Abstract
Ordinary common knowledge is formally expressed by strong probabilistic common belief. How strong exactly? The question can be answered by drawing from the similar equivalence, recently explored, between plain and probabilistic individual beliefs. I argue that such a move entails that common knowledge displays a double fragility: as a description of a collective state and as a phenomenon, because it can respectively disappear as group size increases, or more worryingly as the epistemic context changes. I argue that despite this latter fragility, the effects of common knowledge on action are robust. Unfortunately, this in turn leads to a third fragility, that of the concept of common knowledge, which threatens to collapse on probabilistic common belief. This also reveals a disanalogy between the individual and the collective cases. I finally pinpoint the subtle difference entailed by the two concepts, expressed in terms of the attitude towards contrary evidence or of the agents’ awareness. As a result, common knowledge can be defended as a concept, which refers to a fragile yet distinct collective attitude.
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Notes
Or after multiple typical events have taken place, such as when several agents have been exposed to similar sources of information. This is the basis for common knowledge based on what is called a distributed indicator’ for Cubitt and Sugden (2003:191), or equivalently for indirect common knowledge for Paternotte (2011). The distinction between public and typical event-based common knowledge will not matter here—although I will later appeal to the fact that much common knowledge is indirect.
See for instance Fagin et al. (1995). For now we can do without any formal apparatus though.
In particular, the question of the relation between common knowledge and common belief does not hinge on a particular concept of knowledge, and is independent from the debate concerning the links between individual knowledge and belief.
So p-common belief is just partial common belief for a given strength of belief p.
Belief operators need not refer to beliefs. For instance, Paternotte (2011) uses them to formally express Lewis’ notions of “having reason to believe” and of indication. Even alternative frameworks of Lewis’ concept of common knowledge, such as Cubitt and Sugden’s (2003), are based on an individual epistemic operators.
If this does not strike you, think of a lottery with a million tickets, as in Leitgeb’s example.
Where this equivalence is understood according to the Lockean thesis introduced earlier, rather than to the Humean thesis on belief, which states that “it is rational to believe a proposition just in case it is rational to have a stably high degree of belief in it.” (Leitgeb 2014:145, fn. 9). The Humean thesis involves stability, but not the Lockean one.
Note that any threshold strictly greater than \(\frac{999}{1000}\) could still be chosen here. As a result, the only stable set would be that of all worlds—all the agent would plainly believe is that one of the tickets is a winning one.
Another use of Leitgeb’s account in the context of social epistemology—although on the topic of judgment aggregation—can be found in Cariani (2016).
This characterization of common knowledge in terms of reachable possible worlds harks back to Aumann’s (1976) seminal formalisation of common knowledge.
One may think that agents face a similar problem arises concerning the attribution of underlying probability distributions by others to them–that it is hard for agent A to determine what agent B may think A’s probability distribution over a set of worlds inconsistent with A’s current knowledge is. I do no think this is a strong issue though. This is because Leitgebs as well as traditional epistemic models presuppose a unique underlying probability distribution defined on all possible worlds. So when an agent reflects a set of possible worlds that are inconsistent with her current knowledge (an element E of her epistemic partition that does not contain the actual world), she can in principle infer what her subjective probability distribution would be by conditionalising the general probability distribution on the occurrence of E.
This is a non-technical term, expressing the common knowledge of parameters of the situation that are typically assumed but not formally expressible, such as of the underlying probability distribution or of each agent’s epistemic partitions.
Such similarity may be seen as a presupposition that all share the same “inductive standards” (Lewis 1969), that is (in our vocabulary), draw the same beliefs from a similar epistemic context because they assume they share their awareness contexts. This parallel with inductive standards is justified if one sees Leitgeb’s framework as a way to recover the “inductive link from partial reasons to full reasons”, which Paternotte (2011) claims is needed to fully work out the degrees for which partial common belief may constitute common knowledge.
Each agent believes that the other s-believes that the actual world obtains with a degree \(s'\) slightly greater than s—as it is possible that both mistakenly heard a non-existent siren. But \(s'\)-belief implies s-belief.
For instance, for \(p=0.5\), \(q=0.9\) and r = 0.99, ordinary common knowledge is impossible among more than eight agents.
Note that this does not mean that the phenomenon of common knowledge is fragile, that is, that common knowledge dissipates easily; but that the appearance of common knowledge from a given public event is not robust with respect to certain changes of the number of witnesses.
Because the messages are sent successively, m can only be equal to n or to \(n+1\), as the first agent sent the first message.
Although this would not be classical common knowledge of P, which can be defined iteratively as: all agents know that all know...that all know P. Here, we have a different iterative definition, saying that agent 1 knows that agent 2 knows that...that agent 1 knows P (as well as the same proposition in which 1 and 2 are inverted). These two definitions are equivalent in the case of knowledge if based on certain belief, but not on partial belief (See Morris 1999 for more details about the divergence between various definitions of common partial belief that coincide for certain belief).
Note that there is a difference with Leitgeb’s solution. For Leitgeb, changes in awareness happen with a fixed set of possible worlds: what an agent is aware of does not impinge on what she deems possible. In our solution to the coordinated attack problem, agents can be aware of worlds they deem impossible: if an agent has received 30 messages, she knows that she has not received 31, let alone an infinity. Still, the agents’ interactive set of possible worlds stays unchanged—that is, the set of worlds such that it is possible for an agent that it is possible for another that... that the world is possible for an agent. In interactive epistemic situations, agents have to consider impossibilities that others deem possible.
In particular, certain coarsenings of partitions leave plain beliefs unchanged; partition refinements are what typically result in belief loss. For a detailed discussion, see Leitgeb (2014:155–160).
By this, I mean the definition sought in this paper, based on the partial common belief plus a way to obtain plain from partial common belief, that is, based on a classical epistemic framework supplemented by Leitgeb’s.
Here and earlier, I am indebted to an anonymous referee, who stressed the need for clarifications regarding Barwise’s three fundamental questions.
More specifically, common knowledge guarantees agree to disagree results, iterated deletion of dominated strategies in games and convergence on a Pareto-dominant outcome by best response dynamics in a coordination problem.
Of course, this strategy is impossible in the coordinated attack problem, in which there is no public event to be repeated or multiplied.
Still, note that pluralism, if common, is not universal. See Lyon (ms.) for a recent criticism of pluralism with respect to plain and partial beliefs.
Note that this does not contradict the claim that there can be ordinary common knowledge in the coordinated attack problem although it features no public event. The definition does not exclude a other possible sources for common knowledge.
Kinds of salience-history-based common knowledge that Lewis (1969) already discussed in his truly seminal work.
Here, priority and posterity are to be understood conceptually rather than temporally.
See Schipper (2014) for a short survey.
More precisely, agents are aware or unaware of propositions—worlds are included or excluded from a subjective state space depending on which propositions are expressible in them.
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Acknowledgments
I thank the audience of the Salzburg Social epistemology Workshop (September 2014) for insightful questions and comments, as well as two very helpful anonymous referees. This work was supported by the Alexander von Humboldt Foundation.
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Paternotte, C. The Fragility of Common Knowledge. Erkenn 82, 451–472 (2017). https://doi.org/10.1007/s10670-016-9828-4
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DOI: https://doi.org/10.1007/s10670-016-9828-4