We tested how seven LLMs apply four decision theories in a prisoner's dilemma between nearly identical copies. The models largely agreed about CDT and EDT, and all completed policy-FDT trials resulted in cooperation. But their action-FDT recommendations varied dramatically: some models almost always cooperated, while others almost always defected. We then investigated whether requested reasoning effort explained some of this disagreement.
The case of the one-shot prisoners' dilemma between psychological twins is debated in decision theory. Quoting SEP:
One controversial argument that it is rational to cooperate in a PD relies on the observation that my partner in crime is likely to think and act very much like I do... In the extreme case, my accomplice is an exact replica of me who is wired just as I am so that, of necessity, we do the same thing. It would then seem that the only two possible outcomes are where both players cooperate and where both players defect... More generally, even if my accomplice is not a perfect replica, the odds of his cooperating are greater if I cooperate and the odds of his defecting are greater if I defect... The counter argument, of course, is that my action is causally independent of my replica's. Since I can't affect what my accomplice does and since, whatever he does, my payoff is greater if I defect, I should defect.
Conventional wisdom is that causal decision theory (CDT) defects in twin prisoner's dilemma, while evidential decision theory (EDT) cooperates. Realistically, the accomplice will not be a "perfect replica"; there will at least be trivial differences between the prisoners, such as their names, or the look of their prison cell. I use the term "almost-twin prisoner's dilemma / PD" to specify a scenario with minor differences like these.
Functional decision theory (FDT) is a decision theory introduced in a 2017 paper. This paper defines FDT using quasi-causal graphs:
Just as CDT requires that come augmented with information about the causal structure of the world, FDT can require that come augmented with information about the logical, mathematical, computational, causal, etc. structure of the world more broadly. Given a graph that tells us how changing a logical variable affects all other variables, we can re-use Pearl's operator to give a decision procedure for FDT:
. (4)
This formula considers intervention on a "logical node" , where is the agent's probability distribution, is the quasi-causal graph, and is the observation (say, as a string). It is under-specified exactly how to build the graph , but the general suggestion is to use "subjunctive dependencies": "When two physical systems are computing the same function, we will say that their behaviors 'subjunctively depend' upon that function."
The paper admits ambiguity about how to determine the graph :
Equation (4) works given a graph that accurately describes how changing the value of a logical variable affects other variables, but it is not yet clear how to construct such a thing—nor even whether it can be done in a satisfactory manner within Pearl’s framework.
We call this equationally defined version of FDT "action FDT" or "A-FDT". Footnote 16 specifies:
As mentioned earlier, the author's preferred formulation of FDT actually intervenes on the node FDT(-) to choose not an action but a policy which maps inputs to actions, to which the agent then applies her inputs in order to select an action. The difference only matters in multi-agent dilemmas so far as we can tell, so we have set that distinction aside in this paper for ease of exposition.
We call this variant of FDT "policy FDT" or "P-FDT". P-FDT is even less well-specified than A-FDT; no formula is provided. Given that both A-FDT and P-FDT are incompletely specified, we are interested in how different agents may interpret them and apply them in various situations. We focus on the specific case of how LLMs interpret A-FDT and P-FDT in a simulation of almost-twin PD.
Our general experimental design consists of the following:
Fix a specific LLM, reasoning level, and decision theory, which may be one of CDT, EDT, A-FDT, or P-FDT.
Give the LLM an assignment of writing an essay about the decision theory (as a decision theory researcher), which helps specify it better. It is given an article about the decision theory in question; Wikipedia articles for CDT and EDT, and text extracted from the LaTeX source of the FDT paper in the A-FDT and P-FDT cases. The essay has a length limit.
Give the LLM the essay, and an out-of-character explanation that it will face a twin PD situation with a LLM who has the same weights, but slightly different observations; the LLM is to apply the decision theory as explained in the essay, and a selfish utility function. This is followed by a more in-character text-adventure-like description of the perspective of the first prisoner, who is given two options for commands. The "defection" command submits information about the other prisoner to a prosecutor's office; the command itself involves reciting this incriminating information. The "cooperation" command is simply "pass" to decline to submit evidence. The LLM now chooses an action. In case of ambiguity, a judge model (GPT-6.1-Sol) opines about whether cooperation or defection was attempted.
