It's an old book, I know, and one that many of us have already read. But if you haven't, you should.
If there's anything in the world that deserves to be called a martial art of rationality, this book is the closest approximation yet. Forget rationalist Judo: this is rationalist eye-gouging, rationalist gang warfare, rationalist nuclear deterrence. Techniques that let you win, but you don't want to look in the mirror afterward.
Imagine you and I have been separately parachuted into an unknown mountainous area. We both have maps and radios, and we know our own positions, but don't know each other's positions. The task is to rendezvous. Normally we'd coordinate by radio and pick a suitable meeting point, but this time you got lucky. So lucky in fact that I want to strangle you: upon landing you discovered that your radio is broken. It can transmit but not receive.
Two days of rock-climbing and stream-crossing later, tired and dirty, I arrive at the hill where you've been sitting all this time smugly enjoying your lack of information.
And after we split the prize and cash our checks I learn that you broke the radio on purpose.
Schelling's book walks you through numerous conflict situations where an unintuitive and often self-limiting move helps you win, slowly building up to the topic of nuclear deterrence between the US and the Soviets. And it's not idle speculation either: the author worked at the White House at the dawn of the Cold War and his theories eventually found wide military application in deterrence and arms control. Here's a selection of quotes to give you a flavor: the whole book is like this, except interspersed with game theory math.
The use of a professional collecting agency by a business firm for the collection of debts is a means of achieving unilateral rather than bilateral communication with its debtors and of being therefore unavailable to hear pleas or threats from the debtors.
A sufficiently severe and certain penalty on the payment of blackmail can protect a potential victim.
One may have to pay the bribed voter if the election is won, not on how he voted.
I can block your car in the road by placing my car in your way; my deterrent threat is passive, the decision to collide is up to you. If you, however, find me in your way and threaten to collide unless I move, you enjoy no such advantage: the decision to collide is still yours, and I enjoy deterrence. You have to arrange to have to collide unless I move, and that is a degree more complicated.
We have learned that the threat of massive destruction may deter an enemy only if there is a corresponding implicit promise of nondestruction in the event he complies, so that we must consider whether too great a capacity to strike him by surprise may induce him to strike first to avoid being disarmed by a first strike from us.
Leo Szilard has even pointed to the paradox that one might wish to confer immunity on foreign spies rather than subject them to prosecution, since they may be the only means by which the enemy can obtain persuasive evidence of the important truth that we are making no preparations for embarking on a surprise attack.
I sometimes think of game theory as being roughly divided in three parts, like Gaul. There's competitive zero-sum game theory, there's cooperative game theory, and there are games where players compete but also have some shared interest. Except this third part isn't a middle ground. It's actually better thought of as ultra-competitive game theory. Zero-sum settings are relatively harmless: you minimax and that's it. It's the variable-sum games that make you nuke your neighbour.
Sometime ago in my wild and reckless youth that hopefully isn't over yet, a certain ex-girlfriend took to harassing me with suicide threats. (So making her stay alive was presumably our common interest in this variable-sum game.) As soon as I got around to looking at the situation through Schelling goggles, it became clear that ignoring the threats just leads to escalation. The correct solution was making myself unavailable for threats. Blacklist the phone number, block the email, spend a lot of time out of home. If any messages get through, pretend I didn't receive them anyway. It worked. It felt kinda bad, but it worked.
That was an answer for your question about what should you replace CDT with. If you won't be able to convince other agents that you now run on timeless CDT, you gain a little smaller advantage than otherwise, but that's a separate problem. If you know that your claims of precommitment won't be believed, you don't precommit, it's that easy. But sometimes, you'll find a better solution than if you only lived in a moment.
Also note that even if you do convince other agents about the abstract fact that your decision theory is now timeless, it won't help you very much, since it doesn't prove that you'll precommit in a specific situation. You only precommit in a given situation if you know that this action makes the situation better for you, which in case of cooperation means that the other side will be able to tell whether you actually precommited, and this is not at all the same as being able to tell what decision theory you use.
Since using a decision theory with precommitment is almost always an advantage, it's easy to assume that a sufficiently intelligent agent always uses something of the sort, but that doesn't allow you to know more about their actions -- in fact, you know less, since such agent has more options now.
Yes, I see that your decision theory (is it the same as Eliezer's?) gives better solutions in the following circumstances:
Do you think it gives better solutions in the case of AIs (who don't initially think they're copies of each other) trying to cooperate? If so, can you give a specific scenario and show how the solution is derived?