INTRO GAME THEORY
INTRO GAME THEORY ECON 171
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Econ 171 Fall 2008 Lecture 10 October 28 Finitely Repeated Games Can the players sustain cooperation X Y Z A B 00 21 Figure 1 The stage game from Watson page 259 4 7 Result 1 Any sequence of stage Nash pro les can be sup ported as the outcome of sabgame perfect Nash egal llbrlam De nitionl Player z s mmmazr payo is the solution to the problem rnin Inasxulsi7 5 94654 sie i Q D 0 1 DO co 7 Figure 2 PD 7 In nitely repeated Games if you get a payoff of 1 each period your discounted payoff is s166263 Which we can simplify by noting that s 16s and solving to get 1 s So if your payoff is xin every period your discounted pay Off is 1 Lecture 1 Normal Form Games Econ 171 Fall 2008 De nition 1 Game Theory is a formal way to analyze interaction among a group of rational agents who behave strategically De nition 2 A strategic situation is a situation in which one party s behavior a ect s another party s well being Example 3 You go out to a restaurant with a friend What do you order You each pay for your own meal this is a decision problem You agree ahead of time to split the bill we have a game De nition 4 A strategy is a complete contingent plan for a player in the game Notation si E 51 A strategy pro le speci es a strategy for each player 8 E S Where S 5391 X 52 times X W Example 5 Sequential chicken Player 1 goes rst then Player 2 What are the strategies for each player 0 Player 1 BNl 0 Player 2 De nition 6 A game in normal form also called strategic form consists of o a set ofplayers I 1111N o a strategy space S o payo functions for each player ui S A R for each i E 1 De nition 7 Given a belief 94 6 A54 about the strategies played by the other players Player i s strategy si is a best response if uisit9i 2 uist9i for every 3 6 Si De nition 8 A pure strategy si is dominated if there is another strategy 01 E ASZ such that uiai 371 gt uisisi for all strategy pro les 31 6 51 of other players Player 2 C Player 1 D a Game f Two rms produce the exact same good and choose output levels qi While facing cost function ciqii Demand determines a price depending upon total output pql 12 Suppose both rms are pro t maximizersi Formalize this as a normal form gamer Players Strategy Space Preferences Econ 171 Fall 2008 Notes in place of Lecture 2 September 30 Remmder these rtotes take the place of lecture ort Tuesday September 30 The heat class will meet as scheduled ort Thursday October 2 I will hold OH this week ort Thursday 1030am rtoort 3049 North Hall Overview In these notes we will continue looking at games in the normal or strategic form which is a model in which each player chooses her action once and for all with the choices being made simultaneously We have three goals 0 The rst section aims to shore up your understanding of lterated Elimination of Domi nated Strategies or lterated Dominance for short particularly when it involves mixed strategies 0 The second section seeks to familiarize you with several classical normal form games and the issues that they highlight o The third section seeks to familiarize you with iterated dominance in normal form games by presenting an application to supplement those presented in Chapter 8 of Watson Review of Important Terms and Concepts We begin by returning to the Handout on Normal Form Games First recall game a oth erwise known as the Prisoners7 Dilemma We solved this game by applying the concept of dominance which relies on the assumption that the players are rational Player 2 0 Pl 1 ayer D Figure 1 Prisoners7 Dilemma Next recall game b Applying dominance alone did not yield a unique prediction but iterated elimination of dominated strategies or iterated dominance did In general games for which iterated dominance provides a unique prediction are called dominance soluable In this class of games iterated dominance is equivalent to another solution concept called ratlortallzablllty which uses the assumption that rationality is common knowledge We call the strategies that survive this concept ratlortallzable In class we did not get to game c If you have not already done so please try to nd the set of rationalizable strategies in this game before reading on We begin by noting that A is dominated by B Whether Player 1 believes 2 will play X or Y she is better off playing B Thus we can eliminate A However we cannot eliminate any further strategies B is a best response to the belief that 2 will play X and C is a best response if 2 plays Y Likewise for Player 2 X is a best response to B and Y is a best response to 0 Thus none of the remaining strategies are dominated This game shows that iterated dominance does not always yield a unique prediction ie not all games are dominance solvable Now lets look at game d through