Scandinavian Journal of Educational Research,Vol.46,No.1,2002

Towards a Model of StrategicActions in the Classroom:gamestheory as research heuristic

EYVIND ELSTADDepartment of Teacher Education and School Development,University of Oslo,PO Box 1099,Blindern,N-0317 Oslo,NorwayABSTRACT Why do students adapt to the role of receptive learners despite the fact that theyfeel this to be unpleasant?Why do they behave so differently when they have substitute teachers?Effective classroom management,or the lack thereof,results from the roles or games that areplayed between teacher actors and student actors.The conceptual apparatus and modusoperandi of games theory gives us a research heuristic for understanding the strategic possibilitiesinherent in typical classroom situations where the interests of teacher and students are partlyconcurrent and partly in con ict.The interaction focussed upon here is composed of the ongoingtransactions that take place between teacher and students,each playing different roles.Tacitagreement as implicit contract is rational in iterated play.The preconditions for contracts withself-sustaining properties are discussed,as well as the reasons why contracts between the teacherand the students can come unravelled.Keywords:strategic actions;classroom interaction;games theory;contractsINTRODUCTIONConsider the following examples that draw upon real life school situations.1.In Dif dence and Dignity Norwegian author Dag Solstad describes in detail anexample of classroom interaction in one of Oslo’s most prestigious schools.Theteacher has been giving lessons about Ibsen’s classic work The Wild Duck for over amonth.The students de nitely are not interested:by means of a series of different facial expressions and body language,adense and hostile structure…made it clear to him[the teacher]that it wasa pain to sit there…[they]constituted a structural enmity,directed againsthim and all that he stood for.Nevertheless,they did what he said.(Solstad,1994,p.23.Translation)Here the students adapt themselves to the role of receptive learner despite the factISSN 0031-3831 print;ISSN 1430-1170 online/02/010065-17ó2002 Scandinavian Journal of Educational ResearchDOI:10.1080/00313830120115615

66 E.Elstadthat his lessons do not inspire them at all.Why are they uninterested?Why do thestudents nevertheless cooperate?And what kind of mechanisms and motivations caninduce cooperation,stability and social order in the classroom or contribute to thebreakdown of social order in the classroom?When some students even groan aloudat the teacher’s lessons,as described by Solstad,the teacher does not in the leastseem to nd this unacceptable behaviour.Why not?2.The Norwegian school system issues two types of grades to upper secondaryschool students.The rst,the achievement results,are awarded by teachers and arebased on the year’s work.The second are awarded through national matriculationexaminations.It is interesting to note that at the end of the school year,after theachievement grades have been awarded,remarkable changes can occur in thebehaviour of those students who are graduating from upper secondary school.It isnot uncommon to observe that once studious students become unruly.Why?3.Many who have been substitute teachers in different classes have experienced thateven in classrooms dominated by bright,studious,ambitious students,the classoften tries to‘knock the teacher off his/her perch’.David Hargreaves has given anexample of the substitute teacher’s desperate situation:Ten boys sat before him.‘Good morning,sir’said the one nearest him.‘Good morning’said Paul.‘Good morning,sir’said the next.‘Goodmorning’said Paul.‘Good morning,sir’said the next.‘Oh,shut up’saidPaul.…‘I suppose the rst thing I ought to do is to get your names clear.What is your name?’he asked,turning to the rst boy.‘Tangent,sir.’‘Andyours?’‘Tangent,sir’said the next boy….‘Well,is there anyone who isn’tTangent?’Four or ve voices instantly arose.‘I’m not,sir.I’m notTangent.I wouldn’t be called Tangent,not on the end of a barge pole.’Ina few seconds the room had become divided into two parties:those whowere Tangent and those who were not.(Hargreaves,1983,pp.230–231)Why do students often behave with substitute teachers in a way that is completelyat odds with their behaviour with the usual class teacher?In this article I argue thatan approach based upon mutual dependency between strategic and rational actorsin the classroom can contribute to understanding the phenomena described aboveand,in particular,contribute to a clari cation of the fundamental motivationalforces in classroom interaction.GAMES THEORY AS A FRAMEWORK FOR CLASSROOM INTERACTIONANALYSISContracts,Transactions and Transactional Positions in the ClassroomConsideration of rational choices of action is central to a games theory analysis ofclassroom interaction.The game here has actors,each of whom has the possibilityto make choices about their actions,and their choices have consequences,negativeor positive.This interdependence involves each actor participating in the game,so

