Conput. & Ops Ret, Vol. 2, pp. 109-114. Pergmon Press. 1975. Printed in Great Britain

MARKOV CHAIN THEORY AND JURY DELIBERATION PROCESS

SUBHASH C. NARULA* and VASANT KATDAREt

Scope and purpose-In a recent decision, the Supreme Court ruled that the right to a trial by jury does not imply that the jury have twelve members[l2]. To study the effect of the jury size (12 vs 6) on the justice system, it is important to understand the jury deliberation process. That is, after listening to both sides of a case, how does the jury come to a final verdict of “guilty” or “not guilty.” This paper attempts to model the jury deliberation process and presents results showing one effect of jury size.

Abstract-The paper develops a Markov chain model for the jury deliberation process. Appearance of guilt of the defendant, capability of a juror to present a convincing argument and his/her receptiveness to the views of others are considered inputs to the model. For the simple model developed, the paper also presents results showing the effect of jury size (12 vs 6) on the jury deliberation process and the time required for a final verdict. The scope of the analysis and our ability to compare the model results with the real processes are severely limited by lack of data on the details of actual jury processes.

1. INTRODUCTION The jury is an important component of our criminal justice system and efforts have been made to make the jury system more efficient (i.e., reduce time, cost, and inconvenience) by employing modern management techniques to control the inventory of waiting jurors [6]. Also in response to criticism regarding the efficiency of the jury system, a number of jurisdictions have decreased the size of the jury. But it has been shown that a reduction in jury size does not necessarily provide the anticipated savings unless steps are also taken to reduce the size of the juror panels drawn from the pool[6].

Arguments for the reduction in the jury size are generally qualitative and are based to a large degree on legal precedent or the lack of an appropriate precedent. In a recent decision, the Supreme Court ruled that the right to a trial by jury does not imply that the jury have twelve members [12]. The majority cited six “experiments” which allegedly demonstrated the non-existence of any discernable differences between the results reached by the juries of two different sizes. Zeisel[13] argues that the available data were misinterpreted by the majority and his article provides an indication of the quality of the available empirical evidence regarding the effects of jury size on the resulting decision. The stated purpose of his article is: “to make it clear that the changes imposed on our jury system are more serious than we are led to believe.” Furthermore, as Saari [S] points out, with the current trends we can expect more radical and more widespread changes to occur in the jury system.

The motivation for the research is: if the changes in jury size are to be made then efforts should be made to provide better quantitative arguments and empirical evidence for assessing the magnitude of the effects for guiding future public policy regarding the nature of the jury. Hence, we are focusing on modeling the jury deliberation process as a first step in resolving the issue of jury size.

2. BACKGROUND

A notable study of jury performance was conducted at the University of Chicago in the mid 1950’s. This pioneering effort at gaining an insight into the factors which effect jury decisions attempted to investigate the process along three different avenues. The first involved covert observations of real jury deliberations. This approach was abandoned due to the negative public

*Subhash C. Narula is assistant professor of Industrial Engineering at the State University of New York at Buffalo. He holds the B.E. in mechanical engineering from the University of Delhi, and the MS. and Ph.D. in industrial engineering from the University of Iowa. His primary interest is in applied statistics and Operations Research, and he has published articles in Applied Statistics, Technomekcs, Journal of Quality Technology, Canadian Journal of Statistics, and the htemalional Statistical Review.

Wasant Katdare is engineering stat? supervisor at AT &T Longlines, Sommerset New Jersey. He holds the B.S. in electrical engineering from Poona Engineering India, the M.S. in Industrial engineering from the University of Massachusetts, and the Ph.D. in industrial engineering from the State University of New York at Buffalo. His experience includes statistical and OR analysis of both hospital and court operations.

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110 S. C. NARULA and V. KATDARE

opinion[4]. A second approach involved an extensive structured survey of judges. The responses of the presiding judge with respect to questions regarding a particular case were compared with the jury’s decision. A third approach involved experiments with mock juries [9-l 11. While these efforts provide a wealth of information regarding the inner workings of a jury from a sociological viewpoint, much more needs to be done to provide definitive answers to the question of the impact of jury size.

It should also be noted that an extensive literature exists in the field of small group dynamics. The experiments and analysis conducted in the field for different types of groups provide some insight into factors which may be significant in studying the question of jury size.