which we kind of rushed in class First we7ll use this game to clarify some important terms What is a difference between a strategy strategy space strategy pro le and an outcome A strategy is a plan for one player eg B is a strategy for Player 1 and X is a strategy for Player 2 A strategy pro le is a vector of strate gies one for each player eg BX A player7s strategy space is the set of all strategies available Here 51 ABC and 2 X Y The set of strategy pro les is the set of all possible combinations of strategies or the cartesian product of the strategy spaces S Sl gtlt SQ AX AY BXBYCXCY Players preferences are de ned over strategy pro les and each one can be associated with an outcome in which each player experiences a particular level of utility For example the strategy pro le A X leads to the outcome 3 2 When we apply solution concepts and make predictions we typically talk in terms of strategy pro les as opposed to outcomes For example for game d we say that according to iterated dominance the solution of the game is AX which leads to the outcome 3 2 We do not say that the solution is 3 20 strictly speaking Recall that we looked at the this game and rst observed that none of the strategies appear to be dominated at rst but then noted that B is in fact dominated by a mixed strategy To help us see how this is the case let us go back and clarify some points regarding the calculation of utility in the presence of uncertainty whether it comes from mixed strategies or uncertain beliefs ln standard game theory we assume that people maximize their Expected Utility What this means is that when facing uncertainty about the strategies that will be played a person evaluates her choices according to a weighted average of the utility for each of the possible outcomes that she considers possible where the weight is the likelihood for that outcome Here are two examples from game d that illustrate how Player 1 calculates her expected utility with two different sources of uncertainty her beliefs about the other players action and her own mixed strategy 0 Uncertainty regarding other player s choice Suppose Player 1 is not sure what Player 2 will choose and believes that X and Y are equally likely Notationally we could express this as 01X 61Y We evaluate 17s expected utility for A as the weighted average of u1A X and u1A Y with the weight on each term taken from the corresponding value of 01 Thus 307 437i mmmzmmwiwmmmmmm 15 52 Doing the same for different beliefs eg g yields a different expected utility with more weight on the higher payoff AX strategy pro le U1At91 g 3 0 In general the more weight Player 17s beliefs place on X the more her expected payoff for playing A looks like the payoff for A X the more weight she puts on Y the more her expected payoff looks like the payoff for A Y Her expected utility depends upon these probabilities linearly as is shown by line A in Figure T This gure also shows the expected utility for B and C as a function of the likelihood that 2 will play X versus Y Here the uncertainty is due to uncertain beliefs about 27s strategy but even if 1 were certain of 27s strategy she still may face uncertainty over the outcome if 27s chosen strategy is a mixed strategy The utility calculations work exactly the same way Suppose that 1 know for sure that 2 will play 02X 02Y Then 307 U1A0392 20252u1A 82 2 i 52 Uncertainty stemming from one s own micced strategy Even if Player 1 is certain of Player 27s choice she may be uncertain about the outcome if she herself chooses to play a mixed strategy Suppose 1 plays the mixed strategy given by 01A 010 and 01B 0 and she knows that 2 is playing X Then her expected utility is given W U10391X 20151u151X 0 5u1C 51 A similar calculation gives us U10391 Y 15 which is the exact same value In fact no matter what beliefs 1 has about 27s behavior playing this strategy yields an expected utility of 15 This follows from the linearity of expected utility in probabilites and is illustrated in Figure by the horizontal line labeled i 0 i which characterizes 01 Mr FriZ 0 12 1 Prl713yer 2 plays X Flgure 2 Player 1 s expecled uuhly as a luhcuon bl bellels aboul Player 2 s slralegy slralegy lhal plays A and c wrlh equal probablhly guarahlees ah expecled payo o 1 5 whlle ll should be clear lhal E wlll always yleld onb 1 Thls slralegy dommales E and game d shows us lhal a slralegy may be dommaled by a mlxed slralegy ever ll ll 15 ml dommaled by ary pure slralegy Sn l ll golng lo be prelly hard lo check every posslble slralegy ll w lo lnclude mlxed as u whlch represehls bellefs aboul lhe llkellhood lhal each slralegy is played Pul lhe player s expecled uuhl on lhe vemcel exls and or aeh pu slralegy plol lhe payo on lhe lell exlre e ll lhe opponehl p ay lt2 vs a expecled uuhly lor eaeh bl lhe osr slble bellefs