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by: Miss Sigurd Dicki

GameTheoryandApplications ECON250

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This 136 page Class Notes was uploaded by Miss Sigurd Dicki on Wednesday September 23, 2015. The Class Notes belongs to ECON250 at Drexel University taught by RogerMcCain in Fall. Since its upload, it has received 31 views. For similar materials see /class/212538/econ250-drexel-university in Economcs at Drexel University.


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Date Created: 09/23/15
Economics 250 Game Theory 4 h New Example The Gotham Bats a majorleague baseball team are threatening to move their franchise from Gotham to Metropolis if Gotham does not build them a new stadium If the Bats do move the city government will have political problems and they will be even worse if they have failed to build the stadium On the other hand the stadium is costly and the best outcome for the government is to keep the Bats and not build the stadium Thus we have the following 2X2 game New Example Game Government ibuild IDon t Bats move 32 31 Don t k 54 j 25 3 Determine all Nash equilibria Pure Strategies Government build Don t Batsr move 32 Mil 7 Don t i4 2 3 There are no Nash equilibria in Pure Strategies wyi i I 39 Mixed Strategies Let p be the probability that the government builds and q be the probability that the Bats move Then we have a mixed strategy equilibrium with p13 q12 3 NPerson Games Many of the quotgamesquot that are most important in the real world involve considerably more than two or three players for example economic competition highway congestion overexploitation of the environment and monetary exchange BUT if we have ten players there are 10 3628800 relationships between them Example The Queuing Game Have you had this experience SiX people are waiting at an airline boarding gate awaiting their Chance to check in and one of them stands up and steps to the counter to be the rst in the queue As a result the others feel that they too must stand in the queue 5 Explanation Yes there is a game theory explanation and it can also be applied to research and development innovation and intellectual property Payoffs 1 There is a twopoint effort penalty for standing in line so that for those who stand in line the net payoff to being served is two less that what is shown in the second column Those who do not stand in line are chosen for service at random after those who stand in line have been served Payoffs 2 If noone stands in line then each person has an equal chance of being served rst second sixth and an expected value payoffof16201617 165 125 In such a case the aggregate payoff is 75 But the net payoff to the person rst in line is 18gt125 so someone will get up and stand in line Payoffs 3 Order served Gross Payoff Net Payoff First 20 1 8 Second 17 175 i Third 14 I I Fourth 11 9 7 7 if J 7 quot Sixth 5 3 Equilibrium This game has a large family of Nash equilibria so we proceed by elimination With 4 persons in the queue we have arrived at a Nash equilibrium of the game The total payoff is 67 the expected value payoff is 65 for those who remain Since the fth person in line gets a net payoff of 6 noone else will join the queue Cui Bono Two people are better off the first two in line with the first gaining an assured payoff of 55 above the uncertain expected value payoff she would have had in the absence of queuing and the second gaining 25 But the rest are worse off The third person in line gets 12 losing 05 for example Simplifying Assumptions for NPerson Games In the Queuing Game all of the participants are assumed to be identical to be quotrepresentative agents quot The length of the line is a state variable A state variable is a variable or vector that sums up the state of the game from the point of View of the representative agent 5 Another Example Using these ideas I and my colleagues Richard Hamilton of the Medical School and Frank Linnehan of the Management Department have done some work on the problem of crowding of hospital emergency departments 7 D p The Patients Coordination Game with a Population of 10 Number of ED Patients average average and Position net position net benefit of in line benefit in line benefit alternative 1 10 1 10 5 2 8 15 9 5 3 6 2 8 5 4 4 25 7 5 5 2 3 6 5 6 O 35 5 5 7 2 4 4 5 8 4 45 3 5 9 6 5 2 5 i39 10 8 o 12 5 Analysis 1 The number of people who choose the ED is the state variable for this game For a Nash