Some attempts have been made to develop probabilistic models of group interactions. Davies [I] describes a unifying theory for group decision schemes and uses juries as an illustrative example. In this example he postulates that after hearing the case and prior to deliberations, an individual juror would select one of two alternatives, namely “guilty” and “not guilty”. The number of individuals on a particular jury of size J who select the “guilty” and “not guilty” alternatives are denoted by Jo and JNG( = J - JG), respectively. Davies constructed five gross models to represent the deliberation process. All the models are based on the initial vote only and ignore all other details of the process. These models map the initial dist~bution of preferences (JG,JNG) into the probabilities of a guilty verdict, a not guilty verdict, and a hung jury. For example, his Model 3 assumes that initial votes of (12,O) and (11,l) would ultimately lead to a guilty verdict with a probability of one; initial votes of (10,2) (9,3), . . ., (6,6) would lead to a guilty verdict with a probability J&J and not guilty and hung with a probability of (I - J&)/2 each; initial votes of (5,7), . . ., (2,lO) yield not guilty with probability J&J and guilty and hung with equal probabilities of (I- JNo/J)j2; and initial votes of (1,ll) and (0,12) would lead to a not guilty verdict with probability one. This particular model, although it ignores all the details of the deliberation process, is in good agreement with the mock jury data presented by Simon[9].

Two additional mathematical treatments of the jury process appear in the literature. Both treatments are presented for pedogo~cal purposes, i.e. to provide real and interesting examples for illustrating Type I and Type II errors. Feinberg[2] presents qualitative arguments as to how changes in the jury system might effect Type I and Type II errors in the outcome of the trial. Friedman[3] attempts to go further by providing quantitative examples. He offers an interesting illustration of Type I and Type II errors for jury size (i.e. 6 and 12) and for conviction by unanimous or less than unanimous vote.

All models ignore salient features of the deliberation process whichhave been observed in studies of mock jury and other small groups. That is, the process by which the individual jurors either switch or maintain their initial vote.

3. DELIBERATION PROCESS

The important features of the jury deliberation process can be summarized as follows: The jurors arrive at their initial vote, after hearing the case and prior to deliberation,

independently. The initial vote depends upon a parameter, “appearance of guilt,” which may be considered equivalent to “the probability that an individual juror would consider the defendant guilty.” This parameter depends upon (a) evidence offered by both sides, (b) summation by the opposing attorneys, and (c) final instructions of the judge.

At any stage of the deliberation process, each juror has only two choices available, namely, “guilty” or “not guilty.” Indecision on the part of a juror is not allowed. If the verdict is not reached on the first ballot, the jury deliberation process starts.

During the deliberation process, a juror may change his/her position. Conceptually, there appear to be two forces which may cause a juror to change his/her vote. First, a juror may switch position because another juror put forth a substantive argumeht which convinces the juror to accept the opposite view. For example, Mr. Smith niay have misunderstood a witness or a point of law in the judge’s instructions and when a fellow juror puts forth some argument, Mr. Smith realizes his initial misunderstand~g and changes his position. The second force is one which compels some jurors to change their vote in order to avoid being a minority or to avoid a hung jury. There is a’distfnction between the switch in votes due to substantive arguments and the acquiescent switch in votes for individual convenience. This dichotomy may be an

oversimplification but it is useful for discussion purposes.

Markov chain theory and jury deliberation process 111

Whenever the jury is unanimous, the verdict is reached and the deliberation process stops. Otherwise it continues. For example, for a jury of size 6, if the vote on the first ballot is 0 or 6 guilty votes, the verdict has been reached. If the vote is 1, 2, 3, 4, or 5 guilty votes, the jury deliberation process starts. At the end of the first round the number of guilty votes can be anything from 0 to 6. If it is not 0 or 6, the second deliberation round starts and so on. For a jury of size 6, Fig. 1 represents all the possible states in which the jury may be at the beginning and the end of a given round.

Fig. 1. All possible states and transitions during deliberation for jury size 6.

In the next section, we present a simple Markov chain model of the jury deliberation process. This model is based on our conversations with a judge and a number of persons who have served on juries. It should be noted that the model assumes an individual voter will not switch vote to avoid being a minority or to avoid a hung jury. In this sense it is incomplete.

4. MODEL

Let p denote the initial probability that an individual juror will consider the defendant guilty. Further, let 13 and y denote the probabilities that an individual juror will present a convincing argument in favor of his/her position and that he/she is receptive to the arguments of the others, respectively. We assume that all jurors are identical, i.e. p, 0, and cp are the same for all jurors.