aboul lhe whlch bl lhe opponehus lwo acuons wlll be pl h h eae ol lhe l lhls case lhree pure slraleges avallable lo lhe player Now lry lhls wrlh game e Also helpful is the following interesting result on dominated strategies an action of a player in a nite strategic game is never a best response if and only if it is strictly dominated So in Figure T The fact that the EU line for B never is the highest is suf cient for us to know that it is dominated We dont even have to nd the dominating strategy to eliminate it The mixed strategy we looked at above is not the only one that dominates B As an exercise you should now try to identify all such mixed strategies for Player 1 Try it on your own rst then read through the logic below First we know that all the mixed strategies must involve mixing between A and C It would not make sense to include B in a mixed strategy that dominates B and if A or C were left out we7d then be looking at a pure strategy for which we know there is no dominance Next we look back at Figure and think about what we are looking for A strategy that dominates another has to appear as a line that lies completely above the other without ever touching or intersecting it This is what it means to give higher utility for my belief that 1 may have about 27s action What does a strategy that mixes A and 0 look like It is a combination of the B and C a line that lies between them on the graph The more weight is put on B in the mix the closer it is to the B line the more weight on C the closer the line is to C The mixed strategy i 0 is exactly halfway between the two lines at every point If the strategy puts too much weight on A then it will dip below B on the right where Y is very likely because playing AY gives the payoff 0 On the other hand if the strategy puts too much weight on C if will be worse than B on the left where X is very likely We can see that all mixed strategies of the form q01 7 q where q E g will stay above the B line and thus dominate B Let7s now look at the ip side of this in game e When you drew the diagram you should have noticed that all we have done is add 1 to Player 1 s payoff for B in every case shifting the line up 1 in the diagram Now B is optimal for beliefs in the middle It follows from the result stated above because it is a best response to some beliefs it cant be dominated In general this game shows us that a strategy may be a best response and therefore undomi nated even if it is not the best response to any pure strategy Now we will move on to our challenge games g and h which were 3 gtlt 3 games1 We7ll walk through the iterated elimination of strategies pointing out both the dominating strategy and the hierarchy of beliefs needed to make that step First the solution for game g is CY 1What happened to game f The purpose of that game was to illustrate that normal form games donlt have to be represented with matrices and frequently can t be Because we will see more examples like this in the nal section we wonlt worry about f nowi 0 Initially none of Player 17s strategies are dominated All are consistent with rationality 0 However we can eliminate Z for Player 2 which is dominated by Y Only X and Y are consistent with rationality 0 Following this elimination we can eliminate A which is dominated for Player 1 by 0 Only B and C are consistent with 1 being rational and knowing that 2 is rational 0 Player 2 being rational and knowing that 1 is rational is not suf cient to eliminate any further strategies but we can eliminate X if 2 is rational knows 1 is rational and knows 1 knows she is rational because it is then dominated by Y 0 Finally 1 eliminates B which is dominated by C This relies on 1 being rational knowing 2 is rational knowing that 2 knows she is rational and knowing that 2 knows she knows 2 knows she is rational The solution for h is AX 0 Initially 2 has no dominated strategies as each strategy is a best response to some belief X is a best response to A Y is a best response to B and Z is a best response to 0 Player 1 does not have any strategies dominated by pure strategies but we should look more closely at mixed strategies How do we know this Our big clue is that C is not a best response to any pure strategies Where do we look Well there are two other pure strategies A and B and we can see that their payoffs average at least as high as that of C for any belief Thus a 50 50 mixture of these two will dominate C It follows that only A and B are consistent with rationality for Player 1 0 Following this 2 eliminates Z which is dominated by X Only X and Y are consistent with 2 being rational and knowing that 1 is rational 0 Next 1 eliminates B which is dominated by A Only A is consistent with 1 being rational knowing 2 is and knowing 2 knows that 1 is rational 0 Finally 2 eliminates Y which