DeLong on the Symmetry Argument

by on February 6, 2007 at 12:08 pm in Economics | Permalink

Brad DeLong has an excellent post on the prisoner’s dilemma, the symmetry argument and Newcomb’s problem.  He hits the nail on the head with this:

I am a dominant-strategy guy. If you find the Symmetry Argument
convincing–well, Grasshopper, you have once again failed to snatch the pebble
from my hand. But I feel the force of the other side: If you find the Symmetry
Argument an obvious fallacy–well, Grasshopper, you have once again failed to
snatch the pebble from my hand.

Sandy P February 6, 2007 at 12:50 pm

Sound like the global warming reasoning.

It’s below 0 – global warming – it’s not below 0 in winter in the Midwest – it’s global warming.

Or, I win either way.

theCoach February 6, 2007 at 1:12 pm

Sandy P,

If you think the only evidence for climate change is based on the current tempreature in the Midwest you have gone past being skeptical.

MattXIV February 6, 2007 at 3:11 pm

Is Newcomb’s problem an active controversy? The argument against the applicability of dominance seems pretty clear since the premise specifies a lack of independence between the outcomes of the decision making processes. You can come up with some much more mundane games were the lack of independence would cause a rational actor to not chose the dominant solution. For example, take a standard prisoner’s dilemma, but allow each party to see the other’s answer and change their own answer at will and not end the decision making process until both parties want to. The dominant solution is still to betray, but a rational actor will instead chose to cooperate, since he knows that neither player will consent to an outcome where they are cooperating while being betrayed, making the choice effectively between cooperate/cooperate and betray/betray.

Tyler Cowen February 6, 2007 at 4:17 pm

The obviously correct answer is to take only one box. Don’t be greedy, and who knows what kind of other delusions you are laboring under?

Kieran February 7, 2007 at 11:31 am

> The obviously correct answer is to take only one box.

A philosopher once remarked to me that the main feature of Newcomb’s Problem is that, as a rule, people quickly see either the one-box or the two-box solution as obvious, and then quickly move to the belief that no reasonable person could think otherwise.

MattXIV February 7, 2007 at 8:02 pm

On the symmetry argument for the simple prisoner’s dilemma, I think it depends on how you specify the problem. If you specify the problem as the two players being identical and that they having knowledge of that, then they will be able to deduce that mixed choices will never occur and cooperate. But if you do not require the players to be identical, but merely to both have the same utility functions and information and to proceed rationally, then the symmetry argument doesn’t hold up. Ironically, the way Brad specifies the problem does include grounds for assuming symmetry although I think he’s right that dominance should be considered the governing principle when thinking about this practically.

Back to Newcomb’s problem, I don’t see any other argument than dominance which suggests A+B. No matter how you model the predictor actually works, B comes out ahead of A+B, with the caveat that the decider needs an actual decision art

If you model the predictor as being infalable (predicted choice = choice), then there there will always be agreement so taking B only becomes obvious.

If you model the predictor as having sufficient knowledge to model and evaluate the decision making process of the subject in full detail, the predictor will always be right. A player trying to model the model and chose the opposite runs into the halting problem.

If you model the predictor as a black box having a very high probability of spitting out the correct answer, you still get B.

Expected A+B = 10^3 + (1-p) * 10^6

Expected B = 10^6 * p

Expected A+B > Expected B iff:

10^-3 + 1 > 2 p

p < .5005, which for two choices couldn’t reasonably be described as a high probability of being correct.

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