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Unformatted text preview: More on Sequential and Simultaneous Move Games • So far we have studied two types of games: 1) sequential move (extensive form) games where sequential move (extensive form) games where players take turns choosing actions and 2) strategic form (normal form) games where players simultaneously choose their actions. • Of course it is possible to combine both game forms as for example happens in the game of forms as, for example, happens in the game of football. • The transformation between game forms may change the set of equilibria, as we shall see. • We will also learn the concept of subgame perfection Combining Sequential and Simultaneous Combining Sequential and Simultaneous Moves. • Consider the following 2 player game, where Player 1 the following player game, where Player moves first: Player 1 Stay Out Player 2 Enter A 3,3 Player 1 A B B 2,2 3,0 0,3 4,4 • If player 1 chooses to stay out, both he and player 2 earn a payoff of 3 each, but if player 1 chooses to enter, he plays a simultaneous move game with player 2. Forward Induction • The simultaneous move game has 3 equilibria: (A,A), (B,B) and a mixed strategy equilibrium where both players play A with probability 1/3 and earn expected payoff 8/3. Player 1 Stay Out Player 2 Enter A 3,3 • • Player 1 A B B 2,2 3,0 0,3 4,4 If player 2 sees that player 1 has chosen to Enter, player 2 can use forward induction reasoning: since player 1 chose to forego a payoff of 3, it is likely that he will choose B, so I should also choose B. lik The likely equilibrium of the game is therefore: Enter, (B,B). The Incumbent The Incumbent-Rival Game in Extensive and Game in Extensive and Strategic Form How many equilibria are there in the extensive form of this game? How many equilibria are there in the strategic form of this game? The Number of Equilibria Appears to be Different! There appears to be just 1 pp equilibrium using rollback on the extensive form game. There appears to be 2 equilibria using cell-by-cell inspection of the strategic form game Subgame Perfection Subgame Perfection • In the strategic form game, there is the additional equilibrium Stay Out Fight that is not an equilibrium equilibrium, Stay Out, Fight that is not an equilibrium using rollback in the extensive form game. • Equilibria found by applying rollback to the extensive form game are referred to as subgame perfect equilibria: every player makes a perfect best response at every subgame of the tree. – Enter, Accommodate is a subgame perfect equilibrium. – Stay Out, Fight is not a subgame perfect equilibrium. • A subgame is the game that begins at any node of the th th th decision tree. 3 subgames (circled) are all the games beginning at all tree nodes including the root node (game itself) Imperfect Strategies are Incredible • Strategies and equilibria that fail the test of subgame perfection are called imperfect. • The imperfection of a strategy that is part of an imperfect Th th equilibrium is that at some point in the game it has an unavoidable credibility problem. • Consider for example, the equilibrium where the incumbent promises to fight, so the rival chooses stay out. • The incumbent’s promise is incredible; the rival knows that if he enters, the incumbent is sure to accommodate, since if the incumbent adheres to his promise to fight, both earn zero, while if the incumbent accommodates, both earn a payoff of 2. • Thus, Stay Out, Fight is a Nash equilibrium, but it is not a subgame perfect Nash equilibrium. • Lesson: Every subgame perfect equilibrium is a Nash equlibrium eq is Nash eq but not every Nash equlibrium is a subgame perfect equilibrium! Another Example: Mutually Assured Destruction (MAD) What is the rollback (subgame perfect) equilibrium to this game? A subgame Subgames must contain all nodes in an information set set Another subgame The Strategic Form Version of the Game Admits 3 Nash Equilibria • Which Equilibrium is Subgame Perfect? • Only the equilibrium where the strategies Escalate, Back Down are played by both the U.S. and Russia is subgame perfect. – Why? From Simultaneous to Sequential Moves Si • Conversion from simultaneous to sequential moves from simultaneous to sequential moves involves determining who moves first, which is not an issue in the simultaneous move game. • In some games, where both players have dominant strategies, it does not matter who moves first. – For example the prisoner’s dilemma game. th dil • When neither player has a dominant strategy, the subgame subgame perfect equilibrium will depend on the equilibrium will depend on the order in which players move. – For example, the Senate Race Game, the Pittsburgh LeftTurn Game. The Equilibrium in Prisoner’s Dilemma is the Same Regardless of Who Moves First Wh Fi This simultaneous move game is equivalent to either of the either of the 2 sequential