Abstract
The aim of this book is to contribute to the philosophy of the theory of games by offering internal as well as external investigations of game theory. The internal investigations were reported in the first three chapters, especially in the discussions in Chapters 2 and 3, and in places made use of heavily technical, logical apparatus. Among the outcomes were that the Nash equilibrium implausibly presupposes veridicality of beliefs, that the Dekel–Fudenberg procedure is not necessarily the only consequence of common true belief about payoff-uncertainty, and that both Aumann’s and Reny’s analyses of backward induction make conceptually inconsistent assumptions.
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Notes
- 1.
Recent publications include Margaret1 Archer and Jonathan Tritter (eds.), Rational Choice Theory: Resisting Colonisation (London: Routledge, 2000), Ian Shapiro , The Flight from Reality in the Human Sciences (Princeton: Princeton University Press, 2005), and Michael Taylor , Rationality and the Ideology of Disconnectedness (Cambridge: Cambridge University Press, 2006). Paul Moser (ed.), Rationality in Action: Contemporary Approaches (Cambridge: Cambridge University Press, 1996) collects a number of classic essays.
- 2.
I thank Frank Hindriks for a question about this point. The presentation owes much to Daniel Hausman , ‘Revealed Preference, Belief, and Game Theory’, Economics and Philosophy, 16 (2000), 99–115. Revealed preference theory was developed by Paul Samuelson , ‘A Note on the Pure Theory of Consumers’ Behaviour’, Economica, 5 (1938), 61–71. Defining preferences in terms of dispositions to choose, the idea is that in order to find out what an agent’s preference ordering is, she is confronted with a number of choice problems. The axioms of revealed preference then ensure that under certain conditions the underlying preference ordering of the agent can be derived. The term revealed preference theory is also used to refer to the idea that the only access the theorist has to an agent’s preferences and beliefs is observation of her actual action. Even if this were plausible (which I do not think, given what historians learn from reading diaries of historical figures) it would not follow that one can derive someone’s beliefs and desires merely from information about rationality; an action maximises expected utility relative to a whole lot of combinations of expectations and utility functions.
- 3.
The classic reference is Milton Friedman , ‘The Methodology of Positive Economics’, in ibid., Essays in Positive Economics (Chicago: University of Chicago Press, 1953), 3–43. In game theory, the most explicit phrasing is probably due to Robert Aumann , ‘What is Game Theory Trying to Accomplish?’, in K. Arrow and S. Honkapohja (eds.), Frontiers of Economics (Oxford: Blackwell, 1987), 28–76.
- 4.
Reinhard Selten , ‘The Chain-Store Paradox’, Theory and Decision, 9 (1978), 127–159 pointed out that a straightforward model of market entrance involving backward induction led to predictions that were highly off the mark. 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’, Journal of Economic Theory, 27 (1982), 253–279, and Paul Milgrom and John Roberts , ‘Predation, Reputation, and Entry Deterrence’, Journal of Economic Theory, 27 (1982), 280–312 construct alternative game-theoretic models.
- 5.
Karl Popper , Logik der Forschung (Vienna: Julius Springer, 1935). Philippe Mongin , ‘Le principe de rationalité et l’unité des sciences sociales’, Revue Économique, 53 (2002), 301–323 argues that the principle of rationality Popper speaks about has to be considered as purely metaphysical, because no interesting statements can be deduced from it. On Popper’s views on the principle of rationality in the social sciences, see Popper’s 1963 lecture ‘The Myth of the Framework’, in ibid., Models, Instruments and Truth, ed. M. Notturno (London: Routlegde, 1998), 154–184. A critical appraisal is Boudewijn de Bruin , ‘Popper’s Conception of the Rationality Principle in the Social Sciences’, in I. Jarvie, K. Milford and D. Miller (eds.), Karl Popper: A Centenary Assessment: Selected Papers from Karl Popper 2002: Volume III: Science (Aldershot: Ashgate, 2006), 207–215.
- 6.
Max Weber, Wirtschaft und Gesellschaft, ed. Marianne Weber (Tübingen: Mohr, 1921; 5th ed. 1990), 12–13.
- 7.
John Stuart Mill, ‘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 1844 ed. repr. in ibid., Collected Works of John Stuart Mill, ed. J. Robson (Toronto: University of Toronto Press, 1967), 309–339.
- 8.
Ibid. 326.
- 9.
Weber , op. cit. 12–13 (original German).
- 10.
