,0, 0, 0,3 0,,0,0, 0, 0,3 0,,0,0, 0, 0,3 0,,0 Outcome =(, Weaness of Iteated Elmnaton Concet of Nash Equlbum Assume t s common nowledge that the layes ae atonal All layes ae atonal and all layes now that all layes now that all layes ae atonal. he ocess often oduces a vey mecse edcton about the lay of the game Examle L C R, No stctly domnated stategy was elmnated Each laye s edcted stategy must be that laye s best esonse to the edcated stateges of the othe layes Stategcally stable o self enfocng No sngle wants to devate fom hs o he edcated stategy A unque soluton to a game theoetc oblem, then the soluton must be a Nash equlbum
efnton of Nash Equlbum Examles of Nash Equlbum efnton In the n-laye nomal-fom game G={S,.,S n ; u,.,u n }, the stateges (s,.,s n ae a Nash equlbum f, fo each lay, s s laye s best esonse to the stateges secfed fo the n- othe layes, (s,, s -, s +,.,s n u( s, K, s-, s, s+,, sn u( s, K, s-, s, s+,, sn um Fn um Fn -,- -9,0 0,-9, Oea Fght L C R, fo evey feasble stategy s n S ; that s s solves Oea, 0,0 max u( s, K, s-, s, s+,, sn s S Fght 0,0, Examles of Nash Equlbum Playe s stategy (best esonse functon R (L= R (C= R (R= Playe s stategy (best esonse functon L C R (=L R (=C R (=R R, Alcaton Counot odel of uooly q,q denote the quanttes (of a homogeneous oduct oduced by fm and emand functon P(Q=a-Q Q=q +q Cost functon C (q =cq Stategy sace S = [ 0, Payoff functon π ( q, q = q [ P( q + q c] = q [ a ( q + q c] Fm s decson max π ( q, q = max q[ a ( q + q c] 0 q 0 q Fst ode condton q = ( a q q = ( a q q = ( a q c a c q = q = 3 Counot odel of uooly (cont est esonse functons R( q = ( a q R ( q = ( a q q (0,a-c R (q Nash equlbum (q,q (0,(a-c/ R (q q ((a-c/,0 (a-c,0 Alcaton etand odel of uooly Fm and choose ces and fo dffeentated oducts Quantty that customes demand fom fm s q (, = a + b Pay off (oft functons π (, = q (, [ c] = [ a + b ][ c] Fm s decson max π ( 0, = max[ a + b 0 ][ c] Fst ode condton = ( a + b = ( a + b a + c = = b = ( a + b 3
atchng ennes xed Stateges als als -,,-,- -, In any game n whch each laye would le to outguess the othe(s, thee s no ue stategy Nash equlbum E.g. oe, baseball, battle he soluton of such a game necessaly nvolves uncetanty about what the layes wll do Soluton : mxed stategy efnton of xed Stateges efnton In the nomal-fom game G={S,.,S n ; u,.,u n }, suose S ={s,,s K }. hen the mxed stategy fo laye s a obablty dstbuton =(,,, whee 0 fo =,,K and +,,+ K = Examle In enny matchng game, a mxed stategy fo laye s the obablty dstbuton (q,-q, whee q s the obablty of layng s, -q s the obablty of layng al, and 0 q xed stategy n Nash Equlbum Stategy set S ={s,,s }, S ={s,,s } Playe beleves that laye wll lay the stateges (s,,s wth obabltes (,,, then laye s exected ayoff fom layng the ue stategy s s Playe s exected ayoff fom ayng the mxed stategy =(,, s v J efnton In the two laye nomal-fom game G={S,S ;u,u }, the mxed stateges (, ae a Nash equlbum f each laye s mxed stategy s a best esonse to the othe laye s mxed stategy. hat s v, v ( v(, v(, (, u( s, s = (, = u( s, s v ( =, = = J u( s, s = = Playe xed Stategy - als Playe q -q als -,,-,- -, Playe s exected layoff =q(-+(-q(=-q when he lay =q(+(-q(-=q- when he lay al Comae -q and q- If q</, then laye lays If q>/, then lay lays al If q=/, laye s ndffeent n and al xed stategy n Nash Equlbum: examle Playe - als Playe q -q als -,,-,- -, Playe s exected layoff =q(-+(-q(+ (-q(+(-(-q(- =(q-+(-4q Playe s exected layoff =q(+q(-(-+ (-q(-+(-q(-( =(-+q(-4 = f q</ =0 f q>/ = any numbe n (0, f q=/ q= f </ q=0 f >/ q= any numbe n (0, f =/ xed Stategy n Nash Equlbum: examle (cont (s (als / als (q / q( (,q=(/,/ xed stategy Nash equlbum s q 4
Examle Examle Oea q -q Oea Fght, 0,0 (Oea (q (,q=(, - Fght 0,0, /3 q( (,q=(/3,/3 Playe s exected layoff =q(+(-q(0+ (-q(0+(-(-q( = (3q--(+q Playe s exected layoff =q(+q(-(0+ (-q(0+(-q(-( =q(3-+- (Fght /3 q (,q=(0,0 Fght Oea Paye s mxed stateges (,-=(/3,/3 ue stateges (,-=(,0,(0, Paye s mxed stateges (q,-q=(/3,/3 ue stateges(q,-q=(,0,(0, heoem: Exstence of Nash Equlbum (Nash 950: In the n-laye nomal-fom game G={S,.,S n ; u,.,u n }, f n s fnte and S s fnte fo evey I then thee exsts at least one Nash equlbum, ossbly nvolvng mxed stateges Homewo # Poblem set.3,.5,.,.7,.8,.3(fom Gbbons ue date two wees fom cuent class meetng onus cedt Poose new alcatons n the context of I/IS o otental extensons fom Alcaton -4 5