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I am having difficulties in distinguish them. I kind of understand what is the difference intuitively, but I still fail to put some games in the correct category.

Can you give me two examples of games and show me the difference from a mathematical (notation) point of view?

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Consider enter image description here where $\epsilon_1,\epsilon_2$ are iid random variables with distribution $F$. $a,b,\dots,h$ are constant parameters.

Game of incomplete but symmetric information would be something like: $F$ is common knowledge among the two players.

Game of asymmetric and incomplete information would be something like: before making their respective decision, Player 1 observes the realization of $\epsilon_1$ but not $\epsilon_2$ and Player 2 observes $\epsilon_2$ but not $\epsilon_1$.

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  • $\begingroup$ I am not sure, but I think that asymmetric information implies "imperfect information" and not "incomplete information" $\endgroup$
    – Lex
    Commented Apr 21, 2015 at 23:02
  • $\begingroup$ @Lex: You're right. I've updated my answer. Thanks for catching the error. $\endgroup$
    – Herr K.
    Commented Apr 21, 2015 at 23:08
  • $\begingroup$ Thank you for your reply! Anyway I still have some doubts: in the symmetric case what do the player observe before making their decision? In the asymmetric case is F then not common knowledge? $\endgroup$
    – Lex
    Commented Apr 21, 2015 at 23:15
  • $\begingroup$ @Lex: In the symmetric case, you can assume that they don't observe anything before making a decision. In the asymmetric case, it's okay to allow $F$ to be common knowledge; but what's important here is that each player is certain about her own payoff while uncertain about the opponent's payoff, thus making the information structure asymmetric (at the time of making decision). $\endgroup$
    – Herr K.
    Commented Apr 22, 2015 at 5:11
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Verbally:

PERFECT information: all information available is true (no errors).

SYMMETRIC information: all decision-makers involved have exactly the same Information set

COMPLETE information: the information an agent posses includes all that is known about all aspects of the situation that we examine.

Note that when examining the information set of a single agent, neither of the above three presupposes or requires the other. In some cases though, authors seem to use the word "Complete" in order to mean "we know everything there is to be known" about something, in some "objective" sense. Under this interpretation, it is hard (although not infeasible), to avoid identifying "complete" with "perfect", and this is why I would not suggest this approach, because a clear distinction between "complete" and "perfect" is useful. It is subtle to say (although again, not logically inconsistent) "I know everything there is to know, but some of this knowledge is wrong". It appears better to maintain a "relative" aspect to the concept of "completeness" ("all information available").

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