The preference ordering ranges over uncountably many alternatives, but to uncover an agent’s utility functions and probabilistic expectations, only finitely many questions would have to be asked to her. For a textbook treatment of this fact, see Roger Myerson , Game Theory: Analysis of Conflict (Cambridge, Mass.: Harvard University Press, 1991), 12–17.
- 11.
This is the content of Theorem A.1 (see the Appendix A). These requirements may seem to be hardly ever satisfied in practice, and a plethora of experiments in psychology have attempted to establish exactly that. See the relevant classic essays collected in Paul Moser (ed.), Rationality in Action: Contemporary Approaches (Cambridge: Cambridge University Press, 1996). Yet there is nothing intrinsically problematic with the idea that at least in some cases strategic agents actually do form von Neumann–Morgenstern preference orderings, and rather than criticising expected utility maximisation and its presuppositions in the traditional way, I here examine how far to go under the assumption that they have real bite at least in some cases.
- 12.
If \(\mathsf{P}(\varphi) = 1\) and \(\mathsf{P}(\varphi \rightarrow \psi) = 1\), then it is easy to establish that \(\mathsf{P}(\psi) = \mathsf{P}(\varphi \rightarrow \psi) - \mathsf{P}(\neg \varphi) + \mathsf{P}(\neg \varphi \cap \psi) = 1\) using the Kolmogorov axioms and the observation that \(\mathsf{P}(\varphi \rightarrow \psi) = \mathsf{P}(\neg \varphi \vee \psi)\).
- 13.
Joseph Halpern , ‘The Relationship between Knowledge, Belief, and Certainty’, Annals of Mathematics and Artificial Intelligence, 4 (1991), 301–322 contains a general result relating the logic of certainty (probability one belief) to KD45.
- 14.
A case in point can be found in the epistemic characterisation of the Nash equilibrium. True, given the axioms for linear inequalities the maximisation problem is solved automatically. But it is conceptually clarifying to separate mathematical skills from logical skills, because it shows that if, for whatever reasons, you wanted to study systems in which the logical force of the agents is weakened, you would still need to make sure that they have sufficient mathematical skills. See, e.g., Mikaél Cozic , ‘Impossible States at Work: Logical Omniscience and Rational Choice’, in R. Topol and B. Walliser (eds.), Cognitive Economics: New Trends (Amsterdam: Elsevier, 2007), 47–68. Another reason to set apart logical and mathematical capacities is that in the last decade computer scientists have gained deeper understanding of the computational complexity of solving decision processes. Here it is fruitful not to see maximisation problems as merely requiring logical ability. See Vincent Conitzer and Tuomas Sandholm , ‘New Complexity Results about Nash Equilibria’, Games and Economic Behavior, 63 (2008), 621–641 and references therein. To illustrate the relevance of this, note that as early as 1985 it was suggested that bounds to the complexity of the strategies may result in cooperation in the iterated Prisoner’s Dilemma. See A. Neyman , ‘Bounded Complexity Justifies Cooperation in the Finitely Repeated Prisoners’ Dilemma’, Economics Letters, 19 (1985), 227–229.
- 15.
The Economic Approach to Human Behavior (Chicago: University of Chicago Press, 1976).
- 16.
The clause \(\mathrm{RCT}(D, C)\) states that the decision-theoretic model D represents the choice situation C. The analysis could be made more precise by specifying exactly the structure of D, and by making explicit how D and u relate. The same is true for the game-theoretic analysis below. For the purposes of my critique, however, the level of detail of the proposed analysis is sufficient. Furthermore, a distinction could be made between available actions and actions the agent believes to be available. In Chapter 1, however, I argued that in order for a decision-theoretic (or game-theoretic) model to function properly, these two sets of actions have to coincide. Cf., however, Jaakko Hintikka , The Principles of Mathematics Revisited (Cambridge: Cambridge University Press, 1996), 214–215.
- 17.
Only binding the free variable C with a universal quantifier makes sense here. The existential reading would state that an agent faced a decision problem, and that is quite different from expressing the fact that she is rational in some sense.
- 18.
Because the universal quantifier precedes the existential one, the utility function and the probabilistic beliefs may depend on the action.
- 19.
