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Meet meets join: the interaction between pooled and common knowledge

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Abstract

This paper explores the lattice-theoretic properties of the family of all partitions of an arbitrary state space. I discuss a duality correspondence connecting meets (finest common coarsening) and joins (coarsest common refinement) of families of partitions, which conforms to the natural order-theoretic representations of these two operations. The partition lattice turns out to fail to be a distributive, or even a modular, lattice. I provide intuitive interpretations of these negative results in terms of the interaction between common knowledge and pooling multiple sources of private information, with applications to information exchange within and between different populations and to criminal-procedure law.

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Article 22 July 2014

Jeffrey Sanford Russell, John Hawthorne & Lara Buchak

Notes

  1. That is, “an event … [is] common knowledge (in the population) if all know [it], all know that all know it, all know that all know that all know it, and so on ad infinitum” (Aumann 1999a, p. 270, second emphasis added).

  2. It would be remiss of me not to specifically credit one of the referees with suggesting this pun.

  3. In addition, since the join is much simpler to compute directly from first principles than is the meet, duality makes finding the finest common coarsening of a family of partitions less complicated. This point will be illustrated in Example 1(c).

  4. For additional details on the foundations of this vast order-theoretic subfield, the interested Reader is referred to three excellent treatises: Birkhoff (1967) is an early classic, Davey and Priestley (2002) is a modern textbook, and Topkis (1998) is a monograph specifically tailored to the economist audience.

  5. For more on interactive epistemology, the Reader may wish to consult Brandenburger (1992), Geanakoplos (1992), and Aumann (1999a, 1999b) for relatively concise overviews, as well as Dekel and Gul (1997) and Fagin et al. (2003) for more comprehensive treatments.

  6. Note that a partial order \(\ge \) may leave pairs of elements incomparable, so that there may exist \(x,y\in X\) for which neither \(x\ge y\) nor \(y\ge x\) holds. A partial order that satisfies also completeness (either \(x\ge y\) or \(y\ge x\) holds for any \(x,y\in X\)) is called a total (or linear) order.

  7. If \(Y=\{x_i\,|\,i\in I\}\) is an indexed family of elements, where I is some non-empty index set and \(x_i\in X\) for every \(i\in I\), then the supremum and the infimum can also be denoted as \(\vee _{i\in I}x_i\) and \(\wedge _{i\in I}x_i\), respectively.

  8. The axiom of choice guarantees that \({\mathbf {C}}\) is not empty.

  9. It is interesting to mention that equality in (1) alone would imply equality in (2) and vice versa for complete lattices (Raney 1952). Therefore, either one of (1) or (2) could be taken to be the defining property of a completely distributive complete lattice.

  10. Analogously to the duality between (1) and (2) in the context of complete distributivity (see footnote 10), it can be shown that if either one of (3) or (4) held for any three elements \(x,y,z\in X\) of a lattice, then it would imply the other equality as well.

  11. Aumann (1999a, 1999b) calls this formalism the semantic approach to modeling uncertainty. More precisely, an accurate and consistent epistemological model must involve the full description of each state of the world in such a way that the knowledge of exactly which state has occurred entails the complete resolution of all relevant uncertainty, including uncertainty about the actions and knowledge of other agents. This implicit requirement is without loss of generality, since the definition of precisely what constitutes a state can always be refined upon splitting every \(\omega \in \Omega \) into several further states until all uncertainty is eliminated from the interpretation of each state. As Aumann (1987, p. 6) puts it: “The term ‘state of the world’ implies a definite specification of all parameters that may be the object of uncertainty on the part of any player. … Conditional on a given \(\omega \), everybody knows everything.” For a concise and illuminating informal interpretation, see Aumann (1999a, p. 267, n. 3): “If the reader wishes, he can think of a state \(\omega \) of the world as a complete possible history, from the big bang until the end of the world.” While its relative methodological simplicity explains the popular appeal of the semantic approach, it is not immune to criticism. Indeed, the potential need to capture knowledge about other agents’ knowledge about which state of the world has occurred runs the risk of bringing about circularity (Fagin et al. 1999; Board 2004). The use of the semantic approach as a modeling tool, therefore, requires the analyst to exercise caution in properly specifying the underlying state space—especially in interactive strategic contexts in which the opponents’ states of mind lie at the heart of the problem—and to wade through a treacherous territory teeming with intricate conceptual dilemmas as well as profound philosophical questions.

  12. Equivalently, \({\mathscr {P}}\) is finer than \({\mathscr {Q}}\) if each \({\mathscr {Q}}\)-cell can be expressed as a union of \({\mathscr {P}}\)-cells.