Strategic Actions in the Classroom 67that he/she obtains a result that depends upon both his/her own actions and on theactions of others.This also in uences the outcome for the other participants.Strategic behaviour is about how one can in uence the choices of others with theexpectation of how one’s own behaviour is related to that of the others.Thus theconceptual apparatus and modus operandi of games theory gives us a theoreticalframework for understanding the strategic possibilities inherent in typical classroomsituations.Several studies have shown a typical change in the orientation of motivation inthe course of children’s years in primary and secondary school.This change can becharacterized as a shift from intrinsic to increasingly extrinsic motivation(Harter,1981,1992).There are,however,many factors involved in the question of motiva-tional orientation.Research results stress the complexity of the relationship between,for instance,intrinsic and extrinsic motivation.The same student can be intrinsicallymotivated in some subjects and extrinsically motivated in others,and there are manydiscussions on combined intrinsic and extrinsic motivations(see for example Harter&Jackson,1992).Here we will study typical situations at the upper secondaryschool level where pupils have moved from intrinsic to extrinsic motivation.I takeas an example the teacher from the rst example above,who had as a mandate toteach classical,canonized literature,Old Norse language,folk ballads,etc.Thestudents drag themselves into the classroom with earphones connecting them topounding rock music,and to an inner life often lled with thoughts of intrinsicallymotivated activities,and are often bored by the encounter with the content theteacher has a mandate to impart.In other words,this is a typical con ict of interestbetween what intrinsically motivates the students and what must be imparted by theteacher on the basis of the school curriculum.It is worth keeping in mind that theway in which the teacher approaches learning is itself a subtle feature of teacherbehaviour and can have the effect of creating intrinsic motivation with respect toschool activities(Ryan et al.,1983).Furthermore,teachers’and students’interests,with certain quali cations,are united in a desire for achievement within theeducational system.This is where,I believe,an implicit‘contract’can be establishedbetween a teacher and the students in which contractual relations are based partlyon the relations between each student and the teacher and partly upon aspects thatinvolve the class as a group in its relationship with the teacher.Examples of the latterare a shared understanding of the rules of acceptable behaviour.An example of theformer involves the student’s expectations regarding grading and the teacher’sawarding of grades based on formal and informal assessment during the course ofthe school year.Almost all aspects of life can be conceived in terms of exchange.The interac-tions between students and the teacher take place as transactions(Barth,1981,p.47)or exchanges between the parties(Thibaut&Kelly,1959;Homans,1961;Holt,1964).Social interaction in the classroom includes transactions between the teacherand the students as a group and between the teacher and each individual student,and the rules governing the terms of interaction de ne the possible outcomes in theclassroom regime.Regime is here understood to denote the rules that regulate thetransactions.Different regimes involve the teacher in different transactional positions.

68 E.ElstadIn principle we can identify a large number of transactional positions in theinteraction between teacher and students.In this article I will con ne my focus to the two most common regimes at thesecondary school level and examine two different transactional positions regardingthe teacher’s interaction with the students.The rst is the teacher’s transactionalposition with respect to the year’s achievement grading system,the second theteacher’s transactional position towards the registration of these achievement marks.(At the national level a higher body,the National Board of Education under theMinistry of Education,Research and Church Affairs,supervises the evaluations.)The former case pertains to transactions which happen in an evaluation context,i.e.an ongoing evaluation that gradually accumulates the nal achievement mark for theyear.An example of such a transactional position is where,in a normal secondaryschool,teachers of the various subjects have the authority to set the achievementmarks.The students’transactional behaviour here will be related to their expecta-tions regarding the teacher’s awarding of achievement marks.Thus,the more thestudents are oriented to future achievement,the stronger the teacher’s transactionalposition will be in such a regime.An example of the teacher in a strong transactional position is found in thenovel Grammar School Teacher Pedersen,by Dag Solstad.At the outset the studentsreact to the‘leftist distortions’of the teacher’s lessons by asking:‘Is it on the curriculum?’‘Aha!’I[the teacher]said ingratiatingly,‘so you want curriculum?So that’swhat you want.Of course that’s what you will get’.And they got the curriculum.What was it they hated the most,thesewell-dressed souls?New Norwegian.But New Norwegian was on thecurriculum.A minor subject in Norway was important for anyone whowanted to get an A in his examen artium so he would be eligible to go onto become a petroleum engineer.So I gave them New Norwegian.In thebeginning they giggled and sneered at the New Norwegian they had tolearn,but that soon died away because I had not the slightest scruple aboutsending youngsters in blazers out into the hallway when they disturbed thelessons….‘Repeat after me in a low voice:gut[boy]:a boy—the boy—some boys—all the boys.Right.Stut[ox,oaf]:an ox—the ox—someoxen—all the oxen.Once more.’And one by one the blazers said,‘An ox-the ox—some oxen—all the oxen’.I have never experienced anything morebeautiful in a Norwegian classroom during the1980s.(Solstad,1985,pp.252–254.Translation)The other type of teachers’transactional position that has to do with the actualteaching situation arises in schools where the teacher does not have the authority toaward the year’s achievement marks,but rather gives instruction in preparation forthe of cially graded examinations.Examples of this were found in the old commer-cial colleges at the secondary level and private secondary schools that are often

