Thursday, December 08, 2005
Death Penalty Algebra
(I appear to have a little free time, so I thought I would actually post something. Look for more this weekend as it will be postsemester including pictures of Matt, Mike Parks, and I chowing down on some Mexican food and drinking some beer.)
Implicitly in this post, and more explicitly in the comments Matt raises the following problem:
I come to a different answer taking an approach that considers both the benefits and the costs (both measured in terms of lost innocent lives).
The cost of the death penalty in terms of innocent lives are all lives lost due to wrongful execution. The benefits from the death penalty are all innocent lives whose murder was deterred because of the existence of the death penalty.
Note that both the benefit and the cost are effectively achieved "in my name" under the death penalty. Under the specified formulation of a deterrent death penalty, a government that does away with the death penalty condemns to death all those innocents whose murder might have been deterred. While we may not be able to specifically identify ahead of time who those innocents might be, we are nonetheless condemning them to death. Accordingly, I am assign the same magnitude (though opposite signs for obvious reasons) to an innocent life saved via deterrence and to an innocent life lost to wrongful execution.
In the real world, the current death penalty is not a perfect deterrent as evidenced by the fact that murders still occur. Nondeterrence can occur for a variety of reasons such as an expectation of not being caught, insanity, or believing that your death is actually justified as long as the victim dies (that's the calculus of suicide bombers).
So let us assume that if the death penalty is in effect, a considered murder is deterred [by the death penalty] with probability p, then we can find the break even point by solving
To simplify this equation, assume all murders are "solved" and that there is one convicted murderer for each murder. Note, by considering convicted murderers I am intentionally excluding manslaughter which by definition does not include premeditation and thus cannot be deterred. (Plus the death penalty doesn't apply to manslaughter). Then the number of convicted murderers is given by:
Now define r as the ratio of wrongful executions per convicted murderer, and the first equation can be rewritten as
Noting that we have the same term on both sides, the utilitarian break even point for the death penalty can be expressed in terms of the percentage of dettered (considered) murders [by the death penalty], p, and the ratio of wrongful executions, r, as
To help provide an intuitive feel for this equation, note that if p = 0, then r should be 0. In other words, if there truly is no deterrent effect from the death penalty, then executions should only be permitted if we were absolutely sure that no innocents were executed. As perfect certainty is not possible, we would then do away with the death penalty.
If however, p = 0.5, then it should be acceptable to operate with r = 1. While this may seem counterintuitive, note that this situation corresponds to the situation where an equal number of innocent lives are saved via deterence to lost via wrongful execution (for every two considered murders, one actually happens). If p > 0.5, then by this utilitarian metric, we can actually execute more people than we convict and be morally neutral! Why? We're still balancing innocents saved to innocents lost. For p = 2/3, r = 2 because two murders are deterred for every murder that actually occurs. For p = 1, r = infinity. However this corresponds to an undefined number of wrongful executions as there would be 0 convicts to execute as all murders would be deterred.
Tabulating the numbers we get the following
As many people in the blogosphere have noted, we just had our 1,000 execution since 1976 and approximately some 600,000 murders since 1976. Shoehorning these statistics into our equation, let's assume that every execution was wrongful and that every murder resulted in a conviction. Then these wrongful executions are "acceptable" for a deterrent effect as low as p = 1/600 as r = 1/600.
To fill in actual numbers we need the number of murder convictions. I had a bit of trouble finding this (what I really want is the number of murder convictions since 1976), with the closest being this report (the feds seem to like to only compile federal convictions, the states compile only state convictions and I'm not going to sum all of these up by hand) from which I eyeball a mean US murder conviction rate at about 0.05 of the population from 1981 to 1995. Assuming a mean population of about 200 million from 1973 to the present that gives 10,000 convictions per year or about 300,000 murder convictions from 1976 to present.
Then applying simple arithmentic to statistics provided by wiki results in about 242 death sentences being issued per year with 4.7% of these sentences being overturned via exoneration (much higher percentage are commuted to life for procedural reasons). Being extremely ungenerous to the justice system (intuitively, the following number should be lower), let us assume that 4.7% of executions are wrongful. Then 47 executions since 1976 would be wrongful.
Then r is given by 47/300,000 or 0.00016. Thus by the algebra given above, the death penalty in its current form is acceptable on a utilitarian basis if approximately 0.00016 of all murders are deterred.
That's an exceedingly low bar to satisfy. I suspect that its much higher than that. I actually estimate it to be at least 50%. Why? Because rational humans really do respond to changes in incentives and avoiding death is a pretty big incentive. That humans respond to lesser incentives
was recently noted by Kaus (scroll down to Friday Dec 2  Kaus's permalinks suck) that with respect to postKatrina gas prices it was
Now that being said, it is still worthwhile to utilize virtually every means available to ensure that we don't execute the innocent. Why? Because we typically assign a very high value to human life. While it is common to just say that a life has infinite value, we don't really mean it and economists can and do assign a finite value to a human life.
