Common Knowledge and Games with Perfect Information Author(s): Philip J. Reny Source: PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1988, Volume Two: Symposia and Invited Papers (1988), pp. 363-369 Published by: The University of Chicago Press on behalf of the Philosophy of Science Association Stable URL: http://www.jstor.org/stable/192897 . Accessed: 01/07/2014 13:39 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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Common Knowledge and Games with Perfect Information Philip J. Reny The University of WesternOntario 1. Introduction It is by now ratherwell understoodthat the notion of common knowledge (first introduced by Lewis (1969) and later formalizedby Aumann(1976)) plays a centralrole in the theory of games. (An event E is common knowledge between two individuals,if each knows E, each knows the other knows E, etc...). Indeed, most justifications of Nash's (1951) equilibriumconcept usually include (perhapsonly implicitly) the assumption that it is common knowledge among the players that both the Nash equilibriumin question will be played by all and that all players are expected utility maximizers.1 (We shall henceforthcall expected utility maximizers,"rational".2)We hope to illustratein an informalmannerthat there is in fact a large class of extensive form games, in which each of which it is not possible for rationalityto be common knowledge throughoutthe game. The consequences of this for many well-known extensive form refinementsof Nash equilibriumare quite serious. Consider,for example Selten's (1965) notion of subgame perfect Nash equilibrium. The requirementson a solution here are not only thatthe strategiesform a Nash equilibriumof the game as a whole, but also thatthe strategies induce on every propersubgame a Nash equilibrium.3 If, however there are propersubgames beginning at (singleton) informationsets at which it is not possible for rationality to be common knowledge, then Nash behaviorin that subgamecan no longer be justified on common knowledge grounds.4 At best then, significantmodifications are requiredin our explanationof Nash behaviorin such subgames. At worst, Nash behaviorin such subgames should not be consideredas the only possibility. In either case, a re-evaluation of the subgameperfect equilibriumnotion would be called for. Since extensive form refinementssuch as sequentialequilibria(Krepsand Wilson (1982)) and perfect equilibria (Selten (1975)) involve even strongerrestrictionsupon behaviorin subgames, the comments above apply as well to each of these notions.5 In addition,ourresultimpliesthatargumentssupportingthe backwardinductionsolution basedon rationalitybeing commonknowledgeat every informationset, begin with a false hypothesis. Hence, the elegantargumentsupportingbackwardinductionadvancedby Kreps et. al. (1982) runsinto difficulty. Also, the recentworkof Bernheim(1984) andPearce involves heavy use of the commonknowledgeof rationalityin (1984) on "Rationalizability" both normalandextensive form games. If, in extensiveform games, suchcommonknowlof theirextensiveform edge is not always possible, thenat the very least a reinterpretation analysesis called for. These issues will be exploredfurtherat the end of the paper.

PSA 1988, Volume 2, pp. 363-369 Copyright ? 1989 by the Philosophy of Science Association

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364 Finally, others have also expressed certaindifficulties with a variety of the above equilibriumconcepts (See for instance Basu (1985), Binmore (1985) and Rosenthal (1981)). All of these difficulties essentially appearto involve in one way or anothera problemwith the assumptionthatrationalityis common knowledge. We now illustrate by means of the simplest sortof example that thereare games containinginformationsets at which it is not possible for rationalityto be common knowledge. 2. An Example Considerthe following two-playerperfectinformationgame. A refereecomes equipped with n dollarsandplaces one in frontof playersone andtwo. Playerone can take the dollar therebyending the game, or he can leave it. If he leaves it, the refereeplaces a second dollar in frontof the players. Playertwo now has the opportunityto take the two dollarsand end the game or not, in which case the processrepeats. In general,at the kthstage of the game, the refereeaddsone dollarto the pot bringingits totalto k dollars. If k is odd (even), playerone (two) may take the k dollarsand end the game, or leave it. Players'payoffs are assumedstrictlyincreasingin dollars. Finally,shouldthe game continueuntil the nthstage and the playerwhose turnit is decides to leave the n dollars,it is then given to the other player. Call this game TOL(n) (Takeit or leave it). TOL(n)for n odd is depictedbelow. 1

