Here’s an experiment you can do that will teach you something.  Get a partner.  Think of a famous song and clap out the melody of the song as you sing it in your head.  You want your partner to be able to guess the song.

Out of ten tries how often do you think she will guess right?  Well she will guess right a lot less than that.  This is the illusion of transparency which is very nicely profiled in this post at You Are Not So Smart.  We overestimate how easily our outward expressions communicate what is in our heads.

This should be an important element of behavioral game theory because game theory is all about guessing your partner’s intentions.  As far as I know, biases in terms of estimates of others’ estimates of my strategy is untapped in behavioral game theory.  Its effects should be easily testable by having players make predictions about others’ predictions before the play of a game.

There are games where I want my partner to know my intentions.  For example I want my wife to know that I will be picking up coffee beans on the way home, so she doesn’t have to.  Of course I can always tell her, but if I overestimate my transparency we might have too little communication and mis-coordinate.

Then there are games where I want to hide my intentions.  In Rock-Scissors-Paper it shouldn’t matter.  I might think that she knows I am going to play Rock, and so at the last minute I might switch to Scissors, but this doesn’t change my overall distribution of play.

It should matter a lot in a casual game of poker.  If my opponent has a transparency illusion he will probably bluff less than he should out of fear that his bluffing is too easy to detect.  So if I know about the transparency illusion I should expect my opponent on average to bluff less often.

But, if he is also aware of the transparency illusion and he has learned to correct for it, then this changes his behavior too.  Because he knows that I am not sure whether he suffers from the illusion or not, and so by the previous paragraph he expects me to fold in the face of bluff.  So he will bluff more often.

Now, knowing this, how often should I call his bets?  What is the equilibrium when there is incomplete information about the degree of transparency illusion?

In a long game of course reputation effects come in.  I want you to believe that I have a transparency illusion so I might bluff less early on.