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Department of Statistics Dietrich College of Humanities and Social Sciences

11-1977

Optimal Peremptory Challenges in Trials by Juries:A Bilateral Sequential ProcessArthur Roth

Joseph B. KadaneCarnegie Mellon University, kadane@stat.cmu.edu

Morris H. DeGroot

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Published InOperations Research, 25, 6, 901-919.

OPERATIONS RESEARCH, Vol. 25, No.6, November-December 1977

Optimal Peremptory Challenges in Trials by Juries:A Bilateral Sequential Process

ARTHUR ROTH, JOSEPH B. KADANE, and MORRIS H. DEGROOT

(Received September 1976; accepted February 1977)

This paper explores optimal strategies for the use of peremptorychallenges in iury trials where the prospective iurors are examinedand then either challenged' or seated, one by one. We assume thatthe lawyers for each side do not necessarily agree about the proba­bility that each prospective iuror will vote for conviction, but that theassessment of each side is available to the other. The strategies wedevelop are optimal noncooperative sequential strategies in the sensethat each side maximizes its expected utility at each stage under theassumption that both sides will continue to use these optimal strategiesin all future decisions. Under certain regularity conditions we show thatit is optimal to be the first side to decide whether to challenge anyprospective iuror. Necessary and sufficient conditions are given for anoptimal strategy to be reversible, which means that it does not matterto either side whether it decides first or second. Specifically, the op­timal strategy is reversible if the two sides always agree about theprobability that each prospective iuror will vote for conviction. Wegive an algorithm based on backward induction for finding the op­timal strategies and discuss simple examples.

THE APPLICATION of relatively sophisticated methods such as pollsand market research surveys to the choice of which potential jurors

to challenge peremptorily has raised fears for the future of the jury sys­tem as we now know it. Some of the cases in which these methods havebeen used include the Harrisburg Seven trial (Schulman et al. [10]); theCamden, New Jersey, draft board raid trial; the Mitchell-Stans conspiracytrial (Arnold [1], Zeisel [12], and Zeisel and Diamond [13]); the Gainesville,Florida, vete:rans trial; the Wounded Knee, South Dakota, trials of mili­tant Indians and the Cedar Rapids murder trial arising from the WoundedKnee disturbance; the Buffalo Creek, West Virginia, dam disaster civildamage suit; the Ellsburg-Russo trial; the Joan Little murder trial; andthe Attica trials (Shapley [11]). All of these trials involved highly pub­licized cases of defendants who had taken political positions likely to bevery popular with some and very unpopular with others. Furthermore,the nature of the evidence in at least some of these trials was such as toconfirm the prejudices of the jurors.

901

902 Roth, Kadane, and DeGroot

To date these methods have been used more extensively by the defensethan by the prosecution (Kairys [9] and Ginger [8]). It can be arguedthat this use is close to the intent of the jury system, to protect a defendantunpopular with his government by having a group more politically diversethan the government decide his innocence or guilt.

The fear for the jury system arises from the possibility that, now thatthe defense has blazed the trail, an overzealous prosecution, with the fullfinancial resources of the government, may follow. If this occurs, one mightforesee "hanging juries" carefully chosen by sociological methods to havethe most negative view of the defendant, and the defense, except in rareinstances such as those discussed above, unable to match the resourcesof the government. "District Attorneys or U.S. Attorneys cannot be ex­pected to stand by doing nothing while defendants in the most seriouscases buy themselves a significant edge in trial after trial. The championsof the technique will have to realize that the days when it could be reservedfor their favorite defendants will soon be over" (Etzioni [7]). Conceivably,this response by the government could cause a threat to our civil liberties.

To examine whether this possible threat is to be taken seriously, oneshould first ask what the defense and prosecution would do with informa­tion from relevant polls and surveys if they had it. In this paper we pre­sent a simplified model of the jury selection process and explore some ofits implications. One of our difficulties in undertaking this work is that,while the law of most states is clear about the number of challenges al­lowed to the defense and prosecution in varying circumstances, the pro­cedure for exercising these challenges is typically left to the trial judge.Usually the judge first examines prospective jurors to be sure that theyare qualified and asks questions that might result in dismissal for cause,questions that vary depending on the nature of the trial. In our modeleach side then has the opportunity to peremptorily challenge the nextprospective juror, and, if neither side challenges, the juror is then swornin. The question of which side is to challenge first is left arbitrary in ourmathematics, although in our model it cannot depend on previous uses ofthe peremptory challenge by either side. Furthermore, we assume that theprosecution and the defense each have an opinion about the likelihoodthat the prospective juror under consideration would vote for convic­tion, and that these opinions are known to each other.