(0)

2

2 *~ .*

1

0

(o) (3o)

(2)

(n 1)

~ .*1

(on)

(n (a)

TOL(n) In this particulargame with perfect information,backwardinduction,and the subgame perfect, sequentialand perfect equilibriumconcepts each yield the same equilibrium strategies. We proceed via backwardinduction. At the last stage (n odd) player one's best choice is to take the n dollars. With this in mind, two's best response at the second last stage is to take the n-l dollars and end the game. This process continues with each player choosing to take the money and end the game on his turnif he gets the chance. In the end, backwardinduction(and hence the subgameperfect, sequentialand perfect equilibriumconcepts) yields thatplayer one take the one dollar and end the game in the first round. This is independentof the value of n! That is, no matterhow large the pot may potentially grow, the standardequilibriumnotions indicate thatplayer one will take the one dollar in the first round.6 This sort of paradoxis by no means new and is clearly reminiscentof that associated with the finitely-repeatedprisoners'dilemma. We now move to the problemof common knowledge. It is enough to considerTOL (3) (see figure below). 1

2

1

(3)

(3?)

(0)

(20) (o) TOL(3) We claim thatif playerone does not take the dollar and end the game in the first round, but instead leaves it so thatplayer2 must decide whetheror not to take the two dollars,

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then it is no longer possible for rationalityto be common knowledge. (i.e. At player two's informationset, it is not possible for rationalityto be common knowledge). The argument is really quite straightforwardand proceedsby contradiction. Suppose thatrationality were common knowledge at player two's informationset. Player two, believing thatplayer one is rational(i.e. an expected utility maximizer),must believe that at stage 3, player one will take the threedollars, leaving player two with nothing. A rationalplayer two would respondto this by takingthe two dollarsat the second stage of the game leaving player one with zero. Hence, if at playertwo's informationset it is the case that (i) player two is rational,and (ii) player two believes thatplayer one is rational,then player two will take the two dollars leaving player one with zero. But since rationalityis common knowledge, it must be the case that in particular, (i') player one believes thatplayer two is rational,and (ii') player one believes thatplayer two believes thatplayer one is rational. i.e. Player one believes (i) and (ii) above. Finally, however, this implies that player one believes that player two will take the two dollars leaving player one with zero and rendering player one's choice not to take the dollar in the first round (recall that player two's informationset has been reached)an irrational(non expected utility maximizing) one. Hence, player two must believe thatplayer one is not rationalwhich contradictsour original assumptionand completes the argument.7 A similarargumentshows that in TOL(n), as soon as player one leaves the first dollar it is not possible for rationalityto be common knowledge. This observationand the work of Krepset. al. (1982), suggests an alternativeanalysis of TOL(n). Recall that Kreps et. al. were interestedin explaining cooperationin the finitely- repeatedprisoners' dilemma. They showed that if from the start,there is a positive probabilitythat one of the players is not rational,then cooperationcould emerge as a sequentialequilibriumof a long enough repeatedprisoners'dilemma game suitablymodified to take into account the incomplete informationabout the player's rationality.o It turnsout thatone can apply a generalizedversion of Kreps et. al.'s analysis of the prisoners' dilemma to TOL(n) and achieve similarresults. That is, if from the outset there is a positive probabilitythatone of the players is not rational,or none of the players believes the other is not rational,or..., and n is large enough, then allowing the pot to build for some time can emerge as a sequentialequilibrium. Moreover,any such sequentialequilibriummust yield both players higher expected utilities than they would get if player one took the dollar at the first stage. Hence, there is a sense in which both players are better off in TOL(n) when n is large enough and rationalityis not common knowledge. Among those elements left unexplainedby Kreps et. al.'s analysis and the generalized version we've applied to TOL(n), is how the positive probabilitythat a player is not rational, or that a player believes the other is not rational,or ... arises in the first place. What we've shown above however, is thatplayer one can, by not taking that first dollar,create an environmentin which rationalityis not common knowledge. Since for n large enough, this potentially makes player one betteroff, this furnishesa sound explanationof where the positive probabilitycomes from. Hence, this exogenously introducedpositive probability that one of the players is not rationalor thatone believes the other is not rational etc..., need not be exogenously introducedat all. Our observationthat it is not possible for rationalityto be common knowledge once player one leaves the first dollar, supplemented by Kreps et. al's analysis once this is the environment,shows that this positive probabilitycan arise as the result of expected utility maximizing behavior. Takenas a

