This paper is Robert Aumann’s famous clarification of when common knowledge of rationality will imply that players must play the strategies given by backward induction. It turns out that, at least in 1995, the definitions of “knowledge” and “rationality” were not totally clear. “Know” an event E is defined as a player having E as a union of events in his information partition (denoted KiE for player i), knowledge of E by all players (denoted KE) is the intersection of KiE over i, and common knowledge of E (CKE) is the intersection KE with KKE with KKKE, etc. An agent is rational at event v if he chooses the payoff-maximizing strategy of all those available going forward from v, and rational if he is rational at every v (an event denoted Ri, with the event that all players are rational denoted R). Common knowledge of rational is defined is simply CKR.
The proof that common knowledge of rationality as defined above implies that players must play a strategy found by backward induction is interesting in and off itself – knowledge, even knowledge about my own knowledge, is represented in set notation, which can then be manipulated in a fairly straightforward way to get the desired results. The early 20th century dream of set theory as the base of mathematics is not dead in Aumann’s world. The second half of this paper discusses qualitatively what exactly we are assuming in the definitions of rationality and common knowledge; in particular, as discussed earlier on this blog, common knowledge of rationality does not allow player’s to “learn” that other players are not rational. For instance, if an off-path vertex is reached, I am not allowed to infer that the other agent is playing something other than a backward inductive strategy; rather, I must, in my counterfactuals, always maintain that other players will act with common knowledge of rationality from now on forward.
http://www.ma.huji.ac.il/raumann/pdf/36.pdf (link to the only non-gated version of this paper I could find)