Introduction to Economic Analysis Chapter 16 Games and Strategic Behavior

Chapter 16 Games and Strategic Behavior Competitive theory studies consumers and is , people who can individually affect the transaction prices . The assumption that market participants take prices as given is only when there are many competing participants . We have also examined monopoly , precisely because a monopoly , by , doesn have to worry about competitors . Strategic behavior involves the examination of the intermediate case , where there are few enough participants that they take each other into their actions individually that the behavior of any one participant choices of the other participants . That is , participants are strategic in their choices of action , recognizing that their choices will affect choices made by others . The right tool for the job of examining strategic behavior in economic circumstances is game theory , the study of how people play games . Game theory was pioneered by the mathematical genius John Von ( 1957 ) Game theory has also been very in the study of military strategy and , indeed , the strategy of the cold war between the United States and the Soviet Union was guided by analyses . The theory provides a description that common games like poker or the board game Monopoly , but will cover many other situations as well . In any game , there is a list of players . Games generally unfold over time at each moment in time , players have the current state of play and a set of actions they can take . Both information and actions may depend on the history of the game prior to that moment . Finally , players have payoffs and are assumed to play in such a way as to maximize their anticipated payoff , taking into account their expectations for the play of others . When the players , their information and available actions , and payoffs have been , we have a game . URL books 377

Matrix Games LEARNING OBJECTIVES . How are games modeled ?

What is optimal play ?

The simplest game is called a matrix payoff game with two players . In payoff game , all actions are chosen simultaneously . It is conventional to describe a matrix payoff game as played by a row player and a column player . The row player chooses a row in a matrix the column player simultaneously chooses a column . The outcome of the game is a pair of payoffs where the first entry is the payoff of the row player , and the second is the payoff of the column player . Figure The prisoner dilemma provides an example of a matrix payoff most famous game of is known as the prisoners dilemma . In the game , the strategies are to confess or not to confess . Figure ( Column Row URL books 378

In the prisoner dilemma , two criminals named Row and Column have been apprehended by the police and are being questioned separately . They are jointly guilty of the crime . Each player can choose either to confess or not . If Row confesses , we are in the top row of the matrix ( corresponding to the row labeled Confess ) Similarly , if Column confesses , the payoff will be in the relevant column . In this case , if only one player confesses , that player goes free and the other serves 20 years in jail . The entries correspond to the number of years lost to prison . The first entry is always Row payoff the second entry is Column payoff . Thus , for example , if Column confesses and Row does not , the relevant payoff is the column and the second row . Figure Solving Column Row If Column confesses and Row does not , Row loses 20 years , and Column loses no years that is , it goes free . This is the payoff ( in reverse color in Figure Solving the prisoner dilemma . If both confess , they are both convicted and neither goes free , but they only serve 10 years each . Finally , if neither confesses , there is a 10 chance that they are convicted anyway ( using evidence other than the confession ) in which case they each average a year lost . The prisoner dilemma is famous partly because it is readily solvable . First , Row has a strict advantage to confessing , no matter what Column is going to URL books 379

do . If Column confesses , Row gets for confessing , for not confessing , and thus is better off confessing . Similarly , if Column doesn confess , Row gets for confessing ( namely , goes free ) for not confessing , and is better off confessing . Either way , no matter what Column does , Row should choose to confess . This is called a dominant strategy , a strategy that is optimal no matter what the other players do . The logic is exactly similar for Column No matter what Row does , Column should choose to confess . That is , Column also has a dominant strategy to confess . To establish this , first consider what Column best action is , when Column thinks Row will confess . Then consider Column best action when Column thinks Row won confess . Either way , Column gets a higher payoff ( lower number of years lost to prison ) by confessing . The presence of a dominant strategy makes the prisoner dilemma particularly easy to solve . Both players should confess . Note that this gets them 10 years each in prison , and thus isn a very good outcome from their perspective but there is nothing they can do about it in the context of the game , because for each the alternative to serving 10 years is to serve 20 years . This outcome is referred to as ( Confess , Confess ) where the entry is the row player choice , and the second entry is the column player choice . Figure An entry gauze URL books 380