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whole, this analysis of TOL(n) (unlike backwardinductionand the more traditionalsolution concepts) implies that two expected utility maximizing players can, acting in their own self interests,allow the pot to grow. It is somewhatparadoxicalthatwe can justify in termsof rationalbehaviorplayerone leaving the first dollarin TOL(n),when this veryjustificationrequiresthat when player one does so, player two believes thatplayerone is not rationalor thattwo believes thatone believes thattwo is not rationalor etc.... For if an explanationbasedon the players' rationality is possible, won't playersbelieve one anotherarerational? And won't each then believe thateach believe this etc...? i.e. Won'tthen rationalitybe common knowledge? The answerto the last questionis: "absolutelynot". We have alreadyarguedthatonce player 1 leaves the first dollar,rationalitycannot be common knowledge. This is simply inconsistentwith the structureof the game and the currentposition in it. The answerto each of the otherquestionsis "notnecessarily". Since rationalitycan not be common knowledge when playerone leaves the dollar,some statementof the form: two believes thatone believes that two believes that ... thatone (or two) is not rationalmust be true. Hence the answerto at least one of the otherquestionsmust be no. That is, a formalproof can be constructedto demonstratethis. On the otherhand,althoughsuch beliefs are possible, no formalproof can be constructedto demonstratethatafterplayerone leaves the first dollar,two believes thatone is rationaland two believes thatone believes that two is rational. Otherwise(assumingthis proof is common knowledge) this would imply thatplayer two believes thatrationalityis common knowledge. But this is impossible if players' beliefs arerestrictedto being consistentwith the physical descriptionof the game and the currentposition in it. So, althougha rationalexplanationis available,in a formal sense it cannot be the only availableexplanation. And it is precisely these (necessarily)available alternativeswhich make a rationalexplanationpossible at all. As in TOL(n), one can show thatby cooperatingin a finitely- repeatedprisoners' dilemma, the players can create an environmentin which rationalitycan not be common knowledge. As Krepset. al (1982) have shown, once this is the case both players may be betteroff. Hence one can also explain now more fully perhaps,rationalcooperationin the finitely-repeatedprisoners'dilemma. 3. Is TOL(n) An Isolated Example? We shall now fulfil a promise made in the introductionand illustratethat there is a large class of games within which the problemof common knowledge of rationalityarises. To do so, we shall first restrictour attentionto two-personextensive form games with perfect informationwhere no player is indifferentbetween any two endpoints. Insteadof asking which games within this class contain informationsets at which it is possible for rationalityto be common knowledge, we ask a slightly differentquestion. That is, which games in this class allow rationalityto be common knowledge at every informationset simultaneously? Some clarificationis in order. It turnsout thatit is possible thatrationalityis commonknowledgeat a particularinformationset so long as at thatinformationset it is also commonknowledgethatrationalityis not commonknowledgeat some otherinformationset. Hence in such cases it may be that takenone at a time, it is possible thatrationalityis commonknowledgeat every information set, but takentogether,commonknowledgeof rationalityat one informationset precludesit at another.In the lattercase we say thatit is not possible for rationalityto be common knowledgeat every informationset simultaneously.Wheneverit is not possible for rationality to be commonknowledgeat every informationset simultaneously,issues such as those describedfor TOL(n)ariseandagain,the argumentssupportingthe traditionalequilibrium conceptsno longerapply(Reny (1988) containsmoredetails). It can be shown that:11