equilibrium to occur just six choose the ED while the other four choose their alternative 3 Analysis 2 This example is still not very realistic as the population would be much bigger in the real world patients would be heterogeneous there might be more than one hospital ED to choose from and the penalty of standing in line would 3 probably be less relative to the benefits of treatment AgentBased Simulation 1 Experience of Type 1 25 ED Experience of type 1 ELWL Mimii hzwgmw iiim My Aternate Experience oftype 1 AUUU W39 39 39 n 1 benefits N AgentBased Simulation 2 10000 computersimulated patients choose between the ED and an alternative learning what to expect Where the blue and red lines are equal we have the Nash equilibrium The black line shows the experience the ED patients expected they usually were a disappointed WTHll The agents learn but they also make mistakes Those who mistakenly are too optimistic about the ED are more likely to choose it while those who are mistakenly too pessimistic don t aren t counted as a result All the same we have a Nash equilibrium in the population as a whole i Checking It Out We did a questionnaire study in the Tenet Health System to test these hypotheses On the whole they confirmed our hypotheses based on the game theory model and the agentbased computer simulations 39 We are still working and d l isagreeing ontea ppo rite 39 a 4 4 f n quot W x quot r a 39gt 393397 The Choice of Transportation Modes Car or Bus The commuters are representative agents their payoffs vary in the same way with the number of cars on the road and the state variable is the proportion of all commuters who drive cars rather than riding the bus The larger the proportion who drive their cars the slower the commute will be regardless which transport strategy a particular commuter chooses Payoffs Deminant Strategy Payoffs Perhaps More Realistic The Tragedy of the Commons In general quotthe tragedy of the commonsquot is that all common property resources tend to be overexploited and thus degraded unless their intensive use is restrained by legal traditional or perhaps philanthropic institutions The classical instance is common pastures on which according to the theory each farmer will increase her herds until the pasture is overgrazed and all are impoveriShed Hawk vs Dove Revisited Bird B hawk dove Bird hawk 2525 149 A dove 9 14 55 9 h The ManyCreature Case Expected Value Peayoffs With Rande Matching 000 020 040 060 080 100 Types Tradition Tradition Many traditional models in economics both Classical and Keynesian use the representative agent approach and sometimes overuse it Summary N person games can be complex Many applications in economics political science and other fields rely on two complementary simplifying assumptions Representative agents and a One or more state variables Game Theory and Cooperation Game theory is interactive decision theory Schelling Aumann Game theory has two major branches Noncooperative Example Prisoner s Dilemma Cooperative Coalitions can form for mutual benefit The Emergence of Cooperative Game Theory The founding book The Theory of Games and Economic Behavior originated both noncooperative and cooperative game theory Noncooperative game theory was considered the solution only for twoperson zerosum games For all more complex quotgamesquot their approach was cooperative It was John Nash who extended noncooperative i approaches to winwin and loselose quotgamesquot Ideas from Cooperative Game Theory Cooperative game theory draws extensively on mathematical set theory A B C denotes a set comprising three elements perhaps agents in an interdependent decision problem If A B C form a coalition the coalition s best strategy might leave C for example worse off in the first instance In that case C would receive a side payment to assure a mutual benefit Ideas from Cooperative Game Theory 2 If side payments can be made without any cost for the payment itself then we can focus on the total value the coalition can obtain which can be distributed among the members according to whatever rule they choose This is called the transferable utility TU assumption 7 D Solutions 1 0 A solution for a game in coalition function form should tell us 1 which coalitions will form if any and 2 how each coalition will divide its Winnings among the members Solutions 2 Von Neumann and Morgenstern proposed a complex dominance criterion for solution It was not considered suf ciently speci c A major focus of