Abstract
This book reports both internal and external investigations into game theory. Internally, game theory’s tacit modelling assumptions were investigated carefully representing and interpreting epistemic characterisation results. Externally, game theory’s connections to economic consultancy or normative advice and to descriptions and explanations of rational economic agency were studied, and it emerged that the first connection is meaningless and that the second assumes a particularly narrow epistemology on the part of the modelled individuals. This chapter continues the external study. It is devoted to a view of scientific methodology that declares economic theory, if true at all, only true ‘in the abstract’. In that view, game theory does not so much study strategic interaction of economic agents reasoning on the basis of beliefs, desires and rationality principles, as relate it to a field of vaguely adumbrated ‘truths-in-the-abstract’.
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- 1.
Mary Ann Dimand and Robert Dimand , A History of Game Theory: Volume I: From the Beginnings to 1945 (London: Routledge, 1996) shed interesting light on the origins of game theory prior to von Neumann and Morgenstern ’s opus magnum (its sequel, dealing with postwar game theory, will be published in 2009). Also see E. Roy Weintraub (ed.), Towards a History of Game Theory (ann. supp. to History of Political Economy) (Durham, NC: Duke University Press, 1992).
- 2.
Morton Davis , Game Theory: A Nontechnical Introduction (1970; rev. ed. 1983; repr. 1997, Mineola: Dover), Drew Fudenberg and Jean Tirole , Game Theory (Cambridge, Mass.: MIT Press, 1991), Roger Myerson , Game Theory: Analysis of Conflict (Cambridge, Mass.: Harvard University Press, 1991), Martin Osborne , An Introduction to Game Theory (New York: Oxford UP, 2003), id. and Ariel Rubinstein , A Course in Game Theory (Cambridge: MIT Press, 1994). Cf. Eric Rasmusen , Games and Information: An Introduction to Game Theory (Malden: Blackwell, 1989; 4th ed. 2006).
- 3.
The Royal Swedish Academy of Sciences, press release, 11 October 1994, http://nobelprize.org/nobel_prizes/economics/laureates/1994/press.html (accessed 11 September 2008).
- 4.
‘Correlated Equilibrium as an Expression of Bayesian Rationality’, Econometrica, 55 (1987), 1.
- 5.
The Nash equilibrium does not fare well empirically in games with multiple equilibria and in situations in which inexperienced players play the game only once. If individuals gain experience playing games with unique Nash equilibria, they tend to converge to playing Nash equilibrium strategies over time. Evolutionary game theory tries to explain convergence as a result of trial-and-error tactics. See Larry Samuelson , Evolutionary Games and Equilibrium Selection (Cambridge, Mass.: MIT Press, 1997). On experimental results see Colin Camerer , Behavioral Game Theory (Princeton: Princeton University Press, 2003), Andrew Colman , ‘Cooperation, Psychological Game Theory, and Limitations of Rationality in Social Interaction’, Behavioral and Brain Sciences, 26 (2003), 139–198, and Douglas Davis and Charles Holt , Experimental Economics (Princeton: Princeton University Press, 1993).
- 6.
Several highly respected game theorists have contributed to both programmes, and I therefore define NERP and the Epistemic Programme in terms of content and methods, not people. Furthermore, finding mathematics-driven mathematisation in the contributions of a certain author does not entail that his complete oeuvre betrays overmathematisation. This is especially true of Reinhard Selten , who has shown a deep and sincere interest in empirical work, witness, e.g., his founding in 1984 of the Experimental Economics Laboratory at the University of Bonn, Germany (the first in Europe, it is claimed). See also Reinhard Selten , ‘Comment [on Aumann’s ‘What is Game Theory Trying to Accomplish?’]’, in K. Arrow and S. Hohkapohja (eds.), Frontiers of Economics (Oxford: Blackwell, 1987), 77–87.
- 7.
