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Harold Pinter’s The Caretaker and Cooperative Game Theory: A Reappraisal

Le Gardien d’Harold Pinter et la théorie des jeux coopératifs : un ré-examen
Bertrand Crettez et Régis Deloche
p. 75-90

Résumés

Selon Colman (1988), Le Gardien d’Harold Pinter illustre les limites de l’ensemble stable, un concept de solution relevant de la théorie des jeux coopératif, qui a été proposé par Von Neumann et Morgenstern (1944). Prenant appui sur les interprétations de cette pièce proposées par les spécialistes de Pinter, nous remettons en cause ce résultat et nous montrons que Le Gardien illustre bien la fécondité de l’ensemble de négociation, un autre concept de solution de la théorie des jeux coopératifs, qui a été proposé par Aumann et Maschler (1961 ; 1964).

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Texte intégral

1Game theory, which is “a collection of models of rational decision-making in interactive situations” (Rubinstein, 2012, 93), is traditionally divided into two major subfields: non-cooperative games, or strategic games, and coalitional games, or cooperative games.

  • 1 In this connection, three points are noteworthy. First, non-cooperative game-theoretic analyses of (...)
  • 2 Other plays by Harold Pinter have been analyzed from the perspective of game theory. In Birchler an (...)

2Non-cooperative game-theoretic models have been used to study the actions made by individual players in plays, libretti, novels, short stories and poetry (see, e.g., Brams, 1994a; 2011; Chwe, 2009; 2013) and to illustrate game-theoretic solution concepts.1 By contrast, there is, to the best of our knowledge, a dearth of papers using plays or other live performances to illustrate solution concepts in cooperative game theory (hereafter CGT). An exception to this dearth of papers is Colman’s (Colman, 2008, 169-171) study of Harold Pinter’s The Caretaker2 whose aim is to illustrate the inadequacy of a cooperative solution concept, namely the stable set (later, we shall give a precise definition of this concept introduced by von Neumann and Morgenstern, 1944).

3As pointed out by Per Wästberg in his Presentation Speech for the 2005 Nobel Prize in Literature, “Harold Pinter is the renewer of English drama in the 20th century. ‘Pinteresque’ is an adjective listed in the Oxford Dictionary. Like Kafka, Proust and Graham Greene he has charted a territory, a Pinterland with a distinct topography”.

4For many distinguished Pinter scholars, such as Esslin, Billington, and Taylor-Batty, among others, The Caretaker, which was performed for the first time at the Arts Theatre Club, London, on 27 April 1960, is a masterpiece. It is a play which traces the relationship of three men in a dilapidated house. Its theme is “the fight for a room of one’s own” (Esslin, 2004, 247). The Caretaker “focusses on what it means to take care and offer care, and the implications for interpersonal behavior when the balance of those things is impossible to achieve” (Taylor-Batty, 2014, 42).

  • 3 In his Nobel acceptance speech, Pinter provides some insight into his working method as a playwrigh (...)

5Both these points–three characters3, interpersonal behavior–are key points for developing a cooperative game-theoretic analysis of The Caretaker. This play indeed raises “theoretical questions, chiefly related to what coalitions are likely to form and be stable” (Brams, 2011, 11). Colman’s (2008, 171) pioneering contribution addresses these questions specifically. His contribution builds on the work carried out by Howard (1971, 140-146), who also considered the problem of coalition formation, but from a different perspective than CGT.

6After reviewing the plot of this play, Howard regards it “as a ‘reaction game’ in which the players move from one outcome to another, jockeying for position” (ibid., 141). Howard’s analysis of the play can be summarized as follows: The Caretaker “has three characters and three acts, in each of which one of the three possible two-person coalitions negotiates, forms, and achieves its ends” (ibid., 145). Howard concludes his analysis by highlighting “the circularity of the whole process, whereby the final outcome is the same as the first”, and the possibility that the three acts can “be repeated in sequence again and again” (ibid., 145).

7In contrast to Howard, Colman uses CGT to analyze The Caretaker. But he relies on Howard’s analysis of the instability of the different coalitions to show the inadequacy of the stable set as a solution concept in CGT.

8This paper challenges both Howard’s and Colman’s interpretations. We build on the fact that leading Pinter scholars’ comments on The Caretaker (see, e.g., Esslin, 1970; 2004, and Billington, 2009) are not compatible with the idea that the play has a circular configuration, the opening and the ending reflecting each other. We claim to the contrary that The Caretaker better illustrates another solution concept in CGT, namely the bargaining set (later, we shall define this concept introduced by Aumann and Maschler, 1961; 1964). We make this claim on the basis that, as shown by Maschler (1978), and Kahan and Rapoport (1984), bargaining processes in settings like The Caretaker often end in a finite time and with payoff configurations close to elements of the bargaining set. We therefore show that The Caretaker can be used to illustrate the adequacy of the bargaining set as a solution concept in CGT.