Consider an entry game played by ( the row player ) and ( the column player ) a small company . Both and are thinking about entering a new market for an online service . Figure An entry game provides the payoff structure . In this case , if both companies enter , ultimately wins the market , earning and loses . If either firm has the market to itself , it gets and the other firm gets zero . If neither enters , they both get zero . has a dominant strategy to enter It gets when enters , when doesn , and in both cases it does better than when it doesn enter . In contrast , does not have a dominant strategy wants to enter when doesn , and vice versa . That is , optimal strategy depends upon action or , more accurately , optimal strategy depends upon what believes will do . can understand dominant strategy if it knows the payoffs of . Thus , can conclude that is going to enter , and this means that should not enter . Thus , the equilibrium of the game is for to enter and not to enter . This equilibrium is arrived at by elimination of dominated strategies , eliminating strategies by sequentially removing strategies that are dominated for a player . First , we eliminated dominated strategy in favor of its dominant strategy . had a dominant strategy to enter , which means that the strategy of not entering was dominated by the strategy of entering , so we eliminated the dominated strategy . That leaves a game in which enters , as shown in Figure Eliminating a dominated strategy . Figure ( URL books 381

In this simplified game , after the elimination of dominated strategy , also has a dominant strategy not to enter . Thus , we iterate and eliminate dominated strategies time eliminating dominated wind up with a single outcome enters , and doesn . The elimination of dominated strategies solves the game . Figure A game shows another game , with three strategies for each player . Figure A game Column Row URL books 382

The process of elimination of dominated strategies is illustrated in Figure Eliminating a dominated strategy by actually eliminating the rows and columns , as follows . A reverse color ( white text on black background ) indicates a dominated strategy . Middle dominates Bottom for Row , yielding Figure a dominated strategy Column Row With Bottom eliminated , Left is now dominated for Column by either Center or Right , which eliminates the Left Column . This is shown in Figure Eliminating another dominated strategy . Figure another dominated strategy URL books ) a 383

Column ( With Left and Bottom eliminated , Top now dominates Middle for Row , as shown Eliminating a third dominated strategy . Figure Eliminating a third dominated strategy Column Row Finally , as shown in Figure Game solved , Column chooses Right over Center , yielding a unique outcome after the elimination of dominated strategies , which is ( Top , Right ) Figure Game solved URL books 384

Column Row The elimination of dominated strategies is a useful concept , and when it applies , the predicted outcome is usually quite reasonable . Certainly it has the property that no player has an incentive to change his or her behavior given the behavior of others . However , there are games where it doesn apply , and these games require the machinery of a Nash equilibrium , named for Nobel laureate John Nash ( KEY TAKEAWAYS Strategic behavior arises where there are few enough market participants that their actions individually matter , and where the behavior of any one participant influences choices of the other participants . Game theory is the study of how people play games . A game consists of the players , their information and available actions , and payoffs . In a matrix payoff game , all actions are chosen simultaneously . The row player chooses a row in a matrix the column player simultaneously chooses a column . The outcome of the game is a pair of payoffs where the first entry is the payoff of the row player , and the second is the payoff of the column player . In the prisoner dilemma , two criminals named Row and Column have been apprehended by the police and are being questioned separately . URL books ) a 385

They are jointly guilty of the crime . Each player can choose either to confess or not . Each player individually benefits from confessing , but together they are harmed . A dominant strategy is a strategy that is best for a player no matter what others choose . elimination of dominated strategies first removes strategies dominated by others , then checks if any new strategies are dominated and removes them , and so on . In many cases , elimination of dominated strategies solves a game . Nash Equilibrium LEARNING OBJECTIVE . What is an equilibrium to a game ?

In a Nash equilibrium , each player chooses the strategy that maximizes his or her expected payoff , given the strategies employed by others . For matrix payoff games with two players , a Nash equilibrium requires that the row chosen maximize the row player payoff ( given the column chosen by the column player ) and the column , in turn , maximize the column player payoff ( given the row selected by the row player ) Let us consider first the prisoner dilemma , which we have already seen . Here it is illustrated once again in Figure Prisoner dilemma again . Figure PI again URL books 386

