If I’m the type of person who will always cooperate, what would happen if I went off-the-equilibrium-path and did defect even if my defecting is a zero probability event?
I’m trying to understand the difference between your statement and “1 is not equal 2, but what if it were?” and failing.
A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a “slip of the hand” or tremble, may choose unintended strategies, albeit with negligible probability.
First we define a perturbed game. A perturbed game is a copy of a base game, with the restriction that only totally mixed strategies are allowed to be played. A totally mixed strategy is a mixed strategy where every pure strategy is played with non-zero probability. This is the “trembling hands” of the players; they sometimes play a different strategy than the one they intended to play. Then we define a strategy set S (in a base game) as being trembling hand perfect if there is a sequence of perturbed games that converge to the base game in which there is a series of Nash equilibria that converge to S.
One possible interpretation of “if I always cooperate, what would happen if I don’t” is “what is the limit, as X approaches 1, of ‘if I cooperate with probability X, what would happen if I don’t’?”
Or things I can’t do vs things I don’t want to do.
In my mind “I’m the type of person who will always cooperate” means that there is no difference between the two in this case. Maybe you use a different definition of “always”?
I always cooperate because doing so maximizes my utility since it is better than all the alternatives. I always go slower than the speed of light because I have no alternatives.
I’m trying to understand the difference between your statement and “1 is not equal 2, but what if it were?” and failing.
See trembling hand equilibrium.
Right, as I mentioned in my other reply, CDT is discontinuous at p=0. Presumably a better decision theory would not have such a discontinuity.
One possible interpretation of “if I always cooperate, what would happen if I don’t” is “what is the limit, as X approaches 1, of ‘if I cooperate with probability X, what would happen if I don’t’?”
This doesn’t reasonably map onto the 1=2 example.
Right. There seems to be a discontinuity, as the limit of CDT (p->0) is not CDT (p=0). I wonder if this is the root of the issue.
“1 is not equal 2, but what if it were?” = what if I could travel faster than the speed of light.
Off the equilibrium path = what if I were to burn a dollar.
Or things I can’t do vs things I don’t want to do.
In my mind “I’m the type of person who will always cooperate” means that there is no difference between the two in this case. Maybe you use a different definition of “always”?
I always cooperate because doing so maximizes my utility since it is better than all the alternatives. I always go slower than the speed of light because I have no alternatives.