ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 9

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1 S TO EVEN NUMBERED EXERCISES IN CHAPTER 9 SECTION 9: SIMULTANEOUS ONE SHOT GAMES Exercse 9- Vckrey Bddng Game Bdder values an tem for sale at 7 (thousand) and bdder at 8 (thousand) Each must submt a sealed bd of 5 thousand or hgher n ncrements of thousand Snce no one values the tem above 8 we consder only bds u to and ncludng 8 The bdder who bds the most s the wnner but only has to ay the second hghest bd If the bds are the same, the wnner s selected by a con toss and the wnner ays the common bd Bds are submtted n secret (a) Exlan why the ayoff matrx must be as dected below Bdder Bdder , 5 0, 3 0, 3 0, 3 6, 0 05, 0, 0, 7, 0, 0 0, 05 0, 8, 0, 0 0, 0-05, 0 Fg 9-0: Vckrey aucton (b) Show that (5,8) and (6,8) and (7,8) are all Nash equlbra (c) Are there other equlbra n ure strateges? (d) Show that one of the Nash equlbra s an equlbrum n weakly domnant strateges (a) If there s a te (as s the case on the dagonal) each wns wth robablty ½ The ayoff to the wnner s hs valuaton less hs bd Thus hs exected ayoff s half hs ayoff f he wns Off the dagonal the hgh bdder wns so hs ayoff s hs valuaton less the lower bd Answers to Chater 9 age

2 (b) If bdder chooses 5, t s a best resonse for bdder to choose 8 And gven that bdder chooses 8, bddng 5 s a best resonse An dentcal argument holds for the other two strategy ars (c) If bdder bds 5, then 6 and 7 are also best resonses for bdder If bdder bds 7, bdder s best resonse s to choose 5 so (5,7) s another Nash Equlbrum Smlarly (6,7) and (8,5), (8,6) and (8,7) are Nash Equlbra (d) For bdder the last column weakly domnates the other three columns For bdder the thrd row weakly domnates the other three rows Thus (7,8) s the unque equlbrum n weakly domnant strateges Exercse 9-: Odds and Evens Consder the followng smlfed verson of the wdely used Odds and Evens game Each layer smultaneously holds u one or two fngers and calls ether Odds or evens If the number of fngers s Odd the layer callng odds wns If the total s even the layer callng Even wns If both make the same call no one wns In ths case an acton s a number of fngers and a call Thus each layer has four ossble actons The ayoff matrx s shown below Agent q q r r O O E E O 0,0 0,0 -,,- Agent q O 0,0 0,0,- -, r E,- -, 0,0 0,0 q r E -,,- 0,0 0,0 Fg 9-: Odds and Evens game Ths game was layed by the Romans We have smlfed t her by havng only one round of the game Tycally the game contnues untl one or the other layer s the wnner Answers to Chater 9 age

3 (a) Establsh that there are no weakly domnated strateges and no NE n ure strateges (b) Show that there s a contnuum of NE n comletely mxed strateges (all strateges layed wth strctly ostve robablty) (c) Show that there are at least 6 NE n whch each layer lays strateges wth ostve robablty (d) Use your result from (b) to draw a concluson about a NE of the standard Odds and Evens game where lay contnues untl there s a wnner (a) No weakly domnated strategy For layer the lowest ayoff for any ure strategy s s - Ths occurs f layer makes an ooste call and the total number of fngers s even Let s be the strategy where the number of fngers layer rases s changes Player now gets a ayoff of Thus s does not weakly domnate s A symmetrc argument holds for layer No NE n ure strateges For any ure strategy of layer, layer has a unque best resonse n whch hs ayoff s + Then layer s ayoff s - Gven ths best resonse, t follows from the revous argument that layer s strctly better off changng the number of fngers Thus there s no ar of ure strateges s Sthat are mutual best resonses (b) Gven the symmetry of the game s a NE for both layers to lay the mxed strategy (,, ) Suose that layer s mxed strategy s ( α, βγδ,, ) The ayoffs for layer are strategy Eu { } O δ γ ( ) O ( δ γ ) E α β ( ) E ( α β) Answers to Chater 9 age 3

