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.