MAAEMET SCIECE doi 087/mn0700804e e e7 e-omanion OY AAIABE I EECTOIC OM infom 008 IOMS Eleoni Comanion Call Cene Ououing Cona Unde Infomaion Aymmey by Samee aija Edieal J inke and obe A Sumky Managemen Siene doi 087/mn0700804
e-omanion o aija inke and Sumky: Call Cene Ououing Cona e oof of ooiion and emma Ti doumen onain oof fo all ooiion and emma EC oof of ooiion Te fi ode ondiion fo i ' [ / ] Unde a C-only ona e fi ode ondiion fo i ' Teefoe only if / ien i ondiion e lien ofi i < 0 EC oof of ooiion Condiion i and ii imly a e endo will make a aaiy oie u a Aume a > eefoe Te endo ofi - Teefoe e lien ofi
e-omanion o aija inke and Sumky: Call Cene Ououing Cona e3 Tu e lien maximize ofi EC3 oof of ooiion 3 Te ofi funion of e endo i Teefoe e endo ofi maximizing aaiy deiion i e ame a e eie uly ain ofi maximizing aaiy In addiion e endo ofi Teefoe e lien ofi Tu e lien maximize ofi EC4 oof of ooiion 4 om ooiion we know a a ig oduiiy ye endo will ean a ofi equal o unde ona CS- Aume e ig ye endo ae CS- and ooe a affing leel of Aoding o e SA onain α ow Teefoe i a feaible oluion fo e endo oblem Ti imlie
e-omanion o aija inke and Sumky: Call Cene Ououing Cona e4 Tu e ig oduiiy ye endo will ae CS- Teefoe offeing a oie beween CS- and CS- will no een beween a ig and a low oduiiy ye endo oe a wile e ig oduiiy-ye endo will no ae e oe ona CS- e low ye will ae e oe ona CS- ee ooiion 7 EC5 oof of ooiion 5 om ooiion 3 we know a a ig oduiiy ye endo will ean a ofi equal o unde ona CW- Aume e ig oduiiy ye endo ae CW- Te endo ofi funion i Ti imlie a e endo ofi maximizing aaiy deiion unde ona CW- i equal o Teefoe e endo ofi i Teefoe e ig oduiiy-ye endo will ae ona CW- and ean infomaion en on o of i eeaion alue ene offeing a oie beween CW- and CW- will no een beween a ig and a low ye endo EC6 oof of emma e fax ax-ax We know a f a x a x x ax Teefoe we need o ow a x x ax 0 a x We know a e azad funion fo e andad nomal diibuion i an ineaing funion Ti imlie x x x x x 0 x Teefoe x x fo all x
e-omanion o aija inke and Sumky: Call Cene Ououing Cona e5 Tu i i uffiien o ow a ax a x o y y o omlee e oof Uing oial' ule i an be own a im x x x x ow o omlee e oof i i uffiien o ow a / x < u a xx-x > Aume x < u a xx-x > We know im x x x i imlie a x [ y y y] y < u a yy-y > and < 0 y [ y y y] ow y[ y y y y ] y Ti imlie y y y y < 0 Bu we know yy-y > Ti imlie y-y > 0 Teefoe e inequaliy 0 an be wien a y y y y < y y y By e-aanging em y y y y < 0 Te inequaliy i no ue Ti lead o a onadiion and ene omlee ou oof EC7 oof of ooiion 6 i we oe a oey old Aume i no ue Ti imlie < By e SA onain fo e ig oduiiy ye endo unde ona TS-
e6 e-omanion o aija inke and Sumky: Call Cene Ououing Cona α We now ue e diffuion aoximaion e e Ψ Ψ Teefoe e Ψ Teefoe e inequaliy an be wien a om emma we know a ax-ax i ineaing in a fo all x Alo ax-ax 0 fo a Ti imlie a ax-ax 0 fo all a Teefoe
e7 e-omanion o aija inke and Sumky: Call Cene Ououing Cona Ti lead o a onadiion and ene ou aumion a < i no ue ow we oe a oey old e u aume < 3 om e diffuion aoximaion Te inequaliy 3 an be wien a <
e-omanion o aija inke and Sumky: Call Cene Ououing Cona e8 om emma we know a ax-ax i ineaing in a fo all x Alo ax-ax 0 fo a Ti imlie a ax-ax 0 fo all a Teefoe Ti lead o a onadiion and ene omlee ou oof EC8 oof of emma Aume a e endo a low oduiiy and ae e ona TS- By e definiion of endo will maximize i ofi a and we know a e o all e SA onain i aified ie all ae a feaible oluion o e endo oblem Ti imlie Teefoe EC9 oof of ooiion 7 A low oduiiy-ye endo will ean a ofi equal o unde ona TS- Aume a a low oduiiy ye endo ae CS- e be e aaiy deiion a e low oduiiy ye endo make unde ona CS- ow Ti imlie
e9 e-omanion o aija inke and Sumky: Call Cene Ououing Cona 4 Beaue we aume a e low-oduiiy endo ae CS- α We know a and fo Teefoe Ti imlie a Teefoe a low oduiiy ye endo will no ae ona CS- e u aume a a ig oduiiy-ye endo ae e ona TS- Te ofi eaned by e endo i
