Games, Strategies, and Decision Making Second Edition Solution Manual

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2-11.The countries of Oceania and Eurasia are at war.5As depicted in the figure, Oceania hasfour cities—Argula, Betra, Carnat, and Dussel—and it is concerned that one of them isto be bombed by Eurasia. The bombers could come from either base Alpha, which canreach the cities of Argula and Betra; or from base Beta, which can reach either Carnator Dussel. Eurasia decides which one of these four cities to attack. Oceania doesn’t knowwhich one has been selected, but does observe the base from which the bombers areflying. After making that observation, Oceania decides which one (and only one) ofits four cities to evacuate. Assign a payoff of 2 to Oceania if it succeeds in evacuating thecity that is to be bombed and a payoff of 1 otherwise. Assign Eurasia a payoff of 1 if thecity it bombs was not evacuated and a zero payoff otherwise. Write down the extensiveform game.Building a Model of a Strategic Situation2OceaniaArgulaBetraBetaAlphaDusselCarnatANSWER:In the figure below,Astands for Argula,Bfor Betra,Cfor Carnat, andDfor Dussel.EurasiaEurasiaArgulaBetraCarnatDusselOceaniaOceaniaA01112111BCDA10111211BCDA11011121BCDA11101112BCD

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2-2SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION2.Player1movesinitiallybychoosingamongfouractions:a,b,c,andd.If player 1 chose anything butd,then player 2 chooses betweenxandy. Player 2 gets toobserve the choice of player 1. If player 1 chosed,then player 3 moves by choosingbetweenleftandright. Write down the extensive form of this setting. (You can ignorepayoffs.)ANSWER:abcdxyxyxyLeftRight122233.Consider a setting in which player 1 moves first by choosing among three actions:a,b,andc. After observing the choice of player 1, player 2 chooses among two actions:xandy. Consider the following three variants as to what player 3 can do and what she knowswhen she moves:a.If player 1 chosea, then player 3 selects among two actions:highandlow. Player 3knows player 2’s choice when she moves. Write down the extensive form of this set-ting. (You can ignore payoffs.)ANSWER:abcxyHighLowHighLowxyxy123322b.If player 1 chosea, then player 3 selects among two actions:highandlow. Player 3does not know player 2’s choice when she moves. Write down the extensive form ofthis setting. (You can ignore payoffs.)

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2-3SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATIONANSWER:abcxyHighLowHighLowxyxy12223c.If player 1 chose eitheraorb, then player 3 selects among two actions:highandlow. Player 3 observes the choice of player 2, but not that of player 1. Write downthe extensive form of this setting. (You can ignore payoffs.)ANSWER:abcxyHighLow HighLowHighLow HighLowxyxy1222334.Return to the game involving the U.S. Court of Appeals in Section 2.2.Suppose, at thestart of the game, it is known by all that Judge Z will read only the brief of Ms.Hasenpfeffer. Write down the corresponding extensive form game. You may excludepayoffs.

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2-4EHEHJFJFAXYZ1022210221ZZZZYYHFHFHFHFHFHFHFHFHFHFBIIIIIIHFHFHFHFHFHFHFHFHFHFHFHFHFHFHFHFHFHFXX102120121110122011210111201111101111011210121011221021101212012210122112211222121212121011011010111020120120121022210221102120121110122011210111201111ANSWER TO 4:

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2-5SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION5.The city council is to decide on a proposal to raise property taxes. Suppose Ms. Tuttle isthe chair and the council’s other two members are Mr. Jones and Mrs. Doubtfire. Thevoting procedure works as follows: Excluding the chair, Mr. Jones and Mrs. Doubtfiresimultaneously write down their votes on slips of paper. Each writes eitherfororagainstthe tax increase. The secretary of the city council then opens the slips of paper andannounces the vote tally. If the secretary reports that both slips sayfor, then the taxincrease is implemented and the game is over. If both voteagainst,then the tax increaseis not implemented and, again, the game is over. However, if it is reported that the voteis oneforand oneagainst, then Ms. Tuttle has to vote. If she votesfor,then the taxincrease is implemented, and if she votesagainst,then it is not. In both cases, the gameis then over. As to payoffs, if the tax increase is implemented, then Mrs. Doubtfire andMr. Jones each receive a payoff of 3. If the tax increase proposal fails, then Mrs. Doubtfirehas a payoff of 4 and Mr. Jones’s payoff is 1. As for Ms. Tuttle, she prefers to have a taxincrease—believing that it will provide the funds to improve the city’s schools—but wouldprefer not to be on record as voting for higher taxes. Her payoff from a tax increase whenher vote is not required is 5, her payoff from a tax increase when herforvote is requiredis 2, and her payoff from taxes not being increased is zero (regardless of whether or notshe voted). Write down the extensive form of the game composed of Ms. Tuttle, Mr. Jones,and Mrs. Doubtfire.ANSWER:Mr. JonesMrs. DoubtfireMrs. DoubtfireMs. TuttleMs. TuttleFAFAFAFAFA3321Mr. Jones403351403321406.Consider a contestant on the legendary game showLet’s Make a Deal. There are three doors,and behind two doors is a booby prize (i.e., a prize of little value), while behind one dooris a prize of considerable value, such as an automobile. The doors are labeled 1, 2, and 3.The strategic situation starts when, prior to the show, the host, Monty Hall, selects one ofthe three doors behind which to place the good prize. Then, during the show, a contestantselects one of the three doors. After its selection, Monty opens up one of the two doors notselected by the contestant. In opening up a door, a rule of the show is that Monty is prohib-ited from opening the door with the good prize. After Monty opens a door, the contestantis then given the opportunity to continue with the door originally selected or switch to theother unopened door. After the contestant’s decision, the remaining two doors are opened.

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2-6SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATIONa.Write down an extensive form game ofLet’s Make a Dealup to (but not including) thestage at which the contestant decides whether to maintain his original choice orswitch to the other unopened door. Thus, you are to write down the extensive formfor when (1) Monty Hall chooses the door behind which the good prize is placed; (2)the contestant chooses a door; and (3) Monty Hall chooses a door to open. You mayexclude payoffs.ANSWER:MontyMContestantDoor 1Door 2Door 31322332MMM1323131MMM1322112MMb.For the stage at which the contestant decides whether or not to switch, write down thecontestant’s collection of information sets. In doing so, denote a node by a triple, suchas 3/2/1, which describes the sequence of play leading up to that node. 3/2/1 wouldmean that Monty Hall put the good prize behind door 3, the contestant initiallyselected door 2, and Monty Hall opened door 1.ANSWER:There are six information sets for the contestant at the point when hehas to decide whether or not to switch doors: (1) nodes 1/1/2 and 3/1/2; (2) nodes1/1/3 and 2/1/3; (3) nodes 1/2/3 and 2/2/3; (4) nodes 1/3/2 and 3/3/2; (5) nodes 2/2/1and 3/2/1; and (6) nodes 2/3/1 and 3/3/1. For example, the first information setcomprises nodes 1/1/2 and 3/1/2. At node 1/1/2, Monty put the prize behind door1, the contestant chose door 1, and Monty opened door 2. At node 3/1/2, Monty putthe prize behind door 3, the contestant chose door 1, and Monty opened door 2.The contestant cannot discriminate between those two nodes since they entail thesame sequence of observed actions—the contestant chose door 1 and Montyopened door 2—and differ only in terms of where Monty put the prize. That theinformation set includes both nodes 1/1/2 and 3/1/2 means that the contestantdoesn’t know whether the good prize is behind door 1 or door 3.7.For the Iraq War game in Figure 2.11, write down the strategy sets for the three players.ANSWER:Iraq has three information sets: (1) the initial node; (2) the set in whichit does not have WMD and the UN requested inspections; and (3) the set in whichit does have WMD and the UN requested inspections. A strategy for Iraq is then atriple of actions. At two of those information sets it has two feasible actions and atthe other one it has three actions. The total number of strategies for Iraq is then12 strategies. The United States has four information sets: (1) the UN did notrequestinspections;(2)theUNrequestedinspectionsandIraqrejectedthe