Column Row Given that the row player has chosen to confess , the column player also chooses to confess because is better than . Similarly , given that the column player chooses confession , the row player chooses confession because is better than . Thus , for both players to confess is a Nash equilibrium . Now let us consider whether any other outcome is a Nash equilibrium . In any other outcome , at least one player is not confessing . But that player could get a higher payoff by confessing , so no other outcome could be a Nash equilibrium . The logic of dominated strategies extends to Nash equilibrium , except possibly for ties . That is , if a strategy is strictly dominated , it can be part of a Nash equilibrium . On the other hand , if it involves a tied value , a strategy may be dominated but still be part of a Nash equilibrium . The Nash equilibrium is as a solution concept for games as follows . First , if the players are playing a Nash equilibrium , no one has an incentive to change his or her play or to rethink his or her strategy . Thus , the Nash equilibrium has a steady state in that no one wants to change his or her own strategy given the play of others . Second , other potential outcomes don have that property If an outcome is not a Nash equilibrium , then at least one player has an incentive to change what he or she is doing . Outcomes that aren Nash equilibria involve mistakes for at least one player . Thus , sophisticated , intelligent players may be able to deduce each other play , and play a Nash equilibrium . URL books ( 999 387

Do people actually play Nash equilibria ?

This is a controversial topic and mostly beyond the scope of this book , but we consider two games ( see , for example , and chess . toe is a relatively simple game , and the equilibrium is a tie . This equilibrium arises because each player has a strategy that prevents the other player from winning , so the outcome is a tie . Young children play and eventually learn how to play equilibrium strategies , at which point the game ceases to be very interesting since it just repeats the same outcome . In contrast , it is known that chess has an equilibrium , but no one knows what it is . Thus , at this point , we don know if the first mover ( white ) always wins , or if the second mover ( black ) always wins , or if the outcome is a draw ( neither is able to win ) Chess is complicated because a strategy must specify what actions to take , given the history of actions , and there are a very large number of potential histories of the game 30 or 40 moves after the start . So we can be quite confident that people are not ( yet ) playing Nash equilibria to the game of chess . The second most famous game in game theory is a coordination game battle of the sexes . The battle of the sexes involves a married couple who are going to meet each other after work but haven decided where they are meeting . Their options are a Baseball game or the Ballet . Both prefer to be with each other , but the Man prefers the Baseball game and the Woman prefers the Ballet . This gives payoffs as shown in Figure The battle of the sexes . Figure The battle , es URL books 388

Woman Baseball ( Man The Man would prefer that they both go to the Baseball game , and the Woman prefers that both go to the Ballet . They each get payoff points for being with each other , and an additional point for being at their preferred entertainment . In this game , elimination of dominated strategies eliminates nothing . One can readily verify that there are two Nash equilibria one in which they both go to the Baseball game and one in which they both go to the Ballet . The logic is this If the Man is going to the Baseball game , the Woman prefers the points she gets at the Baseball game to the single point she would get at the Ballet . Similarly , if the Woman is going to the Baseball game , the Man gets three points going there versus zero at the Ballet . Hence , going to the Baseball game is one Nash equilibrium . It is straightforward to show that for both to go to the Ballet is also a Nash equilibrium and , finally , that neither of the other two possibilities in which they go to separate places is an equilibrium . Now consider the game of matching pennies , a childs game in which the sum of the payoffs is zero . In this game , both the row player and the column player choose heads or tails , and if they match , the row player gets the coins , while if they don match , the column player gets the coins . The payoffs are provided in Figure Matching pennies . Figure Matching pennies URL books 389

Column Row You can readily verify that none of the four possibilities represents a Nash equilibrium . Any of the four involves one player getting that player can convert to by changing his or her strategy . Thus , whatever the hypothesized equilibrium , one player can do strictly better , contradicting the hypothesis of a Nash equilibrium . In this game , as every child who plays it knows , it pays to be unpredictable , and consequently players need to randomize . Random strategies are known as mixed strategies because the players mix across various actions . KEY TAKEAWAYS In a Nash equilibrium , each player chooses the strategy that maximizes his or her expected payoff , given the strategies employed by others . Outcomes that are Nash equilibria involve mistakes for at least one player . The game called the battle of the sexes has two Nash equilibria . In the game of matching pennies , none of the four possibilities represents a Nash equilibrium . Consequently , players need to randomize . Random strategies are known as mixed strategies because the players mix across various actions . Mixed Strategies URL books 390

LEARNING OBJECTIVE . What games require or admit randomization as part of their solution ?

