Law games: Defeasible rules and revisable rationality

Law and Philosophy

Volumn 17, Issue 4, 1998, Pages 443-480

  Chapman, Bruce ^a  

^a UNIVERSITY OF TORONTO   (Canada)

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EID: 0032238583     PISSN: 01675249     EISSN: None     Source Type: Journal    
DOI: 10.2307/3505089     Document Type: Article

References (70)

1. The use of the term "common knowledge" seems to have originated in David Lewis, Convention (Cambridge: Harvard University Press, 1969). For good discussion of its importance for game theory generally, see Cristina Bicchieri, Rationality and Coordination (Cambridge: Cambridge University Press, 1993), pp. 39-43.
2. The use of the term "common knowledge" seems to have originated in David Lewis, Convention (Cambridge: Harvard University Press, 1969). For good discussion of its importance for game theory generally, see Cristina Bicchieri, Rationality and Coordination (Cambridge: Cambridge University Press, 1993), pp. 39-43.
3. If a player finds herself at a point which is inconsistent with the theory of the game she is using, then she is deprived of a theory upon which to base her decisions. This has the effect of leaving the other players in the game without a theory as well, since they will not be able to predict what she will do and, therefore, are unable to decide what to do themselves. On this, see Philip J. Reny, "Common Knowledge and Games with Perfect Rationality," Proceedings of the Philosophy of Science Association 1988, 2 (East Lansing: Philosophy of Science Assn., 1989), p. 363.
4. This is to introduce a mistake at the level of first order beliefs. Fully analogous arguments could be provided for a case of mistake at any other order of belief. For example, I might believe that you believe me to be irrational, something which might allow me rationally to make what appears (to you) to be an irrational move.
5. For an indication that non-economists (at least) tend to cooperate more than would be predicted by rational choice theory in a one-shot prisoner's dilemma game, see Robert Frank, Thomas D. Gilovich, and Dennis T. Regan, "Does Studying Economics Inhibit Cooperation?," Journal of Economic Perspectives 1 (1993), p. 159.
6. Ronald Dworkin, Taking Rights Seriously (London: Duckworth and Co., 1977), Chapters 2 and 3.
7. Id., at 24.
8. Id., at 24-25.
9. Claire Finkelstein, "When the Rule Swallows the Exception" (manuscript, School of Law, Berkeley, 1998), passim. I am particularly grateful to Claire Finkelstein for her analysis of the accounts of both Dworkin and Schauer of the relationship between rules and exceptions.
10. See, e.g., Joseph Raz, "Legal Principles and the Limits of Law," Yale Law Journal 81 (1912), p. 823.
11. Frederick Schauer, Playing By the Rules (Oxford: Clarendon Press, 1991).
12. Id., at 52.
13. Id., at 39.
14. Id, at 117-118.
15. Some (including Schauer himself; see id., at 115) will see this as a key difference between Schauer and Dworkin. For Dworkin, principles have weight, but rules do not. For Schauer, rules can also have weight; they simply do not give into background principles or justifications at every opportunity. That they do not is Dworkin's reason for rejecting the model of rules in favor of the direct application of those principles which justify the rules. By contrast, that they do not is Schauer's reason for accepting the model of rules; there are good reasons, he says, for entrenchment. But what Dworkin and Schauer have in common is the idea that rules, at least to some extent, resist the direct application of background principles and justifications.
16. Id., at 115.
17. Id., at 116.
18. See id., Chapter 7, for Schauer's arguments in favor of having entrenched rules.
19. H. L. A. Hart, "The Ascription of Responsibility and Rights", in Anthony Flew (ed.) Essays in Logic and Language (First Series) (Oxford: Basil Blackwell, 1960), pp. 145-166.
20. Id., at 152.
21. Id., at 150.
22. Albeit not sympathetically. The term is usually "path dependent", which refers to partition dependent choices coming in sequence. But the general point is the same: according to the economist, path or partition dependent choices are less than ideal. See infra note 23.
23. The following example is borrowed from James F. Reynolds and David C. Paris, "The Concept of 'Choice' and Arrow's Theorem," Ethics 89 (1979), p. 363. For comparable examples of partition dependent choice, see I. Levi Hard Choices (Cambridge: Cambridge University Press, 1986), p. 105. Also for a more detailed characterization of defeasibility in partition dependent terms, see Bruce Chapman, "Law, Incommensurability, and Conceptually Sequenced Argument" University of Pennsylvania Law Review 146 (1998), p. 1487.