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2-7SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATIONrequest; (3) the UN requested inspections, Iraq acquiesced to the request, andWMD were not found (that is, either Iraq doesn’t have them or has them and hidthem); and (4) the UN requested inspections, Iraq acquiesced to the request,, andWMD were found (that is, Iraq had them and did not hide them). The first threeinformation sets each comprise two nodes, one corresponding to Iraq’s havingWMD and one to its not having WMD. The final information set is a singletonbecause of the implicit assumption that a UN inspection will reveal that Iraq hasWMD when Iraq does not attempt to hide them. A strategy for the U.S. is then a4-tuple of actions. Since at each of its four information sets the U.S. has two feasibleactions—attacking Iraq or not attacking Iraq—the U.S. has 16 strategies. Finally,the UN has one information set. Like the U.S., it has two nodes with one correspond-ing to Iraq’s having WMD and one not. A strategy for the UN is then a single action;its strategy set is composed ofrequest inspectionsanddo not request inspections.8.Derive the corresponding strategic form for the extensive form game in the figurebelow.5222111500Player 10Player 2105c2a2a1b2b1d2c1d120341c2d2ANSWER:Player 1 has two information sets, the initial node and the informationset associated witha1anda2having been played. Letx/ydenote a strategy forplayer 1 that assigns actionxto the initial node and actionyto the other informa-tion set. Player 1’s strategy then contains four elements:a1/c1,a1/d1,b1/c1, andb1/d1. Player 2 also has two information sets, the singleton associated with 1 havinguseda1and the information set with two nodes—one when the path isa1Sa2Sc1(read as “a1is chosen thena2is chosen thenc1is chosen”) and one when thepath isa1Sa2Sd1. If strategyx/yassigns actionxto the first information setand actionyto the second one, then player 2 has four strategies:a2/c2,a2/d2,b2/c2, andb2/d2. The payoff matrix associated with these strategies is shown in thefigure below.5,220,30,00,015,04,10,00,0Player 1Player 210,510,50,00,0a2/c2a1/c1a1/d1b1/c1b1/d1a2/d2b2/c2b2/d210,510,50,00,0

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2-8SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION9.Write down the strategic form game for the extensive form game below.Player1Player2211342abcdxyxyxyxy220631421500ANSWER:Player 1 has only one information set, which is the initial node. Player 2has two information sets. Her first information set is the information set associatedwith player 1 having chosen eitheraorb. Her second information set is associatedwith player 1 having chosencord. Strategyx/yfor player 2 assigns actionxto the firstinformation set and actionyto the second information set. The strategic form gamefor this game is shown in the figure below.4,22,23,11,54,22,24,20,0Player 1Player 21,30,63,11,5x/xabcdx/yy/xy/y1,30,64,20,010.Write down the strategic form game for the extensive form game in the game below.Player1Player2Player32341abcxyxyrs11602310021356312

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2-9SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATIONANSWER:Each player has only one information set. The strategic form game isshown in the figure below.4,1,21,1,12,3,52,3,06,0,32,3,5Player 1Player 2Player 3:xabcyr4,1,21,1,11,0,62,3,06,0,31,0,6Player 1Player 2Player 3:xabcys11.Three extensive form games are shown in the following figure. State which of them, if any,violate the assumption of perfect recall. Explain your answer.(a)211110111abxyxycdcdcdcd001010111001221111(b)0111xyxycdcdcdcd0010101110012111ab1221111

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2-10SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION(c)0111xyxycdcdcdcd001010111001211abANSWER:Only game (b) satisfies perfect recall. In game (a), consider the infor-mation set for player 1 that includes two nodes. One node is associated with 1having chosenaand 2 having choseny. The other is associated with 1 havingchosenband 2 having chosenx. At this information set, 1 is then unsure whethershe choseaorb. That violates perfect recall. As to game (c), the information setfor player 1 which includes four nodes captures the property that, when 1 choosesbetween actionscandd,she doesn’t know what player 2 chose (which is not inviolation of perfect recall) nor what she originally chose at the initial node (whichis in violation of perfect recall). Game (b) satisfies perfect recall. When 1 choosesbetween actionscandd,she cannot discriminate between the nodes in which playwasbSxand play wasaSy,nor between the nodes in which play wasbSxandplay wasbSy. The former reflects 1’s uncertainty over 2’s action but knowledgethat she originally chosea. The latter reflects 1’s uncertainty over 2’s action butknowledge that she originally choseb.12.Alexa and Judd live in Boston and have been dating for about a year and are fairly seri-ous. Alexa has been promoted to Regional Manager and been given the choice of assign-ments in Atlanta, Boise, and Tucson. After she makes her choice (and this is observed byJudd), he’ll decide whether to stay in Boston or follow Alexa. The payoffs associated withthe six possible outcomes are in the accompanying table.a.Derive the extensive form game.ANSWER:MoveStayMoveStayMoveStayAlexaAtlantaBoiseTucsonJuddAlexa’sJudd’sPlayer 1: AlexaPlayer 2: JuddJuddJudd563321137443(Continued )