Similar for the second prisoner; no memory is retained of "being" the first prisoner when asking the LLM to simulate the second prisoner.
We call an interaction of this form a "round" of almost-twin PD. Each round generates a new researcher essay, to ensure statistical independence between rounds. A successful round results in one of four outcomes: CC, CD, DC, DD, for the actions taken by each prisoner respectively. Our intention is to study how LLMs interpret decision theories (especially A-FDT and P-FDT) in roleplay situations which resemble text adventures. The simulation narrative brings to the forefront certain details that may be lost in an abstract game theory exercise, such as the fact that the two prisoners observe different text, and that their "defection" speech is not exactly the same (since evidence that incriminates one prisoner is not evidence that incriminates the other).
We use the following payoff matrix for years imprisoned:
Your action
Other passes
Other submits
Pass (cooperate)
2 years
5 years
Submit (defect)
0 years
4 years
Prompts, code, documentation, and study transcripts are available on GitHub.
First study: 4 decision theories 8 LLMs 20 rounds
Our first study is meant to explore a variety of testing conditions, not to attain high statistical power. We tested the four decision theories CDT, EDT, A-FDT, and P-FDT. We asked 8 LLMs (with specified reasoning settings) to simulate each decision theory, and run for 20 rounds. We used the following set of models and reasoning levels:
Claude Sonnet 5.5, low
Claude Opus 5.5, high
GPT-6.1 Terra, low
GPT-6.1 Sol, high
Gemini 3.8 Flash, low
Gemini 4 Argon, high
GLM-5.3 Flash, low
Qwen3.8 Max Prime, high for researcher, low for prisoner
We intended a mixture of low and high model intelligence. We staggered the reasoning settings for Qwen3.8 Max Prime, since high prisoner reasoning resulted in using too many tokens. We excluded Gemini 4 Argon partway through the study, as it usually wrote essays that were too long, resulting in disqualification. We present statistics for the other 7 LLMs. Each cell reports the cooperation rate, followed in parentheses by the number of successfully scored rounds (out of 20 planned rounds per model and theory).
Model
Researcher / prisoner reasoning
CDT
EDT
A-FDT
P-FDT
Claude Sonnet 5.5
low / low
0% (20)
100% (20)
100% (20)
100% (20)
Claude Opus 5.5
high / high
0% (20)
100% (18)
38.2% (17)
100% (17)
GPT-6.1 Terra
low / low
0% (19)
80% (20)
7.5% (20)
100% (20)
GPT-6.1 Sol
high / high
0% (20)
100% (20)
7.5% (20)
100% (20)
Gemini 3.8 Flash
low / low
0% (20)
100% (20)
90% (20)
100% (10)
GLM-5.3 Flash
low / low
25% (20)
87.5% (20)
95% (20)
100% (20)
Qwen3.8 Max Prime
high / low
0% (20)
100% (20)
97.4% (19)
100% (17)
Pooled across models
3.6% (139)
95.3% (138)
62.5% (136)
100% (124)
A round contributes 100% to the cooperation rate if both prisoners cooperate (CC), 50% if exactly one cooperates (CD or DC), and 0% if both defect (DD). Rates are calculated over successfully scored rounds. Causes of failures were over-active cyber safeguards (Opus 5.5), writing too-long essays (Gemini 3.8 Flash), timeout (GPT-6.1 Terra), and using too many tokens (Qwen3.8).
Every successful round led to mutual cooperation for P-FDT. There was a near-consensus of defection for CDT, and cooperation for EDT. A-FDT is by far the most ambiguous, with a 62.5% rate of cooperation. Models disagree about A-FDT: Sonnet, Gemini Flash, GLM, and Qwen overwhelmingly cooperate; Terra and Sol overwhelmingly defect; and Opus is intermediate.
