there are many different peremptory challenge systems. The American Bar Association Standards describe a system commonly referred to as the Arizona struck system:
Under this practice, jurors are first examined and challenged for cause by both sides. Excused jurors are replaced on the panel, and the examination of replacements continues
until a panel of qualified jurors is presented. The size of the panel at this time is the sum of the number of jurors to hear the case plus the number of peremptories to be allowed all
parties. The parties then proceed to exercise their peremptories, usually alternating or in some similar way which will result in all parties exhausting their challenges at approxi-
mately the same time.
20
The number of peremptory challenges given to each side generally depends on the gravity of the crime. For example, in California each side is given 20 strikes in criminal cases where
the offense is punishable by death or life imprisonment. In other criminal cases, each side is limited to 10 strikes, except in cases where the offense is punishable by a jail term of 90 days
or less. In those cases, only six challenges are allowed per side. While California and many other states give both sides the same number of strikes, some states allot the defense more
strikes than the prosecution is allowed. For example, in Georgia the defense is allotted twice as many strikes as the prosecution is allowed. There are no states in which the prosecution
is given more strikes than the defense is given.
21
Also, individual judges have discretion in awarding peremptory challenges on a case-by-case basis.
3. The model
In this section, we construct a model of the trial process that allows us to analyze the effects of the number of peremptory challenges awarded on the probabilities of a hung jury,
wrongful acquittal, and wrongful conviction. To do this, the model must have several features. First, the truth must be objective, not subjective, in that defendants must be either
guilty or innocent.
22
This is necessary because a defendant must be guilty to be wrongfully acquitted and innocent to be wrongfully convicted. Second, the reasonable-doubt standard is
already in place to protect against wrongful convictions, and to assess the impact of jury selection procedures on the social cost of wrongful conviction it is imperative to incorporate
some notion of reasonable doubt into the model. Third, jurors must possess some readily observable bias that affects their determination of the defendant’s guilt and that can be used
by the attorneys as a basis for peremptory challenges. We assume that jurors are either biased toward acquittal or biased toward conviction, and both the defense and prosecution can
identify any given juror’s bias. Finally, we assume that a two-way unanimity rule governs jury verdicts: a unanimous decision is needed to convict or acquit. If there is not a unanimous
verdict, the jury is considered hung.
Given the peremptory challenge rules, we can calculate the probabilities that the final jury
20
ABA, supra note 17 at pp. 77–78.
21
See Jon M. Van Dyke, Jury selection procedures 1977 at pp. 282–284.
22
We only consider criminal cases in this model. 228
W.S. Neilson, H. Winter International Review of Law and Economics 20 2000 223–250
is entirely biased toward acquittal, entirely biased toward conviction, or balanced i.e., contains jurors biased toward acquittal and jurors biased toward conviction. We then use
these probabilities to calculate the probabilities of a hung jury, wrongful acquittal, and wrongful conviction.
To begin with, assume that nature chooses whether or not the defendant is guilty and the strength s of the evidence against the defendant. Let PG be the probability that nature
chooses guilty so that 12PG is the probability that nature chooses innocent, let Fs?G be the conditional distribution of the strength of the evidence given that the defendant is guilty,
and let Fs?I be the conditional distribution of the strength of the evidence given that the defendant is innocent. Let fs?G and fs?I be the corresponding density functions. The
evidence is used by jurors in the trial to assess the guilt of the defendant.
To model the reasonable-doubt standard, we assume that some evidence is inconsistent with an innocent defendant, so that if a juror observes that evidence, it could only have been
generated by a guilty defendant. More concretely, suppose that an innocent defendant can generate evidence in the interval [0,s
I
], and that a guilty defendant can generate evidence in the interval [s
G
,1], with 0 s
G
, s
I
, 1 as in Fig. 1. The third inequality means that there is some evidence that is so strong that it could not possibly be consistent with an innocent
defendant. This assumption embodies our notion of reasonable doubt: if a juror observes evidence s s
I
, that juror can say that the defendant is guilty beyond a reasonable doubt.