See, e.g., Camerer , op. cit., Samuelson , op. cit., and Jacob Goeree and Charles Holt , ‘Stochastic Game Theory: For Playing Games, Not Just for Doing Theory’, Proceedings of the National Academy of Sciences, 96 (1999), 10564–10567.
- 8.
Nicola Giocoli , Modeling Rational Agents: From Interwar Economics to Early Modern Game Theory (Cheltenham: Edward Elgar, 2003), studying the history of postwar economic theory, suggests that game theory does not aim at empirical adequacy.
- 9.
See, e.g., Christina Bicchieri , ‘Rationality and Game Theory’, in A. Mele and P. Rawling (eds.), The Handbook of Rationality (Oxford: Oxford University Press, 2003), 182–205, Boudewijn de Bruin , ‘Game Theory in Philosophy’, Topoi, 24 (2005), 197–208, Zachary Ernst , ‘Explaining the Social Contract’, British Journal for the Philosophy of Science, 52 (2001), 1–24, Francesco Guala , ‘Has Game Theory been Refuted?’, Journal of Philosophy, 103 (2006), 239–263, Daniel Hausman , ‘Testing Game Theory’, Journal of Economic Methodology, 12 (2005), 211–223, Hans Jörgen Jacobsen , ‘On the Foundations of Nash Equilibrium’, Economics and Philosophy, 12 (1996), 67–88, Harold Kincaid , ‘Formal Rationality and its Pernicious Effects on the Social Sciences’, Philosophy of the Social Sciences, 30 (2000), 67–88, Steven Kuhn , ‘Reflections on Ethics and Game Theory’, Synthese, 141 (2004), Philip Mirowski , ‘When Games Grow Deadly Serious: The Military Influence on the Evolution of Game Theory’ in C. Goodwin (ed.), Economics and National Security: A History of their Interaction (ann. supp. to History of Political Economy) (Durham, NC: Duke University Press, 1991), 227–256, Ahti-Veikko Pietarinen , ‘Games as Formal Tools versus Games as Explanations in Logic and Science’, Foundations of Science, 8 (2003), 317–364, Mathias Risse , ‘What is Rational about Nash Equilibria?’, Synthese, 124 (2000), 361–384, Julius Sensat , ‘Game Theory and Rational Decision’, Erkenntnis, 47 (1998), 379–410.
- 10.
‘Building Economic Machines: The FCC Auctions’, Studies in History and Philosophy of Science Part A, 32 (2001), 453–454.
- 11.
‘On the Definition of Political Economy; and on the Method of Philosophical Investigation in that Science’, London and Westminster Review, 26 (1836), 1–29; citations are from the 1844 ed. repr. in ibid., Collected Works of John Stuart Mill, ed. J. Robson (Toronto: University of Toronto Press, 1967), 309–339. On what follows, see pp. 324, 325 and 330.
- 12.
‘Comments on the Interpretation of Game Theory’, Econometrica, 59 (1991), 909. Other relevant papers include ‘A Subjective Perspective on the Interpretation of Economic Theory’, in A. Heertje (ed.), The Makers of Modern Economics: Volume 1 (New York: Harvester Wheatsheaf, 1993), 67–83, and ibid., ‘Joseph Schumpeter Lecture: A Theorist’s View of Experiments’, European Economic Review, 45 (2001), 615–628.
- 13.
Ibid.
- 14.
Ibid. 919.
- 15.
‘What is Game Theory Trying to Accomplish?’, in K. Arrow and S. Honkapohja (eds.), Frontiers of Economics (Oxford: Blackwell, 1987), 28–76. On what follows, see pp. 25, 34, 36, 39 and 41–42.
- 16.