move games below it. The Senate Race Game has a Different Subgame Perfect Equilibrium Depending on Who moves first. Wh fi In the simultaneous move game, there is only one Nash equilibrium. subgame perfect eq, subgame perfect eq, Similarly, in the Pittsburgh Left-Turn Game Driver 1 Driver 2 Driver 2 Driver 1 Proceed Driver 2 Proceed -1490, -1490 Driver 1 Driver 2 Yield Proceed 5, -5 Proceed Yield -5, 5 Yield -10, -10 Proceed -1490, -1490 1490 Yield Driver 1 Yield Proceed 5, -5 -5, 5 Yield -10, -10 10 These subgame perfect equilibria look the same, but if Driver 1 moves first he gets a payoff of 5, while if Driver 2 moves moves first he gets payoff of while if Driver moves first Driver 1 gets a payoff of –5, and vice versa for Driver 2. Going from a Simultaneous Move to a Sequential Move Game may eliminate the play of a mixed strategy equilibrium • This is true in games with a unique mixed strategy Nash equilibrium. – Example: The Tennis Game Serena Williams DL CC Venus Williams DL CC 50, 50 80, 20 90, 10 20, 80 The Pure Strategy Equilibrium is Different The Pure Strategy Equilibrium is Different, Depending on Who Moves First. There is no possibility of mixing in a sequential move game without any information sets. Empirical Plausibility of Subgame of Subgame Perfection? • Consider the “Centipede Game”: The original game has 100 decision nodes hence, the name. For our purposes, our purposes, four nodes will suffice. Unique Subgame Perfect Equilibrium • • • Player 1 chooses Take at the First Opportunity. Empirically it takes time for players to fi figure this thi equilibrium prediction out. Players are boundedly rational Econometrica, Vol. 60, No. 4 (July, 1992), 803-836 AN EXPERIMENTALSTUDY OF THE CENTIPEDEGAME BY RICHARD D. MCKELVEY AND THOMASR. PALFREY 1 i p We reporton an experiment n whichindividuals laya versionof the centipedegame. In this game, two players alternatelyget a chance to take the larger portion of a continuallyescalatingpile of money. As soon as one person takes, the game ends with that playergettingthe largerportionof the pile, and the other playergettingthe smaller g portion.If one views the experimentas a completeinformation ame, all standardgame theoretic equilibrium oncepts predictthe first movershould take the large pile on the c firstround.The experimental esultsshow that this does not occur. r An alternativeexplanationfor the data can be given if we reconsiderthe game as a game of incomplete informationin which there is some uncertaintyover the payoff functions of the players. In particular,if the subjects believe there is some small o likelihoodthat the opponent is an altruist,then in the equilibrium f this incomplete information ame, playersadopt mixedstrategiesin the early roundsof the experiment, g h with the probability f takingincreasing s the pile gets larger.We investigate owwell a o a versionof this model explainsthe data observedin the centipedeexperiments. a KEYWORDS:Game theory,experiments, ationality, ltruism. r A 1. OVERVIEW FTHEEXPERIMENT ND THERESULTS O T o THIS PAPERREPORTS HE RESULTS f several experimentalgames for which the predictions of Nash equilibrium are widely acknowledgedto be intuitively W unsatisfactory. e explainthe deviationsfromthe standardpredictionsusing an approachthat combinesrecent developmentsin game theorywith a parametric specificationof the errors individualsmight make. We construct a structural econometricmodel and estimatethe extent to which the behavioris explainable by game-theoreticconsiderations. In the games we investigate, the use of backwardinduction and/or the eliminationof dominatedstrategiesleads to a uniqueNash prediction,but there are clear benefits to the playersif, for some reason, some playersfail to behave in this fashion.Thus, we have intentionallychosen an environmentin which we t expect Nash equilibrium o performat its worst. The best known exampleof a game in this class is the finitely repeated prisoners'dilemma.We focus on an even simpler and, we believe more compelling, example of such a game, the closely related alternating-movegame that has come to be known as the "centipedegame"(see Binmore(1987)). The centipede game is a finite move extensiveform two persongame in which each playeralternatelygets a turn to either terminatethe game with a favorable payoff to itself, or continue the game, resultingin social gains for the pair. As 'Supportfor this researchwas providedin partby NSF Grants#IST-8513679and #SES-878650 to the CaliforniaInstituteof Technology.We thank MahmoudEl-Gamalfor valuablediscussions a t e concerning he econometric stimation, ndwe thankRichardBoylan,MarkFey, ArthurLupia,and David Schmidtfor able researchassistance.We thankthe JPL-Caltechoint computingprojectfor j W grantingus time on the CRAY X-MP at the Jet PropulsionLaboratory. e also are gratefulfor f commentsand suggestionsfrom many seminar participants, rom an editor, and from two very thoroughreferees. 