The Psychological Twin Prisoner's Dilemma is a variant of the original Prisoner's Dilemma. From Functional Decision Theory: A New Theory of Instrumental Rationality:
An agent and her twin must both choose to either “cooperate” or “defect.” If both cooperate, they each receive $1,000,000. If both defect, they each receive $1,000. If one cooperates and the other defects, the defector gets $1,001,000 and the cooperator gets nothing. The agent and the twin know that they reason the same way, using the same considerations to come to their conclusions. However, their decisions are causally independent, made in separate rooms without communication. Should the agent cooperate with her twin?
Because the twin's decision is causally independent from the agent's decision, Causal Decision Theory (CDT) says "No". Note that a CDT agent would do poorly in this problem, as she and her twin would both defect, getting the agent only $1000.
Evidential Decision Theory (EDT) says "Yes", as cooperating is correlated with cooperation from the twin and defecting correlates with the twin defecting. An EDT agent would do well in this problem: she and her twin would both cooperate, resulting in a $1,000,000 for the agent.
Functional Decision Theory (FDT) agrees with EDT, but reasons very differently. FDT acknowledges that the agent and her twin are both reasoning the very same way, and are therefore subjunctively dependent on the agent's decision procedure. This is because both the agent and her twin both compute that same decision procedure! Compare this to two calculators which both output "4" on the input "2 + 2". The calculators are causally independent, but if one of them gives an answer to a certain input (like "2 + 2"), you know the other must give the same answer. It's the same in the Psychological Twin Prisoner's Dilemma: if an FDT agent imagines cooperating, she knows her twin would do the same. If she imagines defecting, she again knows her twin would do the same. Of these two possibilities, both the agent and her twin cooperating is clearly the best option for the agent; the FDT agent therefore cooperates.