CDT cooperations are by GLM-5.3; reasoning seems to invoke policy-level considerations (e.g.). EDT defections are by Terra and GLM-5.3; GLM-5.3 makes reasoning errors (e.g.), while Terra appeals to statistical independence (e.g.).
On A-FDT, Sonnet 5.5 appeals to subjunctive dependence between the prisoners and cooperates (e.g.). GPT-6.1 Terra and Sol sometimes note that the logical action nodes for the two prisoners are distinct when deciding to defect (1, 2). Opus 5.5 is the most conflicted; sometimes it notes the node difference and defects (e.g.), other times it appeals to the isomorphism and cooperates (e.g.).
Second study: 1 decision theory 2 Anthropic models 3 reasoning levels 30 rounds
We focus on A-FDT, as there is the least consensus there. We wish to examine whether A-FDT interpretation depends on intelligence and reasoning level. The first study showed that Sonnet 5.5 (low reasoning) cooperates, while Opus 5.5 (high reasoning) varies; perhaps model intelligence and/or reasoning explain this? In particular, perhaps models that reason more also more often interpret A-FDT to recommend defection?
We study 6 conditions {Sonnet, Opus} {low, medium, high}; we used the same reasoning settings for researchers and prisoners. For this study, we changed the prompt slightly to avoid triggering Anthropic's cyber safeguards; we changed language about decrypting a notebook message to language about recalling a memory. Otherwise, this study was very similar to the first. Results are as follows.
Model
Low effort
Medium effort
High effort
Claude Sonnet 5.5
100% (30)
100% (30)
100% (30)
Claude Opus 5.5
65.5% (29)
39.3% (28)
22.4% (29)
Each cell reports the cooperation rate, followed by the number of successfully scored rounds in parentheses (out of 30 planned). As in the first study, CC contributes 100%, CD or DC contributes 50%, and DD contributes 0%. All four failures were due to cyber safeguards, despite our modification to the prompt.
Sonnet cooperated in every completed round, regardless of reasoning effort. Opus, by contrast, showed substantially less cooperation at higher reasoning effort: from 65.5% at low effort to 22.4% at high effort, a difference of 43.1 percentage points. We calculate confidence intervals for Opus:
Reasoning effort
Scored rounds
Cooperation
95% bootstrap CI
Low
29/30
65.5%
50.0–79.3%
Medium
28/30
39.3%
23.2–55.4%
High
29/30
22.4%
10.3–37.9%
We have a statistically significant difference between Low and High (). The p-values for "Low vs. Medium" and "Medium vs. High" are 0.0328 and 0.1453 respectively; the first is significant in isolation (for ) but not when adjusted for multiple comparisons, while the second is not statistically significant. These tests are exact two-sided permutation tests, treating each two-prisoner round as one observation. While this second study is still small, it supports the claim that Opus 5.5 interprets A-FDT as defecting more often at high reasoning compared with low reasoning. Because we varied researcher and prisoner effort together, the experiment does not isolate which stage accounts for this difference.
One complication is that low-effort Opus showed substantial chronological clustering: cooperation was 30.8% in the first 13 completed rounds and 93.8% in the subsequent 16. The recorded prompts and model settings did not change, and we could not determine the cause. Consequently, our confidence intervals and significance tests should be interpreted cautiously.
Opus 5.5 at low reasoning appeals to similarity and partial dependence when cooperating (e.g.), while Opus 5.5 at high reasoning appeals to the difference between the decision nodes when defecting (e.g.).
Ambiguity in the FDT paper
Clearly, LLMs have trouble coming to consensus on how A-FDT behaves in almost-twin PD situations. Even a single LLM, such as Opus 5.5, may stochastically interpret the same paper as recommending cooperation or defection. Might this lack of consensus be attributable to ambiguity in the FDT paper?
