When probabilities don’t matter for proper individual choices but matter for public health policy

Australian health policy has an interesting choice. Vaccinate the population with with a combination of the AstraZeneca and Pfizer vaccine or wait until the supply of Pfizer vaccine is shipped and inoculate everyone with the Pfizer vaccine. The former choice may well make sense from a public health policy point of view.

The issues I want to address here has to do with the difference between what policy makers want and individual choices; and whether we can induce a change in behaviour by simply calculating and publicising the low probabilities of adverse effects of the AstraZeneca vaccine, and calculating and publicising the probabilities of the spread of the disease in Australia.

Can we induce the publicly “desirable” choice by clarifying simple probabilities of adverse events? Or does the nudge need more than educating people about the smallness of the probabilities of adverse events?

Let’s bring some sort of structure to the way we understand choices made by individuals. We envisage a person making a choice between two uncertain lotteries:

1. Take AstraZeneca vaccine now, which is represented by a random variable A (whose values are real) of outcomes, with a probability cumulative distribution FA over outcomes.
2. Wait T months, take the Pfizer jab, represented by a random variable P (whose values are real) of outcomes, with probability cumulative distribution FP over outcomes.

We make two assumption here and explain them afterward:

a. A and P have non-positive support.

b. FP(z) ≤ FA(z) for all z

Assumption (a) simply says that if you are not infected, then all possible outcome are negative. Assumption (b) says that taking Pfizer first order stochastically dominates taking AstraZeneca. That is, all individuals with monotone preferences over outcomes, would prefer P over A. Realistically, we may well be able to get away with a second order stochastic dominance assumption; meaning that risk averse individuals prefer P over A. The analysis and message is not affected.

We go to the first obvious and trivial result written loosely.

L1 A decision maker prefers P over A for all T≥0, regardless of the functional form of the probability distribution A or how low the probabilities of adverse outcomes of A are.

Of course, you would want to delay. This is the basis of vaccine hesitancy, if you think that you wont get the disease and are forced to choose between P and A, you will choose to wait for P as it causes less harm. We make a simple guiding remark arising from the independence axiom in decision theory.

Remark a (technical reminder): P will be preferred to A even when mixing the lotteries with any third lottery if utilities are expected utilities.

Thus, discussion of third alternatives (like flip a coin before making a the choice and on heads choose not to be vaccinated) will generally not be altogether useful.

We now model the community spread of the disease. Our choice of modelling this is in the form of ambient noise represented a latent unobserved random vector X with values in some space, that is statistically independent of the randomness generating P and A:

c. the states of the world generating X are independent of the states of the world generating the outcomes of P and A.

I cannot envisage a situation where community spread is correlated with the adverse effects of the vaccines. Note importantly that this X is latent and unobserved thus you cant condition your choice of which vaccine to take based on information about X. This assumption is hidden and not entirely realistic, for example it is reported that demand for P (and presumably A) went up in Melbourne, Victoria, with reports of community transmission. The assumption however does not greatly affect the narrative here.

We move to utilities and suppose that the individual is faced with two utilities over lotteries:

U(FA,X) and U(FP,X)

these are random utilities and the randomness depends on the state of community spread and on the distributions of the adverse effects of the vaccines. What is a reasonable assumption on this random utility function?

We adopts a version of the properness assumption of Pratt and Zeckhauser (1987) Proper Risk Aversion, Econometrica Vol. 55, №1 (Jan., 1987), pp. 143–154 (12 pages). We argue that the sign of

U(FA,X) — U(FP,X)

does not depend on the independent ambient noise. That is,

sign U(FA,g(X)) — U(FP,g(X)) = sign U(FA,X) — U(FP,X)

for any (measurable) transformation g of the ambient noise X. Noting that g(X) remains independent of P and A for any transformation g. The transformation g may be thought of as describing the individual’s attitude to the risky ambient noise. We get the following trivial result.

L2 A decision maker prefers P over A for all T≥0, regardless of the functional form of the probability distribution A or how low the probabilities of adverse outcomes of A are, and regardless of the transformation g.

This assumption on decision making with independent ambient noise is occasionally called properness. It is closely related to the usual independence of irrelevant alternatives assumption. If it doesn’t hold, then observed attitudes to risk will have parameters with measurement errors that are dependent of non-observed ambient noise, and thus falsely display risk aversion or risk seeking. Examples of where the properness assumption doesn’t hold are provided here.

This properness assumption may well be a strong assumption. But in my view properness is a reasonable assumption to make. It is the natural assumption to make when the mechanisms are complicated. The hidden and strong assumption here is the independence of X from the outcome of choosing P after time T or choosing A at time T. Advocacy for switching choices from P over to A must go well beyond explaining probabilities. The nudge needs something else, for example, a sausage sizzle or ability to win a new Toyota V8 Landcriuser Sahara if you choose AstraZeneca. That is, advocacy needs to be adding uncertainty that is statistically dependent on the choices, probabilities in their own right do not speak to this.

We conclude with the point of the post. Consider the policy maker. Their social welfare function is a function of the aggregate choices C made by all individuals and the background ambient noise:

S(C,X)

Though, the ambient noise is statistically independent from the outcomes of the choices of individuals. It will generally be highly dependent on the aggregate choices. This may seem odd, but it generally is not if the number of individuals is very large!.