FDT and the sacrificial volunteer's dilemma

Consider the "twin sacrificial volunteer's dilemma":[1] 1000 identical agents have the same deterministic decision-making process and the same observations, and can't communicate with each other. They can decide to either do nothing, or pay $1 to volunteer. If at least one agent volunteers, everyone who didn't volunteer gets $1 million[2]. Volunteers never receive anything. All of this is common knowledge. Payoff matrix for clarity: You don't volunteer You volunteer Nobody else volunteers $0 -$1 Someone else volunteers $1M -$1 Because the agents all make decisions the same way, there are only two possible outcomes (in bold in the table): everybody volunteers (everybody gets -$1) or nobody volunteers (everybody gets $0). Not volunteering is the dominant strategy, and all reasonable decision theories don't volunteer. (In this situation the agents can't implement a policy like "volunteer with a small chance", since they are deterministic and don't have access to randomness).[3] Giving agents serial numbersNow imagine the same situation, but instead of every agent having the same observations, every agent has a serial number from 1-1000 and is told their serial number (and this is common knowledge). This serial number doesn't change anything about the payoffs. I call this the "numbered sacrificial volunteer's dilemma", and here the symmetry is broken so it's possible for some agents to volunteer and others to not. Does this change things? CDT and EDT agents still don't volunteer for almost the same reasons as before. FDT is more complicated. (This scenario is somewhat similar to this one from the UDT1.1 description.) FDT is most commonly described as action-based, where you intervene on FDT(P, G, x) (where P and G represent the agent's probability distribution and causal graph, and x is the observations). But you can also formalize it by having it find a policy instead: footnote 16 of the FDT paper says that the authors prefer to formalize it as intervening on FDT(P, G