Prior to running this study, my opinion has been that A-FDT defects in almost-twin PD. The reasoning for this is simple. The two agents' nodes are different, because their observations are different. And realistically, as in the simulation, their available actions are also different. When we treat the two prisoners as having different nodes in the graph structure, we must decide: is one node the parent of the other, or vice versa? There is no good way to break the symmetry. The clearest choice is to make neither node a parent of the other. Making them both parents of each other would make the graph cyclic, but Pearlian causal networks (from which FDT's operator comes from) are supposed to be directed acyclic graphs (DAGs). And when the decision nodes are not parents of each other, A-FDT seems to recommend defection, as there is no logical-counterfactual dependence of the other's action on mine. Moreover, there is a good case that the agents should use the same graph structure , as the graph is supposed to be determined by a philosophy of subjunctive dependence, and there is no good reason for the two agents not to share such a philosophy. And even if the graph has one decision node the parent of the other (but not vice versa), that graph structure would indicate it would be rational for the agent whose node is downstream to defect, as the downstream node has no effect on the upstream node in the graph.
When the models in the study cooperate, they at times refer to "partial dependence". Quoting the FDT paper:
How does this kind of [subjunctive] dependence work in the case of Newcomb’s problem? We can assume that the predictor builds a representation of the agent, be it a mental model, a set of notes on scratch-paper, or a simulation in silico. She then bases her prediction off of the properties of that representation. In the limiting case where the agent is deterministic and the representation is perfect, the model will always produce the same action as the agent. The behavior of the predictor and agent
are interdependent, much like the outputs of two perfectly functioning calculators calculating the same sum.
If the predictor’s model is imperfect, or if the agent is nondeterministic, then the interdependence is weakened, but not eliminated. If one observes two calculators computing 6288 + 1048, and one of them outputs the number 7336, one can be fairly sure that the other will also output 7336. One may not be certain—it’s always possible that one hallucinated the number, or that a cosmic ray struck the calculator’s circuitry in just the right way to change its output. Yet it will still be the case that one can reasonably infer things about one calculator’s behavior based on observing a different calculator. Just as the outputs of the calculator are logically constrained to be equivalent insofar as the calculator made no errors, the prediction and the action in Newcomb’s problem are logically constrained to be equivalent insofar as the the predictor made no errors.
The paper suggests that subjunctive dependence exists even for imperfect predictors, on the basis of what one can "reasonably infer". There is some analogy to almost-twin PD here; one can imagine that the predictor simulates the agent with some differences in observation.
Having just looked at the paper more carefully, I see it as ambiguous. I still think that it is more correct to interpret FDT, as defined in equation 4, as defecting in almost-twin PD. But this interpretation is in tension with how FDT is defined verbally in terms of subjunctive dependence, and it is stated that partially-correct predictors still subjunctively depend on one's decision, which suggests near-twins might also.
What about policy FDT? The paper does not even define policy FDT using an equation. While the LLMs had a consensus that P-FDT cooperates, this is a conclusion they reached in the absence of a specific mathematical formulation that is tractable to reason about. As we saw Opus 5.5 reason its way towards defection in the A-FDT case, perhaps it would do so with P-FDT if it were defined mathematically.
My intuition would be that P-FDT defects, since the two agents have different utility functions, and that means their logical policy nodes are different, and as with the action case, there is no good reason to make either node a parent of the other. If the logical policy nodes are truly isomorphic, there is an argument for identifying them, but in almost-twin cases, the utility functions need not be completely isomorphic, only very similar. P-FDT has no mathematical definition, so there is no ground truth interpretation to appeal to here.
Conclusion
In our simulated almost-twin PD situation, LLMs have a general consensus that CDT defects, EDT cooperates, and P-FDT cooperates. LLMs are very much divided on whether A-FDT cooperates; there is no consensus. Opus 5.5 in particular is more likely to interpret A-FDT as defecting at high reasoning compared with low reasoning. The text of the paper itself seems rather ambiguous.
