Wednesday, December 15, 2010

Risk reduction policies

The following policy pattern is common.  There is a risky behavior which a portion of a target population engages in.  There is no consensus on the benefits of the behavior to the agent, but there is a consensus on one or more risks to the agent.  Two examples:
  • Teen sex: Non-marital teen sex, where the risks are non-marital teen pregnancy and STIs.
  • Driving: Transportation in motor vehicles that are not mass transit, where the risks are death and serious injury.
In both cases, some of us think that the activity is beneficial when one brackets the risks, while others think the activity is harmful.  But we all agree about the harmfulness of non-marital teen pregnancy, STIs, death and serious injury.

In such cases, it is common for a "risk-reduction" policy to be promoted.  What I shall (stipulatively) mean by that is a policy whose primary aim is to decrease the risk of the behavior to the agent rather than to decrease the incidence of the behavior.  For instance: condoms and sexual education not centered on the promotion of abstinence in the case of teen sex; seat-belts and anti-lock brakes in the case of driving.  I shall assume that it is uncontroversial that the policy does render the behavior less risky.  

One might initially think--and some people indeed do think this--that it is obvious, a no-brainer, that decreasing the risks of the behavior brings benefits.  There are risk-reduction policies that nobody opposes.  For instance, nobody opposes the development of safer brakes for cars.  But other risk-reduction policies, such as the promotion of condoms to teens, are opposed.  And sometimes they make the argument that the risk-reduction policy will promote the behavior in question, and hence it is not clear that the total social risk will decrease.  It is not uncommon for the supporters of the risk-reduction policy to think the policy's opponents "just don't care", are stupid, and/or are motivated by something other than concerns about the uncontroversial social risk (and indeed the last point is often the case).  For instance, when conservatives worry that the availability of contraception might increase teen pregnancy rates, they are thought to be crazy or dishonest.

I will show, however, that sometimes it makes perfect sense to oppose a risk-reduction policy on uncontroversial social-risk principles.  There are, in fact, cases where decreasing the risk involved in the behavior increases total social risk by increasing the incidence.  But there are also cases where decreasing the risk involved in the behavior decreases total social risk.  

On some rough but plausible assumptions, together with the assumption that the target population is decision-theoretic rational and knows the risks, there is a fairly simple rule.  In cases where a majority of the target population is currently engaging in the behavior, risk reduction policies do reduce total social risk.  But in cases where only a minority of the target population is currently engaging in the behavior, moderate reductions in the individual risk of the behavior increase total social risk, though of course great reductions in the individual risk of the behavior decrease total social risk (the limiting case is where one reduces the risk to zero).

Here is how we can see this.  Let r be the individual uncontroversial risk of the behavior.  Basically, r=ph, where p is the probability of the harm and h is the disutility of the harm (or a sum over several harms).  Then the total social risk, where one calculates only the harms to the agents themselves, is T(r)=Nr, where N is the number of agents engaging in the harmful behavior.  A risk reduction policy then decreases r, either by decreasing the probability p or by decreasing the harm h or both.  One might initially think that decreasing r will obviously decrease T(r), since T(r) is proportional to r.  But the problem is that N is also dependent on r: N=N(r).  Moreover, assuming the target population is decision-theoretic rational and assuming that the riskiness is not itself counted as a benefit (both assumptions are in general approximations), N(r) decreases as r increases, since fewer people will judge the behavior worthwhile the more risky it is.  Thus, T(r) is the product of two factors, N(r) and r, where the first factor decreases as r increases and the second factor increases as r increases.  

We can also say something about two boundary cases.  If r=0, then T(r)=0.  So reducing individual risk to zero is always a benefit with respect to total social risk.  Of course any given risk-reduction policy may also have some moral repercussions--but I am bracketing such considerations for the purposes if this analysis.  But here is another point.  Since presumably the perceived benefits of the risky behavior are finite, if we increases r to infinity, eventually the behavior will be so risky that it won't be worth it for anybody, and so N(r) will be zero for large r and hence T(r) will be zero for large r.  So, the total social risk is a function that is always non-negative (r and N(r) are always non-negative), and is zero at both ends.  Since for some values of r, T(r)>0, it follows that there must be ranges of values of r where T(r) decreases as r decreases and risk-reduction policies work, and other ranges of values of r where T(r) increases as r decreases and risk-reduction policies are counterproductive.

