MMP.19:5.2 - Expose a coupling that experiments do not determine
Let a randomized binary action A be independent of the response type U=(Y_0,Y_1). Consider two possible type distributions:
| Type (Y_0,Y_1) | Model S | Model T |
|---|---|---|
| (0,0) | 3/8 | 1/8 |
| (0,1) | 1/8 | 3/8 |
| (1,0) | 1/8 | 3/8 |
| (1,1) | 3/8 | 1/8 |
Each action succeeds with probability 1/2 in both models. With a fair random assignment, all four observed (A,Y) combinations have probability 1/4 in each model.
For a case observed with A=1,Y=1, only types (0,1) and (1,1) remain. In S, the probability that it would also succeed under A=0 is (3/8)/(1/2)=3/4. In T it is (1/8)/(1/2)=1/4. Both models fit the supplied observational and intervention laws, so these laws do not identify the answer.
For a future choice scored only by expected success, the actions tie at 1/2 under both models. If A=1 adds a positive cost and no other consequence, A=0 is sufficient for that criterion. Resolving the counterfactual ambiguity would not improve this choice.