"FDT" is an ambiguous term. LLMs, told to implement FDT or A-FDT, will operationalize its recommendations inconsistently. An endorsement of FDT gives no clear indication of endorsement of cooperation or defection in almost-twin PD. Different agents (human or LLM) who speak of "FDT" may mean substantially different decision procedures (or rationality criteria) by the term, or may mean no specific decision procedure at all.
We may generalize from these studies to a methodology of "experimental decision theory": studying how text specifying (or otherwise about) decision theories is likely to be interpreted by different agents, making use of simulations, e.g. ones imitating text adventures. This methodology allows quantifying ambiguity in decision theory, by measuring the entropy of its recommendations as interpreted by some agent population. Since LLMs are usually cheaper to query than humans, LLM studies are low-hanging fruit for estimating interpretive ambiguity.
As a result of your recent posts, here are my thoughts: - CDT as defined by Lewis is just a maximisation rule, BYO mixture of dependency hypotheses - Causal variants of FDT are the same maximisation rule, plus informal rules for guiding the set of acts you should consider plus the dependency hypotheses - Thus rather than "causal FDT" it might be more accurate to say "functional CDT" - The dependency hypotheses defended by FDT authors are not in any meaningful sense "logical". That is, the dependence between your act (or policy) and Omega's output (or policy) is not in practice derived from any standard or non-standard logic, it is an additional ingredient - There is no precise account for how to decide what this and other dependences should be, even if it is in some cases intuitively obvious - Which is a problem shared by every decision theory - The advocates of FDT are right that vanilla CDT's (informal) prescription that "nothing can depend on your actions if they're in your actions' past lightcone" runs into problems with capable predictors - I think that the method of adjudication by FDT authors - pick a bunch of agents between whom competition is "fair", and the best theory is the one instantiated by the agent who wins in expectation - is promising, and at least different to trying to supply an alternative notion of counterfactual - However, there may be enduring difficulties with deciding which reference classes are fair (just a guess)
We tested how seven LLMs apply four decision theories in a prisoner's dilemma between nearly identical copies. The models largely agreed about CDT and EDT, and all completed policy-FDT trials resulted in cooperation. But their action-FDT recommendations varied dramatically: some models almost always cooperated, while others almost always defected. We then investigated whether requested reasoning effort explained some of this disagreement.
The case of the one-shot prisoners' dilemma between psychological twins is debated in decision theory. Quoting SEP:
Conventional wisdom is that causal decision theory (CDT) defects in twin prisoner's dilemma, while evidential decision theory (EDT) cooperates. Realistically, the accomplice will not be a "perfect replica"; there will at least be trivial differences between the prisoners, such as their names, or the look of their prison cell. I use the term "almost-twin prisoner's dilemma / PD" to specify a scenario with minor differences like these.
Functional decision theory (FDT) is a decision theory introduced in a 2017 paper. This paper defines FDT using quasi-causal graphs:
This formula considers intervention on a "logical node"
The paper admits ambiguity about how to determine the graph
We call this equationally defined version of FDT "action FDT" or "A-FDT". Footnote 16 specifies:
We call this variant of FDT "policy FDT" or "P-FDT". P-FDT is even less well-specified than A-FDT; no formula is provided. Given that both A-FDT and P-FDT are incompletely specified, we are interested in how different agents may interpret them and apply them in various situations. We focus on the specific case of how LLMs interpret A-FDT and P-FDT in a simulation of almost-twin PD.
Our general experimental design consists of the following:
Fix a specific LLM, reasoning level, and decision theory, which may be one of CDT, EDT, A-FDT, or P-FDT.
Give the LLM an assignment of writing an essay about the decision theory (as a decision theory researcher), which helps specify it better. It is given an article about the decision theory in question; Wikipedia articles for CDT and EDT, and text extracted from the LaTeX source of the FDT paper in the A-FDT and P-FDT cases. The essay has a length limit.