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Any two-personfinite extensive form game with no player indifferentbetween any two endpoints that allows rationalitycommon knowledge at every informationset simultaneously,must be of the form:

where the arrowsindicate the unique subgame-perfectequilibrium(i.e. every decision node is reachedin equilibrium). In our view, this indicates that the problemof common knowledge of rationalityin extensive form games occurs ratherfrequently. 4. Conclusion Whatare we to makeof all of this? We've shownthatrationalitycannotsimplybe assumedcommonknowledge. The physicaldescriptionof the game and the currentposition in it may precludethis. But once rationalitycan no longerbe commonknowledge,it is not at all clear whatthe theoryshouldtell the playersto believe aboutone another.One thing however, is clear. If we insist thatthe theoryitself is commonknowledgeamongthe players(as has implicitlybeen the traditionalapproach;andeven explicitly in Bemheim (1984) and Pearce(1984)), then the theorycannotindicatethatthe playersarerational,since this would automaticallyrenderrationalitycommonknowledgeandthis is not alwayspossible. But if the theorycannotindicatethatplayersarerational,then the playersmay believejust about anythingaboutthe behaviorof theiropponents,since thereis no obvious substitutefor rational behavior. Clearly,this will lead to a plethoraof possible outcomessince rationalplayers who need not believe theiropponentsarerationalmay have many strategieswhich area best responseto somethingtheiropponentsmightplay. Hence, we can expect a very weak (in termsof predictions)theoryto resultif we insist thatthe theoryitself is commonknowledge amongthe playersat every informationset. (Formoreon this see Reny (1988).) The most promising avenue to pursuethen, is one which explicitly allows the theory not to be common knowledge at every informationset. Indeedone might reach this conclusion simply on the groundsthat once a player has deviated it is impossible that the rationalityof the players and the equilibriumstrategiesremaincommon knowledge. One way to consistently explain the deviation and still hold fast to the rationalityof all players (including the deviator) is to postulatethatthe equilibriumstrategies(i.e. the theory) are no longer common knowledge. We close by mentioningthatthis approachhas been recently undertaken(Reny 1987) in a mannerthat yields relatively strong predictions, indicatingthat some of the issues raised here can be usefully taken into account.

Notes 1Ina two-player normalform game, a pair of strategies(one for each player) is a Nash equilibriumif each player's choice maximizes his expected utility given the choice of the other player. 2This follows Bernheim (1982) and Pearce (1982). 3Thatis, not only must it be that (i) every player has decided what to do in every possible eventuality so thatfrom his perspective at the beginningof the game his choices are