To say anything more precise, we need a model of the target population.  Here is my model.  The members of the population targeted by the proposed policy agree on the risks, but assign different expected benefits to the behavior, and these expected benefits do not depend on the risk.  Let b be the expected benefit that a particular member of the target population assigns to the activity.  We may suppose that b has a normal distribution with standard devision s around some mean B.  Then a particular agent engages in the behavior if and only if her value of b exceeds r (I am neglecting the boundary case where b=r, since given a normal distribution of b, this has zero probability).  Thus, N(r) equals the numbers of agents in the population whose values of b exceed r.  Since the values of b are normally distributed with pre-set mean and standard deviation, we can actually calculate N(r).  It equals (N/2)erfc((r-B)/s), where erfc is the complementary error function, and N is the population size.  Thus, N(r)=(rN/2)erfc((r-B)/s).

Let's plug in some numbers and do a graph.  Suppose that the individual expected benefit assigned to the behavior has a mean of 1 and a standard deviation of 1.  In this case, 84% of the target population thinks that when one brackets the uncontroversial risk, the behavior has a benefit, while 16% think that even apart from the risk, the behavior is not worthwhile.  I expect this is not such a bad model of teen attitudes towards sex in a fairly secular society.  Then let's graph T(r) (on the y-axis it's normalized by dividing by the total population count N--so it's the per capita risk in the target population) versus r (on the x-axis). (You can click on the graph to tweak the formula if interested.)

We can see some things from the graph.  Recall that the average benefit assigned to the activity is 1.  Thus, when the individual risk is 1, half of the target population thinks the benefit exceeds the risk and hence engages in the activity.  The graph peaks at r=0.95.  At that point one can check from the formula for N(r) that 53% of the target population will be engaging in the risky activity.

We can see from the graph that when the individual risk is between 0 and 0.95, then decreasing the risk r always decreases the total social risk T(r).  In other words we get the heuristic that when a majority (53% or more for my above numbers) of the members of the population are engaging in the risky behavior, we do not have to worry about increased social risk from a risk-reduction policy, assuming that the target population does not overestimate the effectiveness of the risk-reduction policy (remember that I assumed that the actual risk rate is known).

In particular, in the general American adult population, where most people drive, risk-reduction policies like seat-belts and anti-lock brakes are good.  This fits with common sense.

On the other hand, when the individual risk is between 0.95 and infinity, so that fewer than 53% of the target population is engaging in the risky behavior, a small decrease in the individual risk will increase T(r) by moving one closer to the peak, and hence will be counterproductive.

However, a large enough decrease in the individual risk will still put one on the left side of the peak, and hence could be productive.  But the decrease may have to be quite large.  For instance, suppose that the current individual risk is r=2.  In that case, 16% of the target population is engaging in the behavior (since r=2 is one standard-deviation away from the mean benefit assignment).  The per-capita social risk is then 0.16.  For a risk-reduction policy to be effective, it would then have to reduce the individual risk so that it is far enough to the left of the peak that the per-capita social risk is below 0.16.  Looking at the graph, we can see that this would require moving r from 2 to 0.18 or below.  In other words, we would need a policy that decreases individual risks by a factor of 11.

Thus, we get a heuristic.  For risky behavior that no more than half of the target population engages in, incremental risk-reduction (i.e., a small decrease in risk) increases the total social risk.  For risky behavior that no more than about 16% of the target population engages in, only a risk-reduction method that reduces individual risk by an order of magnitude will be worthwhile.

For comparison, condoms do not offer an 11-fold decrease in pregnancy rates.  The typical condom pregnancy rate in the first year of use is about 15%;  the typical no-contraceptive pregnancy rate is about 85%.  So condoms reduce the individual pregnancy risks only by a factor of about 6.

This has some practical consequences in the teen sex case.  Of unmarried 15-year-old teens, only 13% have had sex.  This means that risk-reduction policies aimed at 15-year-olds are almost certainly going to be counterproductive in respect of reducing risks, unless we have some way of decreasing the risks by a factor of more than 10, which we probably do not.  In that population, the effective thing to do is to focus on decreasing the incidence of the risky behavior rather than decreasing the risks of the behavior.

