Strategy and Game Theory: Practice Exercises with Answers, Errata in First Edition, Prepared on December 13 th 2016

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1 Strategy and Game Thery: Practice Exercises with Answers, by Felix Munz-Garcia and Daniel Tr-Gnzalez Springer-Verlag, August 06 Errata in First Editin, Prepared n December th 06 Chapter Dminance Slvable Games Page, Intrductin. At the end f the secnd paragraph the index "i" must be in italics. Page, Intrductin. At the end f the third (last) paragraph, the sixth line frm the final lver must be changed by lwer. Exercise 7, Page. Figure.4 shuld delete clumn x rather than z. The figure and subsequent text shuld be changed as fllws: Figure.4. Reduced Nrmal-frm game We can nw mve t player again. Fr him, strategy c strictly dminates b, since it prvides an unambiguusly larger payff than b regardless f the strategy selected by player (regardless f the clumn). In particular, when player chses y (left-hand clumn), player btains a payff f frm selecting strategy c but nly ne frm strategy b. Similarly, if player chses z (in the right-hand clumn), player btains a payff f ne frm strategy c but a payff f zer frm strategy b. As a cnsequence, strategy b is strictly dminated, which allws us t delete strategy b frm the abve matrix, btaining the reduced matrix in figure.. Figure.. Reduced Nrmal-frm game

2 At this pint, nte that returning t player we nte that z strictly dminates y, s we can delete strategy y fr player and finally, cnsidering player always chses z, fr player strategy d strictly dminates c, since the payff f is higher than ne unit derived frm playing c. Therefre, ur mst precise equilibrium predictin after using IDSDS are the slely remaining strategy prfile (d,z), indicating that player will always chse d, while player will always select z. Exercise 8, Page 4. In the third paragraph after x =0 change "yields" by "des nt yield". Similarly, after x =0 change "but the same" by "but lwer" Exercise 0, Page 7. The reference t Fig.. shuld be changed fr Fig... Exercise, Page 0. In the first displayed equatin at the tp f the page, the multiplicative sign shuld have i= in its subscript belw and I in the superscript abve. In the secnd displayed equatin, its secnd line shuld have subscript i everywhere instead f j. In the third displayed equatin, the subscript f the multiplicative sign shuld be jj ii instead f jj =. Exercise, Page. Starting the answer change the wrd abve fr belw. Chapter Nash equilibrium and Simultaneus-mve games with cmplete infrmatin Page, Intrductin. Nash equilibrium: Change the wrd "peple" fr "prfile". Exercise 4. Page. After c>0, add "with n fixed csts" Page. At the end f the secnd paragraph f the answer key, i shuld be in italics. Page. In the figures the best respnse functins are represented by the acrnym BRF. T be cnsistent with the acrnym used in the accmpanying text, all acrnyms in the figures shuld be BR. Exercise 8. Page 44. At the end, befre the last paragraph starting with "Therefre" change the payff f $ fr a payff f $. Page 46. Secnd displayed equatin shuld read mm xx mm xx + 4 (nn mm xx ) The slutin t the quadratic equatin, at the beginning f the secnd paragraph, shuld be mm xx = ± 4( nn). The results f the rest f the exercise d nt change. Exercise 9, Page 48. Add at the end f the first case at the tp f the page (immediately after as depicted in Fig..9 ): Similarly, when the lcatin f the three candidates satisfies xx DD = xx RR = xx II > / each candidate has incentives t deviate twards the left, while if xx DD = xx RR = xx II < / each candidate has incentives t deviate t the right. Secnd case. End f the first line. Replace tw candidates chses fr tw candidates chse Exercise 0, Page. The secnd paragraph in the answer key, the best respnse functin is referred as BRF. Fr cnsistency, it shuld be referred t as BR. Exercise. Page 4,. At the end f paragraph instead f while the prbability f being caught is +xy shuld be while the prbability f nt being caught is +xy.