9The remainder of the paper is structured as follows. In section 1, we present the characters and the plot of The Caretaker. In section 2, we present Colman’s analysis of The Caretaker. In section 3, building on the works made by leading Pinter scholars, we challenge the use of The Caretaker as an illustration of the inadequacy of the stable set. In section 4, we argue that The Caretaker better illustrates the notion of the bargaining set. Section 5 concludes.

1. The Characters and the Plot of The Caretaker

10Before introducing the characters and providing the synopsis of the play, it may be helpful to present its basic story structure and its decor.

11The situation in 1950s England is a significant theme of The Caretaker whose story remains on the firm ground of everyday reality. Pinter explores an idea which often features in his plays: “the disruptive intrusion of an outsider … into an established and safe environment” (Jenkins, 1991, 78). In The Caretaker, the struggle for a safe place is indeed a prime motive. “It is obviously a specific play about three individuals and about the idea of a room as a temporary sanctuary from the outside world. But it is also a play about the domestic nature of power and about the shifting alliances we form as part of our survival tactic …” (Billington, 2009, 201).

  • 4 We take all quotations of The Caretaker from Pinter (2000). The play was published and first perfor (...)

12The entire action of The Caretaker takes place in a single location, a room, which Pinter describes as follows: “A room. A window in the back wall, the bottom half covered by a sack. An iron bed along the left wall. … To the right of the window, a mound: a kitchen sink, a stepladder, a coal bucket, a lawn-mower, a shopping trolley, boxes, sideboard drawers. Under this mound an iron bed”4. [The emphasis is ours].

13In The Caretaker, there are just three characters: “the itinerant Davies, the brain-damaged Aston, and the fly, entrepreneurial Mick” (ibid., 196). Mick and Aston are brothers. Mick, who is the younger of the two brothers, is a businessman in his late twenties. He has bought an old house in London for Aston and talks frequently about his projects and desires to expand.

14Aston is a solitary man in his early thirties. When younger, he was put into a mental hospital against his will and given electroshock treatment, the after effects of which he is suffering. He is a slow-witted man who has difficulty ordering his thoughts. He has been given the task by Mick of taking care of and decorating the house properly, but he has trouble finishing projects. He constantly plays with screwdrivers to fix some electrical appliance or other. He dreams of building a shed out in the garden.

15Davies is an old tramp, prone to get involved in quarrels, and filled with race hatred. He is lazy, ill-tempered, bitter, vain and evasive. Procrastination is one of his main characteristics. He is not only homeless and jobless but also has no identity papers: Davies is his real name, but he has been using the name Jenkins for years. Davies is on stage virtually throughout the play.

  • 5 See Appendix A for a more detailed abstract of the play.

16Here is a brief summary5 of the play provided by Howard (1971, 140-141): One night, Aston brings Davies back home, and invites him to stay until Davies can sort out his life. Davies agrees. Then Mick appears and bullies Davies. This is the first act. In the second act, Mick torments Davies and then he changes his tune: as owner of the house, he offers Davies a job as caretaker. In the third act, Davies turns against Aston, and he appeals to Mick, who withdraws support. Then Davies appeals (apparently) in vain to Aston.

17We turn next to Colman’s use of the play to illustrate a notion of CGT, namely the stable set.

2. Colman’s Contribution: The Caretaker and the Stable Set

18Colman (2008, 169-171) examines The Caretaker through the CGT lens. CGT relates to games in which players are free to negotiate binding and enforceable agreements regarding the formation of coalitions (of players) and the division of the payoffs that result from their coordinated actions. CGT addresses two main questions: which coalitions are likely to form? How will the payoff of each coalition be divided among its members? To answer both these questions, CGT relies on the two concepts of coalition structure and characteristic function.

19A coalitional structure is a partition of the set of players N, i.e., a collection of disjoint and nonempty sets whose union is N. Sometimes the players form the grand coalition N, and sometimes several coalitions are formed instead.

In The Caretaker the set of players is Image 1000020000000065000000188BAFF962.png, where A, M and D correspond respectively to the three characters—Aston, Mick and Davies. The set S of the different coalitions includes the following elements:Image 1000020000000020000000189CCF696D.png Image 100002000000001900000018C031497E.png, Image 100002000000001D000000181A46C9E0.png, Image 100002000000001A0000001889B98C62.png, Image 100002000000002E00000018D84E0D59.png, Image 100002000000002C00000018A8CA71D1.png, Image 100002000000003000000018F0D1E010.png, and the grand coalition Image 10000200000000410000001898549F70.png.

