The Electronic Mail Game
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 Additional Contributors: None
Categories: CategoryModel CategoryGameTheory
 Related Models: None
Created: 20080820
Suggested Citation: See license page.
Contents
Introduction
The 'Electronic Mail Game' model was put forward by Rubinstein in his paper The Electronic Mail Game: Strategic Behavior Under "Almost Common Knowledge" [1] in order to demonstrate how the difference between common knowledge and 'almost' common knowledge regarding the structure of a game (specifically the payoffstructure) can be substantial in terms of the equilibrium outcome.
The model is identical in its essentials to the "Coordinated Attack Problem" of Joseph Halpern.
Model
Two players are to play a simple coordination game. To introduce the issue of common knowledge regarding the game assume that there is uncertainty as to which of two possible games is being played:
Game 1 (prob: (1p)) :

A 
B 
A 
M,M 
0,L 
B 
L,0 
0,0 
Game 2: (prob: p)

A 
B 
A 
0,0 
0,L 
B 
L,0 
M,M 
Thus each game is a coordination game with NE in Game 1 being A,A and in Game 2 being B,B.
Suppose that player 1 knows the state of nature (i.e. whether game A or game B is played) but that player 2 does not. Player 1 can send an electronic mail (email) to player 2. Player 2 can send one back in acknowledgment etc etc (one can even think of this process as being automated without any intervention required of the players).Suppose that there is a probability $$\epsilon$$ that a message is lost. Let $$T_{i]$$ be the number of messages sent by each player. Then the possible states of nature given by pairs $$(T_{1}, T_{2})$$ of the form: $$(0,0), (n+1,n), (n+1,n+1)$$  with respective probabilities $$1p, p\epsilon(1\epsilon)^{2n}, p\epsilon(1\epsilon)^{2n+1}$$.
NB: it is this incomplete information game that is the 'Electronic Mail Game'. Players do have common knowledge regarding the structure of this game but the Electronic Mail Game is a model of a failure of common knowledge in the underlying coordination game.
NB: common knowledge can be thought as $$N=\infty$$. However with probability 1 the players do not have such common knowledge.
Then one can easily show that the only Nash equilibrium of this game of incomplete information and in it both players play A independent of the number of messages sent.
Thus even when player 1 knows the game is B and for any amount of finite knowledge (i.e. any finite number of sent messages) the players play (A,A) (and get 0 each) whereas with common knowledge they would play (B,B) (and get M each).
Refs
[1]: http://ideas.repec.org/a/aea/aecrev/v79y1989i3p38591.html
[2]: Stephen Morris and Hyun Song Shin: Approximate Common Knowledge and Coordination: Recent Lessons from Game Theory, Journal of Logic, Language and Information 6 (1997), 171190.