24. The following example is borrowed from James F. Reynolds and David C. Paris, "The Concept of 'Choice' and Arrow's Theorem," Ethics 89 (1979), p. 363. For comparable examples of partition dependent choice, see I. Levi Hard Choices (Cambridge: Cambridge University Press, 1986), p. 105. Also for a more detailed characterization of defeasibility in partition dependent terms, see Bruce Chapman, "Law, Incommensurability, and Conceptually Sequenced Argument" University of Pennsylvania Law Review 146 (1998), p. 1487.
25. The following example is borrowed from James F. Reynolds and David C. Paris, "The Concept of 'Choice' and Arrow's Theorem," Ethics 89 (1979), p. 363. For comparable examples of partition dependent choice, see I. Levi Hard Choices (Cambridge: Cambridge University Press, 1986), p. 105. Also for a more detailed characterization of defeasibility in partition dependent terms, see Bruce Chapman, "Law, Incommensurability, and Conceptually Sequenced Argument" University of Pennsylvania Law Review 146 (1998), p. 1487.
26. Kenneth Arrow invoked his collective rationality condition, transitivity of the social preference relation, in part to avoid path dependent social choices. Such path dependence, he seemed to think, would make the final choice of some alternative an (historically) arbitrary matter; see K. J. Arrow, Social Choice and Individual Values (New Haven: Yale University Press 1963), p. 120.
27. But not all path dependent choice need be arbitrary path dependent choice; indeed, some "paths" may make much more sense than others, especially if they are "conceptually sequenced". On the idea of a conceptually sequenced path dependent choice, see infra, text following note 25; Chapman, supra note 22;
28. Bruce Chapman, "Pluralism in Tort and Accident Law: Towards a Reasonable Accommodation" in Gerald J. Postema (ed.) Philosophy and the Law of Torts (Cambridge: Cambridge University Press, 1999);
29. and Bruce Chapman, "More Easily Done Than Said: Rules, Reasons, and Rational Social Choice", Oxford Journal of Legal Studies 18 (1998), p. 293.
30. Also, for more explicit discussion of path dependence, see Charles Plott, "Path Independence, Rationality, and Social Choice" Econometrica 41 (1973), p. 1075.
31. For this characterization of defeasibility in some greater detail, see Chapman, "Incommensurability", supra note 22, at 1499-1501, 1507-1514.
32. See supra, text at note 6.
33. George P. Fletcher, "The Right and the Reasonable" Harvard Law Review 98 (1985), p. 949.
34. See R. D. Luce and H. Raiffa, Games and Decisions (New York: Wiley and Sons, 1957), pp. 97-102.
35. Robert Grafstein, Institutional Realism (New Haven: Yale University Press, 1992), p. 141.
36. For a similar argument, see Philip Pettit and Robert Sugden, "The Backward Induction Paradox," Journal of Philosophy 86 (1989), p. 169;
37. and Cristina Bicchieri, "Self-Refuting Theories of Strategic Interaction," Erkenntnis 30 (1989) p. 69.
38. The idea that the rational conclusion of the backward induction argument should be capable of being introduced, without contradiction, into the argument from the beginning was argued for originally by Quine with respect to the "surprise hanging" paradox; see W.V. Quine, The Ways of Paradox (Cambridge: Harvard University Press, 1966), pp. 19-21.
39. Either of these last two possibilities would allow ex ante for cooperative play somewhere in the repeat game by either player. It is, of course, well-known that cooperation can result in the finite repeated prisoner's dilemma when the players' rationality is not common knowledge among them; see David Kreps, Paul Milgrom, John Roberts and Robert Wilson "Rational Cooperation in the Finitely-Repeated Prisoner's Dilemma," Journal of Economic Theory 27 (1982), p. 245. What seems less well-known is the fact that the particular form of rationality which is manifested in the backward induction argument cannot (i.e., without contradiction) be common knowledge amongst the players. But this impossibility of common knowledge of rationality in any finite prisoner's dilemma game where either the players play cooperatively in fact, or where they are hypothesized to play cooperatively for the purposes of, say, a backward induction argument, opens up two quite different ideas, only one of which is exploited by the particular "is not" assumptions of Kreps et al. Their argument suggests that the notion of what is rational play in the game is secure as a notion, but at least one of the players simply does not believe the other to be rational in this way (or does not believe that the other believes him to be rational in this way, etc.). Thus, they want to focus on relaxing the "common knowledge" part of "common knowledge of rationality". However, a different possibility is that the very notion of what it is to play the game rationally is less secure than the conventions of rational choice might suggest. It is this latter idea, which takes a second look at the "rationality" part of "common knowledge of rationality", that is being suggested for analysis in this paper. Unlike the former sort of departure from common knowledge of rationality, the latter can countenance the thought that the players beliefs about each other's rationality in the game might eventually align themselves with what actually is rational for each of them in the game as a matter of fact.