Strategic Actions in the Classroom 69marketed as offering examination-accredited instruction to students wanting toimprove their grades.We have touched upon the case where the achievement marking systemstrengthens the teacher’s transactional position;the opposite would be the absenceof authority to set the achievement marks,implying that the teacher has a weaktransactional position if the students lack motivation for the content of the coursebeing taught.In Agnar Mykle’s novel Lasso Round the Moon such an interactionalbreakdown in the classroom is described where the main character has the task of‘…babysitting a small class of students who needed drilling and invigilation as theydid their assignments’together with answering the students’questions.Just likesubstitute teachers,he has a weak transactional position vis-a`-vis the students,something the students know how to exploit:It was the most gruesome school class that history has ever wit-nessed.…They talked and chatted among themselves,wrote notes to oneanother,tossed paper airplanes around the room,and completely ignoredAsk’s desperate beating of the side of the desk with his foot,and hisinsistent tapping of the desktop with his pencil.…They knew that he satand read his own reading matter.They knew that they could upset hisconcentration if they asked him something.So they took the opportunity toask him questions.‘Sir,what is coitus interruptus?’…But gradually things got worse andworse in the classroom.He did not manage to concentrate on his ownreading,and he did not manage to control the idiots who ought to bestudying their lessons.He begged and implored…he shouted and heyelled…he clowned around and tried to be humorous;nothinghelped.…In the course of four months they managed to undermine hissense of responsibility completely.They had turned him into a quiveringmass of jelly.(Mykle,1960,p.56)What is described here is a dynamic of interaction which leads to‘the sucker’spay-off’(Axelrod,1990;incidence S,T in Figure 1),no matter what the teacher doesin this case to establish a cooperative contract with the students.In teachingsituations where the teacher does not have evaluative authority over the students itis possible for the teacher and students to enter into a mutually agreeable contractualrelationship.If,as in the case above,a teacher has unmotivated students,he couldenter into a cooperative contract with the students,in which he renounced thecontractual assignment he has with the school principal or headteacher with regardto teaching the class.In effect,the cooperative contract here implies that studentsshould sit quietly so that the teacher can read his books in peace while the students,for instance,get to read their detective novels in peace.Thus equilibrium arises inthe interactional pattern between teacher and students:After four months of hell,he nally assessed the situation and gave up,andfrom that moment on,the idiots were the absolute monarchs.In order toobtain some small semblance of peace he had begun to overlook the fact

70 E.Elstadthat they brought newspapers and detective novels into the class,and thatthey sat and read these instead of their lessons.It went so far that heabandoned his desk at the front and sat on the windowsill at the back of theroom,near the door.So situated,he had an excuse not to see what theywere up to,and also,sitting there he could discern the sound of approach-ing footsteps outside in the hallway and give a warning grunt to the class.The pay-off was a secret implicit agreement that the class would not plaguehim with so many questions.This cease re was exceedingly crooked andextremely shameful.(Mykle,1960,pp.56–57)One means of establishing a cooperative contract with students in regimeswhere the teacher does not have instruments of power,such as the right to awardachievement marks,is to emphasize the teacher’s and students’mutual interests,namely goal attainment with respect to the examinations.The relevance of thelessons to the exams will in this manner be of considerable importance for theestablishment of a cooperative contract with extrinsically motivated students.Anillustrative example of this can also be found in Lasso Round the Moon.The maincharacter here is a teacher in a kind of school(a private commercial college)whereachievement marks were not set by the teacher but by centrally set written examsunder the jurisdiction of the educational authorities.The main character’s means ofextrinsic motivation and the students’reaction to this is described as follows:‘Out with your notebooks and write down the example I’m going to put onthe blackboard.There is scarcely a single exam which does not have aquestion on de cit interest.Therefore you should write CURRICULUMbeside it in the margin and underline it twice.That means this is somethingyou have to study.’They sat down quickly.Peace and quiet came over theclass,a strange,expectant calm.(Mykle,1960,p.222)Classroom Interaction as Repeated GamesIn a classroom the primary relationships between parties are the interactions be-tween teacher and student(s)and among the students.The latter is not dealt withhere.Games theory analysis of processes within student groups has been littlestudied.Nevertheless,games theory can make an interesting contribution to thedevelopment of models that simulate group processes(Pitt,2000).The classroomconstitutes a very complex social system distinguished by intertwined processes inwhich a multitude of events are encountered.Classroom situations are on the wholerelatively open situations,i.e.relatively loose rules govern social interaction in theclassroom.This implies that it is dif cult to investigate classroom situations with thehelp of formal analysis.Therefore,the approach taken here is to analyse situationsin the classroom that typically repeat themselves.This lays the foundation forstudying the interactional dynamic as repeated games:a series of repetitions with thesame game between teacher and students.When we look beyond the substituteteacher situation,relations between teacher and students normally extend for the

