I Demand That Cosma Shalizi Write a Weblog Post About What the Real Dyson-Press Contribution to the Theory of Iterated Prisoner's Dilemma Is!
Brad DeLong wrote:
Iterated Prisoner's Dilemma Blogging: Dyson and Press Really Are Very Clever Indeed...: Basically, consider the strategy space of one-period look-back mixed strategies. And take the strategy S = { p(C|CC)=1/2-ε, p(C|CD)=1/4+ε', p(C|DC)=0, p(D|DD)=0 }.
If the opponent cooperates all the time, their average score is a little better than the DD payoff 1--giving them an incentive to choose a strategy that cooperates some time. If they cooperate all the time, then your average score is a little worse than 4--pretty good. And if they do anything other than cooperate all the time, their score falls and is worse than if they cooperate all the time. Thus in this strategy space { P(C )=1 } is a dominant strategy for the opponent if you choose your strategy first. Of course, this is not a Nash equilibrium: if the opponent is playing C you should play C as well
And two opponents each thinking that they move first and committing to S do rather badly: they end up in (D,D) for all time.
And then comes the smackdown:
anon said… I'm hoping someone can enlighten me about why this is particularly interesting. The folk theorem already told us that any outcome above the (D,D) forever payoffs could be implemented, even very extortionary outcomes. There is something interesting about only needing one period of history dependence in the strategies to do it, but I don't think that is what everyone is so excited about. I think I'm probably missing something crucial here, but I'm not sure what it is-any suggestions?
Besttrousers said… Doesn't this conclusion flow out of the folk theorem? It seems like it's just a special case of it, but I haven't really sat down with the paper.
"Besttrousers" and "anon"'s reaction appears to be a very common one, especially among economists. The point is that Press-Dyson is not interesting because (i) the folk theorem already taught us that iterated prisoner's dilemma can support practically anything as an equilibrium; (ii) Press-Dyson simply present us with one of these equilibria; so (iii) why isn't this boring?
What is the "folk theorem"? Wikipedia:
A commonly referenced proof of a folk theorem was published in (Rubinstein 1979).
The method for proving folk theorems is actually quite simple. A grim trigger strategy is a strategy which punishes an opponent for any deviation from some certain behavior. So, all of the players of the game first must have a certain feasible outcome in mind. Then the players need only adhere to an almost GRIM trigger strategy under which any deviation from the strategy which will bring about the intended outcome is punished to a degree such that any gains made by the deviator on account of the deviation are exactly cancelled out. Thus, there is no advantage to any player for deviating from the course which will bring out the intended, and arbitrary, outcome, and the game will proceed in exactly the manner to bring about that outcome.
What response do I have to this?
First, basically, I don't believe that equilibria that exist only because any deviation is a trigger for a GRIM or a nearly-GRIM strategy are really there. To be credible, a strategy has to be chosen to elicit good behavior in the future, and not to punish bad behavior in the past, because sunk costs truly are sunk. If you want to support an equilibrium that requires a GRIM or a near-GRIM trigger, you need to specify that the players in your game like to engage in altruistic punishment--or have a taste for vengeance, which is more-or-less the same thing. Absent a strong taste for engaging in altruistic punishment in your agents, equilibria supported by GRIM out-of-equilibrium just don't count.
Second, I don't believe in equilibria that simply descend from the sky. Some kind of historical process that starts with out-of-equilibrium play or some other set of factors has to be there to produce the coordinated common expectations on which any equilibrium must rest. Folk theorem arguments are absolutely and deliberately and necessarily silent on the disequilibrium foundations of equilibrium game theory.
My problem is that my native code running on my wetware believes these two points strongly, but is not equipped to argue for them. And the Cosma Shalizi emulation module that I could fire up and run on top of my native code is primitive and buggy--and I don't want to crash my entire brain and have to reboot it.
So I demand--I plead for--I beg for Cosma Shalizi to write the weblog post about the value of Dyson-Press that I cannot…
