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Common Knowledge and Convention

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Abstract

This paper investigates the epistemic assumptions that David Lewis makes in his account of social conventions. In particular, I focus on the assumption that the agents have common knowledge of the convention to which they are parties. While evolutionary analyses show that the common knowledge assumption is unnecessary in certain classes of games, Lewis’ original account (and, more recently, Cubitt and Sugden’s reconstruction) stresses the importance of including it in the definition of convention. I discuss arguments pro et contra to argue that, although the assumption might be relevant to a descriptively adequate account of social convention, it is not required for its rational reconstruction. I then point out that Lewis’ account, properly speaking, is of common reason to believe, rather than of common knowledge, and argue that, in order to formalize aptly the distinction between reason to believe and belief, standard formal epistemic models need to be supplemented with so-called awareness structures. Finally, I stress that the notion of knowledge implicit in Lewis’ text involves interesting elements that cannot be captured in the standard propositional formalizations, but need the full expressive force of quantified epistemic logic.

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Notes

  1. The distinction was pointed out a few years before Convention by Schelling, cf. Schelling (1960), p. 84: “If the zero-sum game is the limiting case of pure conflict, what is the other extreme? It must be the “pure collaboration” game in which players win or lose together, having identical preferences regarding the outcome.”

  2. Most authors endorse this interpretation, cf. e.g. Cubitt and Sugden (2003) or Skyrms (2003).

  3. It is difficult to overestimate the influence that the introduction of such an idea has exerted in so many different fields, ranging from economics (Geanakoplos 1992) to computer science (Fagin et al. 1995; Meyer and van der Hoek 1995), from logic (besides the propositional results of Fagin et al. 1995 and Meyer and van der Hoek 1995, cf. also issues of quantification in Wolter 1999 and Sturm et al. 2002, and the proof theoretical analysis of Alberucci and Jaeger 2004) to linguistics (Clark 1996).

  4. For a more detailed analysis of this claim, cf. both Cubitt and Sugden (2003) and Sillari (2005).

  5. The notion of rationality Lewis is referring to here is not the game-theoretical one, but rather, cf. Bicchieri (1993), epistemic rationality. On this subject, cf. also Sillari (2005).

  6. Cf. Vanderschraaf (1998).

  7. The examples given here do not mean to be exhaustive or especially representative of the contributions given by game theorists to the study of convention or to the formalization of epistemic notions. I believe however that they are representative of the game theoretic literature that rejects the notion that conventions need to be common knowledge (other apt examples could have been Ken Binmore, or Peyton Young), as opposed to those authors (for instance, Bob Sugden, or Herb Gintis) who see common knowledge as an essential constitutive trait of social conventions.

  8. Signaling games are a special kind of coordination games in which various equilibria exists. In some of the equilibria, the agents achieve effective communication. Such equilibria are called signaling equilibria. Cf. Lewis (1969), Crawford and Sobel (1982).

  9. Cf. Lewis (1969, p. 97): “conventions may be a species of norms: regularities to which we believe one ought to conform. I shall argue that they are.”

  10. More details of this argument are given in the next section, and in Sillari (2005). Bicchieri argues more extensively against the introduction of a common knowledge clause in the definition, in Bicchieri (2005, pp. 36-37), supporting the conclusion that in a rational reconstruction of the notion of social norm (or convention) the common knowledge assumption is not necessary.

  11. Cf. Lewis (1969, p. 59): “common knowledge of the relevant facts seems to be one [important feature common to our examples of conventions].” For illuminating examples of common knowledge and of its relevance in our social practices and in general for anthropological syntheses, cf. Chwe (2001).

  12. This kind of “regress argument” for the introduction of common knowledge is not unique to the Lewisian definition of social convention. For example Schiffer (1972) justifies his emendation to Grice’s definition of non-natural meaning (1989) through a similar argument. In a nutshell: suppose that the recognition of the speaker’s intention in an act of communication is essential for successful communication. Then the hearer knowing that the speaker, in uttering p, intended x is necessary to establish meaning of p. But the speaker must also have intended that the hearer recognize that the speaker intended x, hence etc. The introduction of a common knowledge clause eliminates the vicious regress. Clark and Marshall (1981) justify the claim that speaker and audience have to have common knowledge of the reference of definite descriptions through a similar regress argument.

  13. This argument is detailed in Sillari (2005).

  14. Whether it is in fact the case that common knowledge functions as a theoretical reason to believe in the coordinative performance of the other party remains an empirical question, and possibly an interesting subject of empirical scrutiny.

  15. The distinction is analyzed in greater detail in (Vanderschraaf and Sillari 2005).

  16. The notion of a public event was probably introduced by Paul Milgrom in (Milgrom 1981). There it is shown that common knowledge can be axiomatically determined, one of the axioms requiring in fact that public events be common knolwedge.

  17. Thus, systems representing knowledge include axiom T, which states that anything known by an agent is actually the case, while systems representing belief allow agents to entertain false belief, while requiring, by axiom D, that the set of belief they entertain be consistent. Axiom D is not required in the system representing knowledge because it is easily derived from T.

  18. Some authors do consider infinitary epistemic systems: cf. e.g. Heifetz (1999), Kaneko and Nagashima (1996), Alberucci and Jaeger (2004).

  19. The equivalence between the two approaches carries over to the semantics of the group knowledge operators, implying that Aumann’s formal definition of common knowledge is equivalent to the fix point characterization.

  20. For a formal proof, cf. Heifetz (1999). The observation was already made, without proof, in Gilbert (1989, p. 468, n. 30.).