I doubt whether it is genuinely possible to classify situations without behaviour. A first possibility would be to describe only the actions available to the agents, their utilities, and so on; this would be a description of the game-playing situation minus the actions actually performed. Different situations could be distinguished game-theoretically along (precisely) three dimensions (for normal form games: the (number of) players, the (number of) actions, and the utility functions), and along a fourth dimension (for extensive games: tree structure). Yet no solution concept is involved, and hence it is questionable whether such classificatory applications are genuinely game-theoretic. A second possibility would be to classify games also along the dimension of solution concepts, or at least to let solution concepts play a significant role in the classification. To be a description of situations without behaviour, solution concepts would have to be used to show possible ways of behaving (e.g., asymmetries in the game, socially suboptimal but individually rational outcomes, coordination problems about multiple Nash equilibria, etc.). Reasonable as this may sound, I doubt that the second option really makes sense without a commitment to some theory of action and motivation. An outcome can only be described as socially suboptimal and individually rational if you have some conception of what it means to be a motivation for a player to perform some action rather than another, and you can only speak about coordination problems if players envisage their choice situation in a way that is sufficiently similar to the description of the game.
- 17.
Rubinstein , art. cit., 919 ironically attributes his characterisation of application as art to Aumann’s 1987 paper. Note furthermore the remark about falsifiability.
- 18.
‘Modeling Rational Players: Part I’, Economics and Philosophy, 3 (1987), 152.
- 19.
‘The Science of Complexity: Epistemological Problems and Perspectives’, Science in Context, 18 (2005), 488.
- 20.
Art. cit. 35.
- 21.
Ibid. The implicit reference is to Herbert Simon , Reason in Human Affair (Stanford: Stanford University Press, 1983), but an explicit bibliographic reference to Simon is lacking in Aumann’s paper. This quotation serves here mainly to illustrate introversion. Yet an inconsistency is lurking in Aumann’s argument. His main objection to Simon’s satisficing is that it ‘ties together’ much less than the principle of maximisation of expected utility. Of course, satisficing does not tie together much of the theory built around the principle of expected utility maximisation, but that is hardly surprising given the fact that that theory is deeply inspired by that very principle of expected utility maximisation rather than by satisficing. To be fair, Aumann would have to compare two fully developed theories: one centred around satisficing, the other around expected utility maximisation. The problem is, however, that while the latter of these theories exists, only the skeleton of a theory around satisficing is available to date. What Aumann should want to do (to compare the merits of two theories) is impossible. Therefore, the comparison Aumann envisages is impossible to carry out.
- 22.
‘The Chain-Store Paradox’, Theory and Decision, 9 (1978), 127–159. See Section 4.1.1.
- 23.
Ibid. 133.
- 24.
David Kreps , et al., ‘Rational Cooperation in the Finitely Repeated Prisoners’ Dilemma’, Journal of Economic Theory, 27 (1982), 245–252, ibid. and Robert Wilson , ‘Reputation and Imperfect Information’, ibid. 253–279, and Paul Milgrom and John Roberts , ‘Predation, Reputation, and Entry Deterrence’, ibid. 280–312. The solution concept used here was developed by David Kreps and Robert Wilson, ‘Sequential Equilibria’, Econometrica, 50 (1982), 863–894.
- 25.
In all fairness it should be noted that Kreps and Wilson , ‘Rational Cooperation in the Finitely Repeated Prisoners’ Dilemma’, 276 seem to have anticipated this kind of critique when they wrote that one may suspect that ‘by cleverly choosing the nature of [some] small uncertainty … , one can get out of a game theoretic analysis whatever one wishes. We have no formal proposition of this sort to present at this time, but we certainly share these suspicions. If this is so, then the game theoretic analysis of this type of game comes down eventually to how one picks the initial incomplete information. And nothing in the theory of games will help one to do this’.
- 26.
I refer to the primary sources from the refinement literature. Eric van Damme , Stability and Perfection of Nash Equilibria: Second, Revised and Enlarged Edition (Berlin: Springer, 1987) contains very similar argumentative strategies for such refinement proposals as the approachable, essential, firm, regular, and strictly and weakly proper equilibrium.
- 27.
‘Non-Cooperative Games’, Annals of Mathematics, 54 (1951), 286–295.