research in the 1950 s and early 1960 s was to narrow the search Nash Shapley and Gillies and later others proposed other solution concepts These all rest on various concepts of bargaining power a and on different judgments as to when agents will reject an offer as a bargaining position 4 h Cooperative Games De nition Cooperative and noncooperatl39ve games and solutions If the participants in a game can make binding commitments to coordinate their strategies then the game is cooperative and otherwise it is noncooperative The solution with coordinated strategies is a cooperative solution and the solution Without coordination of strategies is a noncooperative solution An IT Game User Advanced Proven N0 deal Advanced 5090 00 39 00 Supplier Proven 00 3040 00 N0 deal 00 00 00 9 h E Grand Coalition When the information system user and supplier get together and work out a deal fOr an information system they are forming a quotcoalitionquot A coalition consisting of all both the players in the game is called the grand coalition 3 Necessary Condition A Side Payment Because buying and selling always means that an enforceable agreement is made and on the basis of the agreement a payment changes hands In game theory the payment is called a quotside paymentquot 4 h Payment De nition Transferable utility A game is said to have transferable utility if the subjective payoffs are closely enough correlated with money payoffs so that transfers of money can be used to adjust the payoffs within a coalition Side payments will always be possible in a game with transferable utility but may not be possible in a game Without transferable utility On That Basis We can rule out both quotno dealquot and the proven system as strategies The coordinated strategies advanced advanced yield the most total pro ts The net bene ts to the two participants cannot add up to more than 40 0 Since each participant can break even by going it alone neither will accept a net less in than zero Visualizing Solutions Allowing for side payments all of the points on the solid diagonal line are possible cooperative solutions The range can be narrowed by Competitive pressures from other potential suppliers and users Perceived fairness inl Bargaining Solution Set Concept solution set von Neumann and Morgenstern de ned a complicated solution with many possible solutions called the solution set For a simple game such as this one the set of all ef cient Pareto optimal coalitions and payoffs is the solution set for the game De nition efficient Pareto optimal In neoclassical economics the allocation of resources is said to be ef cient or Pareto optimal if noone can be made better off Without making someone else worse off jar Why do we study non cooperative games at all Noncooperative solutions occur when participants in the game cannot make credible commitments to cooperative strategies Evidently this is a very common dif culty in many human interactions Another Cooperative Game Example Taxi There are three taxi companies in Gotham City each With established customer relations companies that call them kickbacks to hotel concierges and such in different parts of town By merging two or more of them may be able to share costs and customer contacts and so bene t The four are YYellow Cab Co BwB1ackand White Cab Co and BBatmobiles Inc You are to 3 use methods of cooperative game theory to explore such a merger Values of Coalitions We have the data to compute the pro tability of each merged company The values of the potential coalitions are Coalition value YaBWB 10 million YBW 7 YaB BWB Y BW B i NNC xm This table is called the coalition functionquot or characteristic function of the gmne l 39 r l I Coalition Function This example also has a property many game theorists think is correct in general it is superadditive Coalition value YaBWB 10 million YBW 7 YaB BWB Y BW l NNQLII B That is if two companies merge the value of the merged coalition is no less than the sum of the values of the orginal Dali Solutions Characteristic functions are well understood in mathematics and so much of the information we have on solutions is based on this approach The coalition function approach assumes Transferable Utility h Solutions The coalition function approach assumes transferable utility A candidate for solution is a coalition and a payoff schedule Coalition value For example3 YBwB 10 million suppose the grand gig coalition YBWB Elm 6 is formed and pays Y 2 226 This is a BW 2 B 1 candidate