Regardless of what the exact value of life is, it's pretty high, and certainly more than the cost of running a DNA test. So it's to society's benefit to ensure that we take virtually every means available to get a death sentence right. Of course I say "virtually every means available" as some means seem likely to significantly reduce the deterrent effect and thus likely to be of net detriment to society.
Update
In the comments, Tom suggests that many people value the loss of human life differently when it is taken in different manners. Specific to this case, someone may value the loss of an innocent life via execution differently than the loss of an innocent life via murder.
So let's assume that you weight the loss of innocent life via murder a fraction 'f' as much as the loss of innocent life via execution. Then our equation becomes the following:

Implicitly in this post, and more explicitly in the comments Matt raises the following problem:
Assuming the death penalty deters as advertised and is administered impartially, how many executions of innocents are acceptable?Matt says none, thus making the death penalty untenable because he cannot accept executions of innocents being performed in his name.
I come to a different answer taking an approach that considers both the benefits and the costs (both measured in terms of lost innocent lives).
The cost of the death penalty in terms of innocent lives are all lives lost due to wrongful execution. The benefits from the death penalty are all innocent lives whose murder was deterred because of the existence of the death penalty.
Note that both the benefit and the cost are effectively achieved "in my name" under the death penalty. Under the specified formulation of a deterrent death penalty, a government that does away with the death penalty condemns to death all those innocents whose murder might have been deterred. While we may not be able to specifically identify ahead of time who those innocents might be, we are nonetheless condemning them to death. Accordingly, I am assign the same magnitude (though opposite signs for obvious reasons) to an innocent life saved via deterrence and to an innocent life lost to wrongful execution.
In the real world, the current death penalty is not a perfect deterrent as evidenced by the fact that murders still occur. Nondeterrence can occur for a variety of reasons such as an expectation of not being caught, insanity, or believing that your death is actually justified as long as the victim dies (that's the calculus of suicide bombers).
So let us assume that if the death penalty is in effect, a considered murder is deterred [by the death penalty] with probability p, then we can find the break even point by solving
(#of wrongful executions) = p x (#considered murders)
(#convicted murderers) = (#murders) = (1p) x (#considered murders)
r x (#convicted murderers) = [p / (1p)] x (#convicted murderers)
Noting that we have the same term on both sides, the utilitarian break even point for the death penalty can be expressed in terms of the percentage of dettered (considered) murders [by the death penalty], p, and the ratio of wrongful executions, r, as
r = [p / (1p)]
If however, p = 0.5, then it should be acceptable to operate with r = 1. While this may seem counterintuitive, note that this situation corresponds to the situation where an equal number of innocent lives are saved via deterence to lost via wrongful execution (for every two considered murders, one actually happens). If p > 0.5, then by this utilitarian metric, we can actually execute more people than we convict and be morally neutral! Why? We're still balancing innocents saved to innocents lost. For p = 2/3, r = 2 because two murders are deterred for every murder that actually occurs. For p = 1, r = infinity. However this corresponds to an undefined number of wrongful executions as there would be 0 convicts to execute as all murders would be deterred.
Tabulating the numbers we get the following
__p____r__
_0.01__0.01_
_0.05__0.05_
_0.1___0.11_
_0.2___0.25_
_0.3___0.43_
_0.4___0.67_
_0.5___1.00_
_0.67__2.00_
_0.9___9.00_
_0.01__0.01_
_0.05__0.05_
_0.1___0.11_
_0.2___0.25_
_0.3___0.43_
_0.4___0.67_
_0.5___1.00_
_0.67__2.00_
_0.9___9.00_
As many people in the blogosphere have noted, we just had our 1,000 execution since 1976 and approximately some 600,000 murders since 1976. Shoehorning these statistics into our equation, let's assume that every execution was wrongful and that every murder resulted in a conviction. Then these wrongful executions are "acceptable" for a deterrent effect as low as p = 1/600 as r = 1/600.
To fill in actual numbers we need the number of murder convictions. I had a bit of trouble finding this (what I really want is the number of murder convictions since 1976), with the closest being this report (the feds seem to like to only compile federal convictions, the states compile only state convictions and I'm not going to sum all of these up by hand) from which I eyeball a mean US murder conviction rate at about 0.05 of the population from 1981 to 1995. Assuming a mean population of about 200 million from 1973 to the present that gives 10,000 convictions per year or about 300,000 murder convictions from 1976 to present.