Since each juror may either vote “guilty” or “not guilty” and if the state of the system is defined as the number of “guilty” votes cast, for a jury of size J, the system can be in one of the (J + 1) possible states.

Since p is assumed same for each juror and the jurors vote independently, the initial state probability vector can be determined as follows:

P {X=x}= ; 0 p"(1 -pp x=01 J. 5 ,***,

If x is not equal to 0 or J, the deliberation process starts. For modeling purposes, it is reasonable to divide the deliberation process into stages or rounds. In every round, a juror has an opportunity to present his/her arguments only once. As a result of arguments offered during a given round by other jurors, each juror is allowed to switch his/her position. However, this switch may take place only at the end of a given round, when a poll is taken to determine the number of votes in favor of conviction and acquittal. We assume that the probability that a given juror will maintain his/her position at the end of a given round is proportional to the ratio of the number of arguments in favor of the juror’s position to the total number of arguments offered.

Let G(m; x,8) denote the probability that m jurors out of x who voted “guilty” at the beginning of the round will produce a convincing argument in their favor is

G(m;x,e)=(~)e”(l-e)‘-m, m=o,i ,..., X.

Similarly, NG(r ; J -x,0), the probability that r jurors out of J - x jurors who voted “not guilty” will produce a convincing argument in their favor is

v(i-e)J-x-r r=o,i ,..., J--X.

112 S. C. NARULA and V. KATDARE

Expressions for S(i,jlx), the probability that a juror will switch his/her vote from i to j (i, j = G for “guilty” and N for “not guilty”) at the end of the round given that the system was in state x at the beginning of the round, are given by:

I--x x--l

S(G,Glx) = (I- cp) + cp (1 - 0)‘-, + x 2 NW; J -x, e)G(m ; x - 1, B)&) r=, m-1

S(G,Nlx) = 1 - S(G,Glx)

S(N,N~x)=(l-~)+p[(l-8)“+J~~~‘~~,NG(r;J-x-l,~)G(m;x,~)~}

S(N,Glx) = 1 - S(N,Nlx).

Let f(y Ix) denote the probability of transition from state x to state y at the end of a round of deliberation, where x and y specify the number of “guilty” votes cast at the beginning and end of the deliberation round, respectively. Then

f(~ Ix)= ,$ (;) CW,NIX))“-~ {S(G,Glx)}* - I

where x = O,l,. . .,J; y = O,l,. . .,.I

nrl = max [0,x + y -J], and mz = x ifyrx y ifycx’

The transition matrix p can be written as

However,

and

Furthermore,

r f(01~) f(l~o) . . . f(Jlo) f(W) f(ll1) * . * f(m)

p= * . . .

I : . . .

f(OlJ) f(l(J) : : * f(W)

f(Ol0) =f(JlJ) = 1,

f(ll0) = f(2)O) = * . * =f(JlO)=f(l(J)=f(2lJ)=f(J-lIJ)=O.

f(y]x)=f(J-y]J-x),forQ<y<JandOIxsJ.

Since (0) and (J) of P are absorbing states and transition probabilities f(OJx), f(Jlx), for 0 <x <J are strictly positive, P will converge with probability one. Given the initial state probability vector and the transition matrix, the limiting terminal probabilities can be derived by the procedure given in[5].

5. RESULTS Although the model developed in the previous section was solved for various values of 8 from

0.1 to 0.9 in steps of 0.1 and cp from O-1 to O-9 in steps of 0.1, only selected results are given here. It appears that the probability of conviction (acquittal) at the first ballot depends upon the jury

size and the probability of the appearance of guilt (see Table 1).

Markov chain theory and jury deliberation process

Table 1. Probability of conviction (acquittal) on first ballot p = probability of appearance of guilt, J = jury size

113

12 .oooo .oooo .oooo . 0000 .0002 (.2824) (.0687) (.0138) (.0022) (.0002)

6 .oooo .OOOl .0007 .0041 .0156 (.5314) (.2621) (.1176) (.0466) (.0156)

i

If the verdict is not reached on the first ballot, the jury deliberation process starts. If the jury deliberation process is allowed to continue till the verdict is reached, then the probability of conviction seems to be equal to p, the probability of appearance of guilt, irrespective of the jury size and the values of B or cp (# 0).

Table 2 gives the mean number of rounds needed to reach a verdict (acquittal or conviction). It appears to be a function of jury size.