  13. Reflexivity and transitivity follow almost directly from the definition of \(\succsim \). As for antisymmetry, suppose that \({\mathscr {P}}\) and \({\mathscr {Q}}\) are two partitions of \(\Omega \) such that \({\mathscr {P}}\succsim {\mathscr {Q}}\) and \({\mathscr {Q}}\succsim {\mathscr {P}}\). Consider an arbitrary element \(\omega '\in \Omega \) of the state space and let \(\pi (\omega '|{\mathscr {P}})\) and \(\pi (\omega '|{\mathscr {Q}})\) be the \({\mathscr {P}}\)-cell and the \({\mathscr {Q}}\)-cell, respectively, containing \(\omega '\). If a state \(\omega ''\) belongs to the \({\mathscr {P}}\)-cell \(\pi (\omega '|{\mathscr {P}})\), then \(\pi (\omega '|{\mathscr {P}})=\pi (\omega ''|{\mathscr {P}})\). Given that \({\mathscr {P}}\succsim {\mathscr {Q}}\), this implies, in turn, that \(\pi (\omega '|{\mathscr {Q}})=\pi (\omega ''|{\mathscr {Q}})\), so that \(\omega ''\in \pi (\omega '|{\mathscr {Q}})\). That \(\omega ''\in \pi (\omega '|{\mathscr {Q}})\) implies \(\omega ''\in \pi (\omega '|{\mathscr {P}})\) can be proved in an analogous manner. In conclusion, \(\pi (\omega '|{\mathscr {P}})=\pi (\omega '|{\mathscr {Q}})\). Since \(\omega '\) has been an arbitrary element of \(\Omega \), it follows that the two partitions \({\mathscr {P}}\) and \({\mathscr {Q}}\) consist of precisely the same cells—that is, \({\mathscr {P}}={\mathscr {Q}}\).

  14. These equivalences hold because if \(\omega '\in R(\omega '')\), then the definition of \(R(\omega '')\) implies that \(\omega '\in \pi (\omega ''|{\mathscr {P}}_i)\) for some \(i\in I\). That is, \(\omega '\) is contained in the same \({\mathscr {P}}_i\)-cell as \(\omega ''\), so that \(\pi (\omega '|{\mathscr {P}}_i)=\pi (\omega ''|{\mathscr {P}}_i)\). The argument showing that \(\omega ''\in R(\omega ')\) implies \(\pi (\omega '|{\mathscr {P}}_i)=\pi (\omega ''|{\mathscr {P}}_i)\) for at least one \(i\in I\) is analogous. Conversely, if there exists some \(i\in I\) such that \(\pi (\omega '|{\mathscr {P}}_i)=\pi (\omega ''|{\mathscr {P}}_i)\), then one can conclude both that \(\omega '\in \pi (\omega ''|{\mathscr {P}}_i)\) and that \(\omega ''\in \pi (\omega '|{\mathscr {P}}_i)\), which imply \(\omega '\in R(\omega '')\) and \(\omega ''\in R(\omega ')\), respectively.

  15. Rubinstein (1989) demonstrates that the breakdown of common knowledge is not a purely academic concern. It has far-reaching strategic consequences, underlying the importance of determining precisely how much a group of people commonly “know” among themselves.

  16. Reflexivity is immediate from the observation that \(\omega \in R(\omega )\) for every \(\omega \in \Omega \). Symmetry follows because if two states \(\omega ',\omega ''\in \Omega \) are such that \(\omega ''\) is reachable from \(\omega '\), then one can simply reverse the sequence of states leading from \(\omega '\) to \(\omega ''\) to conclude that \(\omega '\) is also reachable from \(\omega ''\). Finally, transitivity can be established along the following lines: if three states \(\omega ',\omega '',\omega '''\in \Omega \) are such that \(\omega ''\) is reachable from \(\omega '\) and \(\omega '''\) is reachable from \(\omega ''\), then one can concatenate the sequence leading from \(\omega '\) to \(\omega ''\) and that connecting \(\omega ''\) to \(\omega '''\) to conclude that \(\omega '''\) is reachable from \(\omega '\).

  17. FRBs are short-lasting, high-energy blasts of radio signals of extraterrestrial origin. They are a recently discovered astronomical phenomenon, whose exact source and cause are, as of late 2020, still unknown (Lorimer et al. 2007; Petroff et al. 2019; Bochenek et al. 2020).

  18. See, for example, Aumann (1999a) for an axiomatic definition of a knowledge operator and how this concept ties in with the semantic approach.

  19. The statement and proof of this result appear also in Bach and Cabessa (2017), but these authors restrict attention to a countable state space and a finite number of agents. By contrast, note that no restrictions are imposed on the cardinalities of \(\Omega \) and I (apart from non-emptiness) in the instant paper. Therefore, Proposition 2 can be regarded as a generalization of Lemma 1 in Bach and Cabessa (2017, p. 1171). See also Barwise (1988) for a comparison of different definitions of common knowledge.

