Yet it's not just a PD, it's an iterative PD because it's ongoing through time. Such problems have been solved, through reciprocity. There's an interesting book on these very problems by Axelrod, that explains that punishing non-cooperators once (immediately after previous defections) in iterative PDs and cooperating if they cooperate leads to the greatest cooperation overall. People instinctively know this, and world leaders certainly do (with few exceptions, like the near-permanent embargo on Cuba).
In other words, throwing our hands in the air and going "it's a prisoner's dilemma - we can't do anything!" doesn't mean that the problem is insoluble. We can and must cooperate with Europe, and they know that too.
Your logic is completely self-defeating, because it applies equally to the development of societies - clearly it didn't stop people though. Assuming that the member states of the EU are out to screw each other over at the expense of others is exactly what the EU is designed to stop, and is exactly what I meant when I said that you were missing the point of the EU. Every leader knows that, in cooperation, the result is greater than the sum of its parts: it's not rational at all to defect from that, because of the possibility of reciprocated punishment in the future, which will happen for the express purpose of ensuring mutual cooperation and therefore mutual prosperity.