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2-11SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATIONb.Derive the strategic form game.Alexa’s choiceJudd’s choiceAlexa’s payoffJudd’s payoffAtlantaMove56AtlantaStay33BoiseMove21BoiseStay13TucsonMove74TucsonStay43ANSWER:The strategy set of Alexa is {A,B,T}. A strategy for Judd is a 3-tuple ofactions; what to do if Alexa moves to Atlanta, what to do if she moves to Boise, andwhat to do if she moves to Tucson. His strategy set is {M/M/M,M/M/S,M/S/M,M/S/S,S/M/M,S/M/S,S/S/M,S/S/S}.ABTM/M/M5,62,17,4M/M/S5,62,14,3M/S/M5,61,37,4M/S/S5,61,34,3S/M/M3,32,17,4S/M/S3,32,14,3S/S/M3,31,37,4S/S/S3,31,34,313.When he released his new novelThe Plant, the best-selling author Stephen King choseto make early chapters downloadable for free on his website www.stephenking.com buthe also asked readers to make voluntary contributions. Furthermore, he stated that hewould not release subsequent chapters unless people contributed: “Remember: Pay andthe story rolls. Steal and the story folds.” In modeling this approach to selling a book,suppose there are just three readers: Abigail, Carrie, and Danny. All chapters have beenreleased except for the final one, which, of course, has the climax. For Abigail or Carrie,if the final chapter is released then each receives a payoff of 5 minus how much moneyshe contributed. For Danny, if the final chapter is released then he receives a payoff of10 minus how much money he contributed. If the final chapter is not released theneach reader receives a payoff of 2 minus how much he or she contributed. Abigail andCarrie are deciding between contributing nothing and $2. Danny is deciding between$2 and $4. For the final chapter to be released, at least $6 must be raised.a.Assume all three readers make simultaneous contribution decisions. Write down thestrategic form game.ANSWER:The set of players is Abigail, Carrie, and Danny. The strategy set forAbigail and Carrie is {0,2}, and for Danny is {2,4}. The strategic form is shownbelow where the first payoff in a cell is for Abigail, the second payoff is for Carrie,and the third is for Danny.AbigailCarrieDanny: 2002,2,02,0,00,2,03,3,822AbigailCarrieDanny: 4002,2,−25,3,63,5,63,3,622

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2-12SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATIONNow suppose Danny contributes first, and then Abigail and Carrie make simultaneouscontribution decisions after observing Danny’s contribution.b.Write down the extensive form game.ANSWER:DannyAbigailAbigail$2$4$0$2$0$2$0$2$0$2$0$2CarrieDannyAbigailCarrieCarrie002833022020–222653$0$2635633Player 1: DannyPlayer 2: AbigailPlayer 3: Carriec.Write down each player’s strategy set.ANSWER:A strategy for Danny is a single action and his strategy set is {2,4}. A strategyfor Abigail or Carrie is a pair of actions; what to do when Danny chooses 2 and whatto do when Danny chooses 4. The strategy set for Abigail and Carrie is {0/0,0/2,2/0,2/2}.d.Write down the strategic form game.ANSWER:AbigailCarrieDanny: 20/00/02,2,02,2,02,2,02,2,00/20/20,2,00,2,02/00,2,00,2,00,2,00,2,02/03,3,83,3,80,2,00,2,02/23,3,83,3,82/2AbigailCarrieDanny: 40/00/02,2,−25,3,63,5,63,3,60/20/22,2,−23,5,62/05,3,63,3,62,2,−23,5,62/02,2,−25,3,63,5,63,3,62/23,5,63,3,62/214.There are four shoppers in a store—Amy, Brian, Cassandra, and David—who sequen-tially show up at two checkout lines: line number 1 and line number 2. Initially, bothcheckout lines are empty. Amy shows up first, sees no one in either line, and chooses oneof the two lines. Brian shows up next, sees which line Amy entered, and chooses betweenthe two lines. Next, Cassandra shows up. She sees Amy and Brian in line and choosesbetween the two lines. Finally, David arrives, sees Amy, Brian, and Cassandra waiting inline, and enters one of the two lines. Assume a player’s payoff equals 5 minus the numberof shoppers in line ahead of him or her.a.Derive the extensive form game.b.Now suppose that by the time that David shows up, Amy has already checked out.Thus, only Brian and Cassandra are in line. Derive the extensive form game.