3 Page. At the end f part (a) it shuld read BBBB GG, xx(yy) = yy cc instead f BBBB GG, xx(yy) = yy. In cc additin, the next sentence shuld read it is cnvenient t slve fr yy which yields yy = cc xx instead f it is cnvenient t slve fr yy which yields yy = cccc. Page. Middle f page, makes a reference t Fig..9 which shuld be changed t Fig... Page. In part (b), please replace the sentence yu can plug in the first expressin int the secnd expressin fr yu can plug in the secnd expressin int the first expressin. Page. The title f Figure. shuld be Incentives and Punishment instead f Lbbying-Best respnse functins and Nash equilibrium. Page 6. The title f Figure. shuld be Incentives and Punishment-Cmparative Statics instead f Lbbying-Cmparative Statics. Exercise. Page 7. The last sentence at the end f part (a) shuld read Hence, equilibrium prices are pp CC = cc cc cc = cc +cc =, and every firm s equilibrium prfits are ππ 4 4 CC = +cc cc cc = ( cc) Page 8. Last paragraph f the page, fifth line frm the bttm, abve this cutff indicate parameters indicate must be changed fr abve this cutff indicate. Chapter Mixed strategies, strictly cmpetitive games, and crrelated equilibria Exercise, Page 69. At the msne listed immediately abve part (c) f the exercise, the secnd parentheses shuld have / fr bth sets f prbabilities. Exercise. Page 76. The first paragraph shuld say Similarly, let pp represent the prbability that player chses T, pp the prbability Page 76. The secnd displayed equatin in the triplet at the center f the page shuld read EEUU (CC) = qq + ( qq) = qq. Page 77. The end f the first paragraph, it shuld read shwing that nly sme f them can be sustained in equilibrium. Page 77. In the sectin Mixing between T and C alne player s indifference cnditin is EEUU (TT) = EEUU (CC), which in the next line becmes + qq = + qq. This displayed equatin shuld actually read + qq = qq. The text t the right-hand side f the displayed equatin shuld be then changed fr which yields qq =. The fllwing paragraph (after the displayed equatin) shuld read Hence, player randmizes between TT and CC, assigning a prbability f qq = t TT and qq = t CC. Last, the last displayed equatin f the page shuld read qq =, which yields qq = 0. Page 78. The displayed equatin at the tp f the page shuld read + qq = qq =. The line immediately after shuld then read Prviding us with tw equatins, + qq = and qq =, which cannt simultaneusly hld, i.e., + qq = entails qq = / while qq = yields qq = 0. Page 78. Delete the last paragraph f Exercise, starting at Hence, the unique msne and ending at prbability n each as well. Exercise 7. Page 8. In the last paragraph f the page, replace this result in Fig..9 by nticing that with this result in Fig.. by nticing that Exercise 8. Page 8. The first paragraph f part (c) shuld end with a parenthesis, as illustrated in Fig..7).

4 Page 86. The last paragraph f Exercise 8 shuld include the fllwing explanatin at the end f the paragraph: The equatin in f the line cnnecting pints (,7) and (7,) is uu = 9 uu. T see this, recall that the slpe f a line can be fund in this cntext with mm = 7 =, while the vertical intercept is fund by inserting either f the tw pints n the equatin. Fr instance, using (,7) we find that 7 = bb which, slving fr bb, yields the vertical intercept bb = 9. It is then easy t check that pint (4.,4.) lies n this line since 4. = 9 4. hlds with equality. Exercise 9. Page 88. The msne displayed at the center f the page shuld read UU, DD, LL, RR. Page 89. Figure. shuld have the labels changed t (,), psne (D,R) in the upper left-hand side f the figure, and t (,), psne (U,L) in the lwer right-hand side f the figure. Page 9. Figure. shuld have the labels changed t (,), psne (D,R) in the upper left-hand side f the figure, and t (,), psne (U,L) in the lwer right-hand side f the figure. Exercise. Page 99. The displayed equatin in part (b) shuld be changed t: we find that player prefers the latter, i.e. uu (CC, NNNN) = 0 < 0 = uu (NNNN, CC) while player prefers the frmer i.e., uu (CC, NNNN) = 0 > 0 = uu (NNNN, CC). Page 00. The third paragraph starting at Hence, this game is shuld have ss = (NNNN, NNNN) rather than ss = (NNNN; NNNN). Exercise. Page 0. The last displayed equatin f the page shuld read qq + = 6 6pp pp = /. Page 0. The displayed equatin at the middle f the page shuld read 6qq = qq + 6( qq) qq = /. (Only the part after the arrw needs t be fixed.) Page 06. The displayed equatin at the tp f the page shuld read TTTTTT, BBBBBBBBBBBB, 4 4 LLLLLLLL, RRRRRRhtt. In part (d) f the exercise, the secnd line f expressin EEUU shuld read = + + = 4. Similarly, the secnd line f 4 expressin EEUU shuld read =.. Last, part (e) f the exercise shuld read 4 Player s expected utility frm playing the msne f the game, 4, cincides with that frm playing his maxmin strategy, 4. A similar argument applies t Player, wh btains an expected utility f. under bth strategies. 7 Chapter 4 Sequential-mve games with cmplete infrmatin Exercise. Page 08. Tw lines befre the displayed equatin, his mnetary payff is lwer than shuld be replaced fr his mnetary payff is higher than Page 09. The equatin mm, immediately after the first displayed equatin, shuld have a minus sign (nt a dash) between the and the mm. Page 09. At the equilibrium payffs at the center f the page (displayed equatin), the payff frm player shuld read +αα. That is, the numeratr shuld be + αα rather than αα. +αα 4