is dominated by X This is consistent with 2 being rational knowing 1 is rational knowing 1 knows 2 is rational and knowing 1 knows 2 knows 1 knows 2 is rational To reiterate the dominance concept relies on the assumption of rationality but iterated dominance depends upon the assumption that this rationality is common knowledge In submitting answers to the challenge from the rst class some people concluded their ex planations by writing things like Player 17s only rational action is to choose 0 ls this a correct statement or not Take a moment to think about this This statement is technically correct but it is slightly misleadingienough so that I feel the need to clarify The statement is technically correct because once we assume that rationality is common knowledge the iterated elimination of dominated strategy follows and leads rationalizable solution which in this case is unique Having done this elimination Player 1 knows that 2 will only play Y to which 0 is the unique best response Thus the only rational thing to do is to play the best response 0 However taken by itself the statement could be confused with saying that all we need to do is assume that 1 is rational then we know she will pick 0 A good clari cation would be to say At this point Player 17s only rational action is to choose 0 Some Classic Normal Form Games The purpose of this section is to familiarize yourself with the battery of classic 2 gtlt 2 normal form games I will be brief because most of what I have to say is already presented in Watsor s section on this on page 30 However I do have a few things to add You may nd it helpful to open up the book to the gure that displays the matrices for these games on page 31 The rst point is general These games are classic because of the general issues and tensions that they highlight They are in no way uniquely represented with the payoffs shown on page 31 For example the Prisoners7 Dilemma game shown would still be a Prisoners7 Dilemma if we added or subtracted a xed amount from each payoff or scaled them up or down A Prisoners7 Dilemma may even by non symmetric or include more than 2 players What makes a game a Prisoners7 Dilemma is that it has a unique solution in dominant strategies that is Pareto inferior to one of the dominated strategies There is another outcome that everyone would prefer If they could all cooperate they could be better off but because all players act individually optimization steers them away from it This illustrates what Watson calls the rst of three strategic tensions77 that pervade game theory the clash between individual and joint interests What precisely does he mean by strategic tensions These are the dilemmas problems that face people in strategic situa tions These are the bogeymen then things you worry about when deciding your actions If everyone acted jointly these wouldnt be problems but because we act individually these tensions remain What anyone in these games would hope for is some sort of way around the problem some way to organize ourselves and our interactions with each other than helps us solve or avoid these dif culties That is what we are talking about when we dis cuss institutions that overcome these tensions We will discuss this more in the days to come In general the payoffs of all of these games can be shifted or scaled without affecting the strategic character of the game This re ects a fundamental feature of Expected Utility the ory namely that utility function representations of preferences are unique only up to af ne transformations the aforementioned shifts and scalings In general matching pennies is any game in which one player wants both players to choose the same thing while the other player wants them to choose differently In this sense their interests are directly opposed ln Chicken the players7 interests con ict but in a different manner They each would like to choose the opposite of the other player but only in one particular way maintain the course while the other swerves Battle of the Sexes Coordination and Pareto Coordination are all variations of the same theme Both players would like to coordinate on the same choice but lack any means of communication In BoS each player has a different pro le on which she would prefer they coordinate ln Pareto Coordination7 one is better for both7 but they still are out of luck if they miscoordinate We will be returning to these games in the days to come An Application of Iterated Dominance and Rationaliz ability to Cournot Duopoly The purpose ofthis section is to present an applications of iterated dominance to supplement those of Chapter 8 of Watson which I recommend you read At the same time we will get our rst good look at a normal form game that is not and indeed cannot be represented with a matrix because the strategy