The characteristic function is a function Image 10000200000000490000001877ADF453.png, which gives to each coalition its worth, i.e., the payoff that it can obtain if it forms. For instance, Image 1000020000000048000000189FA4DADC.png gives the gain that coalition Image 100002000000002E00000018D84E0D59.png can obtain if it forms.

An imputation is an allocation of the value of the grand coalition Image 1000020000000059000000182AC9A88E.png to all of its members. Formally, an imputation is a list of three real numbers, Image 100002000000001100000018D54F2CCD.png, Image 100002000000001500000018F9C517A9.png, Image 100002000000001300000018E0E99045.png, such that: Image 10000200000000D9000000189BA24CC4.png, and Image 100002000000005100000018614E0374.png, i = A, M, D. The last inequality insures that the imputation is individually rational: no agent can do better by acting alone.

We say that an imputation y dominates an imputation x if there is a coalition S such that: Image 100002000000003500000018B0F09725.png for all s in S and Image 100002000000006B00000018D2385EF5.png. In plain English, an imputation y dominates an imputation x if there is a coalition of players which will be better off by leaving the grand coalition. Now, the core is the set of imputations that are never dominated. To put it another way, an imputation is in the core if no coalition of players would be better off by leaving the grand coalition. In the game in coalitional form that depicts The Caretaker, an allocation is in the core if it satisfies the following inequations:

Image 10000200000000D80000001828964EEB.png,

Image 100002000000009B000000187CF4D90A.png

Image 100002000000009600000018449585CA.png

Image 100002000000009E00000018C4474D76.png

Image 10000000000000D80000001A90C99AEB.png

Colman considers that the core of the cooperative game associated with The Caretaker is empty. On this point, we side with Colman because there are only two single beds in the room that is the setting for The Caretaker. This room cannot therefore welcome three people at the same time. That someone will be excluded is beyond dispute. The only question is who it will be. We can therefore depict The Caretaker as a game in coalitional form in which Image 100002000000007A00000018B3E8D6A6.png. On the other hand, any two-player coalitions can achieve a positive value. For instance, when the coalition Image 100002000000002C00000018A8CA71D1.png forms, it is beneficial to both players, because Aston gains a room-mate who treats him as normal and Davies gains a place to live.

20As pointed out by Colman (ibid., 168), “in view of the fact that many games have empty cores, the core fails to provide a general solution for n-person games in characteristic function form. Von Neumann and Morgenstern (1944) proposed a different concept solution, which also relies on the notion of stability but is more generally applicable than the core. That proposal is called the von Neumann-Morgenstern solution or the stable set”. This set of imputations is such that: “(a) no imputation in the set dominates any other in the set; and (b) every imputation outside the set is dominated by at least one imputation in the set” (Colman, 2008, 169).

  • 6 There is a slight abuse of language here since the stable set is a set of imputations and not of co (...)
  • 7 As pointed out by Colman (2008, 189), there are two drawbacks to the stable set as a solution conce (...)

21Colman (ibid., 171) argues that, in The Caretaker, the “intuitively ‘natural’ von Neumann-Morgenstern stable set” includes the three two-person coalitions: {A, D},{A, M},{M, D}.6 He adds that the notion of stable set is flawed.7 Indeed, the “game illustrates the inadequacy of the stable set because the game goes on in circles and nothing is resolved” (ibid., 171).

Colman (ibid.) establishes this assertion in the following analysis of the play. He observes that in Act I coalition Image 100002000000002C00000018A8CA71D1.png forms. It is beneficial to both players, because A gains a room-mate who treats him as normal and D gains a place to live. But in Act II, the coalition Image 100002000000002C00000018A8CA71D1.png dissolves, and is replaced by the coalition Image 10000200000000300000001896965FF8.png. This coalition is better for D (M is a normal human being and the actual landlord) and for M since D’s allegiance is transferred to M. But the coalition Image 10000200000000300000001896965FF8.png is also unstable and is thus replaced, in Act III, by the coalition Image 100002000000002E00000018D84E0D59.png. Both brothers prefer D to leave. Colman adds: “Davies is quick to realize that the (Mick, Aston) coalition is also unstable, in the sense that both he and Aston would benefit by re-establishing the original (Aston, Davies) coalition and they have the power to impose it” (ibid.). He concludes his analysis by saying that “the curtain goes down with the possibility of the whole cycle of coalition formation repeating itself from the beginning” (ibid.).

