Since S points to Y, and Y is the target quantity, according to this coarse selection diagram, the causal relationship is not transportable!!!

Not Transportable

X : X = 1 --> Plays Russian Roulette ;  X = 0 --> Doesn't play Russian Roulette
Y : Y = 1 --> Dies ;  Y = 0 --> Survives
S : S points to Y because of the disparity in the base mortality rate of the two cities.

University of Michigan - Ann Arbor

The city of Los Angeles runs an RCT to determine effect of Russian Roulette on mortality. Death rate amongst Roulette players is 17.5%. Death rate among others is 1%.
The mayor of New York now wants to determine what the city's mortality rate would be if everyone plays Russian Roulette. New York starts with a mortality rate of 5% (vs LA's 1%).
How do we use the data from the LA RCT to determine this answer?

Motivating Example : Russian Roulette

Cinelli, Carlos, and Judea Pearl. “Generalizing Experimental Results by Leveraging Knowledge of Mechanisms.” Technical Report - UCLA, 10 Dec. 2019.

https://ftp.cs.ucla.edu/pub/stat_ser/r492.pdf

Generalizing Experimental Results by
Leveraging Knowledge of Mechanisms

X : X = 1 --> Plays Russian Roulette ;  X = 0 --> Doesn't play Russian Roulette
Y : Y = 1 --> Dies ;  Y = 0 --> Survives


Coarse Causal Diagram

Coarse Selection Diagram

Must take into account existing health condition disparities that cause difference in base mortality rate.
Thus we define : 
1. B : B=1-->Losing at Russian Roulette i.e. bullet in the barrel ; B=0--> No bullet in the barrel.
2. H : H=1-->All other health factors that cause death ; H=0--> other factors that cause survival


Building Correct Structural Diagram

We know, given the nature of the problem that 
$$Y = H \vee (X \land B) $$
This can be reframed as a more concrete counterfactual :
$$PS_{01} = P(Y_1 = 1 | Y_0 = 0 )$$
Where $Y_1$ is the event $H \land B$ and $Y_0$ is the event $H$. 
$PS_{01}$ is the probability that a death is caused by Russian Roulette alone.
Now let, 
$$PS_{10} = P(Y_1 = 0 | Y_0 = 1 )$$
By definition, since $Y_1$ contains $Y_0$, $PS_{10} = 0$
The law of total probability gives us : 
$$P(Y_1 = 1) = (1 - PS_{10})(P(Y_0 = 1)) + PS_{01}(1 - P(Y_0 = 1)) $$
We can then solve for $PS_{01}$ to get :
$$\frac{P(Y_1 = 1) + P(Y_0 = 1)}{1 - P(Y_0 = 1)} = \frac{1}{6}$$
Now we use the equation derived from the law of total probability and plug in values for NYC, to get the estimated mortality rate as $20.8%$

Thus using the structural diagram(s) and functional constraints, we've managed to solve this problem using only the two assumptions we began with.

Using Functional Constraints

While having a monotonic function/probability allows for precise computations, the absence of monotonicity does not hinder the calculations of estimates. Instead of a precise value, we then arrive at a tight bound for the estimate.

Effect of Monotonicity

If we have data from multiple trials, irrespective of monotonicty, we can always arrive at a point estimate.

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