Table 2. Mean number of rounds for jury size 12 (jury size 6) and cp, the probability of willingness to listen = 0.5, p = probability of appearance of guilt, 6 = probability that an individual juror will present a

convincing argument

.l 9.79 16.30 (5.55) (9.61)

.2 7.50 12.47 (3.00) (5.01)

I I

20.51 22.89 23.66 (12.39) (14.01) (14.54)

15.68 17.49 18.08 (6.44) (7.27) (7.54)

1 I

Table 2 gives values for rp = 0.5, 0 = 0.1 and 0.2. For fl? 0.2, the mean number of rounds differ only in the second decimal place for each jury size. Further, the mean number of rounds decreases as the value of cp increases. Also, the mean number of rounds needed to reach a verdict increases as the probability of appearance of guilt gets closer to 0.5.

Figures 2(a, b) show the probability of reaching a verdict (conviction or acquittal) as a function of number of deliberation rounds for p = O-1 and p = O-5 respectively (0 = O-5 and cp 0.5).

From Fig. 2(a, b), it seems that the probability of reaching a verdict on a given round depends upon the jury size. For 0 z 0.2, the curves do not change. For 0 = 0.1, the shape of the curve is the same except that the probabilities of reaching a verdict are slightly lower than for 0 2 0.2. The value of cp effects the probability of reaching a verdict without changing the basic shape of the curve. For a given round, as the value of cp increases, the probability of reaching a verdict seems to decrease.

6. DISCUSSION

Because of our inability to get data on p, 0 and cp for real jury trials, it has not been possible to validate our model. It appears that traditionally such data are not kept and at present it seems difficult, if not impossible, to get such data.

The present model is based on some simplifying assumptions. Therefore, before drawing any conclusions with respect to the effect of jury size on the justice system, it is desirable to develop a model which takes into consideration the individual differences among jurors in terms of p, 0 and rp. Also, the present model assumes that a juror may switch his/her vote as often as desired-not a realistic assumption. Further, it does not take into account the fact that an individual juror may switch vote to avoid being a minority or to avoid a hung jury. A simulation model which takes these points into consideration is presently under investigation.

114 S. C. NARULA and V. KATDARE

I.” - . .

.9 -

.8 -

I-

.5 -

6-

.1 -

.?-

2-

1,

o-

Fig. 2. (a) Probability of appearance of guilt = 0.1, (b) Probability of appearance of guilt = 0.5. Probability of reaching a verdict vs number of deliberation rounds.

Acknowledgements-The authors are thankful to Messrs. Eusanio and Mogavero for useful discussions and to Suhas Apte for programming assistance.

Special thanks are due to Dr. Sol Kaufman for constructive criticism and suggestions which have improved the presentation of the paper.

REFERENCES

1. J. H. Davies, Group decisions and social interaction: a theory of social decision schemes, Psycho/. Rev. 80,97-125 (1973). 2. W. E. Feinberg, Teaching the Type I and Type II errors: the judicial process, The American Statistician 25,30-32 (1971). 3. H. Friedman, Trial by jury: criteria for convictions, jury size, and Type I and Type II errors, The American Statistician,

26, 21-23 (1972). 4. H. Kalven, Jr., H. Zeisel, The American Jury. The University of Chicago Press, Chicago (1966). 5. J. 0. Kemeny and J. L. Snell, Finite Markou Chains. Van Nostrand, Princeton (1960). 6. W. R. Pabst, Jr., Juror waiting time reduction, The National Technical Information Service, No. PB-20142, (June 1971). 7. W. R. Pabst, Jr., What do six-member juries really save, Judicature 57, 6-1 I (1973). 8. D. J. Saari, The criminal jury faces future shock, Judicature 57, 12-17 (1973). 9. R. J. Simon, The Jury and the Defense of Insanity, Little & Brown, Boston (1967).

IO. F. L. Strodtbeck and R. D. Mann, Sex role differentiation in jury deliberation, Sociomelry 19, 3-11 (1956). Il. F. L. Strodtbeck, R. M. James, and C. Hawkins, Social status in jury deliberation, Readings in Social Psychology (Eds.

E. E. Maccoby et al.), Henry Holt, New York (1958). 12. Williams us Florida. 399 U.S. 78 (1970). 13. H. Zeisel, The warning of the American jury, Am. Bar Ass. J. 58, 367-370 (1972).

(Paper received 20 January 1975)