40. Ken Binmore, "Modeling Rational Players. Part I," Economics and Philosophy 3 (1987), p. 193.
41. John Nash developed the equilibrium idea which now bears his name more than forty years ago in J. Nash, "Non-Cooperative Games," Annals of Mathematics 54 (1951), p. 286. Shaun Hargreaves Heap and Yanis Varoufakis, Game Theory (London: Routledge 1995), p. 37, refer to it as "the most famous concept that game theory has produced for dissecting games."
42. John Nash developed the equilibrium idea which now bears his name more than forty years ago in J. Nash, "Non-Cooperative Games," Annals of Mathematics 54 (1951), p. 286. Shaun Hargreaves Heap and Yanis Varoufakis, Game Theory (London: Routledge 1995), p. 37, refer to it as "the most famous concept that game theory has produced for dissecting games."
43. David Kreps, Game Theory and Economic Modeling (Oxford: Clarendon Press, 1990), p. 28.
44. Compare Hans Jorgen Jacobsen, "On the Foundations of Nash Equilibrium," Economics and Philosophy 12 (1996), p. 68: It does not follow directly from the definition of a Nash equilibrium that equilibrium strategies, or only equilibrium strategies, are sensible candidates for actual play. . . . Consider a two-player, simultaneous-move game. Why should Player 1 play a best reply to the strategy actually played by Player 2, when Player 1 does not know Player 2's choice at the time Player 1 decides his own strategy? . . . In a non-cooperative game each player must make up his mind on what to play and expect others to play on a purely individual basis. On the other hand, a Nash equilibrium is by definition something collective, a collection of strategies with a specific cross player property. The problem of justifying the Nash equilibrium concept is the same as explaining how purely individual considerations may lead each player to play or conjecture his part of such a collective plan. (Emphasis in the original; footnote omitted).
45. Kreps, supra note 33, at 113-114.
46. Binmore, supra note 31.
47. Reny, supra note 2. Compare Reny's characterization (at note 3 in his article) of one of the best known of the Nash refinements, the so-called "subgame perfect Nash equilibrium", with the remarks provided by Kreps in the text: That is, not only must it be that (i) every player has decided what to do in every possible eventuality so that from his perspective at the beginning of the game his choices are best given what the others have planned to do, it must also be that (ii) whenever any possible eventuality becomes a reality, no player will wish to change his previously decided upon choice, (emphasis added to the original)
48. Robert Sugden, "Inductive Reasoning in Repeated Games", in Reinhard Selten (ed.), Rational Interaction: Essays in Honour of John C. Harsanyi (New York: Springer-Verlag, 1992).
49. See, e.g., Kreps et al., supra note 30.
50. Even this way of putting the point equivocates between a player (i) being uncertain about what rationality is and (ii) being certain of that in general, but being uncertain of what that implies for her own rational play in a (any?) particular game.
51. Richard Rorty has characterized the rational person as someone willing to listen and be open to the views of others in his "Science and Solidarity" in John S. Nelson et al. (eds.), The Rhetoric of the Human Sciences (Madison: University of Wisconsin Press, 1987), p. 38.1 prefer to think of such a person as "reasonable", reserving the term "rational" for the more conventional (and unbending) chooser of rational choice theory.
52. It is worth noting that this new account of rationality is just as much at odds with a Kantian approach, which recommends unconditional cooperation (i.e., recommends it as a "categorical imperative") as it is with dominance thinking, which, just as unconditionally, recommends defection.