Let us consider the matching pennies game again , as illustrated in Figure Matching pennies again . Figure pennies again Column Row Suppose that Row believes Column plays Heads with probability Then if Row plays Heads , Row gets with probability and with probability ( for an expected value of . Similarly , if Row plays Tails , Row gets with probability ( when Column plays Heads ) and with probability ( for an expected value of . This is summarized in Figure Mixed strategy in matching pennies . If , then Row is better off , on average , playing Heads than Tails . Similarly , if , then Row is better off playing Tails than Heads . If , on the other hand , then Row gets the same payoff no matter what Row does . In this case , Row could play Heads , could play Tails , or could a coin and randomize Row play . A mixed strategy Nash equilibrium involves at least one player playing a randomized strategy and no player being able to increase his or her expected URL books 391

payoff by playing an alternate strategy . A Nash equilibrium in which no player is called a pure strategy Nash equilibrium . Figure ed in pennies Column Heads ( Row Tails ( Note that randomization requires equality of expected payoffs . If a player is supposed to randomize over or strategy , then both of these strategies must produce the same expected payoff . Otherwise , the player would prefer one of them and wouldn play the other . Computing a mixed strategy has one element that often appears confusing . Suppose that Row is going to randomize . Then Row payoffs must be equal for all strategies that Row plays with positive probability . But that equality in Row payoffs doesn determine the probabilities with which Row plays the Various rows . Instead , that equality in Row payoffs will determine the probabilities with which Column plays the various columns . The reason is that it is Column probabilities that determine the expected payoffs for Row if Row is going to randomize , then Column probabilities must be such that Row is willing to randomize . Thus , for example , we computed the payoff to Row of playing Heads , which was , where was the probability that Column played Heads . Similarly , the payoff to Row of playing Tails was . Row is willing to randomize if these are equal , which solves for . URL books 392

Now let try a somewhat more challenging example and revisit the battle of the sexes . Figure Mixed strategy in battle of the sexes illustrates the payoffs once again . Figure Mixed stI in battle of the sexes Woman Baseball ( Man This game has two pure strategy Nash equilibria ( Baseball , Baseball ) and ( Ballet , Ballet ) Is there a mixed strategy ?

To compute a mixed strategy , let the Woman go to the Baseball game with probability , and the Man go to the Baseball game with probability . Figure Full computation of the mixed strategy contains the computation of the mixed strategy payoffs for each player . Figure Full computation strategy Woman Baseball ( Ballet ( Man Payoff ) Woman ( Man URL books 393

For example , if the Man ( row player ) goes to the Baseball game , he gets when the Woman goes to the Baseball game ( probability ) and otherwise gets , for an expected payoff of ( The other calculations are similar , but you should definitely run through the logic and verify each calculation . A mixed strategy in the battle of the sexes game requires both parties to randomize ( since a pure strategy by either party prevents randomization by the other ) The Man indifference between going to the Baseball game and to the Ballet requires , which yields . That is , the Man will be willing to randomize which event he attends if the Woman is going to the Ballet of the time , and otherwise to the Baseball game . This makes the Man indifferent between the two events because he prefers to be with the Woman , but he also likes to be at the Baseball game . To make up for the advantage that the game holds for him , the Woman has to be at the Ballet more often . Similarly , in order for the Woman to randomize , the Woman must get equal payoffs from going to the Baseball game and going to the Ballet , which requires , Thus , the probability that the Man goes to the Baseball game is , and he goes to the Ballet of the time . These are independent probabilities , so to get the probability that both go to the Baseball game , we multiply the probabilities , which yields . Figure Mixed strategy probabilities fills in the probabilities for all four possible outcomes . Figure ' stra tag URL books 394

Woman Baseball Man Note that more than half of the time ( Baseball , Ballet ) is the outcome of the mixed strategy and the two people are not together . This lack of coordination is generally a feature of mixed strategy equilibria . The expected payoffs for both players are readily computed as well . The Man payoff is , and since , the Man obtains . A similar calculation shows that the Woman payoff is the same . Thus , both do worse than coordinating on their less preferred outcome . But this mixed strategy Nash equilibrium , undesirable as it may seem , is a Nash equilibrium in the sense that neither party can improve his or her own payoff , given the behavior of the other party . In the battle of the sexes , the mixed strategy Nash equilibrium may seem unlikely and we might expect the couple to coordinate more effectively . Indeed , a simple call on the telephone should rule out the mixed strategy . So let consider another game related to the battle of the sexes , where a failure of coordination makes more sense . This is the game of In this game , two players drive toward one another , trying to convince the other to yield and ultimately swerve into a ditch . If both swerve into the ditch , we call the outcome a draw and both get zero . If one swerves and the other doesn , the driver who swerves loses and the other driver wins , and we give the winner one point . The only remaining question is what happens when neither yield , in which case a crash results . In this version , the payoff has been set at four URL books 395