Then applying simple arithmentic to statistics provided by wiki results in about 242 death sentences being issued per year with 4.7% of these sentences being overturned via exoneration (much higher percentage are commuted to life for procedural reasons). Being extremely ungenerous to the justice system (intuitively, the following number should be lower), let us assume that 4.7% of executions are wrongful. Then 47 executions since 1976 would be wrongful.
Then r is given by 47/300,000 or 0.00016. Thus by the algebra given above, the death penalty in its current form is acceptable on a utilitarian basis if approximately 0.00016 of all murders are deterred.
That's an exceedingly low bar to satisfy. I suspect that its much higher than that. I actually estimate it to be at least 50%. Why? Because rational humans really do respond to changes in incentives and avoiding death is a pretty big incentive. That humans respond to lesser incentives
was recently noted by Kaus (scroll down to Friday Dec 2  Kaus's permalinks suck) that with respect to postKatrina gas prices it was
"almost as if insidious law was at workas prices rise, demand declines!"People changed their behavior just to save a few bucks. Surely they would do the same to save their own lives (except for pirates with booty  I'll let Thason explain that one). Surely, this deterent effect is greater than 0.00016, so surely the death penalty is acceptable on a utilitarian measure of innocent lives.
Now that being said, it is still worthwhile to utilize virtually every means available to ensure that we don't execute the innocent. Why? Because we typically assign a very high value to human life. While it is common to just say that a life has infinite value, we don't really mean it and economists can and do assign a finite value to a human life.
Regardless of what the exact value of life is, it's pretty high, and certainly more than the cost of running a DNA test. So it's to society's benefit to ensure that we take virtually every means available to get a death sentence right. Of course I say "virtually every means available" as some means seem likely to significantly reduce the deterrent effect and thus likely to be of net detriment to society.
Update
In the comments, Tom suggests that many people value the loss of human life differently when it is taken in different manners. Specific to this case, someone may value the loss of an innocent life via execution differently than the loss of an innocent life via murder.
So let's assume that you weight the loss of innocent life via murder a fraction 'f' as much as the loss of innocent life via execution. Then our equation becomes the following:
r = f x [p / (1p)]
For instance suppose your f is 1/10. Then based on the statistics cited above where we found an r of 0.00016, you should require the deterrent effect of the death penalty to be at leastp = 0.0016. If f = 100, then p would need to be p=0.016. Rewriting this as its own equation relating p and f for an r of 0.00016 gives the following.
f = 0.00016 x [(1p) / p]
Or for a small value of p approximation, use
f = 0.00016 / p.
I encourage the motivated reader to estimate his/her own value for f. Figure out how much more you value not killing an innocent via execution over saving an innocent from murder and then solve for the p you would require. Woo hoo hoo.
Update 2
More rational behavior in response to changing prices. (h/t DailyPundit)
Update 3
As it's come up more than once in the comments, apparently it wasn't clear that p refers to the deterrence effect assumed for the death penalty. If a murder is deterred for another reason, such as life in prison or because it rained, then it doesn't count in the calcuation. If p was intended to mean any deterred murder as two commenters have suggested, then instead of writing
In the hopes of limiting further confusion, I've added this update and a couple phrases to the original post in square brackets clarifying the fact that I really am considering the benefit of the death penalty as innocent lives deterred because of the death penalty and not attempting to include some lesser deterrent.
Update 4
And here's supply responding to increased demand in a way that is very similar to the phenomenon noted in Update 2.
Or for a small value of p approximation, use
f = 0.00016 / p.
I encourage the motivated reader to estimate his/her own value for f. Figure out how much more you value not killing an innocent via execution over saving an innocent from murder and then solve for the p you would require. Woo hoo hoo.
Update 2
More rational behavior in response to changing prices. (h/t DailyPundit)
Update 3
As it's come up more than once in the comments, apparently it wasn't clear that p refers to the deterrence effect assumed for the death penalty. If a murder is deterred for another reason, such as life in prison or because it rained, then it doesn't count in the calcuation. If p was intended to mean any deterred murder as two commenters have suggested, then instead of writing
The benefits from the death penalty are all innocent lives whose murder was deterred because of the existence of the death penalty.I would've written
The benefits from the death penalty are all innocent lives whose murder was deterred.This second formulation would lead to misleading results as more benefit would be assigned to the death penalty than it should claim while costs that are solely attributable to the death penalty would remain the same.
In the hopes of limiting further confusion, I've added this update and a couple phrases to the original post in square brackets clarifying the fact that I really am considering the benefit of the death penalty as innocent lives deterred because of the death penalty and not attempting to include some lesser deterrent.
Update 4
And here's supply responding to increased demand in a way that is very similar to the phenomenon noted in Update 2.