22In the next section, we challenge this use of The Caretaker as an illustration of the inadequacy of the stable set.

3. The Caretaker as an Illustration of the Inadequacy of the Stable Set: A Literary Critique

  • 8 It should be noted that the conclusion of Howard’s analysis regarding the possibility of endless ne (...)

23The illustration of the inadequacy of the stable set proposed by Colman can be criticized on literary grounds. Indeed, it is not clear at all that the end of the play suggests the possibility of an endless cycle of negotiations between the characters.8

24Evidence to support this criticism of Colman’s conclusion may be found in Esslin (1970; 2004) and Billington (2009).

  • 9 Pinter’s interview with Tynan.

25First, as pointed out by Esslin, Pinter has revealed the following: “The original idea … was … to end the play with the violent death of the tramp …. It suddenly struck me that it was not necessary. And I think that in this play … I have developed, that I have no need to use cabaret turns and blackouts and screams in the dark to the extent that I enjoyed using them before. I feel that I can deal, without resorting to that kind of thing, with a human situation … I do see this play as merely … a particular human situation, concerning three particular people and not, incidentally … symbols”9 (Esslin, 2004, 249). If Pinter ends The Caretaker with A turning away (“aston remains still, his back to him [=davies], at the window”) rather than with the violent death of D, this is only because he can deal with the human situation he depicts in the play, without resorting to D’s violent death: for Pinter, A's silence is equivalent to D’s death. According to Esslin, at the end of The Caretaker, D is “certain to leave” (Esslin, 1970, 101). The story of the play is therefore simple: an outsider is invited in, fails to get on with the occupant, and is asked to leave. “Indeed, Pinter himself revealed in an article in New Theatre Magazine in 1961: At the end of The Caretaker, there are two people alone in one room, and one of them must go in such a way as to produce a sense of complete separation and finality.” (Billington, 2009, 228).

26Second, as pointed out by Billington, The Caretaker was “triggered, to a greater or lesser degree, by personal experience” (ibid., 306). This play is related to Pinter’s own biography. It has its origin in the specific circumstances of Pinter’s life in Chiswick, South-West London. Pinter and his wife, Vivien Merchant, were living in a clean first-floor flat whose owner was a builder. The owner’s brother lived in the house. He was a handyman, who was able to do only occasional domestic repairs and minor renovations. He was very introverted. He had been in a mental home some years before and had had some kind of electrical shock treatment. His “real name, according to the actor Kenneth Cranham, was Austin” (ibid., 199). He did bring a tramp back one night. After staying in the house some weeks, the tramp was eventually expelled from the house. Thus, the three characters of The Caretaker all had their antecedents in life. Therefore, if The Caretaker is a transcription of experience, something like a slice of Pinter’s life, its end is not an open end: in Chiswick, the expulsion of the tramp was definitive.

27It is debatable that the opinions of literary critics can provide an ultimate criterion to decide whether or not the end of The Caretaker suggests the possibility of an endless cycle of negotiations between the characters, and therefore whether or not this play is, as pointed out by Colman (2008), an illustration of the inadequacy of the stable set. This is to a large degree a matter of opinion and predilection, and readers can and should make up their own minds.

  • 10 See Appendix B for a justification of this assertion.

It is undebatable, however, that, for an endless cycle of negotiations to appear, the last coalition Image 1000020000000027000000153A94F6FA.png must be challenged. This is unlikely, or at least difficult, since a careful reading of the play indicates that we should have Image 1000020000000088000000150DC498E7.png, Image 100002000000008C00000015106FD15D.png, because the fraternal bond of understanding that A and M share is stronger than the bonds between D and either of them.10

28The above arguments show that The Caretaker may be an unconvincing example to illustrate the inadequacy of the notion of stable set by suggesting that the coalitions in this set may all be unstable. We argue, by contrast, that The Caretaker is a convincing example to illustrate the adequacy of the notion of bargaining set by suggesting that, in a context where the core is empty, the negotiation cycle can come to an end in a finite time.

4. The Caretaker as an Illustration of the Bargaining Set

29In CGT there are many solution concepts, but for understanding the ambiguous ending of The Caretaker a key feature of the bargaining set–it does not presume that the grand coalition (the coalition of all players) always forms–makes it particularly relevant as an alternative to the stable set.

30The bargaining set captures the idea that for an agreement to be stable, any objection to this agreement must be met by a counter-objection (whose existence thus deters the first objection).