4 Suose at least one ayoff s strctly ostve WOLOG suose δ > γ and α β Then layer wll never chose anythng but O and E and so the best resonse s not comletely mxed Then all ayoffs must be zero and so δ γ = α β = 0 A symmetrcal argument hold for layer The new ayoff matrx and mxed strateges are therefore as shown below Agent α α α α O O E E α O 0,0 0,0 -,,- Agent α O 0,0 0,0,- -, α E,- -, 0,0 0,0 α E -,,- 0,0 0,0 Fg 9-: Equlbrum n comletely mxed strateges Exected ayoffs are zero for all strateges Thus the dected strateges are mutual best resonses The strateges are comletely mxed for all α (0, ), =, (c) For any ( α, α ) {(0,0),(0, ),(,0),(, )}, the argument above establshes that these are NE strateges It s easly confrmed that (,0,,0) and (0,,0, ) are also NE mxed strateges (d) It s natural to focus on the symmetrc equlbrum Instead of the game endng f both call Odds or both call Evens, the game contnues If the symmetrc equlbrum s used n all future rounds the contnuaton ayoff s zero Thus the ayoff matrx on the one round game s the reduced form of the mult round game where t s beleved that both layers wll use the symmetrc comletely mxed strategy n all future rounds Answers to Chater 9 age

5 SECTION 9: MULTI-ROUND GAMES Exercse 9-: Alternatng offer game A e of value v s to be dvded between two layers, but only f they agree on how much each should get In each round one layer makes a demand and the other ether accets or reects the offer Player moves frst and makes a demand of x n round Then layer has v x If the offer s acceted the game ends If the offer s reected layer demands x and so on Between each round one quarter of the e s lost Thus after rounds there s nothng left (a) Assume that f a layer s ndfferent he or she wll accet the other s demand wth robablty Consder the sub-game begnnng n round Only one quarter of the e remans If layer makes any offer x < v t wll be acceted by layer snce ths gves layer u = v x > and f he reects there s no e left By hyotheses a 0 layer accets f he or she s ndfferent so the best resonse of layer s x = v Aly backwards nducton to show that the ntal offer wll be v and that t wll be acceted (b) Show that f layer accets n the last stage wth robablty < there s no best resonse by layer Use ths argument to establsh that the equlbrum of art (a) s the unque sub-game erfect equlbrum (d) For any nteger T, solve for the sub-game erfect equlbrum ayoff f v/ T of the e s lost each round (e) What Nash equlbrum s most favorable to layer? What Nash equlbrum s most favorable to layer (a) In any odd round the strateges of the two layers can be wrtten as t t ( s, s) = ( xt, t) where s the robablty that layer accets In even numbered rounds the strateges can be wrtten as (, x ) Consder round There s no best offer t x less than v So suose x = v Player s ndfferent so a best resonse s to accet, that s = Then the sub-game erfect equlbrum ayoffs n round are t Answers to Chater 9 age 5

6 ( u, u ) = (0, v) Makng the same argument n round three, the sub-game erfect t t equlbrum ayoffs are ( u, u ) = ( v, v) Contnung to work backwards, the t t equlbrum ayoffs are ( u, u ) = ( v, v) (b) Consder the round strategy ar (, v ) where < These are not mutual best resonses Note that layer s ayoff s v If layer demands x < v t wll be acceted Thus layer s strctly better off choosng x suffcently close to v But there s no best resonse x( ) for < Exercse 9-: To take or not to take, that s the queston Two artcants, A and B have agreed to ay a game at a well known casno Any wnnngs wll be n the form of a casher s check, to be maled the followng day The game s as follows A rze of $500 wll be offered to A If he takes t the game wll be over A s also told that f he declnes, then after he leaves, a total of $000 +x wll be offered to B She can ether take t or declne If B also declnes each wll be sent a check for $000 (a) If x > 0 exlan why A s unque SPE s for each layer to take the money (b) Suose ( ) x= $500 ( ) x = $000 If you were contestant B what would you do n each case? (a) Consder the sub-game layed by B For x > 0, B s best resonse s take the money Then at the frst stage of the game layer A s best resonse s take the money (b) Many eole are dstrustful f they are layer A But f layer A does choose to declne the money, a sense of farness often leads to B choosng to declne as well Ths sense of farness s more than offset f the ayoff to takng the money s suffcently hgh (c) If layer B has to face the erson she cheats, she s more lkely to share (d) Whle some fnd ths a maor refutaton of the theory of games, most economsts argue that such case are relatvely rare and when they do occur t s ossble to ncororate the dsutlty of embarrassment nto the ayoffs Answers to Chater 9 age 6

