Kaplan and Cullison: Jury Decision Model
The concept of “reasonable doubt” is one of the most difficult to quantify In the whole of legal theory. Suppose you’re on a jury and you’ve seen all the evidence, and you’ve come to some conclusion about the probability of the defendant: 75%, say, or 90%. Should you vote for a conviction?
The concept of “reasonable doubt” is one of the most difficult to quantify In the whole of legal theory. Suppose you’re on a jury and you’ve seen all the evidence, and you’ve come to some conclusion about the probability of the defendant: 75%, say, or 90%. Should you vote for a conviction?
In this post we are going to discuss a mathematical model proposed by John Kaplan and Alan Cullison, which is supposed to help jurors, who have already assessed the probability of guilt of the defendant, to decide what verdict to return. The novelty and importance of their idea is that in order to reach a decision, the juror must make some kind of measurement of his own personal degree of repugnance at the idea of acquitting a guilty person, a degree which can obviously vary greatly according to the crime being judged, and the danger of its being repeated if the culprit is acquitted.
Our explanation of the model comes from a seminal 1971 article by Laurence Tribe:Trial by Mathematics, Precision and Ritual in the Legal Process (84 Harvard Law Rev. 1329, 1971). Tribe’s rebuttal of the model, and his discussion of the use of mathematics in trials in general, is complex and fascinating, and will be the subject of a series of future posts.
Kaplan and Cullison’s model for jury decision-making.
Let's say that the trial is over, all the evidence has been seen, and the trier (i.e. the jury member) assesses the probability of guilt of the accused as some probability value P between 0 and 1.
Now the trier must decide on a verdict by choosing between two acts: convict or acquit.
There are four possibilities for the outcome of a trial:
There are four possibilities for the outcome of a trial:
C_G (conviction of a guilty person)
C_I (conviction of an innocent person)
A_G (acquittal of a guilty person)
A_I (acquittal of an innocent person).
The trier will assign numerical values between 0 and 1 to each of these possibilities. He begins by taking C_G=1 (most desirable) and C_I=0 (least desirable).
Next, the trier must assign values to A_G and A_I according to the following procedure. Start with A_G.
The trier asks himself the following question: "Would I rather have a result that I know to be A_G, or take a 1/2 - 1/2 chance between C_G and C_I?”
If he realizes that he would prefer A_G, then he knows that the value of A_G will be greater than 1/2. Now he will try to see if it's greater than 3/4 by asking himself: "Would I rather have a result that I know to be A_G, or take a 3/4 - 1/4 chance between C_G and C_I?” And so on, until he closes in on an actual value for A_G. He then uses the same procedure for A_I.
There are many things to keep in mind when making the decisions of what one would prefer. The choice of a value for A_G is particularly delicate, because if the outcome of the trial is the acquittal of the culprit, then the jury may feel some responsibility if, for example, the crime was a type of brutal murder which then occurs again after the criminal's acquittal. In such a situation, the trier may well feel that A_G is not preferable to a 50-50 chance between C_G and C_I. The value for A_G is never likely to be less than 1/2, since few people if any would take a chance between C_G and C_I if the likelihood of C_I is actually perceived as greater than C_G. But 0.5 may be an acceptable value, or, if the crime is not likely to re-occur, A_G may end up being a high value such as 0.9, in order to minimize the chance of convicting an innocent.
The same procedure is used to determine a value for A_I, but the meaning is different. On the whole, A_I is a better outcome than A_G, because at least the trier has not erred in his work. Therefore the trier is unlikely to take much of a risk of convicting an innocent, in comparison to acquitting an innocent, and the value of A_I will tend to be significantly higher than 0.5. On the other hand, the trier may deeply dislike the outcome A_I because if the real culprit is in fact free, then there is a danger that he will continue his crimes, so he may not want to simply set A_I=1; he may prefer to have a good chance of having convicted the guilty party than to be certain of having acquitted someone innocent. Still, on the whole, A_I is likely to be quite high, higher than A_G.
Now that the numbers P, A_G and A_I have been fixed subjectively by the trier, the Kaplan-Cullison model suggests the following calculation to decide between conviction and acquittal.
The trier asks himself the following question: "Would I rather have a result that I know to be A_G, or take a 1/2 - 1/2 chance between C_G and C_I?”
If he realizes that he would prefer A_G, then he knows that the value of A_G will be greater than 1/2. Now he will try to see if it's greater than 3/4 by asking himself: "Would I rather have a result that I know to be A_G, or take a 3/4 - 1/4 chance between C_G and C_I?” And so on, until he closes in on an actual value for A_G. He then uses the same procedure for A_I.