53. See Binmore, supra note 31, at 203: "One might say that she is subject to trembles in what she supposes to be the correct way to play." It is worth noting that these are "trembles" which go to the definition of what is rational. Thus, they are very different from the trembles introduced by Reinhard Selten, "Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games," International Journal of Game Theory 4 (1975), p. 25. According to Selten, trembles are a kind of mistake which go merely to the execution of choices which rationality properly identifies, a bit like pushing the wrong button. But this means that the probabilities of mistakes in different plays of the game will be independent of one another, something which makes reaching the final iterations of a repeat game under the termination rule increasingly unlikely. On this, see Jon Elster, The Cement of Society (Cambridge: Cambridge University Press, 1989) p. 6. Thus, a forceful explanation of actually reaching these iterations requires a tremble over something much more systematic, like the notion of rationality itself. This is what is being emphasized by Binmore, and by the argument in the text.
54. See Binmore, supra note 31, at 203: "One might say that she is subject to trembles in what she supposes to be the correct way to play." It is worth noting that these are "trembles" which go to the definition of what is rational. Thus, they are very different from the trembles introduced by Reinhard Selten, "Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games," International Journal of Game Theory 4 (1975), p. 25. According to Selten, trembles are a kind of mistake which go merely to the execution of choices which rationality properly identifies, a bit like pushing the wrong button. But this means that the probabilities of mistakes in different plays of the game will be independent of one another, something which makes reaching the final iterations of a repeat game under the termination rule increasingly unlikely. On this, see Jon Elster, The Cement of Society (Cambridge: Cambridge University Press, 1989) p. 6. Thus, a forceful explanation of actually reaching these iterations requires a tremble over something much more systematic, like the notion of rationality itself. This is what is being emphasized by Binmore, and by the argument in the text.
55. See Binmore, supra note 31, at 203: "One might say that she is subject to trembles in what she supposes to be the correct way to play." It is worth noting that these are "trembles" which go to the definition of what is rational. Thus, they are very different from the trembles introduced by Reinhard Selten, "Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games," International Journal of Game Theory 4 (1975), p. 25. According to Selten, trembles are a kind of mistake which go merely to the execution of choices which rationality properly identifies, a bit like pushing the wrong button. But this means that the probabilities of mistakes in different plays of the game will be independent of one another, something which makes reaching the final iterations of a repeat game under the termination rule increasingly unlikely. On this, see Jon Elster, The Cement of Society (Cambridge: Cambridge University Press, 1989) p. 6. Thus, a forceful explanation of actually reaching these iterations requires a tremble over something much more systematic, like the notion of rationality itself. This is what is being emphasized by Binmore, and by the argument in the text.
56. This will sound like the structure of the assurance game; see Amartya Sen, "Isolation, Assurance, and the Social Rate of Discount," Quarterly Journal of Economics 80 (1967), p. 112. However, it is important to appreciate that the structure of the game being discussed in the text is still that of a prisoner's dilemma. What gets changed, according to the argument, is what counts as rational behavior within that game. For the same reason, the solution being proposed in the text differs from that offered in the theory of psychological games;
57. see, e.g., Peter Huang and David Malueg, "Psychological Remorse in the Prisoner's Dilemma and Folk Theorems for Psychological Games" (manuscript, Department of Economics, Stanford University, 1994). In the theory of psychological games, expectation-dependent remorse (or regret) for having defected (cooperated) when the other player cooperated (defected) has the effect of changing the outcomes in the game so that they approach that of an assurance game. This is also a convenient point to emphasize that what is being proposed here is different from what evidential decision theorists propose for the prisoner's dilemma. Evidential decision theorists argue that the probability assigned to an outcome or a strategy choice by another player need not be independent of the first player's chosen strategy even though there is causal independence between the two. (This causal independence, of course, is exactly what causal decision theorists emphasize as good reason for assuming probabilistic independence and, therefore, for choosing the dominant strategy in the prisoner's dilemma.) Evidential decision theorists would allow the first player to assign, say, a higher probability to the other player choosing to cooperate in the prisoner's dilemma if the first player cooperates, something which can then make it rational to cooperate on conventional expected utility grounds. For good discussion of evidential and causal decision theory, and for an all-embracing decision-theoretic analysis which finds a place for both,
58. see Robert Nozick, The Nature of Rationality (Princeton: Princeton University Press, 1993), pp. 41-63. However, in the argument being proposed in the text, just as the outcomes are not changed (so that the prisoner's dilemma becomes, say, an assurance game or a game where defection is associated with remorse), so the probabilities assigned to the outcomes also remain unchanged. Rather, what is changed under the proposed analysis is the status of the rules of rational choice which are brought to bear on the original prisoner's dilemma game. In the face of apparently irrational cooperation, a player reconsiders what constitutes rational choice.
