Electronic_Mail_Game_Model - Atlas of Economic Models
 

The Electronic Mail Game

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 payoff-structure) 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: (1-p)) :

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 $$1-p, p\epsilon(1-\epsilon)^{2n}, p\epsilon(1-\epsilon)^{2n+1}$$.

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/v79y1989i3p385-91.html

[2]: Stephen Morris and Hyun Song Shin: Approximate Common Knowledge and Co-ordination: Recent Lessons from Game Theory, Journal of Logic, Language and Information 6 (1997), 171-190.