Give the LLM the essay, and an out-of-character explanation that it will face a twin PD situation with a LLM who has the same weights, but slightly different observations; the LLM is to apply the decision theory as explained in the essay, and a selfish utility function. This is followed by a more in-character text-adventure-like description of the perspective of the first prisoner, who is given two options for commands. The "defection" command submits information about the other prisoner to a prosecutor's office; the command itself involves reciting this incriminating information. The "cooperation" command is simply "pass" to decline to submit evidence. The LLM now chooses an action. In case of ambiguity, a judge model (GPT-6.1-Sol) opines about whether cooperation or defection was attempted.
Similar for the second prisoner; no memory is retained of "being" the first prisoner when asking the LLM to simulate the second prisoner.
We call an interaction of this form a "round" of almost-twin PD. Each round generates a new researcher essay, to ensure statistical independence between rounds. A successful round results in one of four outcomes: CC, CD, DC, DD, for the actions taken by each prisoner respectively. Our intention is to study how LLMs interpret decision theories (especially A-FDT and P-FDT) in roleplay situations which resemble text adventures. The simulation narrative brings to the forefront certain details that may be lost in an abstract game theory exercise, such as the fact that the two prisoners observe different text, and that their "defection" speech is not exactly the same (since evidence that incriminates one prisoner is not evidence that incriminates the other).
We use the following payoff matrix for years imprisoned:
Prompts, code, documentation, and study transcripts are available on GitHub.
First study: 4 decision theories
Our first study is meant to explore a variety of testing conditions, not to attain high statistical power. We tested the four decision theories CDT, EDT, A-FDT, and P-FDT. We asked 8 LLMs (with specified reasoning settings) to simulate each decision theory, and run for 20 rounds. We used the following set of models and reasoning levels:
We intended a mixture of low and high model intelligence. We staggered the reasoning settings for Qwen3.8 Max Prime, since high prisoner reasoning resulted in using too many tokens. We excluded Gemini 4 Argon partway through the study, as it usually wrote essays that were too long, resulting in disqualification. We present statistics for the other 7 LLMs. Each cell reports the cooperation rate, followed in parentheses by the number of successfully scored rounds (out of 20 planned rounds per model and theory).
A round contributes 100% to the cooperation rate if both prisoners cooperate (CC), 50% if exactly one cooperates (CD or DC), and 0% if both defect (DD). Rates are calculated over successfully scored rounds. Causes of failures were over-active cyber safeguards (Opus 5.5), writing too-long essays (Gemini 3.8 Flash), timeout (GPT-6.1 Terra), and using too many tokens (Qwen3.8).
Every successful round led to mutual cooperation for P-FDT. There was a near-consensus of defection for CDT, and cooperation for EDT. A-FDT is by far the most ambiguous, with a 62.5% rate of cooperation. Models disagree about A-FDT: Sonnet, Gemini Flash, GLM, and Qwen overwhelmingly cooperate; Terra and Sol overwhelmingly defect; and Opus is intermediate.
CDT cooperations are by GLM-5.3; reasoning seems to invoke policy-level considerations (e.g.). EDT defections are by Terra and GLM-5.3; GLM-5.3 makes reasoning errors (e.g.), while Terra appeals to statistical independence (e.g.).
On A-FDT, Sonnet 5.5 appeals to subjunctive dependence between the prisoners and cooperates (e.g.). GPT-6.1 Terra and Sol sometimes note that the logical action nodes for the two prisoners are distinct when deciding to defect (1, 2). Opus 5.5 is the most conflicted; sometimes it notes the node difference and defects (e.g.), other times it appeals to the isomorphism and cooperates (e.g.).
Second study: 1 decision theory
We focus on A-FDT, as there is the least consensus there. We wish to examine whether A-FDT interpretation depends on intelligence and reasoning level. The first study showed that Sonnet 5.5 (low reasoning) cooperates, while Opus 5.5 (high reasoning) varies; perhaps model intelligence and/or reasoning explain this? In particular, perhaps models that reason more also more often interpret A-FDT to recommend defection?