From a formal viewpoint, these ideas of objection and counter-objection can be understood as follows. Consider a payoff configuration Image 100002000000005700000018EC4643E7.png namely a sharing of the gains realized by different coalitions, such that Image 10000200000000970000001852BDE92E.png, Image 100002000000005B0000001827E21862.png, Image 100002000000009800000018FAD94C84.png We say that M has an objection to this payoff configuration if there is another payoff configuration Image 100002000000005800000018E51FCD61.pngsuch that: Image 100002000000009B00000018F77A222C.png, Image 100002000000005900000018E6F9B87D.png, and Image 1000020000000040000000184C56D423.png, and Image 100002000000005F000000189DA97BDE.png Now we say that A has a counter-objection, if there is a payoff configuration Image 10000200000000500000001835B4FC06.png such that: Image 100002000000009100000018C12DABEB.png, Image 100002000000005F00000018215A19E3.png, Image 1000020000000038000000185BFE24B3.png and Image 100002000000003B00000018825477B9.png. In that case, it is unlikely that M will object Image 10000200000000540000001888D29E29.png in the first place.

  • 11 Interestingly, the bargaining set is always non-empty, even if the core is empty.

31The bargaining set is the set of payoff configurations that satisfy the following property: whenever any agent belonging to a coalition has an objection to another member of the coalition, this last member has a counter-objection to the first agent. Thus, the payoff configurations belonging to the bargaining set are unlikely to be challenged.11

  • 12 Actually, if a coalition does form, it should stand since by assumption agreements are binding in C (...)

32As pointed out by Colman (2008, 170), in The Caretaker each of the three two-person coalitions forms and dissolves. This does not, however, imply that, e.g., in Act II, D has an objection (according to the definition introduced above) to A.12 Nor does this imply that in Act III A has a counter-objection to D. The play only displays a bargaining process between the three characters.

33Experiments conducted using games similar to the game in coalitional form that depicts The Caretaker indicate that the results of bargaining processes in such a game are obtained in a finite time and are very close to the payoff configurations in the bargaining set (Maschler, 1978, and Kahan and Rapoport, 1984, see also Maschler et al., 2013, 783).

34The fact that The Caretaker ends when the coalition {A, M} forms suggests, as shown in experiments, that the brothers have agreed to a payoff configuration (or an approximation thereof) belonging to the bargaining set and that the negotiation period is terminated. The precise formulae for the payoff configuration can be found in the appendix C.

  • 13 More precisely we assume that: v({A}) = v({M}) = v({D}) = v({A, M, D}) = 0, v({AM}) ≥ v({AD}) ≥ (...)
  • 14 It can be shown that the bargaining set is given by: (13,7,0), (13,0,3), (0,7,3), where each payoff (...)

As an example of such payoff configuration, assume that Image 10000000000001170000001819B9ABD1.png, Image 100002000000007200000018902864B1.png.13 It is easily verified that, in this game, the core is an empty set and the payoff configuration in the bargaining set associated to the coalition {AM} is (13,7).14 For the coalitional structure ({AM}, {D}) the imputation (13, 7, 0) is actually the only imputation in which every objection of one player against another player in the same coalition can be met with a counter-objection. Interestingly, A receives the highest part of the configuration.

Conclusion

35In this paper, which still leaves plenty of room for further debate about The Caretaker, we have challenged the use by Colman (1988) of this play to illustrate the inadequacy of the stable set, a solution concept for CGT introduced by von Neumann and Morgenstern (1944). We have seen that this illustration relies on the idea that the play seems to have a circular configuration, the opening and the ending reflecting each other. This idea, however, is in line neither with the comments on the play made by Pinter scholars nor with our definition of the different characteristic functions. In contrast, we have argued that The Caretaker better illustrates the adequacy of the bargaining set, a solution concept of CGT proposed by Aumann and Maschler (1961; 1964).

36There are three natural topics for further research. Firstly, to shed new light on the formation of coalitions in The Caretaker, a natural line of reasoning to pursue could consist in using non-cooperative (rather than cooperative) solution concepts (for a survey of the relevant literature on this subject, see, e.g., Bloch, 1997, and Ray, 2007). Secondly, it would be interesting to see if other Pinter plays could be used to illustrate the notion of bargaining set. This is likely to be the case, since, as we have mentioned above by quoting Jenkins (1991), the disruptive intrusion of an outsider in an established and safe environment features often in Pinter’s plays. Lastly, we may also rely on the bargaining set to study dramas in which no cooperation between all the characters is possible, meaning that the core is empty.

Appendix

A) The Summary of The Caretaker

37The play opens, after a brief and silent inspection of the room by Mick, with Aston bringing in Davies, whom he has rescued from a fight at some cafe where Davies was working. Aston invites Davies to stay with him for a few days until he gets fixed up or until he can sort out his life. He gives him a bit of tobacco for his pipe (p.3), a pair of good shoes (p. 13), and five shillings (p.23). The next morning Aston goes out. He gives Davies the keys of the house (p.33), so that Davies doesn’t have to leave just because Aston goes out. The final scene of Act I is violent: after Aston is gone, Mick slides into the room, and treats Davies as though he is a burglar.

