(I'll review some motivations for decision theories in the context of Counterfactual Mugging, leading to the answer.)

Precommitment in the past, where it's allowed, was a CDT-style solution to problems like this. You'd try making the most general possible precommitment as far in the past as possible that would respond to any possible future observations. This had two severe problems: it's not always possible to be far enough in the past to make precommitments that would coordinate all relevant future events, and you have to plan every possible detail of future events in advance.

TDT partially resolves such problems by implementing coordinated decisions among the instances of the agent within agent's current worlds (permitted by observations so far) that share the same epistemic state (or its aspects relevant to the decision) and decide for all of themselves together, so arrive at the same decision. (It makes sense for the decision to be a strategy that then can take into account additional information differentiating the instances of the agent.) This is enough for Newcomb's problem and (some versions of) Prisoner's Dilemma, but where coordination of agents in mutually exclusive counterfactuals are concerned, some of the tools break down.

Counterfactual Mugging both concerns agents located in mutually exclusive counterfactuals, and explicitly forbids the agent to be present in the past to make a precommitment, so TDT fails to apply. In this case, UDT (not relying on causal graphs) can define a common decision problem shared by the agents from different counterfactuals, if these agents can be first reduced to a shared epistemic state, so that all of them would arrive at the same decision (which takes the form of a strategy), which is then given each agent's particular additional knowledge that differentiates it from the other agents within the group that makes the coordinated decision.

In the most general case, where we attempt to coordinate among all UDT agents, these agents arrive, without using any knowledge other than what can be generated by pure inference (assumed common among these agents), at a single global strategy that specifies the moves of all agents (depending on each agent's particular knowledge and observations). However, when applied to a simple situation like Counterfactual Mugging, an agent only needs to purge itself of one bit of knowledge (identifying an agent) and select a simple coordinated strategy (for both agents) that takes that bit back as input to produce a concrete action.

So this takes us the whole circle, from deciding in a moment, to deciding (on a precommitment) in advance, and to deciding (on a coordinated strategy) in the present (of each instance). However, the condition for producing a coordinated strategy in the present is different from that for producing a precommitment in the past: all we need is shared state of knowledge among the to-be-coordinated agents, and not the state of knowledge they could've shared in the past, if they were to attempt a precommitment.

So for this problem, in coordinating with the other player (which let's assume abstractly exists, even if with measure 0), you can use your knowledge of the millionth digit of pi, since both players share it. And using this shared knowledge, the strategy you both arrive at would favor the world that's permitted by that value, in this case the paperclip world, the other world doesn't matter, contrary to what would be the case with a coin toss instead of the accessible abstract fact. And since the other player has nothing of value to offer, you take the whole pie.