Suppose you are a a timeless decision theory agent playing this modified Prisoner's Dilemma with an actor that will always pick "defect" no matter what your strategy is. Clearly, your best move is to cooperate, gaining you 1 util instead of no utility, and giving your opponent his maximum 3 utils instead of the no utility he would get if you defected. Now suppose you are playing against another timeless decision theory agent. Clearly, the best strategy is to be that actor which defects no matter what.

Here is what I believe to be the standard explanation.

Unfortunately, you don't have the option of playing the same strategy as a "perfect defector" since you are currently a hypothetical TDT agent. You can of course play the strategy of being a hypothetical TDT agent that turned itself into a perfect defector. However, from the point of view of your TDT opponent this is a different strategy. In particular, a TDT will cooperate when confronted with a "true" perfect defector but defect§ when faced with an ex-TDT that turned itself into one. Therefore, even though the perfect defector would gain 3 utils, there is no strategy you as a TDT can follow that will mimic the perfect defector so you might as well act like a true TDT and agree to cooperate.

This does, however, raise interesting questions about why you aren't winning.

BTW, the standard name for this prisoner's dilemma variant is chicken.

§ Edit: Actually after thinking about it I realized that what a TDT would do is cooperate with probability 2/3-ε and defect with probability 1/3+ε. This gives him a higher utility, 2/3-ε instead of 0, and still leaves you with a utility of 2-3ε, which is still enough to make you wish you had played a strait TDT strategy and cooperated.