38Act II takes up the action immediately at the point where it left off at the end of Act I. Mick cross–examines Davies, alternating between brutality and politeness. He says that Davies can rent the room if he wants. Then Aston returns with a bag of Davies’s belongings. Mick grabs the bag repeatedly so that it passes ceaselessly between the three characters. Mick leaves again. Aston tells Davies that he is refurbishing the room for Mick whose house it is (p.62). Davies now knows that ownership and power are lodged with Mick, not with Aston. Then, Aston reveals that the bag is not really Davies’s: he has bought another bag for Davies together with a bundle of clothes. Finally, Aston offers Davies a job as caretaker (p.65). The same evening, Davies returns to the room. He tries to switch on the light, but it does not work, because Mick, who is using a vacuum cleaner, has plugged it into the lamp socket. When the light is on again, Davies is discovered holding a knife in his hand, ready to strike any attacker. But Mick has become polite and considerate to Davies. He asks Davies’s advice concerning Aston’s slowness. A little after that, he offers Davies the position of caretaker (p. 80), and he tells Davies that he is the owner of the house (p.82). Mick’s offer to Davies of the job as caretaker parallels the offer Aston has already made, except that Mick wants references. Davies mentions having plenty of references in Sidcup. The next morning, Aston, in the longest speech of the play, tells Davies the story of his life. He reveals that he once received electric-shock treatment in a mental hospital. Aston depicts himself as a helpless victim, betrayed not only by an uncaring society, but also by his mother who gave permission for the operation to be performed. As Aston finishes his speech, the curtain falls on Act II.

39As Act III opens–two weeks later–Davies is full of complaints against Aston, delivering them to Mick. Aston mentions that he and Mick could make a success of the flat together but Mick remarks that he and Aston will live there (p.97). Aston returns. He has a pair of shoes for Davies, but Davies complains that they do not fit properly. Mick exits. Next the relationship between Aston and Davies deteriorates. Davies turns against Aston. He even menaces Aston with a knife. Aston eventually says that the time has come when Davies should find somewhere else to live (p.108). Davies retaliates that it might be Aston who will have to go. Davies goes out. The same evening, Davies returns in Aston’s absence and denounces Aston to Mick. Davies suggests that Aston should go back to the mental hospital. When Mick hears about these developments he springs to his brother’s defense, shouts at Davies, and finally tells him to go. Aston returns. Aston and Mick exchange smiles. These smiles suggest a bond of understanding between the two brothers: Mick assumes fatherly responsibility for Aston. Then Mick leaves the room and Davies makes an attempt to regain Aston's favor, but Aston is adamant. The play ends with Davies pleading for the home he has lost, while Aston stands by the window with his back turned to Davies.

B) The Characteristic Functions in The Caretaker

40“The particular needs being dramatized in The Caretaker attach to the characters as follows: Davies needs a refuge; Aston needs mental and physical order; and Mick needs an outlet for his bourgeois aspirations” (Wyllie and Rees, 2017, 97).

Each player alone cannot realize his dream. We have therefore Image 10000200000000E900000018B3D72803.png.

41The whole dynamic of the play is generated by the way in which the different characters/dreams marry or clash with each other.

42Despite their lack of communication, Aston and Mick share a bond of understanding. This emerges, for an example, in a brief scene in Act III (p. 97) between Mick and Davies, when they are discussing the improvements which can be made in the room:

mick: It wouldn’t be a flat it’d be a palace.

davies: I’d say it would, man.

mick: A palace.

davies: Who would live there?

mick: I would. My brother and me.

Pause.

davies: What about me?

mick: (Quietly.) All this junk here. It's no good to anyone....

Aston and Mick act in tandem with each other. If the Image 1000020000000027000000153A94F6FA.png coalition forms, both Aston and Mick realize their dreams.

43For both Aston and Davies, the world beyond the room is dangerous and threatening. Aston was subjected to the horror of the mental hospital, and he knows that, as pointed out by Davies in Act III (p.107), it can start again at any time: “They can put the pincers on your head again, man! They can have them on again! Any time”. The life of Davies is filled with a lot of menaces. For instance, he is frightened that people representing authority might question him about having only four stamps on his false insurance card. In Act I (p.25), Davies describes this fear to Aston in the following terms:

Aston: What name have you been going under?

davies: Jenkins. Bernard Jenkins. That’s my name. That’s the

name I’m known, anyway. But it’s no good me going on

with that name. I got no rights. I got an insurance card

here.