23
23
Put another way, we adopt as a reasonable-doubt standard the condition that Prob{G?s}51. A less stringent reasonable-doubt standard might use, say, the 95-sure rule of Prob{G?s}0.95. Such a change would raise the
probability of wrongful conviction and reduce the probability of wrongful acquittal. Fig. 1. Probability distributions for evidence.
229 W.S. Neilson, H. Winter International Review of Law and Economics 20 2000 223–250
On the other hand, if the juror observes s , s
I
, that juror cannot conclude that the defendant is guilty beyond a reasonable doubt. The middle inequality, s
G
, s
I
, allows for analysis of medium strength cases, that is, cases in which the defendant is neither certainly innocent nor
certainly guilty.
24
The third crucial component of the model is juror bias. The strength of the evidence is observed by the jurors as the case is litigated. Jurors do not observe s directly; rather, they
observe s with bias. Furthermore, the bias is predictable based on an observable binary characteristic, such as race or gender. We assume that all jurors are biased, but that not all
of them are biased in the same direction.
25
We assume, without loss of generality, that jurors with the characteristic are biased toward conviction and that those without the characteristic
are biased toward acquittal. Consistent with the assumption of bias being determined by a binary characteristic, we assume no variation in the magnitude of bias. The bias is opera-
tionalized by assuming that jurors do not observe the strength of evidence that nature selects; instead, jurors “observe” the true strength of the case plus some bias parameter. So, for
example, when the true strength of the evidence is s, a juror j biased toward conviction observes evidence of strength s
j
5 s 1 x, where x is a positive number. A juror biased toward acquittal would perceive the same evidence to be of strength s
j
5 s 2 y, where y is a positive number. Since higher values of s are more likely to have been generated by a guilty
defendant, these biased perceptions of the strength of the evidence cause jurors with the binary characteristic to be more likely to find the defendant guilty, and jurors without the
characteristic to be more likely to find the defendant not guilty.
We make the assumption that jurors are oblivious to their own bias and do not compensate for it in any way when making judgments. Justice Scalia, in his dissenting opinion in J.E.B.
v. Alabama, argues that:
The biases that go along with group characteristics tend to be biases that the juror himself does not perceive, so that it is no use asking about them. It is fruitless to inquire of a male
juror whether he harbors any subliminal prejudice in favor of unwed fathers.
26
24
Joel Schrag and Suzanne Scotchmer, Crime and prejudice: the use of character evidence in criminal trials, 10 J Law Econ Org 319 1994 use a similar strength of evidence model. In their model, however, the
reasonable-doubt standard is determined by a jury made up of identical jurors, whose objective is to minimize the sum of expected wrongful acquittal and wrongful conviction costs. The jurors’ estimates of these costs are a
function of the evidence presented. Schrag and Scotchmer’s jury process is a means to the ultimate social goal of crime deterrence. The social goal we consider is to minimize the expected social loss of the jury process itself.
25
These are strong simplifying assumptions. First, we have assumed that all jurors are biased, ruling out the possibility of unbiased jurors. Second, there is only one source of bias: the observable characteristic being
discussed. Weakening either of these assumptions would make the model more realistic, but at the expense of ease of exposition. With different sources of bias, or, equivalently, different magnitudes of bias, it would be necessary
to keep track of the order in which jurors should be struck, greatly complicating the mathematical analysis. The model developed here provides a thorough analysis of the simplest setting that has all of the necessary
components. A brief analysis of how weakening these assumptions affects the probabilities of hung juries, wrongful acquittals, and wrongful convictions is contained in the Appendix.