Strategic Actions in the Classroom 71period of a school year or longer.As we shall see later,the time aspect oftransactions has clear implications for the players’classroom behaviour.The analysis which follows depends on a number of presuppositions.First,there is the analytic assumption that the strategic actors seek to do the best they canto realize their goals,given the limits they have in relation to possible choices(valuemaximization).Second,they are aware of one another’s consciousness.Third,theteacher and students are two rational players in the interaction process.Here it isimportant to note that the treatment of students as one rational player is highlyproblematical from what we know about group behaviour(see for example thetheoretical discussion of the problem in relation to collective preferences in Arrow,1963).Furthermore,it is problematical in relation to the principle that socialphenomena ought to be explained on the basis of the individual actor(methodolog-ical individualism;Elster,1989c),rather than as a collective entity,as they will betreated here.Nevertheless,my decision to do so despite these problems is forreasons of parsimony(Occam’s razor)in the generation of theory.A study ofinteraction between a normal number of students in a school class would quicklybecome extremely complicated to deal with analytically.The games theoreticalviewpoint is useful in settings with a small number of players:moving away from asystem with two participants means imprecise deductions.It is therefore importantto nd a balance between simpli cation and prediction when we are building modelsof reality.What is presented here is a simpli ed model that I use with the aim ofcapturing fundamental aspects of the motivational forces in classroom interaction.Self-sustaining Contractual Relations between Teacher and StudentsIn the introduction I outlined a typical school situation described in Dif dence andDignity.The students attend one of Oslo’s prestigious schools.We may assume thatthey are extrinsically motivated to obtain good marks,even if they lack a clearintrinsic motivation to learn about Henrik Ibsen’s classic work The Wild Duck:The class had been at it for over a month and were still only in the middleof Act Four.…It struck him[the teacher]once more how hostile they weretowards him…even though they did what he said[emphasis added].(Sol-stad,1994,p.6 Translation)The question is why did they do what the teacher said if they had such a hostileattitude towards him and the content of the lessons?This is what I shall discuss withthe help of a repeated game model,in which each round consists of a repeated gameof the Prisoner’s Dilemma.The story behind the Prisoner’s Dilemma game is thattwo persons are arrested for a crime.The police lack evidence to convict eithersuspect.The suspects are prevented from communication with each other.Thepolice tell each of them that if he testi es against the other he will be released.Hewill receive a reward for testifying,provided that the other suspect does not testifyagainst him.If none of them testi es,both will be released,but no reward will bepaid.If one testi es,the other will go to prison.If both testify,both will go to prison,but both will collect a reward for testifying.Classroom interactions are here analysed

72 E.ElstadFIG.1.A Prisoner’s Dilemma game.The following notations are employed:T,the temptation to defect;R,the reward for mutual cooperation;P,the punishment for mutual defection;S,the sucker’s payoff.as though they comprise a repeated Prisoner’s Dilemma game between two players,teacher and students,where the interests of teacher and students are often partlyconcurrent and partly in con ict.The presentation is based upon Axelrod(1990),Taylor(1987),Midgaard(1997),Fudenberg&Tirole(1991,Ch.4)and Hovi&Rasch(1993).In the phenomenology of classroom interaction between the teacher and thestudents,four typical incidences stand out as important.Figure 1 presents the fourincidences and the pay-off matrix of the Prisoner’s Dilemma game,where thepay-offs to the students are listed rst.In the following I will give an example ofparadigmatic cases of the pay-offs in the matrix.One example of the(P,P)type can be found in a newspaper article with the title‘Mechanics pupils at Sogn Upper-Secondary risk failing exams because teachershave spent their time sorting out con icts and social problems instead of teaching’:When the reporter asked the pupils‘What do you think about going to thisschool?’they answered‘It’s boring!’‘In the beginning we were unruly everysingle moment and didn’t learn anything.’One pupil felt it was unfair thatthe teacher had not gone through the entire syllabus before the exams.‘Iwant to become a printer,and that means I really need OK grades from thefoundation course’said Larsen,another pupil(Aftenposten,23 February1998,my translation).What has been described here is an example of the dynamics of pursuing individualrationality in the absence of a cooperative contract coordinating behaviour in anequilibrium(P,P)which is clearly Pareto sub-optimal.An example of the teacher managing to establish a cooperative contract of the(R,R)type with the pupils is to be found in the following:The Consul(a nickname)was a rather unusual character.He was anaffable,pompous man with a mood to match…we(the pupils)felt goodjust to see him.He had a heat of gold,not the least before and during theexaminations.When from behind the master’s desk he loudly proclaimed‘Now listen up boys,because you don’t really know what you will need forthe examination’we did as he said.However,this did not mean that all oranything we heard was of any help to us.The Consul’s compassion seemedwarm and heartfelt when he saw in‘despicable examination papers’dan-gers that could cause trouble for the weaker students and ruin their chancesof a good start in life.If only all grading commissions,outside examiners