but IS 1t a solution Domination 1 A candidate fails as a solution if it is dominated That means that members of the coalition can shift to another coalition and all be better off Coalition value Fo xample Y and YBWB 10 million YDBW 7 BW can drop out and YB 5 form their own BWB 6 1t f 7 Y 2 coa1 lon or 13w 2 paymg 35 35 That B 1 dominates the GC if 7 With Z g Domination 2 The Grand Coalition can dominate any other coalition because of superadditivity if the payouts are right Conditions Coalition value y wz7 i0 mllllon YBZS YB 5 BwBZ6 g 2YBBw218 Bw 2 YB BWZ9 B 1 4 3 3 will do it 3 The Core The core of a game in coalition function form comprises all candidate solutions that are undominated In this game the Coalition value YDBWDB 10 million core 1ncludes the YBw 7 grand coalition With 13 5 ff h d 1 BWDB 6 any payo sc e u 6 Y 2 that sat1s es the BW 2 inequalities shown B 1 before Marginal Contribution 1 A solution method suggested by Shapley uses the concept of a marginal contribution Since the grand coalition is efficient we assume it will be formed Coalition value YaBWB 10 million YBW 7 YaB BWB Y BW B l NNC xm 9 h Su pose it is formed by adding B BW and Y in that order and each gets What it adds to the value of the coalition Marginal Contribution 2 Thus since B forms a singleton coalition the value it adds is l The value of B BW is six so BW adds 5 to the value of the coalition gal31153 Ygl illion Since the value of Yng 7 B BWY is 10 Y YB 5 adds 4 to the value EEZV B of the coalition 13w 2 Then the payoffs B 1 would be 1 5 4 for i 7 BtiVTY in l V 39 l 39 39 Order Order But the order B BW Y is arbitrary B might object Why can t1 come last ThenI would add 5 that 5 What I m worth Coalition value ACQrdingly the YBWB 101111111011 h 1 YDBW 7 S ap ey Values are Y1B 5 computed by BWB 6 11 Y 2 avereglng over it 13w 2 p0551b1e orders 111 B 1 which the players rr1ight b eadded Shapley Value YBwB YBBw BYBw BwYB BwBY BBwY total average B 1 3 1 3 4 1 15 2 5 Bw i 5 5 4 239 5 24 4 Y 2 2 4 5 4 Z 21 3 5 Accordingly the Shapley values for B BW and Y are 25 4 35 Note that Shapley showed that this is the only solution that has some nice properties including symmetry lIn this game the Shapley solution is Within the 3 core but that is not always so Shapley Value as Cooperative Solution The Shapley value has many of the properties we want in a solution When we have spoken of cooperative solutions before as in the Social Dilemmas the Shapley Value ts as the cooperative solution we mean However it does not always agree with the Core Moreover there are other cooperative solution concepts that may disagree with either i r This is a problem for cooperative game theory A Stag Hunt and a Problem Let s return to the Stag Hunt Game Without looking at strategies we assume that there are 3 hunters A B C and any 2 can catch a stag 3 can catch both a stag and a rabbit 6 AB 39 C 3 D 5 l l 1 A stag is worth ve and a rabbit is worth 1 Therefore any 2 person coalition is worth 5 the grand coalition with 6 while a singleton is worth only 1 Stag Hunt Core 1 This Coalition Function is superadditive Suppose the grand coalition forms with payoffs 221 Then any 2pcrson coalition Stag Hunt Core 2 Conditions for the stability of the grand coalition ARgtlt C AB C not nossib le 3 Stag Hunt Core 2 Now suppose coalitionAB forms With payoffs 25 25 C who is left out approaches A with the following proposition Let s form coalltlon AC C AB C B C A B Slmllal ly all 2person coalitions are dominated 5 Empty Core Singletons are also dominated by all 2 or 3 person coalitions There are no undominated coalitions The core in this case is the null set it is an empty core This is a recognized limitation of the core concept Note by the way that if a stag is taken this game really has no individual payoffs until the payoff schedule is cooperatively determined The noncooperative solution is not really determined unless the cooperative solution is Problem Stag Hunt Shapley However we can calculate a Shapley value for this game as we can for any superadditive game in coalition function form 0 AA 11 1 Hum Ayn ABC AB AaC BC A B C 6 5 5 5 l l l 5 One More Cooperative Game Example A Real Estate Development Jay the realestate developer wants to put together two or more parcels of property in order to develop them jointly He is considering properties owned by Kaye Laura and Mark Jay s Problem Jay wants to propose a deal that Will be stable in the sense that none of the three property