For games with only one player (which includes the absent-minded driver, newcomb's problem, counterfactual mugging and parfit's hitchhiker) I think we can characterize it like this:
CDT gives the correct decisions for Sam Beckett assuming he's just 'leaped' into you (and for some reason shares your utility function).
EDT gives the correct "decisions" for a spectator who shares your utility function but is unable to actually make any decisions. (It tells the spectator which decision to hope for.)
TDT gives the correct decisions for a God who shares your utility function and when you pray to it, can control the flow of mathematical logic as it deduces your actual decision from your predetermined prior mental state (of TDT-prayer). (Meanwhile, 'you' are just carried along by the force of mathematical truth, which must be consistent everywhere and at all times, including at 'other instances of yourself' if there are any.)
TDT is the correct decision theory for a God who shares your utility function and when you pray to it, can control the flow of mathematical logic as it deduces your actual decision from your predetermined prior mental state (of TDT-prayer).
Assuming the God must answer all possible prayers, and can't deduce, for example, that Parfit must have picked you up or you wouldn't have made that prayer.
I couldn't find any concise explanation of what the decision theories are. Here's mine:
A Causal Decision Theorist wins, given what's happened so far.
An Evidential Decision Theorist wins, given what they know.
A Timeless Decision Theorist wins a priori.
To explain what I mean, here are two interesting problems. In each of them, two of the decision theories give one choice, and the third gives the other.
In Newcomb's problem and you separate people into groups based on what happened before the experiment, i.e. whether or not Box A has money, CDT will be at least as successful in each group as any other strategy, and notably more successful than EDT and TDT. If you separate it into what's known, there's only one group, since everybody has the same information. EDT is at least as successful as any other strategy, and notably more successful than CDT. If you don't separate it at all, TDT will be at least as successful as any other strategy, and notably more successful than EDT.
In Parfit's hitchhiker, when it comes time to pay the driver, if you split into groups based on what happened before the experiment, i.e. whether or not one has been picked up, CDT will be at least as successful in each group as any other strategy, and notably more successful than TDT. If you split based on what's given, which is again whether or not one has been picked up, EDT will be at least as successful in each group as any other strategy, and notably more successful than TDT. If you don't separate at all, TDT will be at least as successful as any other strategy, and notably more successful than CDT and EDT.
There's one thing I'm not sure about. How does Updateless Decision Theory compare?