Optmzaton and Control o Ntork Potntal Gam and th Inny o Equlbrum Ljun Chn 3/8/216
Outln Potntal gam q Rv on tratg gam q Potntal gam atom and nonatom Inny o qulbrum q Th pr o anarhy and lh routng q Rour alloaton q Ntork dgn gam and th pr o tablty 2
Stratg gam q D: a gam n tratg orm a trpl q Notaton q q q N G { N S N u N } q th t o playr agnt q S th playr tratgy pa q u : S R th playr payo unton S S : th t o all prol o playr tratg 1 S2! S N 1 2! N : prol o tratg 1 2! 1 +1! N : th prol o tratg othr than playr 3
Potntal gam atom Dnton: A unton a potntal unton or gam or Whn t th gam alld a potntal gam. Dnton: A unton an ordnal potntal unton or gam or Whn t th gam alld an ordnal potntal gam. 4 S R Φ : G S S. u u Φ Φ. > Φ Φ > u u S R Φ : G S S Φ Φ
Eampl Gam Potntal 11 3 2 1 3 22 1 5
Equlbrum a pur tratgy Nah qulbrum or ordnal potntal gam Proo: I a potntal unton a pur tratgy Nah qulbrum 6 G. S Φ Φ Φ. Φ Φ u u u u. Φ Φ
I Φ ha a mamum at thn a pur tratgy Nah qulbrum o th ordnal gam. Evry nt ordnal potntal gam ha a pur tratgy Nah qulbrum. Contnuou ordnal potntal gam ha a pur tratgy Nah qulbrum th tratgy pa ompat and potntal ontnuou. 7
Congton gam Dnton: A ongton modl dnd a ollo q q N th t o playr M th t o alt or rour S { N M S N j M } q th t o th rour that playr an u q j k j th ot to ur ho u th rour j hn ur ar ung t Dnton: A ongton gam aoatd th a ongton modl a gam th ot k j j j { N S N N } k j 8
Evry ongton gam a potntal gam th potntal Congton gam hav many applaton q Ntork dgn Φ j j k 1 k j k. 9
Potntal gam nonatom Nonatom gam: th ur numbr nnt q N r la o nntmal playr q th ma o la playr q th raton o la playr that hoo tratgy q u ; th payo or a playr o la th Dnton: an qulbrum or all S > u Dnton: A nonatom gam a potntal gam thr t potntal unton Φ uh that u ; ; Φ. u ;. 1
Eampl: lh routng Condr a multommodty lo ntork q N our-dtnaton par ommodt V E q Eah ommodty ha a total rat and an u a t o path P q Th aggrgat tra among lnk : q lnk ot a nonngatv ontnuou nondrang unton o tra q Th ot ; r 11
Wardrop qulbrum: th ot o all th path atually ud ar qual and l than tho hh ould b prnd by a ngl ur on any unud path. { V E; r } a potntal gam th potntal Φ d. 12
Inny o qulbra Equlbra o tratg gam ar typally nnt Eampl: Pronr Dlmma D C 33 4 D C 4 11 13
Pgou ampl 1 t On ommodty th rat 1 q A unqu Wardrop qulbrum th all tra routd on th lor dg q A bttr lo: rout hal o th tra on ah o th to dg 14
Quton ar Th tn o qulbrum Quanty th nny o qulbrum Th onvrgn o qulbrum 15
Quanty th nny Pr o anarhy: quanty nny th rpt to om objtv unton pr o anarhy obj n valu o th ort qulbrum optmal obj n valu <1 or mamzaton; >1 or mnmzaton Intrtd n tuaton n hh an bound th pr o anarhy 16
Slh Routng At Wardrop qulbrum Th abov th KKT optmalty ondton or 17 mn ; ; r Φ r d :.t. mn
A lo or a Wardrop qulbrum and only t a global mnmum o th potntal unton Dn th objtv unton.. th ot o lo a Dnton: An optmal lo or th lo that mnmz. 18 } ; { r E V. Φ d C } ; { r E V C
