perfectness.txt

Septmeber 25, 2001
Eric Rasmusen

The diagram perfectness.pdx shows two game that together illustrate
the difference between perfect equilibrium and precommitment.

First,look at the top diagram. An example of a Nash but non-perfect
equilibrium is (b, d|a).  This is Nash because if player 1 deviates he
gets 0 instead of 4 and if player 2 deviates he gets 2 instead of 4.
It is non-perfect because if player 1 did choose a and the game
arrived at player 2's node, player 2 would choose c|a.

 It is true in the top diagram that if player 2 could move first, or
could precommit,  he would choose  d|a.  But that is not the same
thing, quite, as perfectness.

Look now at the bottom diagram. In that game, if player 2 could move
first or precommit he would choose d|a, and the result would be an
equilibrium of (a, d|a). But that is not even a Nash equilibrium, much
less a perfect one. It is not Nash because if player 2  unilaterally
deviated to c|a he would increase his payoff from 1 to 2. To be sure,
player 1 would not choose a if he foresaw that happening, but Nash
equilibrium only  tests for unilateral deviations from equilibrium
strategies.

   The top diagram is a game in which player 2 would like to commit to
his behavior   off the equilibrium path,  a threat he never actually
has to use. The bottom diagram is a game in which player 2 would like
to commit to his behavior    *on*  the equilibrium path, a promise  he
must keep.