(Site note: apologies for the recent slow rate of posting. In my defense, this is surely the first post in the economics blogosphere to be sent from Somalia, where I am running through a bunch of ministerial and businessman meetings before returning to the US for AEA. The main AEA site is right down the street from my apartment, so if you can’t make it next week, I will be providing daily updates on any interesting presentations I happen across. Of course, I will post some brief thoughts on the Somali economy as well.)
We economists know common knowledge via the mathematical rigor of Aumann, but priority for the idea goes to a series of linguists in the 1960s and to the superfamous philosopher David Lewis and his 1969 book “Conventions.” Even within philosophy, the formal presentation of Aumann has proven more influential. But the economic conception of common knowledge is subject to some serious critiques as a standard model of how we should think about knowledge. One, it is equivalent to an infinite series of epistemic iterations: I know X, know you know that I know X, and so on. Second, and you may know this argument via Monderer and Samet, the standard “common knowledge is created when something is announced publicly” is surely spurious: how do I know that you heard correctly? Perhaps you were daydreaming. Third, Aumann-style common knowledge is totally predicated on deductive reasoning: every agent correctly deduces the effect of every new piece of information on their own knowledge partition. This is asking quite a bit, to say the least. The first objection is not too worrying: any student of game theory knows the self-evident event definition of common knowledge, which implies that epistemic iteration definition. Indeed, you can think of the “I know, know that you know, know you know that I know, etc.” iterations as the consequence of knowing some public event. Paternotte gives the great example of any inductive proof in mathematics: knowing X holds for the first element and X holding for element i implies it holds for i+1 is not terribly cognitively demanding, but knowing those two facts implies knowledge of an infinite string of implications. The second objection, fallibility, has been treated with economists using p-belief: assign a probability distribution to the state space, and talk about having .99-common belief rather than common knowledge. The third, it seems, is less readily handled.
But how did Lewis think of common knowledge? And can we formalize his ideas? What is then represented? This paper is similar to Cubitt and Sugden (2003, Economics and Philosophy), though it strikes me as the more interesting take. Lewis said the following:
It is common knowledge among a population that X iff some state of affairs holds such that
1: Everyone has reason to believe that A holds
2: A indicates to everyone that everyone has reason to believe that A holds, and
3: A indicates to everyone that X.
Note that the Lewisian definition is not susceptible to the three arguments noted above. Agents don’t necessarily believe something, but rather just have reason to do so. They know how each other reason, but the method of reasoning is not necessarily deductive. Let’s try to formalize those conditions in a standard state space world. Let B(p,i)E be the belief operator of agent i: B(.7,John):”It rains today” means John believes with probability .7 that it will rain today. Condition 1 in Lewis looks like claiming that all agents believe with p>.5 that A holds (have a “reason to believe A”). The word “A indicate X” should mean that there is a reasoning function of agent i, f(i), such that if A is believed with p>.5, then so is X (we will need some technical conditions here to ensure the function f(i) is defined uniquely for a given reasoning standard).
What is interesting is that this definition is tightly linked to standard Monderer-Samet common p-belief. For every common p-belief, p>.5, there are a set of parameters for which Lewisian common knowledge exists. For every set of parameters where Lewisian common knowledge exists, there is at least .5-common belief. Thus, though Lewisian common knowledge appears to be not that strict, it in fact is in a strong sense equivalent to common p-belief, and thus implies any of the myriad results published using that simpler concept. What an interesting result! I take this to mean that many common complaints about common knowledge are not that serious at all, and that p-belief, quite standard these days in economics, is much more broadly applicable than I previously believed.
http://www.springerlink.com/content/n81219v23334n610/ (GATED. Philosophy community: you have to do something about the lack of working papers freely accessible! Final version in Synthese 183.2 – if you are a micro theorist, you should definitely be reading this journal, as it is definitely the top journal in philosophy publishing analytic, formal results in theory of knowledge.)
A Fine Theorem 