Strategic Actions in the Classroom 73and teachers could have been so understanding and sympathetic.(Stok-stad,1944,pp.161–162.My translation)Above we have given an example of the teacher experiencing the sucker’spay-off,i.e.(S,T)in Figure 1.An example of the opposite,of the studentsexperiencing the same(T,S),can be seen in the following example:Arnesen taught Latin to the senior classes.Now the tables were turned,wehad to learn Latin whether we wanted to or not.He was the most perfectexample of an utter despot I have ever met,and ruled through violence andrandom scoldings.He would slam in through the door like a hurricane,barge up to the master’s desk where Fredrik Holst(later professor ofmedicine)sat,and start the lesson by manhandling him,hitting him andcalling him an utter dunce and blockhead.A few others were also assaultedregularly,Morten Hartman even quit school,and Otto Ottesen(later avicar at Toten),who was every bit as big as Arnesen,had his ears boxed tillthe day he graduated and left school.(Lange,1905,p.22.My translation)The order of preference T.R.P.S is assumed to be typical for both sidesin the following analysis.It is further understood that the inequality R.(T 1 S)/2pertains.This games theory mode of analysis is deductive,which implies that pointsof equilibrium are deduced from rst principles.The deferment of this pay-offmatrix rests upon a somewhat simpli ed set of assumptions.First we assume thatthe results experienced by the players are compatible,which in reality is rarely thecase.Second,the pay-off matrix is symmetrical,though there is nothing in real lifewhich would anticipate such symmetrical properties.In other words,the currentanalysis is based upon simpli ed assumptions for reasons of analytical convenience.Even though further developments in games theory analysis in the direction of morerealistic patterns can be drawn from the analysis and yield more nuanced conclu-sions,they nevertheless do not contradict the deduced conclusions about the drivingforces of classroom interaction based upon the simpli ed model.We shall thereforeanalyse this truce in the classroom(described above)as the entering into a cooper-ative contract by teacher and students.Such a contract will typically be self-sustain-ing in the sense that it can last throughout the school year,despite partly con ictinginterests.In what follows we will go more deeply into the terms and conditionsnecessary for a contract to have such self-sustaining properties.In addition,we willalso discuss reasons why a cooperative contract between teacher and students cancollapse.As a one-time game and as a game in which players do not have an agreed uponcontract of the type of concern here,the dominant strategy from the students’sidewould be rebellion against the teacher’s lessons,given the assumed preferencestructure laid out here.The dominant strategy from the teacher’s side,on the otherhand,would be to suppress unacceptable behaviour.Interaction between substituteteachers and students can often shape up like a one-time game that results in aPrisoner’s Dilemma equilibrium.The point of equilibrium in such an interactionbecomes(P,P),not a Pareto optimal equilibrium.A considerable number of obser-

74 E.ElstadFIG.2.Prisoner’s Dilemma outcomes and their convex envelop.The shaded area is the zone of contract.vations from secondary schools show,however,that the most frequently occurringinteraction between students and teachers is that described in the introductoryExample 1.As mentioned,we can analyse this as if the players had entered into acooperative contract.Every contract can come unravelled.The point where oppo-sition can be unleashed is point B in Figure 2.The feasible sets are A,B,C,D andtheir convex envelope,but the zone of contract is the convex set F,B,E,D(theshaded area).The dominant strategy in a game with an agreed upon contract is to cooperateright up to the point where the opposing player breaks off the collaboration.Thisstrategy has been shown to be the best and most intentioned in experimentalsituations(Axelrod,1990).In order to discuss what kind of advantage each actorcan experience in such a repeated Prisoner’s Dilemma,we must familiarize ourselveswith the actors’weighing of the present against the future,what we call timediscounting.Time DiscountingThe future is considered here as being less important than the present because,inrelation to the present,the actors have a tendency to place much less value on futurewins.One way to model this is to consider that the pay-off for the next move willcount for less than the pay-off of the current move.That is to say,the resultsaccumulate over time so that the next play is evaluated as a part of the previousgame(Shubik,1970).This is af rmed analytically by considering the actors’timediscounting,designated w,to be between zero and one.The discounting parametergives expression to the actors’relative evaluation of future winnings in relation tothose in the present.The numerical magnitude of the discounting parameter is,aswe shall see,extremely important because the greater the discounting parameter thegreater the probability that the contract between teacher and students is self-sustain-