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368 best given what the others have plannedto do, it must also be the case that (ii) whenever any possible eventuality becomes a reality,no player will wish to change his previously decided upon choice. 4A physical descriptionof chess for instance,includes the initial position, the set of all possible first moves and subsequentpositions etc.... A subgameis simply the description of what the currentposition is and which positions are possible duringsubsequentplay. A propersubgamemust begin at a point in the game where every playerhas full knowledge of the past choices made by others. Hence, every subgamein chess is proper. In other games (like poker) one must take a turnin ignoranceof what othershave done previously. Those aspects known to a player when it is his turnto move are embodiedin his "information set". Correspondingto each turnthen is an informationset, and so "reaching"a particularinformationset of player 1 say, indicatesthatit is a particularturnof player 1. 50ne might arguethat since Selten's (1975) theory allows players to make "mistakes",one can always preservethe common knowledge of rationalityby explaining arrivalat any particularinformationset througha sequence of independent"errors".Our response to this is thatplayers who make "mistakes",independentor not, arbitrarilysmall or not, are by definition not expected utility maximizers(i.e. not rational). Expected utility maximizers always and everywheremake decisions that maximize theirexpected utility. Hence any explanationbased on mistakesor tremblesis one thatembracesthe lack of common knowledge of rationality. 6In fact, taking the first dollar is the unique Nash equilibriumoutcome. 7A formal version is containedin Reny (1988). The formal definition given there uses infinite hierarchiesof beliefs as in Mertensand Zamir(1985) and Tan and Werlang (1985). It should be pointed out that what here we've called common knowledge should more appropriatelybe called common belief since we never requireany player to actually be rationalonly thatplayers believe that one anotherare rationaletc.... Since an event which is common knowledge (in Aumann's(1976) sense) must also be common belief (see Tan and Werlang(1985)) we have actuallyobtainedthe strongerresult that at player two's informationset rationalitycannot be common belief. 8Theparticularkindof irrationalityimposeduponthe playerwhose behavioris not completely knownis important.Krepset. al. assumein particularthatthereis a positive probability thatthis playeris one who uses the TIT-FOR-TAT strategy(i.e. cooperatein the first periodand in every subsequentperiodcopy the previousperiodchoice of youropponent). TIT-FOR-TAT is not rationalsince it sometimesdictatescooperationin the last period. 9Again, the particularkind of irrationalityinvolved is important. 10Recallthat a game is one of perfect informationif at every stage of the game both players know all past choices made by theiropponent. A player is not indifferent between any two endpointsif whenever asked to choose which of two ways he would like the game to end, he always strictlyprefersone over the other. (Chess does not satisfy the lattercondition since for instance, there are many ways that the game can end with a win for white, and white is indifferentbetween all of these.) Note thatthe perfect information restrictionrules out the finitely- repeatedprisoners'dilemma. The no indifference between endpointsformally rules out TOL(n),but replacingthe payoffs of zero at stage k by 1/k leaves all relevantfeaturesof TOL(n)intact and this new version is a memberof the class of games described. 11Fora proof, see Reny (1988) pp. 72-79.

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369 References Aumann,R. (1976), "Agreeingto Disagree",The Annals of Statistics 4: 1236-1239. Basu, K. (1985), "StrategicIrrationalityin Extensive Games",mimeo, Institutefor Advanced Studies, Princeton. Berheim, D. (1984), "RationalizableStrategicBehavior",Econometrica52: 1007-1028. Binmore, K.G. (1985), "ModellingRationalPlayers",mimeo, London School of Economics and University of Pennsylvania. Kreps,D., Milgrom,P., Roberts,J. andWilson, R. (1982), "RationalCooperationin the Finitely RepeatedPrisoner'sDilemma",Journalof EconomicTheory27: 245-252. Kreps, D., and Wilson, R. (1982), "SequentialEquilibria",Econometrica50: 863-894. Lewis, D. (1969), Convention: A Philosophical Study.Cambridge:HarvardUniversity Press. Mertens,J.F., and Zamir,S. (1985), "Formulationof Bayesian Analysis for Games with Incomplete Information",InternationalJournal of Game Theory 14: 1-29. Nash, J. (1951), "NoncooperativeGames",Annals of Mathematics54: 286-295. Pearce, D. (1984), "RationalizableStrategicBehaviourand The Problemof Perfection", Econometrica52: 1008-1050. Reny, P.J. (1987), "ExplicableEquilibria",mimeo, PrincetonUniversity and the University of WesternOntario. __ __

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Rosenthal, R.W. (1981), "Gamesof Perfect Information,PredatoryPricing and the Chain-StoreParadox",Journal of EconomicTheory25: 92-100. Selten, R. (1965), "SpieltheoretischeBehandlungeinesOligopolmodellsmit Nachfragetragheit", Zeitschriftfurdie GesamteStraatiswissenschaft121: 301-324. _ _ _ __. (1975), "Reexaminationof the PerfectnessConcept for EquilibriumPoints in Extensive Games",InternationalJournal of Game Theory4: 25-55. Tan, T. and Werlang,S. (1985), "On Aumann'sNotion of Common Knowledge - An AlternativeApproach",mimeo, University of Chicago Business School and PrincetonUniversity.

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