times the loss of swerving , as shown in Figure Chicken , but you can change the game and see what happens . Figure Column Row This game has two pure strategy equilibria ( Swerve , Don ) and ( Don , Swerve ) In addition , it has a mixed strategy . Suppose that Column swerves with probability Then Row gets ( from swerving , from not swerving , and Row will randomize if these are equal , which requires . That is , the probability that Column swerves in a mixed strategy equilibrium is . You can verify that the row player has the same probability by setting the probability that Row swerves equal to and computing Column expected payoffs . Thus , the probability of a collision is in the mixed strategy equilibrium . The mixed strategy equilibrium is more likely , in some sense , in this game If the players already knew who was going to yield , they wouldn actually need to play the game . The whole point of the game is to out who will yield , which means that it isn known in advance . This means that the mixed strategy equilibrium is , in some sense , the more reasonable equilibrium . Figure Rock , paper , URL books ( 999 396

Column Rock Paper ( Row Rock , paper , scissors is a child game in which two children use their hands to simultaneously choose paper ( hand held ) scissors ( hand with two protruding to look like scissors ) or rock ( hand in a ) The nature of the payoffs is that paper beats rock , rock beats scissors , and scissors beats paper . This game has the structure that is illustrated in Figure Rock , paper , scissors . KEY TAKEAWAYS A mixed strategy Nash equilibrium involves at least one player playing a randomized strategy and no player being able to increase his or her expected payoff by playing an alternate strategy . A Nash equilibrium without randomization is called a pure strategy Nash equilibrium . If a player is supposed to randomize over two strategies , then both must produce the same expected payoff . The matching pennies game has a mixed strategy and no pure strategy . The battle of the sexes game has a mixed strategy and two pure strategies . The game of chicken is similar to the battle of the sexes and , like it , has two pure strategies and one mixed strategy . URL books 397

EXERCISES . Let be the probability that Row plays Heads . Show that Column is willing to randomize , if and only if . Hint First compute expected payoff when Column plays Heads , and then compute expected payoff when Column plays Tails . These must be equal for Column to randomize . Show that in the rock , paper , scissors game there are no pure strategy equilibria . Show that playing all three actions with equal likelihood is a mixed strategy equilibrium . all equilibria of the following games Figure Column Column Left Right Left Right Up ( 11 , Up ( Down ( Down ( Column ' Column Left Right Left Right UP ( UP ( Down ( Down ( Column Column Left Right Left Right UP ( UP ( Down ( Down ( If you multiply a player payoff by a positive constant , the equilibria of the game do not change . Is this true or false , and why ?

URL books 398 Examples LEARNING OBJECTIVE . How can game theory be applied to the economic settings ?

Our example concerns public goods . In this game , each player can either contribute or not . For example , two roommates can either clean their apartment or not . If they both clean , the apartment is nice . If one cleans , then that roommate does all of the work and the other gets half of the . Finally , if neither cleans , neither is very happy . This suggests the following payoffs as shown in Figure Cleaning the apartment . Figure Cleaning the Column Row Do ( 15 , You can verify that this game is similar to the prisoner dilemma in that the only Nash equilibrium is the pure strategy in which neither player cleans . This is a Version of the tragedy of the though both roommates would be better off if both cleaned , neither do . As a practical matter , roommates do solve this problem , using strategies that we will investigate when we consider dynamic games . URL books 399

Figure Driving on the right Column Row As illustrated in Figure Driving on the right , in the driving on the right game , the important consideration about which side of the road that cars drive on is not necessarily the right side but the same side . If both players drive on the same side , then they each get one point otherwise , they get zero . You can readily verify that there are two pure strategy equilibria , Left , Left ) and ( Right , Right ) and a mixed strategy equilibrium with equal probabilities . Is the mixed strategy reasonable ?