Strategic Actions in the Classroom 75ing(because the expected pay-off that the players accumulate will also be consider-able).The superior strategy in such a game involves choosing cooperation overnon-cooperation as long as the opposing side also chooses cooperation(Axelrod,1990).If we let V signify cardinal value,the utilities can be measured using aninterval scale.As long as both players operate as though they have entered into acooperative contract,i.e.choice A,(R,R)will be the equilibrium and the cumulativepay-off for each of the players will be V(A:A)5ont 51 wt 21Rt,in which n signi es thenumber of interactions.When w P(0,1)and o`t51wt 21Rt the range will convergetoward R/(1 2w).This result gives us the possibility of studying the game’s dynamicproperties,i.e.we utilise an analytical convention even though it is understood thatsimpli cation,with n approaching in nity,implies no loss of generality.If one playerreneges on the contractual relationship(choice B)and the other player undertakesthe contract(choice A),the player who has broken the contract experiences T solelyin that period(because the other player complies with the contract)but experiencesP in all subsequent periods.The accumulated value of utilities for the actor whobroke away then becomes V(B:A)5T 1 wP/(1 2w).If V(A:A)$V(B:A),thencompliance with the contract relationship will represent a perfect Nash equilibriumsub-game.(We reach this point by working through reverse induction.)Conversionof the formula yields the following requirement that the contract be strong:w$(T 2R)/(T 2 P).If this requirement is met it will also be rational for bothplayers to abide by the cooperative contract until the opposing player breaks it.Theless that can be gained by breaking the contract,the less,as well,(T 2R)will be.The greater the loss if the contract terminates,the greater(T 2P)will be and thestronger the contract.[Here we utilise the fact that(T 2 P)5(T 2R)1(R 2 P)].Thus,we nd that the contract has stronger self-sustaining properties the higher the valueof w.Empirical measurements giving insights into future time orientation are suchas how important a pupil considers a good grade to be,how important school isconsidered,if value is placed upon further education,etc.Empirical ndingsreported in Nordahl(2000)support the supposition of importance of future timeorientation for problematical classroom behaviour.Self-sustaining Contracts that Unravel as a Consequence of the Constitution of RulesThe formal conditions allowing a contract to be self-sustaining are(R 2 P)$(1 2w)(T 2R)/w.This can be interpreted as follows.In cases wherethere is no future to the interaction between teacher and students(such as in thesporadic instances of substitute teachers or such as occur at the end of the schoolyear when the achievement marks have already been awarded)the cooperativecontract will not be self-sustaining.This is because to the extent that the cumulatedbene ts on the right side of the disparity are greater,the less w is and the greater(T 2R)is.In other words,student behaviour incorporates a system of evaluationthat re ects an incentive structure.This analysis underlines how important thesystem of evaluation(as a constitution of rules implied in the classroom regime)isto the enforcement of the cooperative contract between teacher and students.Letme mention an example from my personal teaching experience.This event occurred