7 SECTION 93: DUOPOLY GAMES Exercse 93-: Cournot roducton game Suose that the market demand curve s = a q There are two frms The outut of frm s q Each has the same unt cost of roducton c The game s layed once (a) Solve for the Nash Equlbrum and comare t wth the symmetrc collusve outcome (b) For n dentcal frms show that the frst order condton for a best resonse s Π n = a c b( s + q ) = 0, where s = q q Sum over the n frms and hence show that na ( c) ( n+ ) s= 0 n n Hence show that the equlbrum rce s = a+ c n+ n+ (d) Comare the outcome wth the Walrasan equlbrum outcome as n becomes large = (a) The roft of frm s U ( q) = ( a c q q ) q = ( a c q ) q q Gven q, frm has a best rely q BR = arg max U ( q) Solvng, q ( ) = a c q Argung BR q BR symmetrcally for frm, q ( ) = a c q For a Nash Equlbrum both strateges must be best resonses Solvng these two N equatons, ( ) N q = 3 a c hence U ( ) = 9 a c Total roft s U+ U = ( c) q = ( a c q) q Ths takes on ts maxmum at q= ( a c)/ Thus f frms share equally, the collusve C roft of frm s U ( ) = 8 a c (b) Let n s = q be the total outut of the n frms The roft of frm s = U ( q) = ( a c s) q = ( a c q ) q q Frm chooses q to maxmze U ( q, q ) Dfferentatng, the FOC s Answers to Chater 9 age 7

8 Rearrangng, U ( q) = ( a c q) q = 0 q q = ( a c s), =,, n n Summng over the n frms, s = n( a c s) hence total outut, s = ( a c) n + equlbrum rce s n = a s= ( ) a+ ( ) c n+ n+ and the Exercse 93-: Producton game wth sequental entry Demand s gven by = q, where q= q+ q Frm has a cost C = q Frm s cost functon s C = 8q (a) Suose that each frm makes ts roducton decson wthout observng ts comettor s decson Solve for the best resonse functons for each frm Hence solve for the smultaneous move equlbrum (b) Suose frm makes the frst move Utlze the fact that frm wll choose a best resonse to get an exresson for frm s roft n terms of q alone (c) Solve for frm s otmal outut decson and comare outut levels and rofts wth those n the smultaneous move equlbrum (a) Frm chooses q to maxmze ts roft U ( q) = ( c q q) q The FOC s U q = q ( ) c + q = q ( ) c q = 0 q It follows that q+ c = q + c hence q = q + Substtutng back nto the FOC for N commodty f follows that q = (8,) (b) From art (a), frm chooses q to satsfy U q = q ( ) 8 q = 6 q q = 0 Answers to Chater 9 age 8

9 BR Thus q = 8 q Then U ( q ) = ( c q q ( q )) q = ( q ) q BR N N (c) Frm chooses q to maxmze U From the FOC, q =, hence q = Total outut s hgher then n the smultaneous move game thus the equlbrum rce s lower Frm must be better off movng second snce one of ts feasble alternatves s ts choce n the smultaneous move game Exercse 93-6: Exstence of equlbrum Consder two frms roducng an dentcal roduct The unt cost for frm s c where c < c All consumers urchase from the frm offerng the lowest rce If the two frms set the same rce frm gets a fracton f of the market and frm a fracton f = f Consumer demand for the two roducts s as follows ( a,0), < ( q, q) = ( f( a ), f( a ), = (0, a ), > The acton set for each frm s the nterval [0, a ] (a) Show that there can be no equlbrum n ure strateges wth ether rce strctly greater than c (b) Suose that frm adots a mxed strategy Let be the frm s maxmum rce That s, μ ( ) = and μ ( ) <, forall < Suose that > c and obtan a contradcton HINT: Argue that frm s best resonse s to announce a rce < and hence that the rce cannot be a best resonse for frm (c) Show that (, ) = ( c, c) s the unque Nash equlbrum f f = Answers to Chater 9 age 9

