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 \noindent   
 September 30, 2009
\\
 Eric Rasmusen, erasmuse@indiana.edu

 
\begin{center} {\bf   A Paradox of  Sequential Rationality    }
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Consider the perfect-information game in the figure below.   Player 1's strategy set is:  $(A, E|AC), (A, F|AC),   (B, E|AC),  (B, F|AC)$.  Player 2's strategy set  is: $C|A,   D|A$.  

 Using backwards induction, Player 1 uses $F|AC$, Player 2 uses $C|A$, and Player 1 uses $B$ in the    subgame perfect Nash equilibrium. 

The other  Nash equilibrium is for Player 1 to use $(A, E|AC)$ and for Player 2 to use $D|A$. 

  Player 1 prefers the non-perfect equilibrium with its (3,1) instead of (2,1) payoffs.  The story justifying subgame perfectness that  we tell ourselves is that before the game starts, if Player 1 threatens Player 2, saying: "I am going to choose A, so you'd better choose D in response, or I'll go ahead and choose E and we'll both get 0," then Player 2 will respond, "I don't believe you. I'm calling your bluff. You are rational, and so I know you would never choose E instead of F." Then Player 1 would give up and choose B instead.  

But what if Player 1 makes his little speech, and then actually does choose A? What should Player 2 think? In equilibrium, choosing A isn't supposed to happen. It seems to  refute  the assumption of   Player 1 being rational.  So maybe Player 2 should respond by choosing D after all. If he does that, however, then Player 1's action has turned out to be rational after all. 

\includegraphics[width=4in]{paradox1.pdf}



(This is inspired by Section 6.4 of Osborne and Rubinstein's  1994 {\it A Course in Game Theory}, which has a similar but not identical game.) 

 





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