# Time for counterfactuals

I have just discovered Scriven’s stimulating (if grim) challenge to a counterfactual understanding of causation (see the debate recorded in Cook et al., 2010, p. 108):

“The classic example of this is the guy who has jumped off the top of a skyscraper and as he passes the 44th floor somebody shoots him through the head with a .357 magnum. Well, itβs clear enough that the shooter killed him but itβs clearly not true that he would not have died if the shooter hadnβt shot him; so the counterfactual condition does not apply, so it canβt be an essential part of the meaning of cause.”

I love this example because it illustrates a common form of programme effect and summarises the human condition – all in a couple of sentences! Let’s reshape it into an analogous example that extends the timeline by a couple of decades:

“A 60 year old guy chooses not to get a Covid vaccine. A few months later, he gets Covid and dies. Average male life expectancy is about 80 years.”

(I guess jumping is analogous to being born!)

By the end of the first sentence, I reason that if he had got the vaccine, he probably wouldn’t have died. By the end of the second sentence, I am reminded of the finiteness of life. So, the vaccine didn’t prevent death – similarly to an absence of a gunshot in the skyscraper example. How can we think about this using counterfactuals?

In a programme evaluation, it is common to gather data at a series of fixed time points, for instance a few weeks, months, and, if you are lucky, years after baseline. We are often happy to see improvement even if it doesn’t endure. For instance, if I take a painkiller, I don’t expect its effects to persist forevermore. If a vaccine extends life by two decades, that’s rather helpful. Programme effects are defined at each time point.

To make sense of the original example, we need to add in time. There are three key timepoints:

1. Jumping (T0).
2. Mid-flight after the gunshot (T1).
3. Hitting the ground (T2).

When considering counterfactuals, the world may be different at each of these times, e.g., at T0 the main character might have decided to take the lift.

Here are counterfactuals that make time explicit:

• If the guy hadn’t jumped at T0, then he wouldn’t have hit the ground at T2.
• If the guy hadn’t jumped at T0, then he wouldn’t have been shot with the magnum and killed at T1.
• If the guy had jumped, but hadn’t been shot by the magnum, he would still have been alive at T1 but not at T2.

To assign truth values or probabilities to each of these requires a model of some description, e.g., a causal Bayesian network, which formalises your understanding of the intentions and actions of the characters in the text – something like the DAG below, with conditional probabilities filled in appropriately.

So for instance, the probability of being dead at T2 given jumping at T0 is high – if you haven’t added variables about parachutes. What happens mid-flight governs T1 outcomes. Alternatively, you could just use informal intutition. Exercise to the reader: give it a go.

Using the Halpern-Pearl definitions of causality on this model (Halpern, 2016), jumping caused death at both T1 and T2. The shooting caused death at T1 but not T2. (R code here – proper explanation to be completed, but you could try this companion blog post and citation therein.)

Back then to the vaccine example, the counterfactuals rewrite to something like:

• If the guy hadn’t been born at T0, then he wouldn’t have died at T2.
• If the guy hadn’t been born at T0, then he couldn’t have chosen not to get a vaccine and died at T1.
• If the guy had been born, but had decided to get the vaccine, he would still have been alive at T1 aged 60, but possibly not at T2 aged 80.

### References

Cook, T. D., Scriven, M., Coryn, C. L. S., & Evergreen, S. D. H. (2010). Contemporary Thinking About Causation in Evaluation: A Dialogue With Tom Cook and Michael Scriven. American Journal of Evaluation, 31(1), 105β117.

Halpern, J. Y. (2016). Actual causality. The MIT press.