5 Page 09. As a cnsequence, the derivative tw lines belw shuld read ( mm ) = (+αα). This derivate is still negative, s all the subsequent intuitin at the end f page 09 remains unaffected. Exercise 4. Page. In the sixth line, it shuld read player has the pprtunity t give any, all, r nne. Exercise. Page 7: instead f QQ = qq + qq it shuld be QQ = qq LL + qq FF. Page 8. At the bttm f the line, after which simplifies int the displayed equatin shuld be [( + cc FF) qq LL ]qq LL cc LL qq LL. Hence, the last displayed equatin f page 8 shuld read ( + cc FF ) qq LL cc LL = 0. All subsequent calculatins in this exercise are crrect. Page : in the figure, the Curnt utput shuld be ( cc). Exercise 6. Page. Bth prfits at the tp f the page shuld be crrected t ππ = 07.7 and ππ = 8.6, respectively. Exercise. Page 4. In the seventh line, it shuld read: If, instead, the new iphne is intrduced Chapter Applicatins t industrial rganizatin Exercise (Strategic advertising and prduct differentiatin). Page 6. After the sentence Plugging firm j s best respnse functin int firm i s, we find the expressin shuld have a d j in the last numeratr n the right-hand side (end f parenthesis) rather than a d i. Page 6. Expressin. shuld nt have a in the secnd term f the denminatr. Page 6. Expressin q j* in the secnd line shuld nt have a in the secnd term f the denminatr. Page 6. The derivative f utput q j* with respect t d i shuld nt have a in the secnd term f the denminatr; neither in the first nr in the secnd expressin. Page 6. The derivative f prfit ππ jj with respect t AA ii shuld nt have a in the secnd term f the denminatr; neither in the first nr in the secnd expressin. Page 6. The derivative f dd ii with respect t AA ii shuld nt have a in the secnd term f the numeratr. Exercise 7, page 6. The utput qq ii = qq jj in the secnd paragraph shuld nt have a in the secnd term f the denminatr. Exercise 9, page 7. The price pp shuld g in italics. Chapter 6 Repeated games and crrelated equilibria Exercise 8 (Cllusin and Imperfect Mnitring). Page 09. The plus sign in dd(aa+ddqq ii ) 4bb (firm i s best respnse functin at the center f page 09) shuld have a minus sign - s it reads dd(aa ddqq ii ) 4bb. mm Page 0. The first tw 4 s in the denminatr f the prfits frm the cllusive agreement, ππ ii (at the center f the page), shuld be s, s the expressin reads ππ mm aa ii = aa bb (bb + dd) dd aa (bb + dd) aa 4(bb + dd) = aa 8(bb + dd) DD Page. The δδππ ii in the right-hand side at the bttm f the page shuld read ( + δδ) ππ DD ii.

6 Page. The secnd displayed equatin shuld have an 8 in the denminatr rather than a aa discunt factr δδ; reading rather than aa. 8(bb+dd) δδ(bb+dd) Page. The previus t the last displayed equatin shuld have a greater than r equal sign rather than =; reading aa (bb+dd) Chapter 7 Simultaneus-mve games with incmplete infrmatin Exercise. Page 0. Table 7. shuld have a payff pair (,-) in the cell crrespnding t (Bf, Bet), lcated in the secnd rw, left clumn. Page 0. In the first bullet pint, it shuld read Fr player, his best respnse when player bets (in the left-hand clumn) is t play Bf since it yields a higher payff, i.e.,, than any ther strategy, i.e., BR (Bet)=Bf. Page 0. In the secnd bullet pint, the secnd sentence shuld read If, instead, player chses Bf (in the secnd rw), player s best respnse is t fld, BR (Bf)=Fld, since his payff frm flding, -/, is larger than frm betting, -. Page. Table 7. shuld have a payff pair (,-) in the cell crrespnding t (Bf, Bet), lcated in the secnd rw, left clumn. In additin, player s underlined payff in the secnd rw shuld be -/ (that crrespnding t Fld in the right-hand clumn) rather than that f betting. Exercise, page. In the paragraph with title Player s best respnses, the sixth line shuld read i.e., 4+p>, 4+p>4-p, and 4+p>+4p, which hld fr all values f p. Exercise, page 4. The displayed equatin after rearranging yields in the middle f the page shuld read BB(pp ββββ) > LL(pp ββββ). All subsequent results are crrect. Chapter 8 Auctins Exercise, page 48. The expressins in the sectin with title Direct apprach shuld nt have N- in the expnents. This applies t the expressin f the prb(win), the EEEE ii (vv ii ), and its rearranged representatin at the bttm f the page. Chapter 9- Perfect Bayesian equilibrium and signaling games Exercise. Page 6. The last sentence f the page shuld read with U, which prvides him a payff Page 6. Part (b) f the exercise shuld be preceded by space t separate it frm the answer key f part (a). Page 6. Ftnte. Strategy R shuld read RR, fr cnsistency with the primes in previus parts f the exercise. Exercise. Page 74. The paragraph labelled Case shuld start as fllws When qq >, the The end f this case (immediately befre Case ) shuld read beliefs satisfy qq >. The paragraph labelled Case shuld start as fllws When qq, the firm : Exercise. 6

