Problem Set 4 Solutions

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1 4 Eoom Altos of Gme Theory TA: Youg wg /08/0 - Ato se: A A { B, } S Prolem Set 4 Solutos - Tye Se: T { α }, T { β, β} Se Plyer hs o rte formto, we model ths so tht her tye tke oly oe lue Plyer kows tht the gme oe s lyed whe hs tye s β, d the gme elow s lyed whe hs tye s β - Belef: Plyer s elef µ t t s the rolty tht lyer s tye s t odtol o tht Plyer s tye s t I ths model, se t s ssumed tht the tyes re deedet, µ β α µ β α /, µ α β µ α β - NM utlty futo:, ; t, t s the NM utlty whe Plyer s to s, Plyer s to s, Plyer s tye s t, d Plyer tye s t B, β ; B, β, B, β 0 ; B, β 0, S, β 0 ; S, β 0, S, β ; S, β, B, β ; B, β 0, B, β 0 ; B, β, S, β 0 ; S, β, S, β ; S, β 0 rst osder Plyer s ete Se she does t kow the gme whh s to e lyed, she wts to mmze her eeted yoff If she lys B, wth rolty of ½ the to gme s lyed d Plyer hooses B d thus she gets yoff of, d wth rolty of ½ the ottom gme s lyed d Plyer hooses S d thus she gets yoff of 0 Therefore her eeted yoff s If she lys S, wth rolty of ½ the to gme s lyed d Plyer hooses B d thus she gets yoff of 0, d wth rolty of ½ the ottom gme s lyed d Plyer hooses S d thus she gets yoff of 0 Therefore her eeted yoff s ½ Therefore, B s tully Plyer s est resose gst Plyer s strtegy

2 Net osder Plyer s ete Whe he kows tht the to gme s eg lyed, B s the est resose ge tht Plyer s hoosg B Whe he kows tht the ottom gme s eg lyed, S s the est resose ge lyer s hoosg B Therefore, hoosg B whe the to gme s eg lyed d hoosg S whe the ottom gme s eg lyed s tully Plyer s est resose gst Plyer s to Se oth lyers re tkg ther est resoses to eh other, the strtegy rofle osttutes Byes Nsh Eulrum Gos 3 rms tos re the hoe of uttes, d the mout of outut tke y oegte lues Therefore, the strtegy se s R for eh frm Ad se the formto out demd s rte to frm, we model ths ft s t hs two tyes hgh or low Oe the other hd, frm hs oly oe tye et s fd the Byes Nsh eulrum for ths gme rst, osder the rolem for frm It kows the mrket demd, d wts to mmze ts yoff for eh stte, rg m yeldg, / / whe whe or frm, se t s uert out the mrket demd, t would wsh to mmze ts eeted yoff, } { m rg or, The, the eulrum e foud y solg the oe est resoses smulteously 6 3, 6, 3

3 Se the outut leel s lest se for, we eed to ssume order for ll eulrum uttes to e oste > 3 Gos 33 Eh lyer s to s the hoe of re A re tke y oegte rel umer Therefore, the to se s R for oth lyers Plyer s tye s her rte formto I ths model, s lyer s tye, d t s ether or Therefore, the tye se for eh lyer s {, } Plyer s elef µ s the rolty tht lyer s tye s odtol o tht lyer s tye s I ths model, se t s ssumed tht the tyes re deedet, µ f f NM utlty ths model s the roft of eh lyer ssumg tht frms re rsk eutrl s futo of the tos d tyes of oth lyers:, ;, Plyer s strteges sefy wht tos to tke for y relzto of her tye I ths model, t s two dmesol etor,, where s the re whe ts tye s d s the re whe ts tye s The strtegy se s R for eh A strtegy rofle {,,, } osttutes Byes Nsh eulrum f eh s est resose, e, mmzer of lyer s eeted yoff, odtol o tht her tye s d the ooet s hoosg strtegy, Tht s, rg m rg m rg m rg m Tkg the frst order odtos,,,

4 , Se the gme s symmetr, let s look for symmetr eulrum where, d The the odtos re redued to Solg these eutos, we get,, 4 Gos 36 et,, K, e the de of dders, dder s luto of the good, d lyer s d We deote lyer s strtegy y futo of, meg tht lyer ds whe her luto s We wt to show tht the strtegy rofle for ll osttutes Byes Nsh eulrum Se the gme s symmetr d strtegy rofle s ll symmetr, t s suffet to hek oe lyer s ete euse eery lyer s fg the sme ete rolem We wll show tht f lyer s luto s, d ll other lyers re tkg the strtegy the the d whh mmzes her eeted yoff s

5 rst, osder the rolty of wg the uto s She ws f d oly f ll other lyers d re less th, e, for ll Ths s eulet to for ll Se Pr for ll euse s uformly dstruted oer [0,], Pr wg Pr, Pr Therefore, the eeted yoff from the ddg s Pr wg Tkg the frst order odto, ' 0, or Therefore, the strtegy of lyer, s tully the est resose to other lyers lyg

6 5 Gos 37 Bdder would hoose hs d B to mmze hs eeted yoff, π Pr < Pr Pr < B, where reresets the umulte dstruto futo of lutos ot e would hoose suh tht 0 By dfferettg π wth reset to, we dπ d Note ths s tully the Eeloe Theorem d d Thus, otmlly hose d must stsfy d π π B d Now, together wth the symmetry ssumto f two dders wth the sme luto wll sumt the sme d, the eulrum odto mles tht dder s otml d must e the d mled y the deso rule B other words, t eulrum, B Whe we susttute ths eulrum odto to the oe euto, we get dπ d We sole the oe dfferetl euto for π y tegrtg usg the oudry odto, B 0 0, π d The, omg ths wth the defto of eeted yoff euto we ot eh dder s strtegy 0

7 d 0 or 0 d B for I,

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