If doctors were to fine tune their prescriptions to take maximal advantage of the placebo effect, what would they do?  It’s hard to answer this question even with existing data on the strength of the placebo effect because beliefs, presumably the key to the placebo effect, would adjust if placebo prescription were widespread.

Indeed, over the weekend I saw a paper presented by Emir Kamenica which strongly suggests that equilibrium beliefs matter for placebos.  In an experiment on the effectiveness of anti-histamines, some subjects were shown drug ads at the same time they took the drug.  The ads had an impact on the effectiveness of the drug but only for subjects with less prior experience with the same drug.  The suggestion is that those with prior experience have already reached their equilibrium placebo effect.  (It appears that the paper is not yet available for download.)

So we need a model of the placebo effect in equilibrium.  Suppose that patients get a placebo a fraction $p$ of the time and a full dose the remaining $1-p$ fraction of the time.  And let $q(p)$ be the patient’s belief in the probability the prescription will work.  Then the placebo effect means that the true probability that the prescription will work is determined by a function h which takes two arguments:  the true dosage (=1 for full dose, 0 for placebo) and the belief $q$.  And in equilibrium beliefs are correct:

$q = p \cdot h(q, 0) + (1-p) \cdot h(q,1) \equiv \hat h(q,p)$

This equilibrium condition implicitly defines a function $q(p)$ which gives the equilibrium efficacy as a function of the placebo rate $p$.

The benefit of the model is that it allows us to notice something that may not have been obvious before.  If instead of using placebos by varying $p$, an alternative is to just lower the dose, deterministically.  Then if we let $d$ be the dosage (somewhere between 0 and 1), we get

$q = h(q,d)$

as the equilibrium condition which defines effectiveness $q(d)$ now as a function of the fixed dose $d$.

The something to notice is that, if the function $h$ is continuous and monotone, then the range of $q$ is the same whether we use placebos $p$ or deterministic doses $d$.  That is, any outcome that can be implemented with placebos can be implemented by just using lower doses and no placebos.  This follows mathematically because the placebo model collapses to the determistic model at the boundary: $\hat h(q,p=0) = h(q, d=1)$ and $\hat h(q,p=1) = h(q,d=0).$

Now this is just a statement about the feasible set.  The benefit of placebo may come from the ability to implement the same outcome but with lower cost.   In terms of the model this would occur if the $d$ that satisfies $q(d) = q(p)$ is larger than $1- p$.  That boils down to a cost-benefit calculation.  But I doubt that this kind of calculation is going to be pivotal in a debate about using placebos as medicine.