Can a 1-D risk score (binary outcome) be sensibly used to create multiple treatment groups? This question concerns predicted probabilities of a binary outcome, and the (I believe) misguided practice of making multiple cutpoints along a one-dimensional risk continuum -- cutpoints that create three or more groups viewed as deserving different treatments. 
My contention is that using estimated risk of an event occurring – say, of catching a dangerous virus vs. not catching it—to cluster people into any more than two groups will be misguided and will result in inefficient treatment plans.  Whatever is the favored treatment to reduce negative outcomes among those with the highest scores, that treatment should be applied to everyone above a certain cutpoint, determined by resources.  E.g., if we have the resources to treat only one tenth, then we set a single cutpoint to include one tenth.  
Being one-dimensional, the risk score, I contend, can tell us nothing about the best alternative for each person among some set of multiple treatment options.  If we truly want to reduce incidence of infection, we offer the best treatment we have, to everyone we have the resources to treat. 
To put it another way, I think it would be illogical to use two cutpoints to create three risk groups (high, medium, and low); to give the high group the most reliable treatment; and to give the medium group some alternative treatment.  What could we expect—-that for the medium group we could get away with partly preventing infection?  (Or if infection isn't the best example, think of pregnancy, or mortality.)  Either they will or they won’t catch the virus.  If they do, in this example, they will experience just the same consequences as if they had been in the group marked as high-risk.  The question is, how can limited resources be applied with the greatest effect—-which suggests again that we give the single best treatment to all those with scores above some single cutpoint.  
Is my thinking sound?  If not, why not?  If so, what would be the most convincing argument for a layperson to hear?
 A: 
What would be the most convincing argument for a layperson to hear?

Different types of treatments carry different risks.
Here is an example:
A sample model:
For sick patients:


*

*No treatment has a success rate (spontaneous recovery) of 0.01

*Treatment A has a success rate of 0.80

*Treatment B has a success rate of 0.95


For healthy patients:


*

*Treatment A can kill a healthy patient with probability 0.01

*Treatment B can kill a healthy patient with probability 0.03


Also assume:


*

*We can't give both treatments to the same patient

*The risk score is calibrated, hence risk score = p(sick).


Our goal: 


*

*Find a strategy that maximizes the expected percentage of lives saved, by assigning a treatment to each patient, given his risk score.


Optimization:


*

*The probability of a patient with risk score p ending up healthy with no treatment is 
p*0.01 + (1-p)

*The probability of a patient with risk score p ending up healthy with treatment A is
p*0.8 + (1-p)*0.99

*The probability of a patient with risk score p ending up healthy with treatment B is
p*0.95 + (1-p)*0.97
Now, let's plot these three probability functions as a function of p:

It is easy to see the rationale for using 2 cutoff values.
The best treatment is not the same for every patient.
A: Consider you have two treatments available:
1. costs 1000 but has a 99% chance of helping
2. costs 10 but has a 90% chance of working
Would you rather treat 1 with the first, or 100 with the second?
Assume your risk distribution is 1, 0, ..., 0 then you should treat only the first.
If your risk distribution is 0.60,0.599,0.598,0.597,... then you could save over 45 by using the second drug.
There is two points in this model that you may have overlooked:


*

*Treatments are not guaranteed to work

*Prediction will barely ever be 100% correct


Assume the real risk is 1, 0, ..., 0 as before. But your method did a mistake, and produced the risk scores 0.9, 1.0, 0, ... 0. If you would bet everything on one treatment, you would be treating the wrong person. If you treat the top 100, you have a 90% success chance of curing the one that was really sick in this toy example.
A: Suppose there were three kinds of patient:


*

*Patients with risk 0 will never catch the virus

*Patients with risk 1 will catch the virus only if untreated

*Patients with risk 2 will always catch the virus


Then the optimal strategy would be to use multiple cutpoints and only treat patients of risk 1.

I came up with that pathological example by thinking about decision trees. The model I had in mind was 


*

*A patient of risk r arrives

*If they're treated, the probability of infection is $t(r)$. 

*If they're not treated, the probability of infection is $u(r)$.


I originally had costs attached to each outcome, but it turns out those aren't important. What's important is that even if you insist that the risk $r$ is 'honest' in that $t(r), u(r)$ are both increasing functions - so higher risk makes for a higher probability of infection - that doesn't mean that the 'benefit' of the treatment $t(r) - u(r)$ has to be increasing with respect to risk.