spaces are not nite Many of you may have already studied duopoly in an intermediate microeconomic theory class7 and you likely applied Nash equilibrium to solve various duopoly models We will be getting to Nash equilibrium in the next class For now7 we will return to Cournot duopoly because it will give us a chance to formalize the situation as a normal form game and because in some versions it turns out to be dominance solveable2 For this example we will use a bit of calculus Consider a linear demand function p a 7 ql q and constant marginal cost7 c such that the total cost of producing 11 units is cqi Let us begin by writing this down in the normal form The set of players is 17 27 for Firms 1 and 3 The strategy space for each rm is any non negative quantity7 ie 51 E R and the preference functions are with 1739 1091397 9739 39 1i i 0 0 Qi QjQi 0 We will calculate the best response function BRq for each rm 239 to the quantity qj chosen by the other rm We do this using the rst order conditions7 which we get by differentiating the pro t function and setting it equal to zero Because of symmetry7 we get the same thing for both rms iciqj lt 170 BRQgt 2 2 0 otherwise Now lets begin our rst round of elimination Note that the best response function is de creasing in Firm is belief about the Firm j7s action If Firm j produces enough more than they will drive the price so low that any production leads to negative pro ts It would better to stay out of the market and produce 11 0 ln fact7 the best response of each rm must lie in the interval gtj 07 All other strategies make negative pro ts and therefore are dominated by some strategy inside this interval7 eg ql 07 and can be eliminated In the second stage of elimination7 only the strategies that are best responses to those in the initial interval survive7 ie BRQ7BRq This is the set of all possible best responses to some strategy that is consistent with theother player being rational In the third stage7 the same process allows us to restrict the strategies to BRBRqBRBRQl Though it is 2For a related problem on the Bertrand duopoly model7 see Problem 85 in Watsoni 13 Contract Law and Enforcement in Static Settings This chapter presents the notion of contract Much emphasis is placed on how con tracts help to align beliefs and behavior in static settings It carefully explains how players can use a contract to induce a game whose outcome differs from that of the game given by the technology of the relationship Further the relationship between those things considered veri able and the outcomes that can be implemented is care fully explained The exposition begins with a setting of full veri ability and complete contracting The discussion then shifts to settings of limited liability and default damage remedies Lecture Notes You may nd the following outline useful in planning a lecture De nition of contract Self enforced and externally enforced components Discuss Why players might want to contract and Why society might want laws Explain Why contracts are fundamental to economic relationships Practical discussion of the technology of the relationship implementation and how the court enforces a contract De nition of the induced game Veri ability Note the implications of limited veri ability Complete contracting Default damage rules expectation reliance restitution o Liquidated damage clauses and contracts specifying transfers Ef cient breach 0 Comments on the design of legal institutions Examples and Experiments 1 Contract game A contract game of the type analyzed in this chapter can be played as a classroom experiment Two students can be selected to rst negotiate a contract and then play the underlying game You play the role of the external enforcer It may be useful to do this once with full veri ability and once with limited veri ability This may also be used immediately before presenting the material in Chapter 13 andor as a lead in to Chapter 18 38 Instructors39 Manual for Strategy Copyright 2002 2008 by Joel Watson An Introduction to Game Theory For instructors only do not distribute CONTRACT LAW AND ENFORCEMENT 39 2 Case study Chicago Coliseum Club 1 Dempsey Source 265 Ill App 542 1 32 Ill App This or a di erent case can be used to illustrate the various kinds of default damage remedies and to show how the material of the chapter applies to practical matters2 F irst7 give the background of the case and then present a stylized example that is based on the case Facts of the Case defendant William Harrison D p ey7 known as Jack Dempsey7 to recover damages for breach of a written contract executed March 137 19267 but bearing date of March 6 of that year Plainti was incorporated as an Illinois corporation for the promotion of general pleasure and a etic purposes and to conduct boxing7 sparring and wrestling