Proportionality and deterrence

There are many contexts where a necessary condition of the permissibility of a course of action is a kind of proportionality between the goods and bads resulting from the course of action. (If utilitarianism is true, then given a utilitarian understanding of the proportionality, it’s not only necessary but sufficient for permissibility.) Two examples:

• The Principle of Double Effect says it is permissible to do things that are foreseen to have a basic evil as an effect, if that evil is not intended, and if proportionality between the evil effect and the good effects holds.

• The conditions for entry into a just war typically include both a justice condition and a proportionality condition (sometimes split into two conditions, one about likely consequences of the war and the other about the probability of victory).

But here is an interesting and difficult kind of scenario. Before giving a general formulation, consider the example that made me think about this. Country A has a bellicose neighbor B. However, B’s regime while bellicose is not sufficiently evil that on a straightforward reading of proportionality it would be worthwhile for A to fight back if invaded. Sure, one would lose sovereignty by not fighting back, but B’s track record suggests that the individual citizens of A would maintain the freedoms that matter most (maybe this is what it would be like to be taken over by Alexander the Great or Napoleon—I don’t know enough of history to know), while a war would obviously be very bloody. However, suppose that a policy of not fighting back would likely result in an instant invasion, while a policy of fighting back would have a high probability of resulting in peace for the foreseeable future. We can then imagine that the benefits of likely avoiding even a non-violent takeover by B outweigh the small risk that despite A’s having a policy of armed resistance B would still invade.

The general case is this: We have a policy that is likely to prevent an unhappy situation, but following through on the policy violates a straightforward reading of proportionality if the unhappy situation eventuates.

One solution is to take into account the value of follwing through on the policy with respect to one’s credibility in the future. But in some cases this will be a doubtful justification. Consider a policy of fighting back against an invader—at least initially—even if there is no chance of victory. There are surely many cases of bellicose countries that could successfully take over a neighbor, but judge that the costs of doing so are too high given the expected resistance. But if the neighbor has such a policy, then in case the invasion nonetheless eventuates, whatever is done, sovereignty will be lost, and the policy will be irrelevant in the future. (One might have some speculation about the benefits for other countries of following through on the policy, but that’s very speculative.)

One line of thought on these kinds of cases is that we need to forego such policies, despite their benefits. One can’t permissibly act on them, so one can’t have them, and that’s that. This is unsatisfying, but I think there is a serious chance that this is right.

One might think that the best of both worlds is to make it seem like one has the policy, but not in fact have it. A problem with this is that it might involve lying, and I think lying is wrong. But even aside from that, in some cases this may not be practicable. Imagine training an army to defend one’s country, and then having a secret plan, known only to a very small number of top commanders, that one will surrender at the first moment of an invasion. Can one really count on that surrender? The deterrent policy is more effective the fiercer and more patriotic the army, but those factors are precisely likely to make them fight despite the surrender at the top.

Another move is this. Perhaps proportionality itself takes into account not just the straightforward computation of costs and benefits, but also the value of remaining steadfast in reasonably adopted policies. I find this somewhat attractive, but this approach has to have limits, and I don’t know where to draw them. Suppose one has invented a weapon which will kill every human being in enemy territory. Use of this weapon, with a Double Effect style intention of killing only the enemy soldiers, is clearly unjustified no matter what policies one might have, but a policy to use this weapon might be a nearly perfect protection against invasion. (Obviously this connects with the question of nuclear deterrence.) I suppose what one needs to say is that the importance of steadfastness in policies affects how proportionality evaluation go, but should not be decisive.

I find myself pulled to the strict view that policies we should not have policies acting on which would violate a straightforward reading of proportionality, and the view that we should abandon the straightforward reading of proportionality and take into account—to a degree that is difficult to weigh—the value of following policies.

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.

A heuristic about contraceptive policy

Here is a quick heuristic about social policy. Suppose it is in the social interest to decrease pregnancy rates in some population of teenagers. Then my quick heuristic is this:

• One cannot expect the promotion of a contraceptive to lower the pregnancy rate very much below the typical-use failure rate of that contraceptive for that population.

For instance, one set of estimates pegs the typical failure rate of condoms for the first year of use at 15% and the pill's failure rate for the first year of use at 9%. In the case of condoms, failure rates decrease with use, and I've seen 5% cited elsewhere for the pill.

What is the reason for the heuristic? It is obviously difficult to simultaneously provide contraception, and instruction on their use, while promoting abstinence. Intuitively (and I'm just giving a heuristic) we do not expect mixed messages to work well. But the only way we're going to get a pregnancy rate below the typical-use failure rate of the contraceptive is by having some people abstain totally or to decrease their sexual frequency significantly below the number used in the typical-use failure rate calculations. But we would expect, for rational choice reasons, the availability of contraception to decrease the rate of total abstinence by reducing the costs of intercourse, and we would also expect it to increase sexual frequency. Here is a rough estimate. Suppose Sally disvalues pregnancy in her circumstances at somewhere between 1.2 and 20 times the value of a year of sex (very roughly: she'd be willing to abstain about 1.2 years to avoid a pregnancy in her circumstances but she wouldn't be willing to abstain 20 years to avoid it). It seems intuitively right to me that many teens will fall in this range. Then if pregnancy is the only consideration, it is decision theoretically rational for Sally to max out her sexual activity if she is on the pill but to abstain totally if she is not using any contraceptive (this is a pretty easy calculation using an 85% no-contraceptive annual conception rate).

The U.S. teen pregnancy rate in 2006 was 7%. Promotion of the pill could imaginably lower that somewhat, if the 5% figure is correct, but not if the 9% figure is. However, unless these teenagers use both the pill and condoms, there will be significant health risks for sexually transmitted infections. Because of these, any contraceptive-based policy would likely involve condoms as well. But intuitively one does not expect all that many teens to double up and use both the pill and condoms. If condoms are promoted, we'd expect a significant percentage of the sexually active population to use only condoms. If we have a half-and-half mix between condoms and the pill, and no overlap, there'll be a failure rate around 10-12%. Which is significantly higher than the pregnancy rate, at least with 2006 data.