In game theory, the electronic mail game is an example of an "almost common knowledge" incomplete information game. It illustrates the apparently paradoxical[1] situation where arbitrarily close approximations to common knowledge lead to very different strategical implications from that of perfect common knowledge. Intuitively, it shows that arbitrarily long but finite chains of "I know that you know that I know that you know..." are fundamentally different from infinite ones.

It was first introduced by Ariel Rubinstein in 1989.[2]

The game

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Setup

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The electronic mail game is a coordination game of incomplete information. Players 1 (she) and 2 (he) can choose between actions   and  . There are two states of the world   and  , which happen with respective probabilities   and  , with  . The payoffs for each action profile in each of those states are:

   
   ,    ,  
   ,    ,  
State  
 
   
   ,    ,  
   ,    ,  
State  

where  . Players would like to coordinate to play   in state of the world  , and to play   in  . If they coordinate in the wrong state, they only get   payoff; but if they choose different actions, the player who chose   gets a negative payoff of  .

Player 1 knows the true state of nature, whereas Player 2 does not. Without communicating, the highest expected payoff they can achieve is  , by always choosing  . If the state of the world were common knowledge, both players would be able to achieve payoff  .

Email communication

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Now assume that the players communicate via emails. Once Player 1 discovers the state of nature, her computer automatically sends an email to Player 2 informing him of the true state; Player 2's computer then automatically replies with a confirmation that he received the information; Player 1's computer then automatically replies with a confirmation that she received the information that he received the information, and so on. This mimics the idea of a "I know that you know that I know that you know..." chain.

However, there is an arbitrarily small probability   that some technical failure will happen and one of those emails will not arrive at its destination, after which communication will cease. If that happens, the last player to send the message does not know if 1) the other player did not get the last message, or 2) the other player got the last message, but could not send the confirmation email due to the technical failure.

Types and strategies

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Let   be the number of messages that were sent by Player  's computer — since that information is only observed by Player  , we can think of   as their Harsanyi type. In terms of choice, players only observe   and then must choose an action  . A strategy in the electronic mail game is thus defined as a function from   to  .

The distribution of types   is given by the following probabilities  :

  •  : the true state is   and no email is sent
  •  : the true state is   and the failure happens in Player 2's computer after Player 1 sent   emails
  •  : the true state is   and the failure happens in Player 1's computer after Player 1 sent   emails

Equilibrium

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The equilibrium concept to be used is that of a Bayesian Nash Equilibrium (BNE). Rubinstein showed that, no matter how small the chance of failure   and no matter how many confirmation emails were sent, both players always choose to play  , even if they know that the state of nature is  .

Proposition: There is only one BNE where Player 1 plays   when the state of nature is  . In this equilibrium, both players play  , independetly of their types.[2]

The result is counterintuitive, since both know that the true state is  , and they can have arbitrarily precise knowledge of "knowing that the other player knows that they know that the other player knows..." that the state is  . Still, since this chain of information eventually stops, their equilibrium best response still is to always play  .

References

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  1. ^ Morris, Stephen (2002). "Coordination, Communication, and Common Knowledge: a Retrospective on the Electronic-mail Game". Oxford Review of Economic Policy. 18 (4): 433–445.
  2. ^ a b Rubinstein, Ariel (1989). "The Electronic Mail Game: Strategic Behavior Under "Almost Common Knowledge"". American Economic Review. 79 (3): 385–391.