For a discussion of the normativity of rationality, see Raymon Boudon , ‘La rationalité axiologique: Une notion essentielle pour l’analyse des phénomènes normatifs’, Sociologie et societés, 31 (1999), 103–117, Francesco Guala , ‘The Logic of Normative Falsification: Rationality and Experiments in Decision Theory ’, Journal of Economic Methodology, 7, 59–93, Philippe Mongin , ‘L’optimisation est-elle un critère de rationalité individuelle?’, Dialogue, 33 (1994), 191–222, Oswald Schwemmer , ‘Aspekte der Handlungsrationalität: Überlegungen zur historischen und dialogischen Struktur unseres Handelns’, in H. Schnädelbach (ed.), Rationalität: Philosophische Beiträge (Frankfurt am Main: Suhrkamp, 1984), 175–197, Wolfgang Spohn , ‘Wie kann die Theorie der Rationalität normativ und empirisch zugleich sein’, in L. Eckensberger and U. Gähde (eds.), Ethik und Empirie: Zum Zusammenspiel von begrifflicher Analyse und erfahrungswissenschaftlicher Forschung in der Ethik (Frankfurt am Main: Suhrkamp, 1993), 151–196, and ibid., ‘The Many Facets of Rationality’, Croatian Journal of Philosophy, 2 (2002), 249–264.
- 20.
William Hait , David August and Bruce Haffty (eds.), Expert Consultations in Breast Cancer: Critical Pathways and Clinical Decision Making (New York: Marcel Dekker, 1999). The determination of the utilities was, quite understandably, left to the patients.
- 21.
Roderick Chisholm , ‘Contrary-to-Duty Imperatives and Deontic Logic’, Analysis, 24 (1963), 33–36 shows that a similar question arises in propositional deontic logic.
- 22.
This clause was avoided in explanatory settings so as to make sure that what is to be explained (the choice of an action) does not figure in its explanation.
- 23.
My choice here is arbitrary to some extent. The deontic operator could even be appended to \(\mathrm{Choose}(S, C, a)\) only. This very strict reading would entail the existence of a maximising action and recommend its performance only. In what follows, what is true of the small scope reading is equally true of the strict reading.
- 24.
‘Equilibrium Distributions of Sales and Advertising Prices’, Review of Economic Studies, 44 (1977), 465–491.
- 25.
The Nash equilibrium is used throughout as an example, but other solution concepts can be easily accommodated.
- 26.
An objection that could be raised to this analysis is that it seems to ignore games with incomplete information where players are modelled to be uncertain about the utility functions. This objection would miss the point, though. Such games can easily be transformed into a normal form game for which the logical analysis proffered here is unproblematic. More importantly, if player S is incompletely informed about the utility function of T, then this is modelled by having the game Γ start with some probability measure on a number of subgames in which the utilities of T are different. Player S does not know in which subgame of Γ she is; player T does know that. In order for Γ to figure in an explanation of the actions of S and T, they still have to carve up their choice situation in exactly the way of Γ. In particular, player S is informed about the utility structure of Γ, even the part of T. Accordingly, S concurs with T as well as with the theorist in the assignment of T’s utility values to all leaves of the game tree. The incompleteness of S’s information about T’s utility function enters in once the game has started. In other words, because she knows what the game looks like, player S knows the kinds of incomplete information about T’s utilities she will encounter during game-play.
- 27.
It is not necessary that the spaces of options and the utility functions be commonly known. Something like that could be the case, and depending on whether it is the case or not, one or other solution concept is more or less appropriate in the explanation (as the epistemic characterisation theorems reveal). Yet for Γ just to model the choice situation C there need not be any higher level beliefs about the epistemic states of the other players.
- 28.
See, e.g., Larry Samuelson , Evolutionary Games and Equilibrium Selection (Cambridge, Mass.: MIT Press, 1997) focuses on this topic in particular. I will return to this briefly below.
- 29.
‘The Nash Equilibrium: A Perspective’, Proceedings of the National Academy of Sciences, 101 (2004), 3999–4002. John Nash , ‘Equilibrium Points in n-person Games’, ibid., 36 (1950), 48–49. Other contemporary calls for a normative conception of game theory appear in Robert Aumann and Jacques Drèze , ‘When All is Said and Done, How Should You Play and What Should You Expect?’, CORE Discussion Paper 2005-21 (University of Louvain, 2005), and Martin Dufwenberg , et al., ‘The Consistency Principle for Set-Valued Solutions and a New Direction for Prescriptive Game Theory’, Mathematical Methods of Operations Research, 54 (2001), 119–131. The advice is always directed at a single agent, as recommending a Nash equilibrium to a group of players is often just wrong. This point will be reinforced below in Section 4.3.1. Moreover, what is meant by normative here is something truly deontic or prescriptive (‘Keep left!’). Game-theoretic explanations of conventions or efficient norms are not the target of the present argument (‘In Great Britain the norm is to keep left’). See, e.g., Peter Vanderschraaf , ‘Convention as Correlated Equilibrium,’ Erkenntnis, 42 (1995), 65–87, and Martin van Hees , ‘Liberalism, Efficiency, and Stability: Some Possibility Results’, Journal of Economic Theory, 88 (1999), 294–309.