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ABBPart(a)1ABCD5432DD12121212121212121212121221122CCCC543554535454554355445544554334435545545545545454545354355432Part(b)ABCDA: AmyB: BrianC: CassandraD: David5433543554545454554455445545554335445545545545545454545454355433DDDDDDANSWER:2-13

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15.Kickstarter(www.kickstarter.com)providesaplatformforraisingventurecapitalthrough crowdsourcing. A project creator sets a funding target and posts the project atKickstarter. People then decide how much money to pledge. If the total pledges are atleast as great as the funding target, then the pledges are converted into contributionsand the project is funded. Though the contributors do not own a share of the project,they can receive rewards from the project creator. If the pledges fall short, then the proj-ect is not funded. Assume there are three players: one project creator and two potentialcontributors. The project creator chooses between a funding target of $1,000 and $1,500.With the funding target posted at Kickstarter, the two contributors simultaneouslydecide whether to pledge $250 or $750. Assume the project creator’s payoff equals threetimes the amount of funding (which is zero if contributions are less than the fundingtarget). A contributor’s payoff is zero when the project is not funded (irrespective of thepledge made), and is two times the total amount of pledges minus three times the con-tributor’s own pledge when it is funded.a.Write down the extensive form game.ANSWER:30001250002507502507502507502507502507502507503000–25045007500000004500750Project CreatorContributor 1Contributor 210001500–2500Project CreatorContributor 1Contributor 21250750000750b.Write down each player’s strategy set.ANSWER:The project creator’s strategy is a target level of funding target and thestrategy set is {1000,1500}. A contributor’s strategy is a pair of actions that speci-fies an amount of pledge for each funding target chosen by the project creator. Acontributor’ strategy set is {250/250,250/750,750/250,750/750}.c.Write down the strategic form game.ANSWER:Contributor #1Contributor #2Project creator: 1000250/250250/250250/750250/750750/250750/2500,0,00,0,0−250,1250,3000−250,1250,30000,0,00,0,0−250,1250,3000−250,1250,30001250,−250,30001250,−250,3000750,750,4500750,750,45001250,−250,30001250,−250,3000750,750,4500750,750,4500750/750750/7502-14SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION

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2-15SOLUTIONS MANUALCHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATIONContributor #1Contributor #2Project creator: 1500250/250250/250250/750250/750750/250750/2500,0,00,0,00,0,00,0,00,0,0750,750,45000,0,0750,750,45000,0,00,0,00,0,00,0,00,0,0750,750,45000,0,0750,750,4500750/750750/75016.Consider drivers who commonly traverse a major highway. Each driver is decidingwhether to buy E-ZPass. E-ZPass electronically charges a driver for going through a toll,which avoids having to stop and hand over money. E-ZPass costs $4 and allows a driverto go through the E-ZPass lane. Without E-ZPass, a driver goes through the Cash lane.With either lane, the toll is $6. The average time it takes for a car to get through theE-ZPass line is 10 seconds multiplied by the number of cars in the E-ZPass lane (whichis assumed to equal the number of cars with E-ZPass). For the Cash lane, the averagetime it takes for a car to get through is 30 seconds multiplied by the number of cars inthe Cash lane (which is assumed to equal the number of cars without E-ZPass). Thevalue of a driver’s time is 30 cents per minute. Assume there are 100 drivers, each ofwhom has a payoff equal to 20 minus the value of time spent in line minus expenditure(the latter is $4 without E-ZPass and $10 with E-ZPass). Drivers make simultaneousdecisions about whether or not to buy E-ZPass.a.The strategy set for a driver is (E-ZPass, No E-ZPass). Derive a driver’s payoff functiondepending on his choice and the choices of the other 99 drivers.ANSWER:LetmdenotethenumberofotherdriversthatchooseE-ZPass.A driver’s payoff from buying E-ZPass is20.3(m16)10,and from notbuying E-ZPass is 20.3(100m2)6b.Now suppose a driver with E-ZPass can use either lane. Assume that it takes the sameamount of time to go through the Cash lane whether a driver has E-ZPass or not. Driverswithout E-ZPass can still go through the Cash-only lane. The strategy set for a driver is(E-ZPass & E-ZPass lane, E-ZPass & Cash lane, No E-ZPass & Cash lane). Derive a driver’spayoff function, depending on her choice and the choices of the other 99 drivers.ANSWER:Letmdenote the number of other drivers that choose E-ZPass & E-ZPasslane andndenote the number of other drivers that choose E-ZPass & Cash lane.A driver’s payoff from E-ZPass & E-ZPass lane is 20.3(m16)10, fromE-ZPass & Cash lane is 20.3(100m6)10, and from not buying E-ZPass is20.3(100m2)6(Continued )

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