There are many things to keep in mind when making the decisions of what one would prefer. The choice of a value for A_G is particularly delicate, because if the outcome of the trial is the acquittal of the culprit, then the jury may feel some responsibility if, for example, the crime was a type of brutal murder which then occurs again after the criminal's acquittal. In such a situation, the trier may well feel that A_G is not preferable to a 50-50 chance between C_G and C_I. The value for A_G is never likely to be less than 1/2, since few people if any would take a chance between C_G and C_I if the likelihood of C_I is actually perceived as greater than C_G. But 0.5 may be an acceptable value, or, if the crime is not likely to re-occur, A_G may end up being a high value such as 0.9, in order to minimize the chance of convicting an innocent.
The same procedure is used to determine a value for A_I, but the meaning is different. On the whole, A_I is a better outcome than A_G, because at least the trier has not erred in his work. Therefore the trier is unlikely to take much of a risk of convicting an innocent, in comparison to acquitting an innocent, and the value of A_I will tend to be significantly higher than 0.5. On the other hand, the trier may deeply dislike the outcome A_I because if the real culprit is in fact free, then there is a danger that he will continue his crimes, so he may not want to simply set A_I=1; he may prefer to have a good chance of having convicted the guilty party than to be certain of having acquitted someone innocent. Still, on the whole, A_I is likely to be quite high, higher than A_G.
Now that the numbers P, A_G and A_I have been fixed subjectively by the trier, the Kaplan-Cullison model suggests the following calculation to decide between conviction and acquittal.
If the trier chooses to convict, he will get C_G with a probability of P, and C_I with a probability of 1-P. Defined the "expected utility" UC of the choice “convict” by the standard weighted formula
UC = P C_G + (1-P) C_I, which in fact is always just equal to UC = P.
Similarly, if the trier chooses to acquit, there's a probability of P that he'll actually get A_G and (1-P) that he'll actually get A_I, so we can defined the "expected utility" UA of the choice A by
UA = P A_G + (1-P) A_I.
Both UC and UA are numbers, and the model says all we have to do is compare them.
If UC > UA, then choose to convict. If UA > UC, then choose to acquit.
Examples : Suppose the trier is only 75% convinced of the guilt of the accused. The trier has done all of the above calculations and has fixed A_G at 0.5 and A_I at 0.9.
Then according to the formulas, UC = .75 and UA = .75 x .5 + .25 x .9 = .375+.225=.6.
Thus UC > UA so in this situation, the trier should vote to convict.
This result may seem really bizarre in view of the injunction to vote for a conviction only if convinced of guilt "beyond a reasonable doubt". Clearly a 75% probability of guilt is not beyond a reasonable doubt. Yet the Kaplan-Cullison model leads to a recommendation to convict.
UC = P C_G + (1-P) C_I, which in fact is always just equal to UC = P.
Similarly, if the trier chooses to acquit, there's a probability of P that he'll actually get A_G and (1-P) that he'll actually get A_I, so we can defined the "expected utility" UA of the choice A by
UA = P A_G + (1-P) A_I.
Both UC and UA are numbers, and the model says all we have to do is compare them.
If UC > UA, then choose to convict. If UA > UC, then choose to acquit.
Examples : Suppose the trier is only 75% convinced of the guilt of the accused. The trier has done all of the above calculations and has fixed A_G at 0.5 and A_I at 0.9.
Then according to the formulas, UC = .75 and UA = .75 x .5 + .25 x .9 = .375+.225=.6.
Thus UC > UA so in this situation, the trier should vote to convict.
This result may seem really bizarre in view of the injunction to vote for a conviction only if convinced of guilt "beyond a reasonable doubt". Clearly a 75% probability of guilt is not beyond a reasonable doubt. Yet the Kaplan-Cullison model leads to a recommendation to convict.
On the other hand, the choices for A_G and A_I above are not necessarily the most normal choices. There are many other possible attitudes. In the case of an unimportant crime, the trier may set A_G=1 and A_I=1, meaning that he would rather acquit the accused, whether innocent or guilty, than accept even the smallest chance of convicting an innocent person. If so, then he will find that UA=1, meaning no matter what the value P of his conviction that the accused is guilty, even if P=99%, he should vote to acquit.
In essence, what the model is suggesting is that the concept of "beyond a reasonable doubt" be replaced by the concept of "utility", with the trier taking into account the negative aspects of acquitting a guilty person or convicting an innocent. This marks a deep rift with respect to the tradition underlying the way trials are conducted and jury decisions are made. It is an important question and one well worth considering.
In a series of upcoming posts, I will introduce the work of Laurence Tribe, and in particular explain his reasons for rejecting this model. These reasons are deep and fascinating, and go far beyond the realm of mathematics or even legal theory, into the domain of psychology.
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