Basic Prisoner's Dilemma
------------------B Cooperates B Defects
A Cooperates- (2, 2) --------- (-5, 3)
A Defects-------- (3,-5)--------- (-4,-4)
Ordinary solution: always defect because no matter what the other person does, you are better off defecting.
Symmetry Argument: if my opponent is exactly like me (or very similar), then they will do what I do. I should, therefore, cooperate: my partner will cooperate; we both win!
The symmetry argument appears to run into a problem in that it assumes what you think will determine how the other person thinks. But, who believes that what you decide causes other people like you to change their mind? The other option is that you don't really have a choice in what you will decide: your reasoning is determined by sparks, atoms, and molecules interacting inside your mind like a machine. Thus, your decision is already made. You can then hope that you and your prisoner partner are both "irrational" and will choose to cooperate. Alleluia!
Assuming the person is not just like you, the first thing you should ask is, "has my partner taken an economics class?" If so, defect. You might have better luck with a sociologist or communist.
I have not heard the Symmetry Argument before, but I have thought along these lines with respect to voting. Voting is (nearly) completely irrational. You have an almost infinitesimal chance of changing an election outcome. It is certainly not worth the time to walk to your voting booth (much less wait in line or read the ballot propositions). However, if you assume that there are a bunch of people out in the world that think like you, they will (irrationally) decide to vote if you do. You want them to go to the polls because they will vote they way you do. Much like a poll of 1000 Americans will fairly accurately tell you what all 300 million feel about a subject. Therefore, if you decide to go to vote, others like you will decide to vote. The problem is that when a person answers a poll question, they do not cause millions of Americans to believe what they believe. The key, then, is to try to convince yourself and people who think like you to make a rational mistake and believe that the following argument is a good one: Go vote because it means that millions of others will vote with you!