  21. Lewis (1969, p. 52) asks “[w]hat premises have we to justify us in concluding that others have certain expectations, that others expect others to have certain expectations, and so on?”; Schiffer (1972, pp. 32 ff.), after introducing common knowledge, which he calls mutual knowledge*, through the iterative definition, gives a finite set of “conditions which must obtain for [common knowledge] to be realized”; Aumann (1976, p. 1237) shows that “the formal definition of ‘common knowledge’ is equivalent to the informal description; Clark and Marshall (1981) also introduce the notion through its iterative description, but then devote a substantial part of their article to dissect a number of variations on the coming about of common knowledge: such variations can in general be subsumed in the Lewisian approach.

  22. To appreciate the similarity with Lewis’ account, consider that for Lewis the inference from the current situation to knowledge that p is warranted by the relation of indication and by the fact that agents share inductive standards and background information. Thus, the assumption of common inductive standards corresponds to Schiffer’s “normality” condition and indication to Schiffer’s “sufficiency for iterated knowing.”

  23. This idea is elaborated in much greater detail in Clark (1996, pp. 92–125).

  24. Cf. Lewis (1969, p. 61), where the claim is made for the particular case of conventions: “to belong to the population in which that convention holds—to be a party to it—is to know, in some sense, that it holds.” A similar, more general claim, can be found in Binmore (1994, p. 140): “[a] community of rational individuals is held together by the pool of common knowledge that I shall call its culture.” But for an argument (purely logical) against this view, cf. Gilbert (1989).

  25. Cf. Clark and Marshall (1981, p. 37).

  26. I have argued elsewhere (cf. Sillari 2005) that, with opportune adjustments, the standard mathematical models for knowledge representation do accommodate the richer epistemic notions that Lewis uses in his essay.

  27. Cf. Cubitt and Sugden (2003, p. 198).

  28. I am of course referring here to (Aumann 1976).

  29. For a number of technical approaches to deal with logical omniscience in epistemic logic, cf. Fagin et al. (1995). For a philosophical discussion of the problem, cf. Stalnaker (1999, pp. 174–241) and Parikh (1987). Further epistemological reflections on the problem logical omniscience are made in the first part of Sillari (2008a).

  30. Indeed, several authors concerned with presenting realistic accounts of common knowledge have addressed the problem, seeking ways to avoid endowing agents with an infinite burden of knowledge attributions. For example, Schiffer, considering possible objections to his definition (cf. Schiffer 1972, p. 36), points out that the definition does not imply that there is common knowledge in the group, but only that, by virtue of the conditions in the definition, each agent in the group may acquire an indefinite amount of knowledge. Gilbert, in putting forth her own definition of common knowledge (cf. Gilbert 1989, p. 189.), uses the notion of a “smooth reasoning counterpart” of an agent i: an ideal agent i* “whose reasoning is untramelled by limits of time, memory capacity, and perseverance.”

  31. Cf. Lewis (1969, p. 52).

  32. The main reference for awareness structures in the economics literature is Heifetz et al. (2006), while for logic and awareness, allow me to refer the interested reader to Sillari (2008a, b) and the references therein.

  33. For a deep analysis of agency and issues of awareness, cf. Newell (1982), discussed in Sillari (2008a).

  34. Among the authors who have taken into account the distinction, see Burge (1975, p. 251) in which it argued that the “non-triviality” characteristic of a conventional equilibrium need not be common knowledge in either sensu diviso or composito; or see Loar (1976, pp. 150–151), in which the distinction is applied in philosophy of language to make sense of the fact that an English speaker can have knowledge of so vast and complex a thing as the English language; or see the more recent (Pettit 1998), in which the distinction is called upon to illustrate the original notion of “practical belief” defended in the article.

  35. In Convention, however, Lewis prefers to refer to Abelard’s terminology and talks about knowledge in sensu diviso and in sensu composito, respectively.

  36. The distinction is akin to different understandings of categorization in cognitive psychology. On the one hand, a category can be seen as defined by a prototypical member to which actual category members resemble under certain important criteria. On the other, membership to a category may be decided through the comparison with sets of exemplars belonging to the category. As Cristina Bicchieri notices discussing this issue (cf. Bicchieri 2005, p. 84) the use of prototypes, as opposed to that of exemplars, may depend on how well the subject knows the category about which she is making judgments. Thus the distinction between categorization through prototypes and through exemplars reflects some kind of learning process from the former to the latter, like the distinction between de re and de dicto knowledge of conventions does, as we shall see in this section.

  37. Notice that Lewis’ definition does not simply require that the agents have knowledge of the convention: they also know that they have such knowledge, they know that they know that they have such knowledge, etc. In other words, there has to be a basis B in the population for common knowledge (or, more precisely, common reason to believe) that the convention holds. But we are considering de re knowledge. Thus, the basis B indicates to the agents that the convention is in place in each particular instance of the coordination problem on which the convention is based.

  38. However, the introduction of cognitive elements in evolutionary analysis—as it is done, for example, in the last chapter of Bicchieri (2005)—is an important point of contact between the two approaches.

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Acknowledgments

The author wishes to thank Cristina Bicchieri, Ken Binmore, Margaret Gilbert, Herbert Gintis, Luca Tummolini, Peter Vanderschraaf and two anonymous referees for valuable comments, suggestions and criticisms.

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    Giacomo Sillari

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Sillari, G. Common Knowledge and Convention. Topoi 27, 29–39 (2008). https://doi.org/10.1007/s11245-008-9030-7

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