Th pr o anarhy Th pr o anarhy ρ C C. Pgou ampl Suppo that γ y dy thn ρ γ. Pgou ampl th dgr-d polynomal ot q q ρ d d +1 + 1 ρ 4 / y dy 3 19
Rour alloaton Condr a mpl ntork: th our ur har a lnk and th ntork lnk managr ant to alloat lnk rat uh that Sytm: ma. t. tlty unton ar not knon to th lnk managr 2
Markt-larng mhanm Eah ur ubmt a bd or llngn to pay th managr k to alloat th ntr lnk apaty and t a pr p uh that q A th ur ha a dmand unton D p p p q Th lnk managr hoo a pr to lar th markt D p / 21
Pr takng ur and ompttv qulbrum Th ur a pr takr: do not antpat th t o h paymnt on th pr It ratonal or th ur to mamz th ollong payo Klly 98 A par a ompttv qulbrum 22 p p u p p p u p u / or any
Thorm Klly 98: thr t a unqu ompttv qulbrum uh that olv th problm ytm. Proo: ondr th Lagrangan q At prmal-dual optmal 23 p p / p p D ' ' > p p p p
q Sn > at lat on potv. So p >. q Thu. q Lt p thn p a ompttv qulbrum and olv th problm Sytm. q In th a th unqun o ollo rom th unqun o p. / p 24
Pr antpatng ur and Nah qulbrum Pr antpatng ur ralz that th pr t aordng to p and ll adjut thr bd / aordngly. Th mak th modl a gam hr ur payo Johar 4 > u Condr Nah qulbrum uh that u u or all or all 25
Thorm Hajk t al: thr t a unqu Nah qulbrum. Morovr th rat ar unqu oluton o th ollong problm 26 t.. ˆ ma Gam: hr. 1 1 ˆ + dz z
Proo: q I a Nah qulbrum at lat to playr hav nonzro bd. u q Thn trtly onav and ontnuouly drntabl n. q Thn at qulbrum ' ' t t 1 t q Th abov ondton alo unt. t t t t t > 27
q Th problm Gam ha a unqu optmal thr t a p uh that ' 1 p > ' p p p. Morovr q Lt / p and p / t t. Thn p at th abov optmalty ondton. 28
Th pr o anarhy Aum hav Thn Sn hav Thn 29 1 dz z ˆ z z z z + 1 2 dz z 2 1 ˆ
Lt and ar th optma o problm Sytm and Gam hav Th pr o anarhy 3 ˆ ˆ 2 1 2 1/ ρ
Tght bound Dn th JT bound by For any thr a rour alloaton gam th th pr o anarhy at mot. q Proo: rt not that an aum. q Dn a gam th q At optmal th ny q At qulbrum 31 ˆ n n n ʹ + β β > ε β + ε & < 2 ˆ 1 1 1 ʹ 1 / 1 ˆ ˆ ˆ ˆ ' 1 1 ' 1 C ʹ ʹ
q Thn a th playr numbr go to nnty. q Thu th ny at qulbrum approahng In vry rour alloaton gam th pr o anarhy at lat. q Proo: lt and ar th optmal and qulbrum 32 1. ˆ ˆ ˆ ˆ 1 C ʹ + ʹ + β + 1 ˆ ' β β
Th bound. q Proo: ttng ho th bound at mot 3/4. q Aum hav 33 4 3/ β 1 2 & 1/ & <. 4 3 / 1 / / 1 / / 1 / 1 ˆ 2 + + + ʹ + ʹ +
Ntork dgn gam Condr a ntork or ah dg E q N our-dtnaton par playr q Eah playr an hoo a path q Th total ot V E th a nonngatv ot P q Lt n dnot th numbr o playr ho path ar ung dg. Eah o tho playr pay a har π o th ot q Th ot or ah playr ; / n / n 34
{ V E; } a potntal gam th potntal unton Φ n j 1 Evry ntork dgn gam ha at lat on Nah qulbrum. j. 35
k t k 1+a playr and a > arbtrarly mall q To Nah qulbra: all hoo th uppr dg or all hoo th lor dg 36
Pr o tablty Pr o tablty pr o tablty objn valuo th bt qulbrum optmal objn valu Sn Φ 1 + 1/ 2 +! tablty at mot 1 + 1/ 2 +! + 1/ k. C + 1/ k C th pr o 37