10 (d) Show that there s no Nash equlbrum f f < (d) For fnte strategy sets show that there s a unque equlbrum (e) Do you thnk t makes sense to model Bertrand rce cometton by assumng that f =? (a) If > c frm slts the market by matchng and doubles ts market share by slghtly under rcng Thus under rcng s always more roftable (b) If frm chooses t maxmum rce > wth strctly ostve robablty the argument of (a) ales The market share rses dscontnuously as frm lowers t s rce below Suose nstead that μ ( ) s contnuous at Then frm wns wth zero robablty f t chooses = and wth strctly ostve robablty f t chooses < Then frm should choose a rce less than or equal to (c) We have ruled out any equlbrum wth rces above c If (, ) = ( c, c) the rofts are ( u, u) = (( a c) c,0) Frm loses money by lowerng rces Frm lowers ts roft by lowerng rces (d) We have ruled out any equlbrum wth rces above c If (, ) = ( c, c) the rofts are ( u, u) = ( f( a c) c,0) Frm loses money by lowerng rces If frm lowers ts rce to = c δ ts roft s u = ( a c δ )( c δ) = ( a c) c δ( c δ) For small δ ths strctly exceeds f( a c) c However there s no best resonse snce δ can be any strctly ostve real number Answers to Chater 9 age 0

11 (e) Suose that c s n the fnte strategy set of both frms Argung as n (d), f the rce dfferences are suffcently small, the equlbrum strategy of frm s to choose the hghest rce (f) Here are two reasons Frst t s the equlbrum that most looks lke the equlbrum wth a fnte strategy set Second, f = = c, frm could urchase from frm rather than roduce to order Then, whle frm s fnal market share s f ts total suly s q= a c Exercse 93-8: Rent seekng contest Two cometng frms send resources on a oltcan The amount sent by frm s x The frm that sends the most on the oltcan s re-electon wns a favor from the oltcan If frm wns the contest, ts roft rses by W and the roft of frm falls by L If frm wns, ts roft rses by W and the roft of frm falls by L If the two frms send the same amount, each wns wth robablty 05 The roft gans and losses are common knowledge Frms choose ther sendng levels smultaneously A frm may also choose not to artcate n whch case ths frm has a zero ayoff and the other frm wns wth any bd (a) Frst consder the secal case ( W, L) = ( W,0) =, Gve an argument as to why there can be no equlbra n ure strateges (b) Show that the mxed strategy wth robablty dstrbuton μ ( x) = Pr{ x x} = x / W s a Nash equlbrum (c) For the general model sketch an argument as to why the equlbrum robablty dstrbutons μ( x), μ( x) cannot have a mass ont at x > 0 (d) Solve for the equlbrum mxed strategy f ( W, L) = ( W, L) =, Answers to Chater 9 age

12 (e) Solve for the equlbrum mxed strateges f W > W > L = L = 0 (a) Consder the ure strategy rofle ( x, x ) where x < x < W Frm s better bddng ust above x than bddng below x Thus the strateges are not mutual best resonses Next consder ( x, x ) where x < W x Frm s bd s not a best resonse because t can bd strctly lower than W and make a roft Snce a symmetrcal argument holds for frm t follows that there s no equlbrum n ure strateges wth x x Suose next that x = x = x < W The two frms make a roft wth robablty But then frm can ncrease ts exected roft by ncreasng ts bd to x + δ where δ s ostve and suffcently small The exected roft rses from W x to W x δ Fnally suose that x = x = x W Both frms have an exected loss Now frm s strctly better off lowerng s bd to zero (b) All we need to do s check that the mxed rofles are best resonses If frm bds x t wns f x x therefore < and wns wth robablty f x = x Frm s wn robablty s x Pr{ x x} = W Frm s exected roft s therefore u ( x) = Pr{frm wns} W x = 0, x [0, W] Thus frm s ndfferent between all bds on [0, W ] (c) If frm bds x wth robablty π, frm s ayoff rses dscontnuously at x = xˆ Answers to Chater 9 age

