Jacques Lacan and game theory: an early contribution to common knowledge reasoning

Abstract

Lacan’s contribution in applying and promoting game theory in the early 1950s is mostly ignored in the history of game theory. Yet his early analyses of logical reasoning made him one of the first social scientists to consider the importance of the hypothesis of common knowledge. By retracing Lacan's path in his discovery of game theory, we show how much he has been a precursor in applying it. While accommodating a narrative approach, he demonstrated rigour and originality. Soliciting mathematicians open to interdisciplinarity, he introduced as early as 1945 modes of reasoning which corresponds to reasoning based on common knowledge.

Keywords:

• Lacan
• common knowledge
• enigma of the three prisoners
• la Lettre Volée
• Interdisciplinarity

JEL CODES:

• B16
• C72

Acknowledgements

We thank the editor and our three anonymous referees for their excellent and constructive comments. We would also like to present our gratitude to Denis Bouyssou, Jean-Louis Chassaing, Bertrand Crettez, Régis Deloche, Antoine Missemer, Olivier Musy, Emmanuel Petit, Philippe Quirion, Gilles Rotillon, Elisabeth Roudinesco, Christian Schmidt, Philippe Solal, and Manon Thobois for their inspiring and insightful remarks on earlier versions of this paper. T. Tazdaït acknowledges the support from Campus France (Cèdre research project No. 42315RA). Obviously, all the remaining errors are ours.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 “Hungary’s famed father of the modern computer, John von Neumann, and his family were on very good terms with Ferenczy, and psychoanalysis was as natural a topic at dinner as economics or mathematics” (Meszaros 2012, 80).

2 Note that it was in order to break out of his isolation that Lacan created in 1953 a public seminary which was held at the Sainte-Anne hospital until 1963 and which, with the help of the philosopher Louis Althusser, was next transferred to the Ecole normale supérieure in the rue d'Ulm. It was later called the “Lacan’s Seminar”.

3 This game is to be distinguished from the “problem of three prisoners” which happens to be a bayesian problem introduced by Frederick Mosteller (1965).

4 Note that this sentence does not appear in the original 1945 version. Lacan added it in the version which appears in his main theoretical work published in 1966, Écrits. We can think that it is after having noticed the link between his article and game theory that he added it.

5 In the French version, see Lacan (1966, 197–198).

6 In the French version, see Lacan (1966, 198).

7 For the Lacan’s proof, see Annex.

8 In the French version, see Lacan (1966, 212–213).

9 We thank Philippe Solal and Bertrand Crettez for making us aware of the existence of Billot’s paper (2008).

10 In the French version, see Lacan (1966, 205).

11 On page 917 of the French version of Écrits published in 1966, Lacan notes “Le temps logique et l'assertion de la certitude anticipée. This paper was written in March 1945. Published in the Cahiers d'Art: 1940-1945”. In the English translation, we find this mention on page 864.

12 Others such as Michel Plon (1976) and Nathalie Charraud (1997) refer also to Lacan's enigma using the 1966 quote and make no mention of the original 1945 version. Jean-Pierre Cléro (2008) even refer to it quoting the 1999 edition of Écrits.

13 See, among others, Robert Bush (1963), Harrison White (1963), Philippe Courrège (1965), Russell Reid (1967) and John Boyd (1969).

14 This reference corresponds to a letter from Lévi-Strauss dated 15 March 1951.

15 This corresponds to a letter from Lévi-Strauss dated 9 January 1952.

16 Note that Schützenberger obtained a doctorate in medicine in 1949 and a thesis in mathematics in 1953 (under the title “Contributions aux Applications Statistiques de la Théorie de l’Information”, i.e. “Contributions to Statistical Applications of Information Theory”). This explains why some of his early work were published in medical journals.

17 Add that Lacan later refined his position to argue that “the unconscious is structured like a language” in his seminar on January 22, 1964 (Lacan, 1998 [1973], 20).

18 For example, Mandelbrot co-edited the book Etudes d’Epistémologie Génétique vol. II with Piaget and the philosopher Leo Apostel (Apostel et al. 1957). Schützenberger contributed with the linguist Noam Chomsky to the new field of free language context by developing the famous Chomsky-Schützenberger theorem (Chomsky and Schützenberger 1963).

19 It is interesting to note that at the same time, social scientists and mathematicians in the United States developed similar interdisciplinary research initiatives. For example, in 1952, in Santa Monica, a two-month seminar supported by the Ford Foundation, the RAND Corporation, the Office of Naval Research and the Cowles Commission, was devoted to “The Design of Experiments in Decision Processes” on the initiative of Clyde Coombs, a psychologist, and the mathematician Robert M. Thrall. This seminar brought together psychologists (as, among others, Leon Festinger, David Beardslee and William Estes), mathematicians (as for instance, von Neumann, John Nash, Merrill M. Flood and Frederick Mosteller and Lloyd S. Shapley) and economists (as among others, Morgenstern, Jacob Marschak, Howard Raiffa, Gérard Debreu, Roy Radner and Herbert Simon) (Thrall, Coombs, and Davis 1954). Guilbaud, Mandelbrot, Riguet and Schützenberger were all the more in favour of interdisciplinarity that they were familiar with these different initiatives they took as examples to be followed.

