One of the arguments supporting Logical decision theories over Causal_dt is the assertion that LDT handles a strictly wider problem class and therefore dominates: Arguendo, we lose nothing by moving from CDT to LDT, and instead gain the ability to handle a strictly wider class of problems.
Wei Dai posed the following counterexample: Suppose that 99 LDT agents and 1 CDT agent are playing a tournament of oneshot Prisoners_dilemma. Then the LDT agents calculate that they will lose a great deal by defecting (since this would make them defect against 99 other LDT agents), while the CDT agent cheerfully defects (and does not move in unison with the LDT agents).
To make this example more precise, we might also imagine an LDT agent, versus a CDT agent, playing the game "Me and 98 LDT agents in a PD tournament." (In the original example, the CDT agent is facing 99 LDT agents, and each LDT agent is facing 98 LDT agents and 1 CDT agent, which technically breaks the fairness of the problem.)
There are two counterarguments that this example does not favor CDT as a decision theory:
First, imagine inserting a single logical-style agent with some other slightly variant algorithm, LDT-Prime, such that it no longer sees itself as moving in lockstep with the other LDT agents in the PD tournament. Then LDT-Prime will do as well as the CDT agent in the tournament, and do better on other Newcomblike problems. This argues that the particulars of the CDT algorithm were not what gave the CDT agent its apparent advantage.
Second, an LDT agent would argue against the fairness of the PD tournament. Since the LDT agent is facing off against 98 other agents moving in logical lockstep with itself, it is being faced with an environmental challenge unlike the one an LDT-Prime agent or CDT agent sees. Arguendo, the LDT agent is being presented with different options or consequences for its decision algorithm, compared to the consequences for the LDT-Prime or CDT algorithm.
Suppose that a CDT agent and an LDT agent both encounter a $20 bill in the street. Two days previously, Omega has already bombed an orphanage if the output of the LDT algorithm is to pick up the $20 bill. The LDT agent naturally refuses to pick up the $20. The CDT agent laughs and remarks that Omega is already gone, and chides the LDT agent for not taking the $20 reward that was equally available to any agent passing by.
Since the CDT agent doesn't think the past correlation can affect outcomes now, the CDT agent believes that the CDT agent and the LDT agent would receive just the same payoff for picking up the $20 bill, and thus that this scenario is a fair challenge. The LDT agent thinks that the CDT agent and LDT agent have been presented with different payoff matrices for the same outputs, and thus that this is an unfair challenge. On the LDT view, CDT agents are blind to Newcomblike dependencies, so the CDT agent may believe that a scenario is a fair CDT problem when it is actually an unfair Newcomblike problem.
On the LDT view, something very similar happens when an LDT agent versus an LDT-Prime agent, or an LDT agent versus a CDT agent, are presented with a PD tournament against 98 LDT agents. Being presented with 98 other contestants that will mirror you but not the competing agent doesn't seem very 'fair'.
However, even the orphanage-bombing example taken at face value seems sufficient to technically refute the general statement that the fair problem class on which LDT agents end up rich is strictly larger than the corresponding 'fair' problem class for CDT agents. And the 99LDT/1CDT tournament does seem in some sense like it's posing a natural or a realistic scenario in which a CDT agent could get a higher payoff than an LDT agent; on the CDT view, this is exactly the sort of problem that CDT is good for.