  20. Logicians and computer scientists typically use the synonymous term distributed knowledge instead of pooled knowledge. Fagin et al. (2003, p. 3) describe the contrast between common knowledge and distributed knowledge in an illuminating way: “While common knowledge can be viewed as what ‘any fool’ knows, distributed knowledge can be viewed as what a ‘wise man’—one who has complete knowledge of what each member of the group knows—would know.”

  21. As an alternative interpretation of the join, one can also think of I as a collection of information sources—such as a population of experts from whom a single decision-maker seeks input—and of \({\mathscr {P}}_i\) as representing the granularity of information provided by each source \(i\in I\). The join \(\vee _{i\in I}{\mathscr {P}}_i\) is then a formal characterization of the knowledge the decision-maker possesses upon pooling all the information learned from these different sources.

  22. Indeed, for any given \(i\in I\),

    $$\begin{aligned} \omega '\in S(\omega ')=S(\omega '')\subseteq \pi (\omega ''|{\mathscr {P}}_i), \end{aligned}$$

    where the set inclusion follows from the definition of \(S(\omega '')\) in (9). This means that \(\omega '\) belongs to the same \({\mathscr {P}}_i\)-cell as \(\omega ''\), that is, \(\pi (\omega '|{\mathscr {P}}_i)=\pi (\omega ''|{\mathscr {P}}_i)\).

  23. Note that the family whose join is taken on the right-hand side of (13) is not empty, as it contains the trivial partition \(\{\Omega \}\). Additionally, the family whose meet is taken on the right-hand side of (14) is not empty, as it contains the discrete partition \(\{\{\omega \}\,|\,\omega \in \Omega \}\).

  24. This intuitive story makes it possible to interpret the duality formula (14) from the perspective of reachability. Two states remain indistinguishable by the actual population once its members are actively sharing information among themselves if and only if those two states are reachable from each other within the hypothetical population of learned recluses.

  25. For instance, a labor economist attending a theory talk would presumably have access to a coarser partition than the theorists when it came to theory topics and vice versa. Correspondingly, the requirement of discussing only those topics that are common knowledge among session participants according to this alternative organizational arrangement is intended to guarantee mutual intelligibility.

  26. Every American knows these facts; every American knows that every American knows these facts; every American knows that every American knows that every American knows these facts; and so forth. Therefore, such fundamental elements of U.S. culture are arguably common knowledge among all Americans.

  27. A common condition imposed on probationers and parolees in several U.S. states, such as California and Wisconsin, is that only “reasonable suspicion” needs to apply to uphold the constitutionality of searches by law enforcement, which is a lesser standard than the Fourth Amendment’s usual requirement of “probable cause.” The Supreme Court of the United States upheld the constitutionality of these regulations—see Griffin v. Wisconsin, 483 U.S. 868 (1987); United States v. Knights, 534 U.S. 112 (2001); and Samson v. California, 547 U.S. 843 (2006).

  28. More generally, U.S. Supreme Court case law abounds with related epistemological constraints in the course of criminal prosecutions. As an additional example, Napue v. Illinois, 360 U.S. 264 (1959), disallows the knowing use of untruthful testimony in criminal proceedings, even if such testimony does not directly speak to the defendant’s culpability. Conversely, Brady v. Maryland, 373 U.S. 83 (1963), forbids the prosecution from withholding exculpatory evidence. This requirement can be interpreted as the government being compelled to allow defense counsel access to applying the join operator \(\vee \) whenever doing so might favor the defendant. In a different context, Kamenica and Gentzkow (2011) cite Brady to rationalize the assumption underlying their Bayesian-persuasion framework that the sender is committed to the information structure that has been optimally designed to influence the receiver’s action.

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Acknowledgements

I am very grateful to Shmuel Zamir (Editor-in-Chief of this Journal), the Associate Editor, and two referees for their excellent comments and suggestions, which have led to a substantial improvement of the paper. In addition, I thank Ildikó Magyari for insightful conversations and advice. All remaining errors and omissions are mine. Previous versions of the paper were circulated under the title “Meet Meets Join: The Interaction between Private and Common Knowledge.”

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  1. Maxwell School of Citizenship and Public Affairs, Syracuse University, 129 Eggers Hall, Syracuse, 13244 NY, USA

    Áron Tóbiás

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Tóbiás, Á. Meet meets join: the interaction between pooled and common knowledge. Int J Game Theory 50, 989–1019 (2021). https://doi.org/10.1007/s00182-021-00778-w

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