26
114 S. Ct. 1419 1994 at pp. 1438-1439. 230
W.S. Neilson, H. Winter International Review of Law and Economics 20 2000 223–250
Explicitly, jurors act as if the evidence s
j
is generated by the distributions Fs?G and Fs?I. The bias we are considering involves an honest different interpretation of the
strength of the evidence between different recognizable groups.
27
No juror in our model is biased in such a way as to want to convict a person believed to be innocent. Thus, a
very strong case will be beyond the reasonable-doubt standard for all jurors, regardless of their bias.
28
Using the Arizona struck system, the attorneys for both sides and the presiding judge see all members of the jury pool before making their selections. Let N be the size of the
pool after all the challenges for cause are exhausted, let n
d
be the allotted number of peremptory challenges for the defense, let n
p
be the allotted number of peremptory challenges for the prosecution, and let n
denote the number of individuals in the pool possessing the characteristic. Also, n
d
1 n
p
5 N 2 12 because 12 jurors are required for a sitting jury, and we are assuming there are no alternate jurors, and n
d
n
p
because the institutional rules do not allow for the prosecution to be given more challenges than the defense is allowed.
The defense will use its peremptory challenges to strike jurors possessing the character- istic, since everyone with the characteristic is biased toward conviction, and the prosecution
will use its challenges to strike jurors who do not possess the characteristic.
29
There are three cases:
1. n , n
d
. In this case, the defense has more strikes than there are jurors biased toward conviction and the prosecution has fewer strikes than it wishes to use. The defense
strikes all n pool members possessing the characteristic and uses the rest of its strikes
indifferently, the prosecution strikes n
p
members who do not have the characteristic, and the final sitting jury consists of 12 members, all of whom are biased toward
acquittal. Everyone who is biased toward conviction is eliminated from the jury. 2. N 2 n
, n
p
. In this case, the prosecution has more strikes than there are jurors biased toward acquittal and the defense has fewer strikes than it wishes to use. The prosecu-
tion strikes everyone without the characteristic and uses the rest of its strikes indif- ferently, and the final sitting jury consists of 12 members, all of whom are biased
27
Sheri Lynn Johnson, Black innocence and the white jury, 83 Michigan L Rev 1611 1985 and Nancy J. King, Postconviction review of jury discrimination: measuring the effects of juror race on jury decisions, 92
Michigan L Rev 63 1993 present a thorough discussion of the evidence pertaining to the effect of racial composition on jury decisions.
28
For example, in J.E.B, a paternity case, Justice Scalia notes that the scientific evidence presented at the trial established the defendant’s paternity with 99.92 accuracy. Even an all male jury in this case would have likely
found paternity. 114 S. Ct. 1419 1994 at p. 1437.
29
An alternative to the Arizona struck system is the sequential jury system in which lawyers only observe 12 jurors at a time and any struck juror is replaced by a juror drawn randomly from an unobserved pool. Since our
model assumes that all jurors biased in one direction are identical, behavior is identical in the two systems because the lawyers wish to strike everyone biased against their sides. If jurors were allowed to have different magnitudes
of bias, though, implementing the sequential system would require the use of a search model, which raises several interesting issues of its own beyond the scope of this article.
231 W.S. Neilson, H. Winter International Review of Law and Economics 20 2000 223–250
toward conviction. Everyone who is biased toward acquittal is eliminated from the jury.
3. n
d
, n , N 2 n
p
. In this case, the defense strikes n
d
members with the characteristic and the prosecution strikes n
p
members without the characteristic. Those who are left comprise the sitting jury, of whom n
2 n
d
, 12 are biased toward conviction and the rest are biased toward acquittal. In this case, the jury is
said to be “balanced.” In a criminal case, the reasonable-doubt standard institution is in place, so the correct
comparison for jurors to make is s
j
against the threshold s
I.
