It seems to me that the issues with MSR are pretty much the same as cooperating in PD with a random stranger from the street. If that's right, you could look at empirical studies of strangers playing PD.

I have posted this comment elsewhere recently, but here is edited version of it. It describes how multiverese wide cooperation could be used for resurrection of the dead. Multiverse wide cooperation could be also used for cheating Big Filter in Fermi paradox, to prevent s-risks (not published), and to press the Papercliper to imitate Benevolent AI.

TL;DR: Resurrection of the dead is possible without measure decline by creating just one random mind, but expecting that in infinitely many worlds infinitely many random minds will be also created, and as result, all possible minds will be created.

Almond suggested the following idea about the resurrection of the dead by the use of a quantum random generator, which create a random mind within a computer (Almond, 2006): If the many-worlds interpretation of quantum mechanics is true, when all possible minds will appear in different timelines starting from the moment of random mind creation, which would mean resurrection of everyone from his own point of view. However, this approach will a) not help an outside observer, who wants to resurrect a relative, for instance, as the observe would see only a random mind, and b) the "measure" of existence of each mind will be infinitely small.

The first problem could be overcome by the use of relatives expectations as priors, for example, if I expect to resurrect John, I create all possible "John"s, and use random generator to generate all possible surnames (and all other personal data, which I will here ignore for the simplification of the problem).

However, there is still the problem of quantum measure of existence decline, which, according to some authors, is the real problem with ideas like quantum immortality. Measure is, roughly speaking, the share of the worlds where I exist. If we use some expected utility calculations, measure decline results in declining utility of any useful outcome associated with it, so we could just ignore my copies with infinitely small measures. (This position itself is vulnerable, as it takes in account the absolute, but not relative measure. Absolute measure is my share between all possible observers, and relative measure is share of my copies with some property between all my future copies. For example, the relative share of all my future copies in the moment T=today evening, who will eat an apple is something like 0.001. But absolute share of my copies created by the quantum randomness generator is something like 10 power (-10 power 30). However share of my randomly created copies who will eat an apple from all randomly created copies is still 0.001. The question, which type of measure should I care about, is open for debate).

But acausaul cooperation between worlds could solve even the problem of the absolute measure decline, because there will be other worlds with other quantum random generators, and if we properly account for all them, the total measure of existence of each person could be approximately the same. Here we assume that there are many different worlds and all of them came to the idea of the use of just one random generator is all that is needed for the resurrection. I will illustrate it with the same example about John:

For example, we know that someone’s name was John S—. His last name was either Smith or Simpson. We create a model of John S— and use a quantum generator to choose between either Smith or Simpson. In half of all possible worlds we will get Smith, and in the other half we’ll get Simpson. If the actual name was Smith, this means that the “measure” of Smith declines by half.

However, there is no decline of measure. If we look at a broader picture, there is another possible world, where John S— was named Simpson, and this is the only difference. In this world, where another AI which will try to recreate John S—, also by using a quantum random generator to decide his full name, which will give “Simpson” half of the time. However, if we combine both worlds, at the beginning we have one Smith world and one Simpson world, and at the end we will have four worlds: two with a half measure of Simpson, and two with a half measure of Smith. Thus, the total measures of Smith and Simpson will not change.

I think the biggest hurdle to the concept is showing that the "trade" part is possible, which (I think) means some causal link between decision and experience. That link can be in the future, or uncertain, or conditional, as in many of the test cases for various decision theories. But I don't see any difference between defect or cooperate in the case where there's no causal transmission of the choice, ever.

Another biggest hurdle is to show that "choice" has any meaning in a Tegmark multiverse. I think all agents both cooperate and defect, in different universes. In that case, your decision mechanism is discovery of which universe you're in, not a narrowing of what universes exist.