59. The following discussion borrows from Susan Hurley, "Newcomb's Problem, Prisoners' Dilemma, and Collective Action" Synthese 86 (1991), p. 173. Hurley explicitly distinguishes her attempt to ground cooperative action in the prisoner's dilemma from attempts made within evidential decision theory. On the latter, see supra note 44.
60. Shafir and Tversky have uncovered an interesting behavioral anomaly which may be relevant here; see Eldar Shafir and Amos Tversky, "Thinking Through Uncertainty: Nonconsequential Reasoning and Choice," Cognitive Psychology 24 (1992), p. 449. In a series of one-shot prisoner's dilemmas, their experimental subjects consistently revealed that while they would tend to defect if they knew the other player had defected and, equally, would tend to defect if they knew the other player had cooperated, they would defect less frequently if they were uncertain about which of these two events had occurred. This "disjunction effect", which they find in a variety of contexts, appears to violate what is known as "the sure thing principle", a cornerstone of expected utility theory. The explanation offered by Shafir and Tversky (id., at 457) for the disjunction effect in the prisoner's dilemma is interesting: Our subjects seem to exhibit a change of perspective that may be described as a shift from individual to collective rationality. Once the other's strategy is known, a player is "on her own." Only one column of the [prisoner's dilemma] table is relevant (that which corresponds to the strategy chosen by the other), and the outcome of the game depends upon her and her alone. The individually rational strategy, of course, is to [defect]. In the disjunctive condition, on the other hand, all four cells of the table are in play. The outcome of the game depends on the collective decision of both players, and the collectively optimal decision is for both to cooperate. Thus, the pattern of behavior observed in the [prisoner's dilemma] may be explained, in part at least, by the greater tendency to adopt the collective perspective in the disjunctive version of the game. One can imagine this effect varying a great deal with the context. For example, while many of us know (for all intents and purposes) that our vote cannot make a difference to the outcome of a large political election, and would not bother to vote once the outcome (whatever it is) is determined, we are inclined, nevertheless, to vote when the outcome of the election is still pending. This much is consistent with Shafir and Tversky's results. But in other contexts, for example, where there is an ongoing collective effort that one can continue to be a part of, one can easily imagine someone cooperating even beyond the point where the outcome is still pending, i.e., to the point where cooperation by others has become certain. This may not be consistent with the disjunctive effect observed by Shafir and Tversky, but it is still, arguably, consistent with the collective rationality perspective to which they refer.
61. See Govert Den Hartogh, "The Rationality of Conditional Cooperation," Erkenntnis 38 (1993), p. 405 for further development of this argument. See also Karl Warneryd, "Rationality, Transparency, and Evolutionary Selection", in Manfred E. Streit (ed.), Cognition, Rationality and Institutions (Jena: Max Planck Institute, Symposium Proceedings, forthcoming 1999), for this criticism of conditional cooperation as well as a prescription for avoiding the problem based on the partial transparency of signaling one's intention to adopt the conditional strategy. For sympathetic comment on the latter approach to the problem, see Bruce Chapman, "Rationally Transparent Social Interactions" in id.
62. See Govert Den Hartogh, "The Rationality of Conditional Cooperation," Erkenntnis 38 (1993), p. 405 for further development of this argument. See also Karl Warneryd, "Rationality, Transparency, and Evolutionary Selection", in Manfred E. Streit (ed.), Cognition, Rationality and Institutions (Jena: Max Planck Institute, Symposium Proceedings, forthcoming 1999), for this criticism of conditional cooperation as well as a prescription for avoiding the problem based on the partial transparency of signaling one's intention to adopt the conditional strategy. For sympathetic comment on the latter approach to the problem, see Bruce Chapman, "Rationally Transparent Social Interactions" in id.
63. See Govert Den Hartogh, "The Rationality of Conditional Cooperation," Erkenntnis 38 (1993), p. 405 for further development of this argument. See also Karl Warneryd, "Rationality, Transparency, and Evolutionary Selection", in Manfred E. Streit (ed.), Cognition, Rationality and Institutions (Jena: Max Planck Institute, Symposium Proceedings, forthcoming 1999), for this criticism of conditional cooperation as well as a prescription for avoiding the problem based on the partial transparency of signaling one's intention to adopt the conditional strategy. For sympathetic comment on the latter approach to the problem, see Bruce Chapman, "Rationally Transparent Social Interactions" in id.