76 E.Elstadin the month of May when the students had completed their assignments leading tothe written exam in the subject I was teaching.The course of study had beencompleted.The achievement marks had in fact been set(at least in the sense thataccording to the current rules the achievement results could not be lower than thosefor the rst two completed terms of the school year).Nevertheless,teaching was tocarry right on through the nal month of June.The summer holidays were ap-proaching.The sun had regained its warmth after winter.The weather was glorious.What could we come up with?One of my colleagues took his guitar and the studentsout to a owerclad hillside and devoted the time to singing popular songs.In otherwords,he tried to do something for which the students were intrinsically motivated.However,he received a message from the school’s leadership that this period mustbe devoted to‘normal’instruction.However,the students were very little motivatedfor‘normal’lessons after the established cooperative contract had come to an end.The students knew that further efforts would not in uence their achievement marks.Facing this situation,I made an amendment to the rules:I told the students that,onthe basis of the newsclipping les in the school library,they still had an obligatoryessay to hand in as a further gauge of their course achievement.This was acompulsory component of the achievement mark,a declaration for which I had,inreality,a weak legal basis.In other words,I sought to revive the formerly establishedcooperative contract on an ethically dubious basis.Actions that Can Cause a Contract to Break DownIn the foregoing we have found that the degree of self-sustenance that the coopera-tive contract is likely to possess[the advantage of complying with the contract(R 2 P)]will be higher the greater w is and the lower(T 2R)is,as expressed in theformula w$(T 2R)/(T 2 R)1(R 2P).The expression(R 2 P)re ects,amongother things,the relationship between actual marks awarded to students throughexaminations and the students’expectations,on the one hand,together with theteacher’s distaste for suppressing undesirable behaviour in the classroom,on theother.Students who feel that the discrepancy between the anticipated mark andthe actual mark is too great can,from this perspective,withdraw from the coopera-tive contract.Since the teacher is aware of the students’potential reactions to thereality of the probable marking,this awareness can be factored into the teacher’sactual setting of marks,which means that transactional behaviour can in uence thesetting of marks.For example,an essay written by a resistant student was a vividdescription of sex and violence(Smidt,1996,pp.95–100).It was meant as aprovocation against traditional teacher expectations of expository essays.However,the student achieved a very good mark,and the contract was sustained.We canfurther conclude from the above formula that one side in the transactional relation-ship ought to tread carefully when putting pressure on the other.When interactionsare approaching the point where opposition can be unleashed,each side ought toavoid behaving in such a manner that the exercise of contractual relationshipapproaches the limits of what the person or persons concerned are willing totolerate.The contract is only applicable so long as one side possesses the necessary

Strategic Actions in the Classroom 77instruments of power to enforce it.The teacher described above in Example 1 isobviously close to this point,i.e.the point where students unleash their opposition:…it was he who was the target of their outrage,because he,the teacher,was boring them to death…they were deeply outraged.It was not merelyon the surface of things,it had permeated them through and through,andhad become their predominant and actual attitude toward him.…Whyshould they put up with this?How long should they put up with it?Doeshe have the right to do this to us?He could literally see their thoughts.(Solstad,1994,p.35.Translation)Even some of the academically brightest students‘…not only reacted to thefact that the lessons left them cold,but also reacted to the fact that he indulgedhimself in squandering their precious time—yes,them,the examination-orientedstudents—by coercing them to occupy themselves with a second-rank character inwhat is otherwise an obligatory,curriculum play by Henrik Ibsen’(Solstad,1994,pp.24–25.Translation).At the same time,the dread of unleashing the students’opposition constrained the teacher’s desire to quell the audible groaning in theclassroom,the wary protest among the students:He simply did not have the courage to say anything about it,and thissurrender made him numb,impotent,unable to think straight.…And thiswas not the rst occasion.Every time the class had reached the point whereone or more of them burst out this way and brought their inner frustrationsout into the open,it startled him and he pretended not to notice.…For hewas afraid…he could not provoke them so much that they would rise upin protest.(Solstad,1994,pp.17–36 Translation)The teacher therefore chose to detour into pedantic lecturing and concludedthe class by giving a number of practical messages.The relationship between thestudents’behaviour and the teacher’s means of resistance is also important to theteacher’s compliance with the cooperative contract.Earlier we discussed the exam-ple from the novel Lasso Round the Moon,where the teacher had to overlook the factthat the students were reading detective novels and magazines instead of doing theirhomework so that he could retain the cooperative contract with them:On one occasion Ask[the teacher]almost snapped.With much ado,oneof the girls ung open a new and glossy copy of the magazine Allers,swivelled around in her desk and in a loud,cool voice said,‘Look whatthey’re coming out with now’.Ask’s face turned grey as lead and his lipstrembled.But while the whole class watched him,he blinked,choked downhis feelings…and began to turn the pages in his own book.(Mykle,1960,p.57)Other Games Theory ModelsIn this paper the main focus has been on the Prisoner’s Dilemma and thecooperative game.In the study of strategic interactions in school settings other game