With automobiles , there is little randomization . On the other hand , people walking down hallways often seem to randomize , whether they pass on the left or the right , and sometimes do that little dance where they try to get past each going left and the other going right , then both simultaneously reversing , unable to get out of each other way . That dance suggests that the mixed strategy equilibrium is not as unreasonable as it seems in the automobile application . Figure Bank location URL books 400

LA Consider a foreign bank that is looking to open a main and a smaller in the United States . As shown in Figure Bank location game , the bank narrows its choice for main to either New York City ( or Los Angeles ( LA ) and is leaning toward Los Angeles . If neither city does anything , Los Angeles will get 30 million in tax revenue and New York will get 10 million . New York , however , could offer a 10 million rebate , which would swing the main to New York but then New York would only get a net of 20 million . The discussions are carried on privately with the bank . Los Angeles could also offer the concession , which would bring the bank back to Los Angeles . Figure Political Republican Democrat URL books 401

On the night before an election , a Democrat is leading the Wisconsin senatorial race . Absent any new developments , the Democrat will win and the Republican will lose . This is worth to the Democrat and the Republican , who loses honorably , values this outcome at . The Republican could decide to run a series of negative advertisements ( throwing mud ) against the Democrat and , if so , the Republican loses his honor , which he values at , and so only gets . If the Democrat runs negative ads , again the Democrat wins but loses his honor , so he only gets . These outcomes are represented in the mudslinging game shown in Figure Political mudslinging . You have probably had the experience of trying to avoid encountering someone , whom we will call Rocky . In this instance , Rocky is actually trying to you . Here it is Saturday night and you are choosing which party , of two possible parties , to attend . You like Party better and , if Rocky goes to the other party , you get 20 . If Rocky attends Party , you are going to be uncomfortable and get . Similarly , Party is worth 15 , unless Rocky attends , in which case it is worth . Rocky likes Party better ( these different preferences may be part of the reason why you are avoiding him ) but he is trying to see you . So he values Party at 10 , Party at , and your presence at the party he attends is worth 10 . These values are in Figure Avoiding Rocky . Figure Rocky URL books 402

You Our example involves two competing for customers . These films can either price High or Low . The most money is made if they both price High but if one prices Low , it can take most of the business away from the rival . If they both price Low , they make modest . This description is in Figure Price cutting game . Figure Price cutting game Firm KEY TAKEAWAYS The problem of public goods with two players can be Firm formulated as a game . Whether to drive on the right or the left is a game similar to battle of the sexes . Many everyday situations are reasonably formulated as games . URL books 403

EXERCISES . Verify that the bank location game has no pure strategy equilibria and that there is a mixed strategy equilibrium where each city offers a rebate with probability . Show that the only Nash equilibrium of the political mudslinging game is a mixed strategy with equal probabilities of throwing mud and not throwing mud . Suppose that voters partially forgive a candidate for throwing mud in the political mudslinging game when the rival throws mud , so that the ( Mud , Mud ) outcome has payoff ( How does the equilibrium change ?

Show that there are no pure strategy Nash equilibria in the avoiding Rocky game . Find the mixed strategy Nash equilibria . Show that the probability that you encounter Rocky is 712 . Show that the firms in the game have a dominant strategy to price low , so that the only Nash equilibrium is ( Low , Low ) Perfection LEARNING OBJECTIVE . How do dynamic games play out ?

So far , we have considered only games that are played simultaneously . Several of these the price cutting and apartment cleaning actually played over and over again . Other games , like the bank location game , may only be played once , but nevertheless are played over time . Recall the bank location game , as illustrated once again in Figure Bank location game revisited . URL books 404

Figure Bank location game I If neither city offered a Rebate , then Los Angeles won the bidding . So suppose that , instead of the simultaneous move game , first New York City decided whether to offer a Rebate , and then Los Angeles could decide to offer a Rebate . This sequential structure leads to a game that looks like Figure Sequential bank location ( payoff listed first ) In this game , makes the move and chooses Rebate ( to the left ) or No Rebate ( to the right ) If chooses Rebate , LA can then choose Rebate or None . Similarly , if chooses No Rebate , LA can choose Rebate or None . The payoffs using the standard of ( LA , ordering are written below the choices . Figure bank location ( URL books 405

No Rebate Rebate None Rebate None ( What would like to do depends upon what believes LA will do . What should believe about LA ?