64. Ken Binmore, Playing Fair (Cambridge, Mass.: MIT Press, 1994), pp. 191-194.
65. See, e.g., J. M. Orbell, R. M. Dawes, and A. J. C. van de Kragt, "Explaining Discussion-induced Cooperation," Journal of Personality and Social Psychology 56 (1988), p. 811.
66. For analysis as to why this might be so, see Allan Gibbard, "Norms, Discussion, and Ritual: Evolutionary Puzzles," Ethics 100 (1990), p. 787.
67. See Elster, supra note 43, at 5, 7 for a comparable demand of rationality theory. Amartya Sen also seems to be looking for an inherently ambiguous notion of rationality so that he can better deal with the problem of counterfactuals, that is, instances of choice which do not seem to correspond with a fully reflective rationality; see Amartya Sen, "Rationality and Uncertainty," Theory and Decision 18 (1985), pp. 109, 112-113. In Binmore, supra note 31, at 198, the idea that "imprecisions are necessarily intrinsic to any properly based theory of rational behavioris effectively related to the more general philosophical claim that a background theory, whose parameters can be varied, is needed if we are to account for meaningful counterfactuals; on this see David Lewis, Counterfactuals (Oxford: Basil Blackwell, 1976). Defeasibility is not exactly "imprecision" (one hopes), but its dual aspect does capture something of Lewis's idea that counterfactuals must, nevertheless, be meaningful in their countervailing effect rather than merely non-sensical (i.e., beyond the theory altogether).
68. See Elster, supra note 43, at 5, 7 for a comparable demand of rationality theory. Amartya Sen also seems to be looking for an inherently ambiguous notion of rationality so that he can better deal with the problem of counterfactuals, that is, instances of choice which do not seem to correspond with a fully reflective rationality; see Amartya Sen, "Rationality and Uncertainty," Theory and Decision 18 (1985), pp. 109, 112-113. In Binmore, supra note 31, at 198, the idea that "imprecisions are necessarily intrinsic to any properly based theory of rational behavioris effectively related to the more general philosophical claim that a background theory, whose parameters can be varied, is needed if we are to account for meaningful counterfactuals; on this see David Lewis, Counterfactuals (Oxford: Basil Blackwell, 1976). Defeasibility is not exactly "imprecision" (one hopes), but its dual aspect does capture something of Lewis's idea that counterfactuals must, nevertheless, be meaningful in their countervailing effect rather than merely non-sensical (i.e., beyond the theory altogether).
69. See Elster, supra note 43, at 5, 7 for a comparable demand of rationality theory. Amartya Sen also seems to be looking for an inherently ambiguous notion of rationality so that he can better deal with the problem of counterfactuals, that is, instances of choice which do not seem to correspond with a fully reflective rationality; see Amartya Sen, "Rationality and Uncertainty," Theory and Decision 18 (1985), pp. 109, 112-113. In Binmore, supra note 31, at 198, the idea that "imprecisions are necessarily intrinsic to any properly based theory of rational behavioris effectively related to the more general philosophical claim that a background theory, whose parameters can be varied, is needed if we are to account for meaningful counterfactuals; on this see David Lewis, Counterfactuals (Oxford: Basil Blackwell, 1976). Defeasibility is not exactly "imprecision" (one hopes), but its dual aspect does capture something of Lewis's idea that counterfactuals must, nevertheless, be meaningful in their countervailing effect rather than merely non-sensical (i.e., beyond the theory altogether).
70. See Elster, supra note 43, at 5, 7 for a comparable demand of rationality theory. Amartya Sen also seems to be looking for an inherently ambiguous notion of rationality so that he can better deal with the problem of counterfactuals, that is, instances of choice which do not seem to correspond with a fully reflective rationality; see Amartya Sen, "Rationality and Uncertainty," Theory and Decision 18 (1985), pp. 109, 112-113. In Binmore, supra note 31, at 198, the idea that "imprecisions are necessarily intrinsic to any properly based theory of rational behavioris effectively related to the more general philosophical claim that a background theory, whose parameters can be varied, is needed if we are to account for meaningful counterfactuals; on this see David Lewis, Counterfactuals (Oxford: Basil Blackwell, 1976). Defeasibility is not exactly "imprecision" (one hopes), but its dual aspect does capture something of Lewis's idea that counterfactuals must, nevertheless, be meaningful in their countervailing effect rather than merely non-sensical (i.e., beyond the theory altogether).