matches and exhibitions for prizes or purses Dempsey was well known in the pugl stism world and7 at the time of the m 39ng and execution of the contract in question7 held the title of world s Champion Heavy Weight Boxer Dempsey was to engage in a boxing match with Harry Wills7 another wells own boxer At the signing of the contract7 he was paid 10 Dempsey was to be paid 338007000 plus 50 percent of the net pro ts over and above the sum of 33270007000 in the event the gate receipts should exceed that amoun Further7 he was to receive 50 percent of the net revenue derived from moving picture concessions or royalties received by the plainti Dempsey was not to engage 39n any boxing match after the date of the agreement and before the date of the contest He was also to have his life and health insured in favor of the plainti 39 anner and at a place to be designated by the plainti The Chicago Coliseum Club was to promote the event The contract between the Chicago Coliseum Club and Wills was entered into on March 67 1926 It stated that Wills was to be payed 50000 However7 he was never paid Chicago Coliseum Club7 a corporation7 as plaintile brought its action against em s The Chicago Coliseum Club hired a promoter When it contacted Dempsey con cerning the life insurance7 Dempsey repudiated the contract with the following telegram message BM Colorado Springs Colo July 10th 1926 B E Clements President Chicago Coliseum Club Chgo Entirely too busy training for my coming Tunney match to waste time on insurance representatives stop as you have no contract suggest you stop kidding yourself and me also Jack Dempsey 2For e more detailed discussion of this case see Barnett R Omtmntsx Cases and Deming 2d Ed Aspen 1999 p125 lns Lructurs Manual fur Strategy Cupyright 2mm 2qu by duei Watsun An lntruductiun in Game Theury Fur insimciurs uniy du nut dismbuie CONTRACT LAW AND ENFORCEMENT 40 The court identi ed the following issues as being relevant in establishing dam ages First Loss of pro ts which would have been derived by the plaintiff in the event of the holding of the contest in question Second Expenses incurred by the plaintiff prior to the signing of the agreement between the plaintiff and Dempsey Third Expenses incurred in attempting to restrain the defendant from engaging in other contests and to force him into a compliance with the terms of his agreement with the plaintiff and Fourth Expenses incurred after the signing of the agreement and before the breach of July 107 1926 The Chicago Coliseum Club claimed that it would have had gross receipts of 3000000 and expenses of 14000007 which would have left a net pro t of 1600000 However7 the court was not convinced of this as there were too many undetermined factors Unless shown otherwise the court will generally assume that the venture would have at least broken even This could be compared to the case where substantial evidence did exist as to the expected pro ts of Chicago Coliseum The expenses incurred before the contract was signed with Dempsey could not be recovered as damages Further7 expenses incurred in relation to 3 above could only be recovered as damages if they occured before the repudiation The expense of 4 above could be recovered Stylized Example The following technology of the relationship shows a possible interpretation when proof of the expected revenues is available D C T O P 1600 800 10 1200 N 100 800 00 This assumes that promotion by Chicago Coliseum Club bene ts Dempsey s reputation and allows him to gain by taking the other boxing match The strate gies for Chicago Coliseum are promote and don t promote The strategies for Dempsey are take this match and take other match This example can be used to illustrate a contract that would induce Dempsey to keep his agree ment with Chicago Coliseum Further7 when it is assumed that the expected pro t is zero7 expectations and reliance damages result in the same transfer Instructors39 Manual for Strategy Copyright 2002 2008 by Joel Watson An Introduction to Game Theory For instructors only do not distribute 14 Details of the Extensive Form This chapter elaborates on Chapter 27s presentation of the extensive form represen tation The chapter de nes some technical terms and states ve rules that must be obeyed when designing game trees The concepts of perfect recall and perfect information are registered Lecture Notes This material can be covered very quickly in class as a transition from normal form analysis to extensive form analysis The key simply is to bring the extensive form back to the front of the students7 minds and in a more technically complete manner than was needed for Part I of the book Here is an outline for a lecture Review of the components of the extensive form nodes branches labels infor mation sets and payoffs initial