- 28.
‘Non-Cooperative Games’, Ph.D. diss. (Princeton University, 1950). The note is absent from the published paper with the same title. The appendix to the Ph.D. dissertation (pp. 21–26) also mentions a second ‘interpretation’ of the Nash equilibrium in terms of repeated game-play, foreshadowing applications in evolutionary game theory.
- 29.
Op. cit. 23.
- 30.
Robert Aumann and Adam Brandenburger , ‘Epistemic Conditions for Nash Equilibrium’, Econometrica, 63 (1995), 1161–1180, and Adam Brandenburger , ‘Knowledge and Equilibrium in Games’, Journal of Economic Perspectives, 6 (1992), 83–101.
- 31.
For references to papers showing adherence to the incorrect folk theorem, see Aumann and Brandenburger , art. cit.
- 32.
Nash ’s second interpretation in terms of repeated game-play is not immediately excluded by the epistemic characterisation of the Nash equilibrium.
- 33.
‘Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit: Teil I: Bestimmung des dynamischen Preisgleichgewichts’, Zeitschrift für die gesamte Staatswissenschaft, 121 (1965), 310–324.
- 34.
Ibid. 308 (original German).
- 35.
Selten ’s own term was ‘perfect equilibrium’ (ibid. 308). It coincides with backward induction on games with perfect information.
- 36.
It is common belief up to level 2, to be precise. Larger games naturally require higher levels. See Section 3.1 to the effect that the full infinity of common belief will never be used in any finite game, but a general characterisation statement is most easily phrased using full infinite common belief.
- 37.
Ibid. 308.
- 38.
See Chapter 3 for a discussion of the views of common belief about rationality and utility developed in the Epistemic Programme, attesting that the formalism was inspired by its applications, namely, to study the behavioural consequences of particular epistemic assumptions involving the beliefs, desires and rationality of the players of extensive games.
- 39.
‘Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games’, International Journal of Game Theory, 4 (1975), 25–55.
- 40.
Ibid. 34.
- 41.
‘Refinements of the Nash Equilibrium Concept’, International Journal of Game Theory, 7 (1978), 73.
- 42.
Ibid. 73–74.
- 43.
Results in the Epistemic Programme show that Myerson’s proper equilibrium does not adequately capture the epistemic assumptions he implicitly makes. For further discussion of the form of rationality excluding weakly dominated strategies and a solution concept based on it, see Section 2.3. For an enlightening discussion and a recent alternative to the proper equilibrium, see, e.g., Geir Asheim , ‘Proper Rationalizability in Lexicographic Beliefs’, International Journal of Game Theory, 30 (2002), 452–478.
- 44.
NERP and the Epistemic Programme have very similar aims in that both attempt to determine what players will choose under certain epistemic and rationality assumptions. They share, for instance, the conception of rationality as expected utility maximisation, and they share, as I have argued here, an interest in common belief, etc. But in NERP these assumptions were not explicitly formalised and a lot of theorising remained mathematics-driven rather than application-driven. In the Epistemic Programme, by contrast, these assumptions were formalised with highly technical, yet in the end application-driven mathematical tools.
- 45.
It is interesting to note that TARK started as an acronym for ‘Theoretical Aspects of Reasoning about Knowledge’, and changed to the current name in 1998, reflecting an increase in attention on decision and game theory.
- 46.
See, e.g., E. Roy Weintraub and Philip Mirowski , ‘The Pure and the Applied: Bourbakism Comes to Mathematical Economics’, Science in Context, 7 (1994), 245–272.
- 47.
Selten , ‘Reexamination’.
- 48.
Nash, art. cit. 286.
- 49.
John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior (Princeton: Princeton University Press, 1944), 31–33, 220–221.
- 50.