This structure leads to a bilateral sequential process, in which decisionsare made by each side one by one, without a simultaneous decision bythe other side. Bilateral sequential processes may be a better model formany social phenomena, such as arms races and duopoly (Cyert andDeGroot [4, 5]), than traditional game theory that requires simultaneousmoves by the players.

Both the information available to each side and the particular se-

Peremptory Challenges in Jury Trials 903

quence we have chosen to study limit the applicability of this paper, andboth assumptions need to be relaxed in further work. Nevertheless, theparticular structure we have chosen, although somewhat oversimplified,does represent a starting place for examining how effective these newmethods are likely to be.

After this paper was submitted for publication, we learned of the earlierrelated work of Brams and Davis [2, 3]. The relationship between theirwork and ours will be discussed at the end of Section 1 and immediatelyafter the corollary to Theorem 2.

1. STATEMENT OF THE PROBLEM AND MAIN RESULTS

Prosecution and defense lawyers are about to select a jury of J people.Each prospective juror is interviewed, and each lawyer must then decidewhether to accept or challenge (Le., reject) that person before interviewingthe next prospective juror; this decision cannot later be changed. The prose­cution is allowed at most A challenges, while the defense has at most Bchallenges. After questioning each prospective juror i, the two sides havepossibly different opionions about the probability that he will vote to con­vict the defendant, giving rise to a vector (Pu , P2i) of the opinions of theprosecution and the defense, respectively. These probabilities will be basedon all of the information available to the legal teams on each side, includingsurveys of community attitudes, investigation of the pool of prospectivejurors, or experience with similar trials analyzed by age, occupation, sex,race, etc. Thus, as each prospective juror is questioned, the legal teamson each side will identify a profile of the person comprising various demo­graphic and psychological characteristics. From this profile, each side willthen use the information just described to assign the probabilities Pliand P2i to that person. For example, if a poll of the community carried outby the defense revealed that older women in the community tended tobe more sympathetic than others to the defendant, the defense lawyerswould assign a relatively low value of P2i to an older WOlnan.

We shall assume that for each possible profile that might be identifiedfor a given prospective juror, each side knows the probability that theother side will assign to that person as well as the probability that its ownside assigns. This would be the case, for example, if the lawyers for eachside were familiar with the jury selection tactics of the other side. Fur­thermore, we shall assume that both sides are sufficiently familiar with thepool of prospective jurors that they know the distribution of profiles inthat pool. Hence both sides will know the joint distribution of the bivariaterandom variable (PI, P2 ) in the pool of prospective jurors after dismissalsfor cause. This known joint c.dJ. will be noted F(PI, P2).

Thus, we assume that the observed vectors (PIi, P2i) are independentand identically distributed, each with c.dJ. F(PI, P2), and are observed

904 Roth, Kadane, and DeGroot

one at a time; i.e., they form a sequential random sample. It is implicitin this process that neither side learns about or changes the c.dJ. F(PI , P2)as prospective jur'ors are questioned. The rule that specifies at each stagewhich side must declare first whether it wishes to challenge that juror isassumed fixed at the outset and does not depend on the previous decisionsof the participants.

From the point of view of the prosecution at any stage in the selectionprocess, the outcome of the entire process will be a random vector(PI1 , ••• , P1J ) of the PH values of the members of the final jury. Weassume that there is no interaction between jurors, so that the overall(random) probability of conviction in the opinion of the prosecution ispO) = II~=l PH, where the product is taken over the J people on thefinal jury. (This assumption is probably valid only on the first post-trialballot taken by the jury prior to any discussion.) Similarly, we definethe overall probability of conviction in the opinion of the defense to bep(2) = II~=l P2i •

Let Cl denote the utility to the prosecution if the jury votes for convic­tion and let nl denote the utility to the prosecution if the jury does notvote for conviction. Then the expected utility of the jury selection processto the prosecution is

E[cIp(l)+nl(l-p(l)]= (cl-nl)E[p(l)J+nl, (1.1)

where the expectation is taken over the distribution of possible values ofp(l). Since the prosecution prefers that the jury vote for conviction, itfollows that Cl>nl. Hence, by (1.1), the prosecution will maximize itsexpected utility by using a strategy for selecting jurors that will maximizethe value of E[p(l)].