decision and terminal nodes Terms describing the relation between nodes successor predecessor immediate successor and immediate predecessor 0 Tree rules with examples of violations Perfect versus imperfect recall 0 Perfect versus imperfect information o How to describe an in nite action space Examples and Experiments 1 Abstract examples can be developed on the y to illustrate the terms and con cepts E0 Forgetful driver This one player game demonstrates imperfect recall The player is driving on country roads to a friends house at night The player reaches an intersection where he must turn left or right If he turns right he will nd a police checkpoint where he will be delayed for the entire evening If he turns left he will eventually reach another intersection requiring another rightleft decision At this one a right turn will bring him to his friends house while a left turn will take him to the police checkpoint When he has to make a decision the player does not recall how many intersections he passed through or what decisions he made previously The extensive form representation is pictured on the next page lnstructors39 Manual for Strategy Copyright 2002 2008 by Joel Watson An Introduction to Game Theory For instructors only do not distribute 14 DETAILS OF THE EXTENSIVE FORM 42 0 reaches police checkpoint 1 reaches fiiend s house 0 reaches police checkpoint Instructors39 Manual for Strategy Copyright 2002 2008 by Joel Watson An Introduction to Game Theory For instructors only do not distribute 15 Backward Induction and Subgarne Perfection This chapter begins with an example to show that not all Nash equilibria of a game may be consistent with rationality in real time The notion of sequential rationality is presented followed by backward induction a version of conditional dominance and then a demonstration of backward induction in an example Next comes the result that nite games of perfect information have pure strategy Nash equilibria this result is used in Chapter 17 for the analysis of parlor games The chapter then de nes subgame perfect Nash equilibrium as a concept for applying sequential rationality in general games An algorithm for computing subgame perfect equilibria in nite games is demonstrated with an example Lecture Notes An outline for a lecture follows 0 Example of a game featuring a Nash equilibrium with an incredible threat 0 The de nition of Sequential rationality o Backward induction informal de nition and abstract example Note that the strategy pro le identi ed is a Nash equilibrium 0 Result every nite game with perfect information has a pure strategy Nash equilibrium Note that backward induction is dif cult to extend to games with imperfect information o Subgame de nition and illustrative example Note that the entire game is itself a subgame De nition of proper subgame 0 De nition of subgame perfect Nash equilibrium 0 Example and algorithm for computing subgame perfect equilibria a draw the normal form of the entire game b draw the normal forms of all other proper subgames 0 nd the Nash equilibria of the entire game and the Nash equilibria of the proper subgames and d locate the Nash equilibria of the entire game that specify Nash outcomes in all subgames Examples and Experiments 1 Incredible threats example It might be useful to discuss for example the cred ibility of the Chicago Bulls of the 1990s threatening to re Michael Jordan 43 Instructors39 Manual for Strategy Copyright 2002 2008 by Joel Watson An Introduction to Game Theory For instructors only do not distribute BACKWARD INDUCTION AND SUB GAME PERFECTION 44 2 Grab game This is a good game to run as a classroom experiment immediately after lecturing on the topic of subgame perfection There is a very good chance that the two students who play the game will not behave according to backward induction theory You can discuss why they behave differently In this game two students take turns on the move When on the move a student can either grab all of the money in your hand or pass At the beginning of the game you place one dollar in your hand and offer it to player 1 If player 1 grabs the dollar then the game ends player 1 gets the dollar and player 2 gets nothing If player 1 passes then you add another dollar to your hand and offer the two dollars to player 2 If she grabs the money then the game ends she gets 2 and player 1 gets nothing If player 2 passes then you add another dollar and return to player 1 This process continues until either one of the players grabs the money or player 2 passes when the pot is 21 in which case the game ends with both players obtaining nothing Instructors39 Manual for Strategy Copyright 2002 2008 by Joel Watson An Introduction to Game Theory For instructors only do not distribute
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