Boy , does that rhetorical question suggest a lot of facetious answers . The natural belief is that LA will do what is in LA best interest . This each stage of a dynamic game is played in an optimal called perfection . perfection requires each player to act in its own best interest , independent of the history of the game . This seems very sensible and , in most , it is sensible . In some settings , it may be implausible . Even ifI see a player make a particular mistake three times in a row , perfection requires that I must continue to believe that that player will not make the mistake again . perfection may be implausible in some circumstances , especially when it pays to be considered somewhat crazy . URL books 405

In the example , perfection requires LA to offer a Rebate when does ( since LA gets 20 by 10 ) and to not offer a Rebate when doesn . This is illustrated in the game , as shown in Figure perfection , using arrows to indicate LA choices . In addition , the actions that LA won choose have been in a light gray . Once LA perfection choices are taken into account , is presented with the choice of offering a Rebate , in which case it gets , or not offering a Rebate , in which case it gets 10 . Clearly the optimal choice for is to offer No Rebate , in which case LA doesn either and the result is 30 for LA , and 10 for . Dynamic games are generally solved backward in this way . That is , establish what the last player does , then figure upon the last player expected the penultimate player does , and so on . Figure ) Rebate No Rebate Rebate None ( URL , org books 407

We consider one more application of perfection . Suppose , in the game avoiding Rocky , that Rocky is actually stalking you and can condition his choice on your choice . Then you might as well go to the party you like best , because Rocky is going to follow you wherever you go . This is represented in Figure Can avoid Rocky . Figure Can it avoid Rocky ( 15 ) 20 ) Since Rocky optimal choice eliminates your best outcomes , you make the best of a bad situation by choosing Party . Here , Rocky has a second mover advantage Rocky ability to condition on your choice means that by choosing second he does better than he would do in a simultaneous game . In contrast , a mover advantage is a situation where choosing first is better than choosing simultaneously . First mover advantages arise when going the second mover advantageously . URL books 408

KEY TAKEAWAYS To decide what one should do in a sequential game , one figures out what will happen in the future , and then works backward to decide what to do in the present . perfection requires each player to act in his or her own best interest , independent of the history of the game . A first mover advantage is a situation where choosing first is better than choosing simultaneously . First mover advantages arise when going first influences the second mover advantageously . A second mover advantage is a situation where choosing second is better than choosing simultaneously . Second mover advantages arise when going second permits exploiting choices made by others . EXERCISES . Formulate the battle of the sexes as a sequential game , letting the URL books woman choose first . This situation could arise if the woman were able to leave a message for the man about where she has gone . Show that there is only one perfect equilibrium , that the woman enjoys a first mover advantage over the man , and that she gets her most preferred outcome . payoffs would players receive if they played this sequential game below ?

Payoffs are listed in parentheses , with Player payoffs always listed first . Note that choosing in allows the other player to make a decision , while choosing out ends the game . Figure 409 in in in in ( out out out out ( 20 ) the following game Figure a . Find all equilibria of the above game . What is the perfect equilibrium if you turn this into a sequential game , with Column going first ?

With Row going first ?

In which game does Column get the highest simultaneous game , the sequential game when Column goes first , or the sequential game when Column goes second ?

LEARNING OBJECTIVES . What can happen in games that are repeated over and over ?

What role does the threat of retaliation play ?

URL org books 410 Some situations , like the game or the apartment cleaning game , are played over and over . Such situations are best modeled as a . A is a game that is played an number of times , where the players discount the future . The game played each time is known as a stage game . Generally are played in times , Cooperation may be possible in , if the future is important enough . Consider the game introduced previously and illustrated again in Figure Price cutting game revisited . Figure Price cutting game Firm High Low ( 15 , 15 ) 25 ) Firm ( 25 , The dominant strategy equilibrium to this game is ( Low , Low ) It is clearly a perfect equilibrium for the players to just play ( Low , Low ) over and over again because , if that is what Firm thinks that Firm is doing , Firm does best by pricing Low , and vice versa . But that is not the only equilibrium to the . Consider the following strategy , called a grim trigger strategy , which involves being nice initially but not nice forever when someone else isn cooperative . Price High , until you see your rival price Low . After your rival has priced Low , price Low forever . This is called a trigger strategy because an action of the other player ( pricing Low ) triggers a change in behavior . It is a grim strategy because it punishes forever . URL books 411

If your rival uses a grim trigger strategy , what should you do ?