803 806 R. D. MCKELVEY AND T. R. PALFREY niques. This enables us to estimate the numberof subjectsthat actuallybehave in such a fashion, and to address the question as to whether the beliefs of subjectsare on averagecorrect. c Our experiment an also be comparedto the literatureon repeatedprisoner's dilemmas. This literature (see e.g., Selten and Stoecker (1986) for a review) finds that experienced subjects exhibit a pattern of "tacit cooperation"until shortlybefore the end of the game, when they start to adopt noncooperative behavior.Such behaviorwould be predictedby incompleteinformationmodels like that of Kreps et al. (1982). However, Selten and Stoecker also find that inexperiencedsubjectsdo not immediatelyadopt this patternof play, but that it takes them some time to "learn to cooperate."Selten and Stoecker develop a learningtheory model that is not based on optimizingbehaviorto account for such a learningphase. One could alternatively evelop a model similarto the d one used here, where in additionto incompleteinformation boutthe payoffsof a others, all subjects have some chance of making errors,which decreases over time. If some other subjects might be making errors, then it could be in the interest of all subjectsto take some time to learn to cooperate, since they can masqueradeas slow learners. Thus, a natural analog of the model used here might offer an alternativeexplanationfor the data in Selten and Stoecker. 2. EXPERIMENTAL DESIGN Our budget is too constrainedto use the payoffsproposedby Aumann.So we run a rather more modest version of the centipede game. In our laboratory games, we start with a total pot of $.50 divided into a large pile of $.40 and a small pile of $.10. Each time a playerchooses to pass, both piles are multiplied by two. We considerboth a two round(four move) and a three round(six move) version of the game. This leads to the extensive forms illustratedin Figures 1 and 2. In addition,we consider a version of the four move game in which all payoffs are quadrupled.This "high payoff" condition therefore produced a payoffstructureequivalentto the last four moves of the six move game. 1 1T fT 0.40 0.10 1 2 0.20 0.80 FIGURE 1.-The 2 I . fT 0.40 0.10 P P 14 2 6.40 fT fT 1.60 0.40 P 1.60 0.80 3.20 four movecentipedegame. 1 2 P P0 2 25.60 P P fT fT fT fT fT 0.20 0.80 1.60 0.40 0.80 3.20 6.40 1.60 3.20 I2.80 2.-The six movecentipedegame. FIGURE 6.40 26 808 R. D. MCKELVEYAND T. R. PALFREY TABLE IIA PROPORTION OF OBSERVATIONS AT EACH TERMINAL NODE f f7 .20 .35 .23 .01 .11 .12 .01 .02 .01 .253 .078 .014 N fi f2 f3 f4 f5 1 2 3 (PCC) (PCC) (CIT) 100 81 100 .06 .10 .06 .26 .38 .43 .44 .40 .28 .20 .11 .14 .04 .01 .09 Total 1-3 281 .071 .356 .370 .153 .049 High Payoff 4 (High-CIT) 100 .150 .370 .320 .110 .050 Six Move 5 6 7 (CIT) (PCC) (PCC) 100 81 100 .02 .00 .00 .09 .02 .07 .39 .04 .14 .28 .46 .43 Total 5-7 281 .007 .064 .199 .384 Session Four Move involved the high payoff four move game, and sessions 5-7 involved the six move version of the game. This gives us a total of 58 subjectsand 281 plays of the four move game, and 58 subjectswith 281 plays of the six move game, and 20 subjectswith 100 plays of the high payoff game. Subjectswere paid in cash the cumulativeamount that they earned in the session plus a fixed amountfor showingup ($3.00 for CIT students and $5.00 for PCC students).7 3. DESCRIPTIVE SUMMARY OF DATA The complete data from the experimentis given in AppendixC. In Table II, t we present some simple descriptivestatistics summarizing he behaviorof the subjects in our experiment. Table IIA gives the frequencies of each of the terminal outcomes. Thus fi is the proportion of games ending at the ith terminalnode. Table IIB gives the implied probabilities, i of takingat the ith p decisionnode of the game. In other words,pi is the proportionof games among those that reached decision node i, in which the subjectwho moves at node i chose TAKE. Thus, in a game with n decision nodes, pi =fi/El f1f. E i All standardgame theoretic solutions(Nash equilibrium, terated elimination e of dominatedstrategies,maximin,rationalizability, tc.) would predict fi = 1 if i = 1, fI = 0 otherwise. The requirement of rationality that subjects not adopt dominatedstrategieswould predictthat fn+i = 0 and Pn= 1. As is evidentfrom Table II, we can reject out of hand either of these hypothesesof rationality.In only 7% of the four move games, 1% of the six move games, and 15% of the high payoffgames does the firstmoverchoose TAKE on the firstround.So the subjectsclearlydo not iterativelyeliminatedominatedstrategies.Further,when a game reaches the last move, Table IIB shows that the player with the last move adopts the dominatedstrategyof choosingPASS roughly25% of the time 7The stakes in these games were large by usual standards. Students earned from a low of $7.00 to a high of $75.00, in sessions that averaged less than 1 hour-average earnings were $20.50 ($13.40 in the four move, $30.77 in the six move, and $41.50 in the high payoff four move version). Strategic Moves Strategic Moves • • So far we have supposed that the rules of the game were fixed, e.g., who moves first, the timing of decisions, payoffs, etc. In real strategic situations, players will have incentives to attempt to manipulate the rules of the game/action choices/payoffs available for their own benefit. – E.g. who moves first, what choices remain, etc. • • • A strategic move is an action taken outside the rules of the game effectively transforming the original game, into a two-stage game—such moves are sometimes called “game-changers.” Some kind of strategic move is made in stage one and some version of the original game, possibly with altered payoffs, is then played in stage 2. For strategic moves to work, they must (1) be observable to the other players and (2) irreversible (to the extent this is credible) so that they alter other player’s expectations and make the outcome more favorable to the player making the strategic move. Kinds of Strategic Moves • • • • Commitments: – Irreversibly limit your choice of action thereby forcing the other player to choose his/her best response to your preferred action. hi – E.g., in the Pittsburgh left-turn game, I hit the gas pedal hard and jump the light early to make my left turn, so your best response is to yield. Threats: If you do not choose an action I prefer, I will respond in a manner If you do not choose an action prefer will respond in manner that will be bad for you (in the second stage). – E.g., if you vote for ObamaCare, I will raise money for your opponent this November. Promises: If you choose an action I prefer, I will respond in a manner that will be good for you. – E.g., if you vote for the Senate health care bill, I will send a check for your re-election campaign. Note that both threats and promises are costly if they have to be carried out. However, if a threat works to alter the target player’s behavior, there is no cost to the player of issuing the threat, while if a promise works, there is a cost. th th th if th Wh St Why Strategic Moves? • Freedom of choice / a larger action space can of choice larger action space can be a bad thing. – For example, allowing students to make up a example allowing students to make up midterm exam or perform extra credit is costly – a new exam, extra assignments have to be written/graded. – By announcing an adhering to a no make-up, no extra credit, no exceptions policy and announcing it on day 1 of class, writing it into the syllabus, etc. your professor is made better off. Credibility of Strategic Moves Credibility of Strategic Moves • The problem with strategic moves, especially threats and promises is that they may not be credible and promises is that they may not be credible. – Ex-post, you may not want to carry out a costly threat or follow through with a promised reward. • Strategic moves that are not credible will be ignored. way to make strategic moves credible is to • The way to make strategic moves credible is to either take options off the table completely by making a truly irreversible move, or make it costly for the strategic mover to deviate from his strategic move, i.e., change the payoffs of the game so that following through with the strategic move is a best th th response in the second stage. Illustration of Credibility Issue: Nuisance Lawsuits • Consider a game between a Plaintiff and a Defendant. • Plaintiff moves first, deciding whether to file a lawsuit against Defendant at cost to Plaintiff of k>0 If lawsuit is filed the Defendant at cost to Plaintiff of k>0. If a lawsuit is filed, the Plaintiff makes a take-it-or-leave it settlement offer of s>0. • Defendant accepts or rejects. If Defendant rejects, Plaintiff has to decide whether or not to go to trial at cost c>0 to the Plaintiff and at cost d>0 to the Defendant. • If the case goes to trial, Plaintiff wins the amount w with the case goes to trial Plaintiff wins the amount with probability p and loses (payoff=0) with probability 1-p. • Assume that pw < c, and this fact is common knowledge-a critical assumption. The Game in Extensive Form A Specific Parameterization of th the Game to Play Pl • k, cost of filing a cost of filing lawsuit = 10 • c=d=cost of a trial =30 • Settlement off =50 • Winnings, w=100 • Probability the Plaintiff wins,p=.10 The Lawsuit is Not Credible The Lawsuit is Not Credible • The subgame perfect equilibrium is that ilib th the Plaintiff does nothing, as pw<c, so that pw so that pw-k-c <-k How to Make the Nuisance Lawsuit Credible? • • • • • • • Consider the strategic move by which Plaintiff pre-pays his lawyer the the strategic move by which Plaintiff pre his lawyer the costs of a trial, c, up-front (Plaintiff puts money in a non-refundable