He takes a card from his pocket.

Under the name of Jenkins. See? Bernard Jenkins. Look.

It’s got four stamps on it. Four of them. But I can’t go

Along with these. That’s not my real name, they’d find

out. They’d have me in the nick.

Davies, who is constantly seeking competitive advantage, is locked into tactical struggles from which Aston is isolated. Aston and Davies act in tandem with, rather than in opposition to, each other. If the Image 10000200000000250000001534A4F7F9.png coalition forms, both Aston and Davies can triumph over their fears. Davies gains a (temporary) place to live, and Aston gains someone to talk to about his fears.

44Mick resents Davies’ s entrance into the room, because the room which Davies has entered without Mick’s consent is Mick’s room, and the bed in which Davies sleeps is Mick’s bed. In Act II, Mick emphasizes these facts many a time (for instance, p. 50 and p. 52):

mick: How do you like my bed?

Pause.

That’s my bed. You want to mind you don’t catch a

draught

davies: From the bed?

mick: No, now, up your arse.

mick: I’m afraid you’re a born fibber, an’t you? You’re

speaking to the owner. This is my room. You’re

standing in my house.

davies: It’s his … he seen me all right … he …

mick: (pointing to daviesbed) That’s my bed.

Davies is presenting a threat to Mick’s relationship with Aston. Mick prefers nobody (neither he nor Aston) to support Davies, but, if Davies is supported, Mick prefers to do it himself instead of having Aston do it. Davies prefers Mick to Aston because Mick is the legal landlord.

Davies and Mick act in opposition to, rather than in tandem with, each other. If the Image 100002000000002800000015D429E4B3.png coalition forms, Davies gains a (permanent) place to live, but he puts himself at the mercy of Mick, and Mick gains allegiance from Davies, but he does not realize his dream of sharing the room with his brother.

It seems to us, therefore, that we have the following inequations: Image 10000200000000C500000018060EC5EA.png, and Image 10000200000000C9000000189215EC56.png.

There are only two single beds in the room that is the setting for The Caretaker. We have therefore Image 100002000000007A00000018B3E8D6A6.png.

C) The Bargaining Set of Three-Player Games

  • 15 See, e.g., Aumann and Maschler (1964).

Assume as in Appendix B that Image 10000200000000E500000018BBB2DCEF.png, Image 10000200000000C500000018060EC5EA.png, and Image 10000200000000C9000000189215EC56.png, and Image 100002000000007A00000018B3E8D6A6.png. In addition, suppose that Image 10000200000000FE000000182C0EF852.png. Then the payoff configurations of the bargaining set are:15

  • Image 100002000000003200000018376D945A.png (if there is no coalition with more than two players),

  • Image 100002000000016D00000022A25FB6C4.png if coalition Image 100002000000002E00000018D84E0D59.png forms,

  • Image 100002000000016B00000022FE975C39.png if coalition Image 100002000000002C00000018A8CA71D1.png forms,

  • Image 100002000000016B00000022775D3F5C.png if coalition Image 100002000000003000000018F0D1E010.png forms.

Second, assume that: Image 10000200000000FE000000180A3971AF.png then the bargaining set is given by:

  • Image 100002000000003200000018376D945A.png (if there is no coalition with two players),

  • Image 100000000000020F0000001A145F362F.png if coalition Image 100002000000002E00000018D84E0D59.png forms,

  • Image 100002000000006F00000018C5CB6555.png if coalition Image 100002000000002C00000018A8CA71D1.png forms,

  • Image 100002000000007600000018235C6FA3.png if coalition Image 100002000000003000000018F0D1E010.png forms.

We are grateful for comments received on an earlier version of this paper that was presented at the workshop “The limits and possibilities of narrative explanations”, Berlin, Wissenschaftskolleg, 2016, March 18th, co-organized by Michael Gordin and Philippe Mongin. We also thank Susan Crettez and two referees for very helpful remarks on this paper.

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Bibliographie

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Aumann, Robert J. and Michael Maschler. 1964. The Bargaining Set for Cooperative Games. In Melvin Dresher, Lloyd S. Shapley and Albert W. Tucker (eds), Advances in Game Theory (Annals of Mathematical Studies, 52: 443-476). Princeton, NJ: Princeton University Press.

Billington, Michael. 2009. Harold Pinter. London: Faber and Faber Limited.

Birchler, Urs and Monika Bütler. 2007. Information Economics. London: Routledge.