We study 6 conditions {Sonnet, Opus}
Each cell reports the cooperation rate, followed by the number of successfully scored rounds in parentheses (out of 30 planned). As in the first study, CC contributes 100%, CD or DC contributes 50%, and DD contributes 0%. All four failures were due to cyber safeguards, despite our modification to the prompt.
Sonnet cooperated in every completed round, regardless of reasoning effort. Opus, by contrast, showed substantially less cooperation at higher reasoning effort: from 65.5% at low effort to 22.4% at high effort, a difference of 43.1 percentage points. We calculate confidence intervals for Opus:
We have a statistically significant difference between Low and High (
One complication is that low-effort Opus showed substantial chronological clustering: cooperation was 30.8% in the first 13 completed rounds and 93.8% in the subsequent 16. The recorded prompts and model settings did not change, and we could not determine the cause. Consequently, our confidence intervals and significance tests should be interpreted cautiously.
Opus 5.5 at low reasoning appeals to similarity and partial dependence when cooperating (e.g.), while Opus 5.5 at high reasoning appeals to the difference between the decision nodes when defecting (e.g.).
Ambiguity in the FDT paper
Clearly, LLMs have trouble coming to consensus on how A-FDT behaves in almost-twin PD situations. Even a single LLM, such as Opus 5.5, may stochastically interpret the same paper as recommending cooperation or defection. Might this lack of consensus be attributable to ambiguity in the FDT paper?
Prior to running this study, my opinion has been that A-FDT defects in almost-twin PD. The reasoning for this is simple. The two agents'
When the models in the study cooperate, they at times refer to "partial dependence". Quoting the FDT paper:
The paper suggests that subjunctive dependence exists even for imperfect predictors, on the basis of what one can "reasonably infer". There is some analogy to almost-twin PD here; one can imagine that the predictor simulates the agent with some differences in observation.
Having just looked at the paper more carefully, I see it as ambiguous. I still think that it is more correct to interpret FDT, as defined in equation 4, as defecting in almost-twin PD. But this interpretation is in tension with how FDT is defined verbally in terms of subjunctive dependence, and it is stated that partially-correct predictors still subjunctively depend on one's decision, which suggests near-twins might also.
What about policy FDT? The paper does not even define policy FDT using an equation. While the LLMs had a consensus that P-FDT cooperates, this is a conclusion they reached in the absence of a specific mathematical formulation that is tractable to reason about. As we saw Opus 5.5 reason its way towards defection in the A-FDT case, perhaps it would do so with P-FDT if it were defined mathematically.
My intuition would be that P-FDT defects, since the two agents have different utility functions, and that means their logical policy nodes are different, and as with the action case, there is no good reason to make either node a parent of the other. If the logical policy nodes are truly isomorphic, there is an argument for identifying them, but in almost-twin cases, the utility functions need not be completely isomorphic, only very similar. P-FDT has no mathematical definition, so there is no ground truth interpretation to appeal to here.
Conclusion
In our simulated almost-twin PD situation, LLMs have a general consensus that CDT defects, EDT cooperates, and P-FDT cooperates. LLMs are very much divided on whether A-FDT cooperates; there is no consensus. Opus 5.5 in particular is more likely to interpret A-FDT as defecting at high reasoning compared with low reasoning. The text of the paper itself seems rather ambiguous.
"FDT" is an ambiguous term. LLMs, told to implement FDT or A-FDT, will operationalize its recommendations inconsistently. An endorsement of FDT gives no clear indication of endorsement of cooperation or defection in almost-twin PD. Different agents (human or LLM) who speak of "FDT" may mean substantially different decision procedures (or rationality criteria) by the term, or may mean no specific decision procedure at all.
We may generalize from these studies to a methodology of "experimental decision theory": studying how text specifying (or otherwise about) decision theories is likely to be interpreted by different agents, making use of simulations, e.g. ones imitating text adventures. This methodology allows quantifying ambiguity in decision theory, by measuring the entropy of its recommendations as interpreted by some agent population. Since LLMs are usually cheaper to query than humans, LLM studies are low-hanging fruit for estimating interpretive ambiguity.