owners Will want to renegotiate With some other property owners not included in the deal Having studied game theory Jay recognizes that the property consolidations are coalitions in a cooperative game and that the solution to his problem is a coalition structure in the core of the game Possible Coalitions quot Coalitions 39 Payoffs 1 39KLM 391 10 2 KLXM 6X4 3 mmm ma 4 I LMXK 4X4 awmwmiemmgt 9 h Unorthodox Note that in this example the payoffs to the singletons KLM depend on whether the other two players form a coalition or not That could be because a development on land held by the other two will generate a positive externality A coalition function cannot capture that The table is a partition function that shows how the payoff to each coalition depends on the whole coalition structure Analysis Solution set The Core KLM KLM is not stable KLM Any player can drop Total payoffs are 10 0L and ear 4 as a singleton so the total payments have to be 12 in both cases Further Analysis KLM is stable provided K and L are each paid 3 If K or L is paid less than 3 that player can drop out and go singleton This shifts to KLM with payoffs 333 KML or LMK don t help anyone Jay s Solution Jay makes offers to Kaye and Laura 9 h Another Problem Ifthe TU assumption doesn t apply we have another sort of problem Anna Bob Carole and Don are all employed at the University of West Philadelphia UWP and commute by car from their homes in the western suburbs of Philadelphia to UWP They are interested in forming one or more carpools to commute together We Will treat the carpools as coalitions in a cooperative game Payoffs are in miles adjusted for gas saving the objective is to minimize The Coalition Function V Coalition Payoffs V ABCD V 7777 V ABC V 6659 V7 ABD 6576 V ACD V 887 V BCD V 65656 V AB V 87 V CD 78 9H 89 79 87 Dominance I Coalition Payoffs Suppose have a I ABCD um carpool and propose to I ABC 67659 add Bob Addrng Bob w111 make Anna and Carole l ABD i 6395 7 6 better off lower overall l ltACDgt l 887 time and gas costs and BCD 656526 leave Don no worse off AB 8 7 Thus the Grand Coalition I lt lt weakly dominates ACD CD l 78 Core Reasoning in this way we nd that the core of the carpool game consists of coalitions ABD and BCD meaning either Anna or Don is out of luck In fact the grand coalition despite its overall advantages is not in the core It is dominated both by ABD which makes A and D better off while B is no worse off and by line 5 which makes BCD all better off Shapley The Shapley value as we have de ned it only works if utility is transferable There are proposals for extension of the Shapley value to NTU games but they are little used and will be beyond our scope Summary 1 When players can commit themselves credibly to coordinate their strategies they can often improve their payoffs 0 There are a number of solution concepts for this case Summary 2 Some payoffs may be ruled out if we also consider stability against defection by smaller coalitions focusing on quotthe corequot However this may rule out all of them or leave more than one to choose among after all Nevertheless the core seems to describe some key economic phenomena Economics 250 Game Theory 4 h Sequential Games While all games can be treated in quotnormal formquot there are other important interactions in which the agents have to choose their strategies in some particular order and in which commitments can only be made under limited circumstances or after some time has passed These are quotsequential gamesquot Strategic Investment to Deter Entry The entry of new competition can reduce the pro ts of established rms Accordingly we would expect that companies might try to nd some way to prevent or deter the entry of new competition into the market even if it is costly to do so Here is an example of that kind The Chips are Down Spizella Corp produces specialized computer processing chips for workstations A plant to fabricate these chips costs 1 billion and will produce 3 million chips per year at a cost of 1 billion per year and so at an average cost of 33333 per chip Demand for Chips Table 1 I Q price per chip 3000000 m 700 6000000 m 400 9000000 m 200 W A Challenge Spizella39s management have learned that Passer Ltd are considering building a quotfabquot to enter this market in competition with SpizellaBut if a second fab comes on line output will increase to 6 million chips the price per chip will drop to 400 reducing Spizella s pro ts Threat to Profits Table 2 I Q