The electronic mail game: Strategic behavior under “almost common knowledge”

Abstract

A very basic assumption in all studies of game theory is that the game is “common knowledge.” Following John Harsanyi (1967), situations without common knowledge are analyzed by a game with incomplete information. A player's information is characterized by his “type.” Each player “knows” his own type and the prior distribution of the types is common knowledge. Jean-Francois Mertens and Samuel Zamir (1985) have shown that under quite general conditions one can find type spaces large enough to carry out Harsanyi's program and to transform a situation without common knowledge into a game with incomplete information in which the different types may have different states of knowledge. Harsanyi's method became the cornerstone of all modern analyses of strategic economic behavior in situations with asymmetric information (i.e., most of the theoretical Industrial Organization literature). What does it mean that the game G is “common knowledge”? Following David Lewis (1969), Stephen Schiffer (1972), and Robert Aumann (1976), this concept has been studied thoroughly by relating it to concepts of “knowledge” and “probability” (for a recent presentation of this literature see Ken Binmore and Adam Brandenberger, 1987). Intuitively speaking, it is common knowledge between two players 1 and 2 that the played game is G , if both know that the game is G , 1 knows that 2 knows that the game is G and 2 knows that 1 knows that the game is G , 1 knows that 2 knows that 1 knows that the game is G , and 2 knows that 1 knows that 2 knows that the game is G , and so on and so on.