Mutual Knowledge of Rationality in the Electronic Mail Game

This paper reexamines the paradoxical aspect of the electronic mail game (Rubinstein, 1989). The electronic mail game is a coordination game with payoff uncertainty. At a Bayesian Nash equilibrium of the game, players cannot achieve the desired coordination of actions even when a high order of mutual knowledge of payoff functions obtains. We want to make explicit the role of knowledge about rationality of players, not only that of payoff functions. For this purpose, we use an extended version of the belief system model developed by Aumann and Brandenburger (1995). We propose a certain way of embedding the electronic mail game in an belief system. And we show that for rational players to coordinate their actions, for any embedding belief systems, it is necessary that the upper bound order of mutual knowledge of payoff functions exceeds the upper bound order of mutual knowledge of rationality. This result implies that under common knowledge of rationality, the coordination never occurs, which is similar to Rubinstein’s result. We point out, however, that there exists a class embedding belief systems for which the above condition is also sufficient for the desired coordination. JEL Classification— C72, D81, D82 Keywords— electronic mail game, mutual knowledge, common knowledge, rationality, interactive belief system. ∗This paper is a largely revised version of our paper entitled “The role of small irrationality in communication and strategic decisions: An example with the electronic mail game” (2002). An earlier version was presented at the International Conference on Game Theory and Its Applications held at the Taj Mahal hotel in Mumbai, India on 8–10 January of 2003. We thank participants of the above conference, and the seminars at Hokkaido University and Otaru University of Commerce. And the first author thanks Akihiko Matsui for motivating the research. †Corresponding author. Institute of Social and Economic Research, Osaka University. 6-1 Mihogaoka Ibaraki Osaka 567-0047 JAPAN. E-mail: [email protected]. ‡Division of Computer Science, Graduate School of Information Science and Technology, Hokkaido University. Kita14 Nishi9 Sapporo 060-0814 JAPAN.