retainer account guaranteeing the attorney’s future availability for trial). trial). In this case the payoff to Give Up changes from –k to –k-c. Plaintiff goes to trial if pw>0, i.e. if there is any positive probability p of winning the amount of winning the amount w. This further implies that Plaintiff prefers settlement to trial only if s>pw. Defendant is now willing to settle for any amount s<pw+d. So the settlement range is [pw, pw+d]. If the Plaintiff can make a take-it-or-leave-it offer, what is the settlement amount s? Outcome is Plaintiff sues, offers to settle, and a settlement is reached. Further Considerations Further Considerations • If Plaintiff settles, her payoff is s-k-c, where k is the cost of filing the lawsuit (threat), c is the retainer cost. • If Plaintiff can make a take-it-or-leave-it offer of s=pw+d, then the question remains as to whether the nuissance then the question remains as to whether the nuissance lawsuit should be brought in the first place. • Since doing nothing yields a payoff of 0 (this is a nuisance lawsuit, no real harm done), then we must have that: s-k-c>0 and if s=pw+d, then we require that pw+d-k-c>0. • Can incorporate malice, whereby d enters positively in the incorporate malice whereby enters positively in the plaintiff’s payoff from going to trial Another Example: Criminal Law • • • • • • • • • Accused criminal can plead guilty or not. If he pleads not guilty, the prosecutor can offer a full sentence, valued at s or a reduced sentence valued at r. For the prosecutor, s>r, but for the accused, the payoffs are losses: payoffs are losses: -s, -r, so -r>-s; the accused strictly prefers a reduced sentence. so r> the accused strictly prefers reduced sentence Accused can accept prosecutor’s offer or reject it and go to trial. Probability of being convicted, p, is assumed known. Cost to the prosecutor of taking the case to trial=c. to the prosecutor of taking the case to trial Cost to the accused of defending himself at trial=d. Suppose d/s > 1-p, probability of an acquittal. In this case, in equilibrium the prosecutor offers no reduction, and the accused pleads guilty and gets sentence s. Strategic move: Accused requests a public defender (free lawyer), so that d=0. In response, the prosecutor offers a reduced sentence, r=ps-c<s; Accused accepts. Logic: By eliminating the cost to the accused of going to trial, the accused will always go to trial rather than plead guilty since ps<s if p<1. Knowing this, prosecutor will cut a deal offering a reduced sentence of r¥ ps-c, greater than or equal to the prosecutor’s expected payoff from going to trial. The accused will only accept the lowest possible reduced sentence r= only accept the lowest possible reduced sentence, r=ps-c. Prosecutor agrees to Prosecutor agrees to this deal as he is made indifferent between accepting it and going to trial. Other Means of Acquiring Credibility • Delegation of your decision to a disinterested /non-emotional third party. – If you do not pay by this date, a repo man is scheduled to take your car. • Legally binding contracts. bi – If you violate parole, you will go back to prison. • Reputation may be a substitute for commitment in a repeated eputat be subst co game setting –more on this next week. – You are known to get angry if you don’t get your way. So you get your way way. • Rational irrationality. – You know I’m so crazy that I will follow through with the incredible threat. Ways to Counter Strategic Moves Ways to Counter Strategic Moves • Counter sunk costs with sunk costs. – In the nuisance lawsuit game the defendant can pay his lawyer in advance for the cost of trial Better yet hire in house lawyers to deter nuisance for the cost of a trial. Better yet, hire in house lawyers to deter nuisance suits in the first place. • Eliminate communication/observation – if a strategic move is not communicated/observed it is useless communicated/observed, it is useless. – For example, I’m not going to look at your sign that reads “homeless, please help.” • Undermine an opponent’s need to carry out threats. an opponent need to carry out threats – I won’t tell if you make an exception for me; it can be “our little secret.” • Salami tactics: Reduce an opponent’s threat by complying to a very small degree especially repeatedly over time like cutting off and small degree, especially repeatedly over time, like cutting off and offering thin slices of salami. In this way, the threat is never triggered, though you may fail to completely comply. – “Sorry for the delay. Times are tough for me right now. I can repay you this amount now and will send the rest of what I owe you shortly.” ...
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This note was uploaded on 12/26/2011 for the course ECON 171 taught by Professor Charness,g during the Fall '08 term at UCSB.

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