In Case 1 above, everyone on the jury is biased toward acquittal, so every member of the jury observes evidence of
strength s
j
5 s 2 y. The verdict is guilty if s s
I
1 y, and the verdict is not guilty if s , s
I
1 y, as shown in Fig. 2. In Case 2, everyone on the jury is biased toward conviction and observes evidence of strength s 1 x, so the verdict is guilty if s s
I
2 x, and the verdict is not guilty if s , s
I
2 x. As one would expect, if the jury is biased toward acquittal, a guilty verdict is more difficult to obtain; and if the jury is biased
toward conviction, a guilty verdict is less difficult to obtain. Finally, in Case 3, some members of the jury are biased toward conviction and some are biased toward acquittal.
Fig. 2. Juror biases and trial outcomes. 232
W.S. Neilson, H. Winter International Review of Law and Economics 20 2000 223–250
The verdict is guilty if s s
I
1 y so that both types believe the defendant is guilty beyond a reasonable doubt. The verdict is not guilty if s , s
I
2 x, so that neither type believes the defendant is guilty beyond a reasonable doubt. The jury is hung if s
I
2 x s , s
I
1 y, so that the jurors who are biased toward conviction judge the defendant to be guilty beyond a reasonable doubt, while those who are biased toward acquittal do
not.
30
Note that the only way to get a hung jury in this framework is if the final jury contains members with both types of biases and if the evidence is sufficiently close to
the reasonable-doubt standard threshold s
I. 31
One more parameter is needed in order to compute the probabilities of a wrongful acquittal, wrongful conviction, and hung jury: the proportion of the population exhib-
iting the identifiable characteristic. Let p denote the probability that an individual drawn randomly from the population exhibits the characteristic associated with bias toward
conviction. It is now straightforward to compute the probabilities of a wrongful acquittal P
WA
, a wrongful conviction P
WC
, and a hung jury P
HJ
. Formal constructions are contained in the Appendix. All three probabilities of interest can be affected by a judge’s
actions. A judge has discretion over the size of the jury pool, N, by choosing how many peremptory challenges each side is allowed. In a symmetric strikes case, the judge sets
n
d
5 n
p
. In an asymmetric strikes case, the judge sets n
d
. n
p
. To summarize, the model takes as primitives the biases of the population and the
conditional distributions of the strength of the evidence against the defendant. Jurors are biased in their perceptions of the strength of the evidence, so that jurors who are biased
toward conviction vote to convict a defendant on weaker evidence than jurors who are biased toward acquittal. The initial jury is drawn randomly from the population at large, and the
parameter p measures the fraction of the population that is biased toward conviction. Lawyers use peremptory challenges to remove jurors biased against their case, making it
possible to calculate the three probabilities of interest: P
WA
, the probability of wrongful acquittal; P
WC
, the probability of wrongful conviction; and P
HJ
, the probability of a hung jury.
32
30
In the Appendix, we briefly examine a more general setting in which there are more than two juror types.
31
One referee correctly points out that a key reason for there being a relatively small number of hung juries is because of a subtle process of accommodation among jurors. Accommodation in our model only occurs in a
trivial sense when the evidence is strong enough for both types of jurors to convict, or when the evidence is weak enough for both types of jurors to acquit. We have no mechanism that allows for one type of juror to change the
bias of another type of juror.
32
Schwartz and Schwartz, supra note 6 also formally model the peremptory challenge process. Although there are several similarities between our model and theirs, including the use of a variable to measure population bias
and a concern with hung juries, the models have several important differences. Specifically, they do not consider wrongful acquittals or wrongful convictions; they do not explicitly consider the guilt or innocence of the
defendant, but instead focus on how jurors differ in their view of how the defendant should be punished; they do not use the size of the jury pool as a choice variable in their model; and, they consider only one composition of
the population from which the jury pool is drawn. Their main focus is on the ability of peremptory challenges to reduce the probability of a hung jury, and they conclude that this probability can best be minimized by moving
to a system of majority as opposed to unanimous verdicts. 233
W.S. Neilson, H. Winter International Review of Law and Economics 20 2000 223–250
4. A numerical example