Basically , your only choice is when to price Low because , once you price Low , your rival will price Low , and then your best choice is also to price Low from then on . Thus , your strategy is to price High up until some point , and then price Low from time on . Your rival will price High through , and price Low from on . This gives a payoff to you of 15 from period through , 25 in period , and then in period on . We can compute the payoff for a discount factor ( If 206 , it pays to price Low immediately , at , because it pays to price Low and the earlier that one prices Low , the higher the present value . If 206 , it pays to wait forever to price Low that is , oo . Thus , in particular , the grim trigger strategy is an optimal strategy for a player when the rival is playing the grim trigger strategy if . In other words , cooperation in pricing is a perfect equilibrium if the future is important enough that is , the discount factor is high enough . The logic of this example is that the promise of future cooperation is valuable when the future itself is valuable , and that promise of future cooperation can be used to induce cooperation today . Thus , Firm doesn want to cut price today because that would lead Firm to cut price for the indefinite future . The grim trigger strategy punishes price cutting today with future Low . offer more scope for cooperation than is illustrated in the cutting game . First , more complex behavior is possible . For example , consider the game shown in Figure A variation of the game Here , again , the unique equilibrium in the stage game is ( Low , Low ) But the difference between this game and the previous game is that the total profits of URL books 412

Firms and are higher in either ( High , Low ) or ( Low , High ) than in ( High , High ) One solution is to alternate between ( High , Low ) and ( Low , High ) Such alternation can also be supported as an equilibrium , using the grim trigger is , if a firm does anything other than what it is supposed to do in the alternating solution , the instead play ( Low , Low ) forever . The folk theorem says that if the value of the future is high enough , any outcome that is individually rational can be supported as an equilibrium to the . Individual rationality for a player in this context means that the outcome offers a present value of profits at least as high as that offered in the worst equilibrium in the stage game from that player perspective . Thus , in the game , the worst equilibrium of the stage game offered each player , so an outcome can be supported if it offers each player at least a running average of . The simple logic of the folk theorem is this . First , any infinite repetition of an equilibrium of the stage game is itself a perfect equilibrium . If everyone expects this repetition of the stage game equilibrium , no one can do better than to play his or her role in the stage game equilibrium every period . Second , any other plan of action can be turned into a perfect equilibrium merely by threatening any agent who deviates from that plan with an infinite repetition of the worst stage game equilibrium from that agent perspective . That threat is credible because the repetition of the stage game equilibrium is itself a perfect equilibrium . Given such a grim type threat , no one wants to deviate from the intended plan . The folk theorem is a powerful result and shows that there are equilibria to that achieve very good outcomes . The kinds of coordination failures that we saw in the battle of the sexes , and the failure to cooperate in the prisoner dilemma , need not arise and cooperative solutions are possible if the future is sufficiently valuable . URL books 413

However , it is worth noting some assumptions that have been made in our descriptions of these that matter but are unlikely to be true in practice . First , the players know their own payoffs . Second , they know their payoffs . They possess a complete description of the available strategies and can calculate the consequences of these just for themselves but also for their rivals . Third , each player maximizes his or her expected payoff they know that their rivals do the same they know that their rivals know that everyone maximizes and so on . The economic language for this is the structure of the game , and the players preferences are common knowledge . Few games will satisfy these assumptions exactly . Since the success of the grim trigger strategy ( and other strategies we haven discussed ) generally depends upon such knowledge , informational considerations may cause cooperation to break down . Finally , the folk theorem shows us that there are lots of equilibria to but provides no guidance on which ones will be played . These assumptions can be relaxed , although they may lead to wars on the equilibrium path by accident a need to recover from such that the grim trigger strategy becomes suboptimal . KEY TAKEAWAYS A is a game that is played over and over again without end , where the players discount the future . The game played each time is known as a stage game . Playing a Nash equilibrium to the stage game forever is a perfect equilibrium to the . A grim trigger strategy involves starting play by using one behavior and , if another player ever does something else , switching to Nash behavior forever . URL books 414

The folk theorem says that if the value of the future is high enough , any outcome that is individually rational can be supported as an equilibrium to the . Individual rationality for a player means that the outcome offers a present value of profits at least as high as that offered in the worst equilibrium in the stage game from that player perspective . If players are patient , full cooperation is obtainable as one of many perfect equilibria to . Consider the game in Figure A variation of the game , and consider a strategy in which Firm prices High in numbered periods and Low in periods , while Firm prices High in periods and Low in periods . If either deviates from these strategies , both firms price Low from then on . Let be the discount factor . Show that these firms have a payoff of or , depending upon which period it is . Then show that the alternating strategy is sustainable if . This , in turn , is equivalent to . URL books 415