Your strategy should be to defect unless you have good enough evidence that cooperating will cause Omega to also cooperate.
Omega will predict this, and will give you good enough evidence. Whether or not this actually leads to Omega cooperating depends on the strength of evidence given. Money placed in escrow would be enough.
Of course, if you can't receive any communication from Omega before placing your decision, Omega is going to defect on you (since Omega's decisions can't affect what you do). This is still assuming it's the only prisoner's dilemma Omega plays in your light cone, however.
Unless otherwise stated, the games theory game of Prisoner's Dilemma takes place as the only event in the hypothetical universe; in this example, prior communication and credible precommitment are not permitted.
Instead of generating a strategy to use against a copy of yourself, consider what the best strategy would be against the optimized player who knows what your strategy is.
TDT permits acting as though one had precommitted, the result being that one never wishes one's opportunities to precommit were different. Consider a perfectly reasoning person and ...
Sometimes I see new ideas that, without offering any new information, offers a new perspective on old information, and a new way of thinking about an old problem. So it is with this lecture and the prisoner's dilemma.
Now, I worked a lot with the prisoners dilemma, with superrationality, negotiations, fairness, retaliation, Rawlsian veils of ignorance, etc. I've studied the problem, and its possible resolutions, extensively. But the perspective of that lecture was refreshing and new to me:
The prisoner's dilemma is resolved only when the off-diagonal outcomes of the dilemma are known to be impossible.
The "off-diagonal outcomes" are the "(Defect, Cooperate)" and the "(Cooperate, Defect)" squares where one person walks away with all the benefit and the other has none:
Facing an identical (or near identical) copy of yourself? Then the off-diagonal outcomes are impossible, because you're going to choose the same thing. Facing Tit-for-tat in an iterated prisoner's dilemma? Well, the off-diagonal squares cannot be reached consistently. Is the other prisoner a Mafia don? Then the off-diagonal outcomes don't exist as written: there's a hidden negative term (you being horribly murdered) that isn't taken into account in that matrix. Various agents with open code are essentially publicly declaring the conditions under which they will not reach for the off-diagonal. The point of many contracts and agreements is to make the off-diagonal outcome impossible or expensive.
As I said, nothing fundamentally new, but I find the perspective interesting. To my mind, it suggests that when resolving the prisoner's dilemma with probabilistic outcomes allowed, I should be thinking "blocking off possible outcomes", rather than "reaching agreement".