U pro t per fab l 3000000 U 11 billion m 6000000 H 200 million I 9000000 U 400 million Response Nevertheless Spizella is considering investing in a second fab 3 Why 1 If Spizella builds before Passer makes their decision Passer will realize that their plant would be the third one and that if they build it everyone will lose 400 million per plant per year So Passer will not build and Spizella will retain 400 million a year of pro t 2 If Spizella doesn39t build the second plant Passer will and Spizella will be left with only 200 million of pro ts on their one present fab Strategic Investment In building the new plant to keep Passer out Spizella would be engaging in quotstrategic investment to deter entryquot If In Extensive Form Another quot Subgame 000 84 Subgames We see two subgames of this game of strategic investment to deter entry In game theory the whole game is also considered a subgame of itself The others are proper subgames Perfect Equilibrium In order for a game with one or more proper subgames to be in equilibrium every subgame must be in equilibrium This sort of equilibrium we call a quotsubgame perfect equilibriumquot amp Solution by Backward lnduc on We start with the last decision in each sequence determine the equilibrium for that decision and then move back determining the equilibrium at each step until we arrive at the first decision Rad Recapitulating Concepts De nition Subgame A subgame of any game consists of all nodes and payoffs that follow a complete information node De nition Proper Subgame A proper subgame is a subgame that includes only part of the complete gmne 9 h The Prisoner s Dilemma Has No Proper Subgames More Concepts De nition Basic and Complex Nodes A node is quotbasicquot if each of its branches leads to just one set of payoffs This means in effect that there are no further decisions to be made A node is complex if it is not basic De nition Subgame Perfect Equilibrium A game is in subgame perfect equilibrium if and only if every subgame is in a Nash equilibrium 4 h Backward Induction Method Backward induction To find the subgame perfect equilibrium of a sequential game first determine the Nash Equilibria of all basic subgames Next reduce the game by substituting the equilibrium payoffs for the basic subgames Repeat this procedure until there are no proper subgames and solve the resulting game for its Nash Equilibrium The sequence of Nash Equilibria for the proper subgames of the original game constitutes the subgame perfect equilibrium of the Whole game The Spanish Rebellion Red used The Centipede Reduced Centipede pass qEJ 41 34 2J5 The game has just one proper subgame shown by the gray oval and it is basic In that subgame B will choose quotgrabquot and the payoff will be 2 for A and 6 for Barb With this solution we reduce the Centipede game to the game shown here In this game Anna chooses quotgra quot for 4 rather than quotpassquot for 2 and that is the subgame perfect solution to the game Lessons Backward reduction can give us a solution to a game in extensive form In some cases people may want to give up their freedom of action by making an early commitment Early commitment may improve payoffs or not it o A Defense Application During the cold war period the Soviet Union had superior numbers of ground troops in Europe and it was generally believed that if they chose to attack they would have been able to overrun West Germany very rapidly To prevent this US troops were stationed on German soil But the American troops were not strong enough to defeat an allout Soviet attack Thus the U S stationed in Germany a force that the Soviets could surely defeat if they wished Why Answer People at the time said that the troops were a tripwire 3 The Cold War on the European Frontier Without Troops in Germany attack counterattack SU US 2 2 D n o o 3 H H 33 51 h g The Cold War on the European Frontier With Troops in Germany attack counterattack SU US 23 n D o o 5 l H H 32 65 amp Nested Part of the idea was that the American electorate would not permit a government that would fail to retaliate if American troops were a acked Thus the US changed the rules by nesting the game against the SU in a larger game Nested and Imbedded Nested games If a game is part of a larger game then equilibrium strategies in the smaller game may depend on the larger game The smaller game is said to be nested Within the larger Imbedded games If a nested game is a proper subgame of the larger game then the nested