In higher age groups, the results may be different.  But even there, a one-size-fits-all policy is not optimal.  The sexual activity rates differ from subpopulation to subpopulation.  The effectiveness with regard to the reduction of social risk depends on details about the target population.  This suggests that the implementation of risk-reduction measures might be best assigned to those who know the individuals in question best, such as parents.

In summary, given my model:
  • When a majority of the target population engages in the risky behavior, both incremental and significant risk-reduction policies reduce total social risk.
  • When a minority of the target population engages in the risky behavior, incremental risk-reduction policies are counterproductive, but sufficiently effective non-incremental risk-reduction policies can be effective.
  • When a small minority--less than about 16%--engages in the risky behavior, only a risk-reduction policy that reduces the individual risk by an order of magnitude is going to be effective;  more moderately successful risk-reduction polices are counterproductive.


Heath White said...

This general phenomenon is well-known to economists under the rubric of "moral hazard"--it turns out, for instance, that Volvo-drivers are among the most accident-prone. I've not seen it worked out in numbers before, though it would surprise me if some economist had not, and I haven't been looking. I think your conclusions are eminently reasonable--in particular, social policy should do the numbers.

Alexander R Pruss said...

I am sure the phenomenon is known to economists. :-)

But googling moral hazard suggests that that's not the same phenomenon. Moral hazard occurs when risks are shifted to another party--insurance is the main case. The kinds of risk-reduction policies I am interested in decrease risk without shifting it.

Moral hazard is a more radical form of the phenomenon I am talking about, where the social cost of the individual risky behavior isn't decreased, but is passed on to others. (Imagine this weird pregnancy insurance case. If you get pregnancy, your baby is beamed by Scotty from your womb into that of a paid insurance company staffer.) In the cases I am interested, the social impact of the individual risky behavior is actually decreased.

Alexander R Pruss said...

A factor that exacerbates the problem is that people in the target population may overestimate the risk reduction.

Jarrett said...

This post reminds me of one of the famous lines in the movie, Jurassic Park. "Some of the worst things imaginable have been done with the best intentions."

Often times social and fiscal policies have the best intentions in mind, but good-intentioned policies can certainly have negative consequences.

I'm sure there's a word for this particular phenomenon. If not we can create one. :)

Alexander R Pruss said...

Some more addenda:

1. The root of the phenomenon is the difference in utility assignments between the policy-makers (or policy-evaluators) in their policy roles and the utility assignments of the individual agents in the target population. Because of the controversial nature of the "intrinsic" utility assignments to the risky activity, the policy makers rely solely on the uncontroversial risks (or so they claim when they present the policy as a risk reduction policy). Thus, they are not supposed to reason: "If we promote condoms, more teens will have sex, and sex is pleasant, so that's a good thing." But of course the target population does take the pleasure of sex into account. If the policy-makers shared the target population's utility assignments, then, assuming decision-theoretic rationality, total utility would indeed go up given the risk reduction policy.

In the political arena, there are always suspicions that those who propose the policy are doing so out of sympathy to the target population's utility assignments and that those who oppose the policy are doing so out of disagreement with these utility assignments. But "officially" the debate is just about the risks, and I take that at face value.

2. A better way to interpret my model would be for the population not to be a population of individual people, but of individuals' opportunities for the risky behavior. That would model the fact that individuals' utility assignments to the risky activity differ from opportunity to opportunity, and that the decision whether to engage in the risky behavior is typically not a once-and-for-all decision to develop a habit of such behavior, but a number of decisions to engage in particular instances of the behavior.

This re-interpretation affects some of the results. When we look at the percentage of the population that engages in the risky behavior, we should look at the percentage of behavior opportunities that are realized.

My intuition is that in the teen-sex case, the percentage of behavior opportunities that are realized will be smaller than the percentage of teens that are sexually active, because few if any of the sexually active teens are accepting every sexual "opportunity" that comes their way (though this is offset by the fact that some of the inactive teens are inactive because they don't have any opportunities). And the smaller the percentage of realized opportunities, the more likely it is for the risk-reduction policy to be counterproductive.

There is, of course, a lot of vagueness with regard to the notion of an opportunity for the behavior.

Alexander R Pruss said...

This is a neat potential application. It could well be that texting increases overall inconvenience in respect of appointment-lateness.

David Parker said...
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