20 None of Lévi-Strauss projects were successful (with Schützenberger and Guilbaud) or took shape (with Mandelbrot and Riguet). For example, in his letter of March 19, 1951, in which he mentions his collaboration with Schützenberger, he states: “we have difficulty finding a common language” (Jakobson and Lévi-Strauss 2018, 137).

21 Cited in Denis Bertholet (2008, 275).

22 This report was primarily a response to the criticisms addressed by some of his colleagues at the Société Psychanalytique de Paris (Paris Psychoanalytical Society; SPP hereafter) about his practice of psychoanalysis with patients. He eventually resigned from the SPP in June 1953 and joined a new institution called the Société Française de Psychanalyse (French Society of Psychoanalysis), created the same year by two other resignees, Daniel Lagache and Françoise Dolto. He will be followed by André Berge who, by the greatest coincidence, will see at the same time, his son Claude Berge (1953a) defend a doctorate in mathematics entirely devoted to game theory entitled Sur une Théorie Ensembliste des Jeux Alternatifs (“On a Set Theory of Sequential Games”). The thesis will be immediately published as it stands in the Journal de Mathématiques Pures et Appliquées (Berge, 1953b).

23 In the French version, see Lacan (1966, 287).

24 In the French version, see Lacan (1966, 287).

25 Behind the notion of mixed strategies is the idea that each individual seeks to protect himself by preventing the other from guessing his intentions.

26 The book was initially published in English in 1954 under the title Compleat Strategyst, Being a Primer on the Theory of Games of Strategy, by McGraw Hill and then in 1966 by Dover Publications, Inc., New York.

27 This is true for almost all authors quoting the seminar on “La Lettre Volée”, including Plon (1976), Charraud (1997) or Cléro (2008). Note that Daniel Read (2020) refers to the seminar on “La Lettre Volée” quoting a paper published in 1972. This corresponds to a translated version published in the journal Yale French Studies. Read says little about the content of the paper and only states that: “Lacan's analysis does not resemble game theory as it currently stands, and is not described here (in fact, I could not describe it)” (Read 2020, 399). It is clear that if Lacan's work is presented in these terms one cannot see what makes it original, and this does not help the diffusion of his ideas.

28 Only Deloche and Oguer (2006, p. 99) install Lacan in the literature by evoking the odd and even game by writing: “Guilbaud [1949; 1954, Ch. Ill; 1997], Lacan [1966], Davis [1970] and Brams [1994a; 1994b] analyzed this game”, but without specifying what makes the originality of his study and by referring (as it is usual) to the 1966 work.

29 This book published in 1933 devoted a short section to “La Lettre Volée” on pages 601–602.

30 It was much later, during his seminar on March 10th, 1965, that Lacan, without mentioning the name of Marie Bonaparte returned to describe her study of Poe's short story as being “pseudo-analytic lucubrations” (Lacan, 1965, 115). However, we cannot ascertain that this assessment is worth of interest as Lacan's view of Marie Bonaparte was negatively biased. Lacan always resented her for being among those who criticized his practice of psychoanalysis and thus contributed to his departure from the SPP in 1953.

31 In the French version, see Lacan (1966, 14).

32 In the French version, see Lacan (1966, 58).

33 It is through the child prodigy's mode of reasoning that Guilbaud and Schützenberger became interested in “La Lettre Volée”, and this can easily be understood given the mathematical dimension underlying it. Nevertheless, it should also be noted that the translator from Poe in France, Charles Baudelaire, helped to highlight this way of thinking. Indeed, having been himself fascinated by this mode of reasoning, he echoed it by repeating word for word the child's argument in his work published (posthumously) in 1869, L'Art Romantique (see pages 274–275).

34 Lacan sees in Poe “a fine precursor of research into combinatorial strategy” (2006a [1966], 46).

35 In the French version, see Lacan (1966, 58).

36 In the French version, see Lacan (1966, 58).

37 Note that explicitly motivating their study on Poe’s child prodigy, Eliaz and Rubinstein (2011) found experimentally a higher probability of winning for the guessing player

38 Read (2020) provides an overview of the different analyses of the child prodigy argument that have been proposed since the 1970s.

39 In the French version, see Lacan (1966, 30).

40 In the French version, see Lacan (1966, 860).

41 Again, and conforming with all authors quoting Lacan’s work on the topic, both Lasry and Levine did not refer to Lacan original works but to Écrits.

42 In the French version, see Lacan (1966, 199–201).