game must be in equilibrium for the larger game to be in a subgame perfect equilibrium Then the smaller game is said to be imbedded in the larger attack counterattack SU US 2 2 a0 0 LUOP LUOD attack cou nterattack SU US 23 LUOP LUOP in 32 6 5 Changing the Rules In effect the U S changed the rules by imbedding the original game in a larger game Planning Dactoral Study 1 The Basic Proper Subgame Nora IR SE IR 37 00 Anna SE 00 73 A New Example 1 This game begins with a decision of a municipal authority M either to build a public parking lot Build or not to build it Don t In either case at the second stage two businesses B1 and B2 decide Whether or open retail sites near the potential parking lot Build or not to do so Don t They have to decide simultaneously The businesses are complementary rather than competitive as for example one offers gourmet groceries and the other Wine and spirits A New Example 2 If the municipal public parking lot has been built then either or both can operate pro tably with a payoff indicated as 4 if the public lot has not been built each Will have to supply parking space for their own customers raising their costs Nevertheless if both build they can both operate pro tably with payoffs at 2 2 If only one builds however it will not attract enough customers to cover the cost of the parking lot and Will operate at a loss indicated by l A New Example 3 A business that does not build has a payoff of zero If both businesses build with their own parking lots the municipality has the further possibility of imposing a regulation that a proportion of all parking places must be public parking and supposing that this public parking will attract more customers away from the rival Mall in a nearby township the pro ts of both businesses are higher at 3 Since the municipality has no sources of revenue other than taxes its payoffs are always nonpositive and the 3 cost of the public parking lot is 2 while the cost of administering a parking regulation is l The payoffs are in the order B1 B2 Proper subga m e s A Nested not Imbedded Game An Imbedded Game 3933391 110 5 The Imbedded Game in Normal Form B2 Build Don39t B1 Builc39 220 100 Don t 010 000 We observe that this subgame has 2 Nash equilibria So has the other imbedded game between the two businesses Concluding the Example Since payoffs for the municipality are always nonpositive its dominant strategy is never to do anything Thus there are two subgame perfect equilibria corresponding to the two solutions we have just seen For a cooperative solution we will require a side payment ie taxes to the municipality A Note though that it is useful to think of private sector decisions as games imbedded in the larger game of public policy Just Fueling Around Transport Equipment Corp TECORP sells busses primarily to urban bus services and wants to convert its busses to fuel cell power since its customers are concerned about air pollution Queen Hill Power QHP has perfected the technology for the fuel cells and TECORP has approached QHP to produce power plants for its busses Queen Hill s Problem However this will require QHP to construct a costly specialized facility for which TECORP will be the only buyer QHP is concerned that TECORP will then demand a renegotiated price which QHP will be forced to grant and thus be a loser Payo s if there is no agreement payoffs are 0 if there is an agreement and no regenotiation payoffs are 100 100 a demand for renegotiation and QHP refuses payoffs are 0 100 a demand for renegotiation and QHP gives in payoffs are 200 50 TECORP first Subgame Perfect How about a merger agree 200 50 ego ate 1 I ree T quottear O 100 3 xii iwxdonlt negotiateLOQ ll8e I Both 5 01 o 0 H agree I 100 100 3 Why Nonprofits1 Nonprofit enterprise is a fastergrowing sector of our economy than investorowned cooperative or government enterprise Nonprofits may actually earn profits but may not distribute the profits Instead any profits must be devoted to the nonprofit corporation s mission Why should such enterprises exist Many nonprofits are supported or founded by dona ons Why Nonpm ts Partly Reduced duq t M 1 a 9 h Partly Reduced duq t M 1 a 9 h Further Reduced 1110 m 9 h Overall Summary Use the concept of subgame perfect equilibrium We use backward induction Think forward and reason backward We first solve all of the basic proper subgames continue step by step Sequential commitment makes a a difference


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