Fix a probability space $(\Omega,\mathcal{F},P)$ and, for $i\in\{0,...,n\}$, random variables $X_i$, $Y_i$, $Y^0_i$, $Y^1_i$ which are all measurable functions $\Omega\to \{0,1\}$. Call $X_i$ "treatment of unit $i$", $Y_i$ "the outcome of unit $i$" and $Y^1_i$, $Y^0_i$ "the potential outcomes of unit $i$".
Rubin, 2005 says that the SUTVA assumption is:
In any case, the depiction in Figure 1 requires assumptions for it to be adequate—in particular, SUTVA (stable unit treatment value assumption) (Rubin 1980), which comprises two subassumptions. First, there is no interference between units (Cox 1958); that is, neither $Y^0_i$ for $Y^1_i$ is affected by what action any other unit received. Second, it assumes there are no hidden versions of the treatments; no matter how unit $i$ received treatment $1$, the outcome that would be observed would be $Y^1_i$ and similarly for treatment 0.
I have tried interpreting this as saying:
- Firstly, $\{Y_i^1,Y_i^0\}\perp \{X_j:j\neq i\}$
- Secondly, for all $\omega\in \Omega$, $X_i(\omega)=x\implies Y^x_i(\omega) = Y_i(\omega)$
The latter being closely related (if not identical) to the assumption of consistency.
I am not confident that this is a correct interpretation of SUTVA.
In particular, it's not obvious how violating either of these would lead to Figure 1 being inadequate. It certainly does not preclude $Y^0_i$ or $Y^1_i$ or functions thereof from being well-defined random variables. The objection seems to be more along the lines that violating SUTVA will mean that we cannot necessarily regard $Y^1_i-Y^0_i$ as a "causal effect". My concern then is that if SUTVA is intended to ensure that this can be regarded as a causal effect then it is perhaps invoking nonprobabilistic assumptions that are not captured by the pair I have supplied above (for example, "unaffected by" is not synonymous with "independent from").