Von Neumann judged Nash’s result to be ‘trivial … [and] just a fixed point theorem’ (Sylvia Nasar , A Beautiful Mind: A Biography of John Forbes Nash (New York: Simon and Schuster, 1998), 94). Note, however, that in view of von Neumann’s rejection of subjective probability and references to players’ beliefs, Nash’s interpretation is perhaps quite radical in a sense. The biographical detail about von Neumann’s life may be read, moreover, as indicating that having considered the mathematical generalisation himself, he (and Morgenstern) found that it did not yield any interesting contribution to the modelling of economic phenomena. The way Nash introduced the generalisation did not change this conviction, because Nash did not show much awareness of economic applications, not even in the interpretative Appendix to his Ph.D. dissertation (op. cit. 21–26).
- 51.
Nash did allow for this insight into the sense that all strategies that receive non-zero probability in a Nash equilibrium that consists of mixed strategies are equally good given that the opponents stick to their equilibrium strategies. This draws me into a discussion about whether a probability distribution over actions (which is what a mixed strategy really is) models anything like actions (as Nash seems to suggest for repeated games), or rather beliefs (as the standard view seems to be nowadays, witness the treatment in most textbooks). See Section 2.1.
- 52.
Using search-term ‘unique* OR unicity’, and 1950–present as the year-range, 15,460 hits were found (July 10, 2010).
- 53.
‘Spieltheoretische Behandlung’.
- 54.
It would be both conceptually and historically inappropriate to describe backward induction as a refinement of the Nash equilibrium, because it already figures in John von Neumann ’s ‘Zur Theorie der Gesellschaftsspiele’, Mathematische Annalen, 100 (1928), 295–320 in which he proves the minimax theorem, as well as in concrete studies of chess by Ernst Zermelo , ‘Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels’, in E. Hobson and A. Love (eds.), Proceedings of the Fifth International Congress of Mathematicians, Held at Cambridge 22–28 August, 1912: Volume II: Communications to Sections II–IV (Cambridge: Cambridge University Press, 1913), 501–504, Dénes König ‘Über eine Schlussweise aus dem Endlichen ins Unendliche’, Mitteilungen der Universität Szeged, 3 (1927), 121–130, Max Euwe (the Dutch world chess champion 1935–1937), ‘Mengentheoretische Betrachtungen über das Schachspiel’, Proceedings van de Koninklijke Akademie van Wetenschappen te Amsterdam, 32 (1929), 633–642, and more fully in John von Neumann and Oscar Morgenstern , op. cit.
- 55.
‘Reexamination’.
- 56.
Ibid. 33.
- 57.
I cannot overemphasise that I do not at all mean to imply anything about Selten ’s or any other game theorist’s general outloo k on science. I focus on the research programmes and some of their key publications, not on the researchers.
- 58.
Art. cit.
- 59.
Wolfgang Spohn, ‘How to Make Sense of Game Theory’, in W. Balzer, W. Spohn, and W. Stegmüller (eds.), Studies in Contemporary Economics: Vol. 2: Philosophy of Economics (Berlin: Springer, 1982), 239–270, B. Douglas Bernheim, ‘Rationalizable Strategic Behavior’, Econometrica, 52 (1984), 1007–1028, and David Pearce, ‘Rationalizable Strategic Behavior and the Problem of Perfection’, Econometrica, 52 (1984), 1029–1050.
- 60.
Art. cit. 1010.
- 61.
Ibid.
- 62.
‘Backward Induction and Common Knowledge of Rationality’, Games and Economic Behavior, 8 (1995), 18.
- 63.
See, e.g., Paul Milgrom , ‘An Axiomatic Characterisation of Common Knowledge’, Econometrica, 49 (1981), 219–222.
- 64.
For further discussion (and qualification), see Section Section 2.3.
- 65.
Ibid. 243 (emphasis in original).
- 66.
See references in Chapters 2 and 3.
- 67.
See Sections Section 2.3 and Section 2.4.
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de Bruin, B. (2010). The Methodology of Game Theory. In: Explaining Games. Synthese Library, vol 346. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9906-9_5
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