- 30.
In an alternative reading the deontic operator only prefixes the statement \(\mathrm{Choose}(S, C, a)\).
- 31.
Larry Samuelson , op. cit.
- 32.
Epistemic characterisation theorems typically have a converse direction to the effect that for any outcome of game-play satisfying the solution concept there exists an epistemic situation satisfying the conditions from the epistemic characterisation theorem. At first sight, the validity of the converse direction seems to be necessary if epistemic characterisation theorems are to furnish beliefs canonically. This appearance is misleading, though. Suppose that for some solution concept the converse does not hold, and that some outcome \((a_1, \ldots, a_N)\) cannot result in an epistemic situation satisfying the relevant conditions. Then, if a game theorist explained i’s behaviour by noting that a i is part of an outcome satisfying the solution concept, he or she would be unable to give an explanation of a i following the belief–desire framework (or the particular solution concept should be abandoned altogether).
- 33.
Anticipating the discussion later, a similar move would not work to reinstall normative interpretations of game theory. The idea would be to take it that a game-theoretic advisor implicitly assumes that the beliefs of the players are such as the relevant epistemic characterisation result specifies. Then, however, the point of the advice would disappear, because expected utility maximisation would suffice.
- 34.
John von Neumann and Oskar Morgenstern , Theory of Games and Economic Behavior (Princeton: Princeton University Press, 1944), 11.
- 35.
Adam Brandenburger and Eddie Dekel , ‘Rationalizability and Correlated Equilibria’, Econometrica, 55 (1987), 1392.
- 36.
See Section 2.1.2.2 for a discussion of the two interpretations of mixed strategies that are paralleled by the two ways to define the correlated equilibrium, and an argument why the Nash equilibrium does not conform to the ban on exogenous information.
- 37.
‘Correlated Equilibrium as an Expression of Bayesian Rationality’, Econometrica, 55 (1987), 2.
- 38.
The game is due to Brandenburger and Dekel , art. cit. 1394. The interpretation I consider is not theirs, though.
- 39.
Art. cit. 15.
- 40.
Adam Brandenburger and Eddie Dekel , ‘Rationalizability and Correlated Equilibria’, Econometrica, 55 (1987), 1391–1402. Robert Aumann and Jacques Drèze, ‘When All is Said and Done, How Should You Play and What Should You Expect?’, to be published in xxx, express similar views. The starting point in the latter paper is a view voiced by Joseph Kadane and Patrick Larkey , ‘Subjective Probability and the Theory of Games,” Management Science, 28 (1982), 113–120 to the effect that players of games should proceed on the basis of subjective probability estimates concerning their opponents’ actions, and that to form such estimates disciplines such as psychology are more suitable than non-cooperative game theory. Aumann and Drèze find this view ‘on its face… straightforward and reasonable’, but note that Kadane and Larkey overlook the ‘fundamental insight of game theory: that a rational player should take into account that all the players are rational, and reason about each other’ (emphasis in original), which they call ‘interactive rationality’. They continue that ‘unlike Kadane and Larkey, we note that the demands of interactive rationality severely restrict the expectations, and go on to characterise precisely what expectations can arise under this restriction’. In other words, where Kadane and Larkey still left room for exogenous information (in fact, they claimed that the only serious source of information is exogenous), Aumann and Drèze decide against it.
- 41.
Cf. Michael Bacharach , Beyond Individual Choice: Teams and Frames in Game Theory, eds. N. Gold and R. Sugden (Princeton: Princeton University Press, 2006) developing a truly collective form of normative game theory (which is very different from traditional non-cooperative game theory, though).
- 42.
A different issue is whether the fact that a certain game ends in an actual Nash equilibrium in fact entails that the players played rationally. Mathias Risse , ‘What is Rational about Nash Equilibria?’ Synthese, 124 (2000), 361–384 shows it does not. My interest here is, however, the reasonableness (as opposed to rationality) of the advice.
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de Bruin, B. (2010). Applications of Game Theory. In: Explaining Games. Synthese Library, vol 346. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9906-9_4
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