e0 e-omanion o aija inke and Sumky: Call Cene Ououing Cona Teefoe e ig oduiiy-ye endo will no ae ona TS- EC0 oof of ooiion 8 A low oduiiy endo will ean a ofi equal o unde ona TW- Aume a a low oduiiy endo ae CW- Te endo ofi funion i Ti imlie a e endo ofi maximizing aaiy deiion unde ona CW- i equal o Teefoe e endo ofi i Teefoe e low oduiiy endo will no ae ona CW- e u aume a a ig oduiiy endo ae ona TW- and ooe a aaiy equal o Te ofi eaned by e endo By e-aanging em 5
e e-omanion o aija inke and Sumky: Call Cene Ououing Cona By e definiion of Uing oey 6 Equaion 5 and 6 imly and eefoe 7 We ae aumed a e ig oduiiy-ye endo ae ona TW- Teefoe we mu ae 0 We an ow a and i imlie a 0 0 Teefoe fom inequaliy 7 we find a Tu e ig oduiiy-ye endo will no ae ona TW- EC oof of ooiion 9
e-omanion o aija inke and Sumky: Call Cene Ououing Cona e A low oduiiy-ye endo will ean a ofi equal o unde ona TS- Aume a a low oduiiy-ye endo ae ona TSA- e be e aaiy deiion a e low oduiiy ye endo make unde ona TS- Te ofi funion fo e endo i AT Teefoe 0 Teefoe a low ye of endo will no ae ona TSA- A ig-ye endo will ean a ofi equal o unde ona TSA- and no ae ona TS- a i would ean a ofi lowe an unde ona TS- ee e oof of ooiion 7 EC oof of ooiion 0 Aume wo endo and wi µ > µ We fi onide a CSA ona e and be e ofi maximizing aaiy deiion and le and be e oeonding ofi fo endo wi eie ae µ and µ eeiely Teefoe e endo ofi ae wee α and wee α i a feaible oluion fo endo Teefoe By definiion and eefoe
e-omanion o aija inke and Sumky: Call Cene Ououing Cona e3 ex we onide a CW ona Again le and be e ofi maximizing aaiy deiion and le and be e oeonding ofi fo endo wi eie ae µ and µ eeiely Teefoe e endo ofi ae: and By definiion EC3 oof of ooiion Aume wo endo and wi µ > µ We fi onide a TSA ona e and be e ofi maximizing aaiy deiion and le and be e oeonding ofi fo endo wi eie ae µ and µ eeiely Teefoe e endo ofi ae: wee α and wee α Aume If i aumion i no ue en
e4 e-omanion o aija inke and Sumky: Call Cene Ououing Cona < Uing e diffuion aoximaion e aboe inequaliy an be ewien a 0 emma and ou aumion a > imly a e aboe inequaliy doe no old Teefoe i ue By definiion α Ti imlie a i a feaible affing leel fo endo Ti imlie Uing oey ex we onide a CW ona Again le and be e ofi maximizing aaiy deiion and le and be e oeonding ofi fo endo wi eie ae µ and µ eeiely Teefoe e endo ofi ae: and By definiion i oimal fo endo
e5 e-omanion o aija inke and Sumky: Call Cene Ououing Cona Uing ooiion 6 EC4 oof of ooiion Te eie uly ain ofi fo a gien affing leel i: Wi e CW ona e endo ofi i: By definiion ewiing e endo ofi Teefoe enue a e CW ona i oodinaing a long a EC 5 oof of ooiion 3
e-omanion o aija inke and Sumky: Call Cene Ououing Cona e6 e u aume a define a oodinaing ay e all ona wi a linea enaly fo waiing e µ l be u a Te lien exeed ofi unde i ona i l M < 8 l l l ow aume a e endo i offeed a oie beween a TWAT o TSAAT ona a maximize e lien ofi if e endo eie ae i µ l and a oodinaing CW ona a maximize e lien ofi if e endo eie ae i oe a if e endo eie ae [ l en e endo will ae e ay e ime ona and ooe o wok a l ee ooiion 8 and If e endo eie ae en e endo will ae e ay e all ona and ooe o wok a ee ooiion 8 and 0 In i ae e eie uly ain l will be oodinaed ee ooiion I an be own a fo i ae e uly ain ofi ineae a and e endo ean an infomaion en of aboe i eeaion alue Te lien exeed ofi i M < > l l l 9 Te oof of e eoem follow by omaing equaion 8 and 9 EC6 oof of ooiion 4 Aume a e lien offe e endo a oie beween a TWAT o TSAAT a maximize e lien ofi if e endo eie ae i µ l we will all i e lowe-t-ona and a TWAT o TSAAT ona a maximize e lien ofi if e endo eie ae i µ we will all i e ige-t-ona oe a if e endo eie ae [ l en e endo will ae e lowe-t-ona and ooe o wok a l ee ooiion 9 and If e endo eie ae en e endo will ae e ige-tona and ooe o wok a ee ooiion 9 and 0 In i ae e eie uly ain
e7 e-omanion o aija inke and Sumky: Call Cene Ououing Cona will no be oodinaed and eefoe e uly ain ean le ofi an unde e ona deibed in ooiion 3 Te lien exeed ofi i e ame a unde e eening ona deibed in ooiion 3 > < l l l M