13 Thus there s some nterval ( xˆ ˆ δ, x] over whch frm wll not bd But then frm s wn robablty s constant over ( x δ, x] and so her ayoff s a strctly deceasng functon of x over ths nterval Ths contradcts our ntal assumton that bddng ˆx wth robablty π > 0 s a best resonse (d) Suose that frm bds x 0 wth robablty G( x ) where G s contnuous for all x > 0 If frm chooses x > 0 her wn robablty s Gx ( ) Hence her exected ayoff s u ( x ) = G ( x ) W ( G( x ) L x = G ( x )( W + L) L x L+ x = ( W + L)( G( x) ) W + L For an equlbrum n contnuous mxed strateges ths must be constant Hence L+ x G( x) = and u( x) = 0, x > 0 L + W L Suose frm does not bd wth robablty Then L+ W L L u (0) = ( ) W ( ) L= 0 L+ W L+ W Then frm s ndfferent between not cometng and makng any bd on [0, W + L] (e) Suose that frm bds x 0 over [0, β ] wth robablty G ( x ) where G s contnuous for all x > 0 If frm chooses x > 0 her wn robablty s Gx ( ) her exected ayoff s Hence u x G x W x W G x ( ) = ( ) = ( ( ) ) W Smlarly, x u x = G x W x = W G x ( ) ( ) ( ( ) ) W x Snce G( β ) = G( β ) = t follows that u( β ) > u( β ) To comete, frm must have a ostve exected ayoff Therefore u ( β ) > 0 It follows that u (0) = WG (0) = u ( β ) > 0 Answers to Chater 9 age 3

14 Argung as above, there cannot be a te at zero wth ostve robablty Therefore G (0) = 0 and so u (0) = 0 Then x u( β ) = W β = 0 and so G( x) = W Also u( x) WG ( x) x W β W W = = = Therefore G( x) = W x W + W Note that the bgger the asymmetry, the hgher the robablty that frm does not comete and hence the hgher the equlbrum ayoff to the hgh value frm SECTION 9: INFINITELY REPEATED GAMES Exercse 9-: Nash equlbrum threats For the data of the revous exercse, consder the strategy q = (3,3) Player can comute hs oonent s maxmum ayoff for each of hs actons, Maxu ( q, q ) Let q M q be the strategy that mnmzes layer s maxmum ayoff M = q q q arg MnMaxu ( q) Ths s known as layer s Mnmax strategy In ths examle, f layer chooses q, layer can do no better than roduce nothng Therefore layer can drve layer s ayoff to zero Of course the same argument holds for layer (a) Exlan why the Mnmax threat by both layers s a Nash equlbrum threat but not a SPE threat (b) For what dscount factors s q = (3,3) a Nash equlbrum outcome of the nfntely reeated game? (a) The roosed strategy s to lay the cooeratve strategy n erod and contnue to lay ths unless the other layer defects If he does so the lanned resonse s to roduce unts n each future erod C Consder the effcent cooeratve strategy q = (3,3) so that u = 8 (b) If layer adots ths strategy and layer defects n the frst round, layer s frst erod ayoff s u = u q x = x = After that layer chooses outut D D (, ) ( ) 8/ Answers to Chater 9 age

15 so that rce s below unt cost f layer roduces any outut So layer s best resonse s to roduce zero Then the resent value of hs ayoff stream s 8/ If he C cooerates forever hs ayoff s u /( δ ) = 8/( δ Thus frm gans by defectng only f 8 8 >, that s δ < /9 δ (c) In a Nash Equlbrum there are no restrctons on behavor off the equlbrum ath In artcular we do not need to consder what s layer s best resonse f layer defects For sub-game erfecton we requre best resonses off the equlbrum ath If layer roduces nothng, layer s best resonse s to roduce the monooly outut of 6 rather than the zero roft outut of Answers to Chater 9 age 5