78 E.Elstadtypes are also relevant.The classic game of Chicken is played in automobiles,twocars head towards each other and the rst driver who swerves to avoid a collisionloses.(This has been well studied in games theory;see Fudenberg&Tirole,1991,pp.119–126).Some instances of classroom observations fall into this category.Oneis of a primary school class that was given a new teacher because their old one hadto take over another class.Here the school principal became the opponent.Thepupils demonstrated by sitting on top of their desks.Some refused to sit on theirchairs even when their new teacher asked them to do so.Among the utterancesheard were‘Up with T!’(the old teacher)and‘Down with S!’(the new teacher).After two weeks of continuous con ict the new teacher went on long-term sick leaveand the principal decided to use the classes old teacher as a substitute.We can also nd situations where students work counter to the expectations theteacher and the school have of them,in a situation characterized by participantshaving radically differing preferences,known as a Zero Sum game.The students canestablish groups and a dominant culture of resistance in a classroom(Furlong,1991).Willis(1977)describes examples where the goal of these boys(it is often amale activity)is to have a laugh at the expense of the teacher.Two person Zero Sumgames can also be relevant in describing the interaction between pupils in the sameclass(I thank an anonymous referee for bringing this to my notice).We often ndthat in classroom interactions students either implicitly or explicitly restrain thoseco-students who express opposition to the established social order:Now there are some among them who are more tolerant than others,andwho,despite sharing their fellow students’feelings of outrage nonethelesstried to put it in perspective and in this way had a dampening effect uponthe others…thanks to these students the bottled-up resentment was in-stead softened,into a more traditional expression of general tedium.(Solstad,1994,p.23.Translation)Expanding the Theoretical ModelsThe games theoretical models of classroom interactions re ect many importantfactors about the mechanisms that can induce cooperation,stability and social orderor the breakdown of social order.They represent theoretical frameworks for under-standing important aspects of classroom interactions.Furthermore,the models canbe expanded so as to capture more of the complexity of classroom interactions.From my perspective there are two routes to enrichment of the games theoreticalframeworks I have outlined in this article.One way is to develop models for a morecomplex view of the conditions for and interactions between the actors,on the onehand,and between the actors,material systems and representational systems,on theother(for example,the content of the subject matter used).Another way is todevelop models in which the actors are portrayed in a more realistic manner.Bothapproaches embody great challenges in relation to generating theory.Theoretical work on social norms represents an analytical re nement of a kindother than that which is based on rationality as a regulative idea(Elster,1989a).

Strategic Actions in the Classroom 79Social norms typically develop in interactions between people and remain stableregulative mechanisms(Elster,1989b)and social norms represent a collectivecomponent in the classroom culture.A means of modelling norms added asepicycles to the games theory model is found in Axelrod(1997).Axelrod(1997)studied norms which were important in political science and used computer simula-tions to understand the conditions under which norms can evolve and prove stable.I nd Axelrod’s approach very interesting for the study of classroom processes.Thisapproach shows that a cooperative contract between teacher and students can bemaintained when some players(some of the students)punish others(other students)who do not adapt to the terms of the cooperative contract.Another way of elaborating the theoretical approach is to present the actors ina paradigmatically more complex manner.In this analysis we have modelled theactors such that the evaluation of virtually identical outcomes depends on when theyare realised in time.An important nding in motivation research is that theawakening of motivation increases as the distance to the goal decreases in real time(Gjesme,1981).In other words,the temporal location of the goal,as perceived andnot necessarily as it occurs in real time,is highly signi cant to creating motivation.This is an example of substantial nuancing found in motivation research which,withsome effort,can be incorporated into games theory modelling.A primitive gamestheory formalization of both participants changing their cooperative strategy ischanging(R,R)to(R 1 Z,R 1 Z)where R 1 Z.T.We can still carry out related,subsidiary reasoning that has to do with good basic empirical correlations.Inaddiction,the assumption of actors’maximiszation of values remains dubious inrelation to empirical research.I am thinking here of different forms of cognitivelimitations found among the participants,limitations which have behavioural conse-quences.Well-documented cognitive biases lead to systematic deviations from thepredictions of pure rationality based models.CONCLUSIONAlthough games theory has had very little signi cance in the development ofpedagogical theory(Boyd et al.,1994;Pitt,2000),my main argument is that gamestheory is a promising tool for understanding the interdependence of conscious partiesin classroom interactions.No assertion is made that the games theoretical approachis the only adequate one for research into social relations in the classroom.However,games theory can contribute by explaining the generation of observed patterns ofclassroom interactions and give them meaning as the result of human relations.Theory should be judged on its truthfulness,but at an early‘rationalistic’stage in the development of simplistic models it is more important to generatefruitful issues than testable hypotheses.A too strict use of falsifying criteria canpreclude useful developments and elaboration.The elaboration of a games theorymodel of classroom interaction lacks the status of a fully edged theory and furtherdevelopment should therefore proceed in a dynamic interplay between the gener-ation of theory and testing of the theory to capture more aspects of reality.Modelswhose regulative ideas are built upon the basis of rationality will intrinsically be

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