Bloch, Francis. 1997. Non-Cooperative Models of Coalition Formation in Games with Spillovers. In Carlo Carraro Carlo and Domenico Siniscalco (eds), New Directions in the Economic Theory of the Environment, 311-352. Cambridge: Cambridge University Press.

Brams, Steven J. 1994a. Game Theory and Literature. Games and Economic Behavior, 6: 32-54.

Brams, Steven J. 1994b. Theory of Moves. Cambridge, MA: The MIT Press.

Brams, Steven J. 2011. Game Theory and the Humanities: Bridging two worlds. Cambridge, MA: The MIT Press.

Chwe, Michael Suk-Young. 2009. Rational Choice and the Humanities: Excerpts and Folktales. Occasion: Interdisciplinary Studies in the Humanities, 1(October 15): 1-15.

Chwe, Michael Suk-Young. 2013. Jane Austen, Game Theorist. Princeton, NJ: Princeton University Press.

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Crettez, Bertrand and Régis Deloche. 2013. On Experimental Economics and the Comparison Between the Last Two Versions of Molière’s Tartuffe. Journal of Economic Behavior & Organization, 87: 66-72.

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Notes

1 In this connection, three points are noteworthy. First, non-cooperative game-theoretic analyses of plays have been made not only by economists but also by literary critics (see, e.g., de Ley, 1988). Second, it is not just standard game theory that has been applied to literature: the theory of moves (Brams, 1994b) and the psychological game theory (Oguer, 2010) have been applied to different strategic situations in fiction. Third, experimental economics has also been used to shed new light on a debate among literary scholars (Crettez and Deloche, 2013).

2 Other plays by Harold Pinter have been analyzed from the perspective of game theory. In Birchler and Bütler (2007, section 8F), Betrayal is used to illustrate the importance of common knowledge in many situations of social life. Pintu (2015) shows that The Birthday Party depicts man’s life in society as a game.

3 In his Nobel acceptance speech, Pinter provides some insight into his working method as a playwright: “I always start a play by calling the characters A, B and C.”

4 We take all quotations of The Caretaker from Pinter (2000). The play was published and first performed in 1960.

5 See Appendix A for a more detailed abstract of the play.

6 There is a slight abuse of language here since the stable set is a set of imputations and not of coalitions.

7 As pointed out by Colman (2008, 189), there are two drawbacks to the stable set as a solution concept. The first drawback is relating to existence: some games have no stable set (as shown by the complex example in Lucas, 1969). The second drawback is relating to uniqueness: other games have more than one stable set.

8 It should be noted that the conclusion of Howard’s analysis regarding the possibility of endless negotiations between characters is ambiguous. On the one hand, Howard points out the following: “In the end, the outcome ‘no relationships’ with which the play began, is apparently stable, though in act 1, it was not stable against the coalition of Aston and Davies. This coalition will not now form, however, as Aston will no longer trust Davies, despite Davies’s offer to go further than before and be loyal to Aston” (Howard, 1971, 144). On the other hand, he highlights “the circularity of the whole process, whereby the final outcome is the same as the first” and the possibility that the three acts can “be repeated in sequence again and again” (ibid., 145).

9 Pinter’s interview with Tynan.

10 See Appendix B for a justification of this assertion.

11 Interestingly, the bargaining set is always non-empty, even if the core is empty.

12 Actually, if a coalition does form, it should stand since by assumption agreements are binding in CGT.

13 More precisely we assume that: v({A}) = v({M}) = v({D}) = v({A, M, D}) = 0, v({AM}) ≥ v({AD}) ≥ 0, v({AM}) ≥ v({DM}) ≥ 0. These conditions are justified in Appendix B.

14 It can be shown that the bargaining set is given by: (13,7,0), (13,0,3), (0,7,3), where each payoff vector is given in the order (A, M, D).

15 See, e.g., Aumann and Maschler (1964).

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Référence papier

Bertrand Crettez et Régis Deloche, « Harold Pinter’s The Caretaker and Cooperative Game Theory: A Reappraisal », Œconomia, 8-1 | 2018, 75-90.

Référence électronique

Bertrand Crettez et Régis Deloche, « Harold Pinter’s The Caretaker and Cooperative Game Theory: A Reappraisal », Œconomia [En ligne], 8-1 | 2018, mis en ligne le 01 mars 2018, consulté le 22 septembre 2018. URL : http://journals.openedition.org/oeconomia/2906 ; DOI : 10.4000/oeconomia.2906

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Auteurs

Bertrand Crettez

Université Panthéon-Assas, Paris II, CRED. Bertrand.crettez@u-paris2.fr

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Régis Deloche

Université Paris Descartes, Paris V, LIRAES. regis.deloche@parisdescartes.fr

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