7 Page 77. The first paragraph f part (a), in the last line, shuld read nly stems frm a sender Page 80. The secnd line after the three expected utility expressins shuld read if and nly if μμ > 4, 4 μμ > μμ (Nw ne f the minus signs lks like a hyphen.) Exercise 4. Page 8. The expressin f pp immediately abve expressin (B) shuld have a in the denminatr, rather than a. As a cnsequence, expressin (B) shuld becme pp = + ff (pp ). Page 86. The expressin f pp in the paragraph starting with Firm anticipates shuld be crrected fr pp = + ff (pp ). A similar argument applies t the expressin plugged at the end f the big parenthesis in the prfit maximizatin prblem immediately after this paragraph. As a cnsequence, the subsequent calculatins shuld be replaced fr the fllwing: Taking first rder cnditin with respect t pp, we btain pp + pp + cc + cc ff ff (pp ) = 0. Simplifying and slving fr pp, we find firm s best respnse functin in the first perid game pp (pp ) = + cc + pp + ( cc )ff ff (pp ) 9 Inserting this expressin f pp int pp = +pp, we btain the ptimal secnd-perid price fr Firm pp = + cc + + ( cc )ff ff (pp ). 7 Plugging this result int the best respnse functin pp (pp ), yields the ptimal first-perid price fr Firm pp = + cc As suggested in the exercise, let us knw. + 4( cc )ff ff (pp ) 7 Page 88. This page shuld be replaced fr the fllwing, starting in the expressin at the tp f the page: AA 0 + AA cc = + cc since AA measures the slpe f the pricing functin (see Fig. 9.0), thus implying AA = ff ff (pp ). Rearranging the abve expressin, we find + AA 4( cc ) 7 which, slving fr AA, yields AA (cc ) = 7 + 8cc 7AA 0 AA = 7( AA 0) + 8cc cc In additin, when firm s csts are nil, cc = 0, the abve expressin becmes 7AA 0 = 7 + AA r, after slving fr AA 0, AA 0 = AA 7

8 Inserting this result int AA = 7( AA 0 )+8cc, yields cc Slving fr AA, we btain AA = 8 becmes AA + 8cc AA = cc Therefre, the intercept f the pricing functin, AA 0, AA 0 = = 9.6. Hence, the pricing functin pp f firm, pp = AA 0 + AA cc, becmes pp = cc. Exercise 6, page 96. The tw maximizatin prblems in the middle f the page shuld have cc in the last term, rather than cc. In additin, the first-rder cnditin with respect t pp TT shuld have the parenthesis crrected, as fllws Chapter 0 Mre advanced signaling games λλ ( λλ) θθ TT θθ BB θθ BB = 0 Exercise. Page 07. The labels belw each term at the center f the page shuld read PP ss, with the apstrphe after the PP. Ftnte, page 07. It reads alpha when it shuld read αα. Exercise. Delete part (b), bth in the questin (page 7), and in the answer key (pages 9-). Exercise 4. Page, last line shuld read they are assciated with a higher wage Exercise, Page 7. The secnd line f the answer key f part (a) shuld read while aa EE is the benefit. Page 8. Figure 0.8 shuld have all subscripts in capital letters, such as ff II, ff EE, aa II, and aa EE. In additin, all ntatins in this exercise shuld g in italics. Exercise 6, page 40. Figure 0.7 shuld have branch Acc shaded in the lwer part f the game tree, since the figure represents the separating strategy prfile (Fight, Acc). 8