See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/4719889 The Electronic Mail Game: Strategic Behavior Under" Almost Common Knowledge" Article in American Economic Review · February 1989 Source: RePEc CITATIONS READS 294 1,780 1 author: Ariel Rubinstein Tel Aviv University 178 PUBLICATIONS 18,904 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: With any requests please contact me directly [email protected] View project Procedures of Decision Making as Revealed through Eye Movements View project All content following this page was uploaded by Ariel Rubinstein on 23 February 2014. The user has requested enhancement of the downloaded file. The Electronic Mail Game: Strategic Behavior Under "Almost Common Knowledge" Ariel Rubinstein The American Economic Review, Vol. 79, No. 3. (Jun., 1989), pp. 385-391. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28198906%2979%3A3%3C385%3ATEMGSB%3E2.0.CO%3B2-U The American Economic Review is currently published by American Economic Association. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/aea.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Thu Mar 27 12:47:26 2008 The Electronic Mail Game: Strategic Behavior Under "Almost Common Knowledge" The paper addresses a paradoxical game-theoretic example which is closely related to the coordinated attack problem. Two players have to play one of two possible coordination games. Only one of them receives information about the coordination game to be played. It is shown that the situation with "almost common knowledge" is very diferent from when the coordination game played is common knowledge. A very basic assumption in all studies (for a recent presentation of this literature of game theory is that the game is "com- see Ken Binmore and Adam Brandenberger, mon knowledge." Following John Harsanyi 1987). Intuitively speaking it is common (1967), situations without common knowl- knowledge between two players 1 and 2 that edge are analyzed by a game with incom- the played game is G, if both know that the plete information. A player's information is game is G, 1 knows that 2 knows that the characterized by his "type." Each player game is G and 2 knows that 1 knows that "knows" his own type and the prior distri- the game is G, 1knows that 2 knows that 1 bution of the types is common knowledge. knows that the game is G, and 2 knows that Jean-Francois Mertens and Samuel Zamir 1 knows that 2 knows that the game is G (1985) have shown that under quite general and so on and so on. conditions one can find type spaces large One of the main difficulties with this intu- enough to carry out Harsanyi's program and itive definition (and with the formal defini- to transform a situation without common tions which capture this perception) is that knowledge into a gaine with incomplete in- even "simple" sentences like "I do not know formation in which the different types may that you do not know that I know that you have different states of knowledge. Har- do not know that I know" are very difficult sanyi's method became the cornerstone of all to visualize, thus malung an assessment of modem analyses of strategic economic be- their validity problematic. Therefore it would havior in situations with asymmetric infor- be interesting to understand whether a mation (i.e., most of the theoretical Indus- game-theoretic informational structure, re- trial Organization literature). ferred to as "almost common knowledge," in What does it mean that the game G is which only a finite (but large) number of "common knowledge"? Following David propositions of the type "1 knows that 2 Lewis (1969), Stephen Schiffer (1972), and knows that 1 knows.. .that the game is G" Robert Aumann (1976), this concept has are true, is very different from the situation been studied thoroughly by relating it to where the game G is common knowledge. In concepts of "knowledge" and "probability" this short paper I will present a simple ex- ample of a situation with "almost common knowledge" of the game. The situation is analyzed using, as a tool, the idea of a game *Department of Economics, The Hebrew University, with incomplete information. It is shown Jerusalem, and the London School of Economics, that the game-theoretic "prediction" for the Houghton Street, London, England W02A2AE. My thanks to Ken Binmore, Edi Dekel, John Geanakopo- "almost common knowledge" situation is los, Avner Shaked, Chuck Wilson, Asher Wolinsky, and very different from the situation with com- a referee of this journal for the very useful comments. mon knowledge. 386 THE AMERICAN ECONOMIC RE VIEW JUNE 1989 The example is similar to the "coordi- cannot achieve an expected payoff hgher nated attack problem" which is well known than (1 - p)M. If the information could be- in the distributed systems literature.' A come common knowledge they would be able description of the problem and a compari- to achieve the payoff M. However, imagine son with ths paper analyzed appears in Sec- that the two players are located at two dif- tion IV. ferent sites and they communicate only by electronic mail signals. Due to "technical 1. Coordination Through Electronic Mail difficulties" there is a "small" probability E > 0,that the message does not arrive at its Two players, 1 and 2, are involved in a destination. At the risk of creating discord, coordination problem. Each has to choose the electronic mail network is set up to send between two actions A and B. There are two a confirmation automatically if any message possible states of nature, a and b. Each of is received, including not only the confirma- the states is associated with a payoff matrix tion of the initial message but a confirmation as follows: of the confirmation; and so on. To be more precise, it is assumed that, when player 1 gets the information that the state of nature The game G, is b, hs computer automatically sends a A B message (a blip) to player 2 and then player A M, M 0,- L 2's computer confirms the message and then B - L,O 090 player 1's computer confirms the confirma- tion and so on. If a message does not arrive, state a then the communication stops. No message probability 1- p is sent if the state of nature is a. At the end of the communication phase the screen dis- The game G, plays to the player the number of messages A B his machne has sent. Let T, be a variable A 0,o 0,- L for the number of messages i's computer B - L,O M, M sent (the number on i 's screen). Notice that sending the messages is not a state b strategic decision by the players. It is an probability p automatic device carried out by the comput- ers. The designer of the system sets up the co~municationnetwork between the players In the state of nature a(b) the players get and they can only choose between A and B a positive payoff, M, if both choose the after the communication phase has ended. action A(B). If they choose the same action If the two machnes exchange an infinite but it is the "wrong" one they get 0.If they number of messages, then we may say that fail to coordinate, then the player who played the two players have common knowledge B gets - L,where L > M. Thus, it is dan- that the game is G,. However, since only a gerous for a player to play B unless he is finite number of messages are transferred, confident enough that hs partner is going to the players never have common knowledge play B as well. The state a is the more likely that the game they play is G,. event: b appears with a priori probability of In choosing between A and B after the p <1/2. end of the communication phase, player 1 The information about the state of nature (and similarly player 2) faces uncertainty: is known initially only to player 1. Without given that he sent T, messages he does not transferring the information, the players know whether player 2 did not get the T,th message, or whether player 2 got the T,th message, but the T,th confirmation has been lost. Any number on the screen corresponds 'I should like to thank John Geanakopolos for refer- to a state of knowledge not only about the ring me to the "coordinated attack problem." state of nature but also about the other VOL.