Similarly, let C2 denote the utility to the defense if the jury votes forconviction and let n2 denote the utility if the jury does not vote for con­viction. Then the expected utility to the defense is

E[C2P (2) +n2(1-p(2»] = (C2- n2)E[p(2)]+n2. (1.2)

Since the defense prefers that the jury not vote for conviction, then c2<n2.Hence, by (1.2), the defense will maximize its expected utility by using astrategy for selecting jurors that will minimize the value of E[p(2)].

We shall show that optimal strategies exist for both sides. We defineour problem to be reversible for our particular values of A, B, and J if,under the optimal strategies, it will never matter at any stage whichside is required to decide first whether or not to use a challenge. Theproblem is called universally reversible if it is reversible for all possible valuesof A, B, and J. Both of these concepts depend on the joint c.dJ. F(Pl, P2).At any stage of the selection process, after some number of prospectivejurors have been acted upon and either mutually accepted or challenged

Peremptory Challenges in Jury Trials 905

by one side or the other, the optimal strategy for the prosecution will beto attempt to maximize the product of the values of Pli for the jurors thatremain to be selected and the optimal strategy for the defense will be toattempt to minimize the product of the values of P2i for those same jurors.Thus, the problem is effectively beginning again with new values for A, B,and J. For any integers a~A, b~B, and j~J, we say that a, b, and jare reachable if there is positive probability when both sides use optimalstrategies that a, b, and j are ever these new values. It follows from itsdefinition that reversibility for A, B, J implies reversibility for any reach­able values of a, b, j.

Our main results can be described briefly as follows. In the next threesections we develop our notation, describe an important regularity as­sumption, and derive some of the basic relations that characterize theoptimal procedure. Then in Section 5 we show that each side can alwaysdo at least as well by making the first decision regarding any juror as itcan when the opposition decides first. In Theorem 1 we give necessary andsufficient conditions under which the problem is reversible; it is univer­sally reversible if and only if these conditions hold for all A, B, J. Inparticular, in Theorem 2 and its corollary, universal reversibility is shownto hold whenever both sides always agree on the p-values of prospectivejurors. In Section 6 we specify an algorithm for finding the optimal strate­gies for both sides. In Section 7 we give examples of problems that are notuniversally reversible. Finally, in Section 8 we present some simple nu­merical examples involving uniform distributions.

Brams and Davis [2] consider a similar problem but restrict themselvesto the case in which both sides always agree on the p-values of prospectivejurors. They present some interesting numerical results for this case andconsider other models in which peremptory challenges can be exercisedaft.er groups of prospective jurors have been questioned.

2. DEFINITION AND PROPERTIES OF THE OPTIMAL PROCEDURE

Before investigating reversibility or finding the form of the optimalprocedure, we must define this procedure and describe the sense in whichit iR optimal. We observe that the jury must be selected after at mostA+B+J people have been interviewed. Thus, the number of decisions inthe selection process is bounded. Clearly, the lawyer who makes the lastpossible decision, when one juror remains to be selected and this lawyerhas one challenge remaining while his opponent has none, has an optimalchoice. Under the assumption that this last possible choice will be madeoptimally, the consequences of the next-to-Iast possible decision areknown. Hence it can also be made optimally. Proceeding by backwardinduction, each decision can be made optimally if the side making thatdecision is willing to assume that both sides will act optimally on all sub-

906 Roth, Kadane, and DeGroot

sequent decisions. The optimal procedure is taken to be the one resultingfrom all these optimal choices by both sides. Since this procedure com­pletely defines the actions of both sides, it determines a pair of values(EP(l) , EP(2»), which represents the best the prosecution and the de­fense, respectively, can expect to do under the assumption that the otherside will proceed optimally according to its own opinions about the prospec­tive jurors.

It is important to keep in mind that the procedure just described isoptimal only in the sense of the assumptions that we have made. It isimplicit in these assumptions that the two sides do not cooperate or colludein any way. If pel) and p(2) are not necessarily equal, then the jury selec-tion process is a non-zero sum, two-person game, and it is possible thatboth sides could increase their expected utilities by cooperating in theiruse of challenges. However, such cooperation or collusion is presumablynot permitted in our jury system, and we shall not consider it further inthis paper.

3. NOTATION AND A BASIC REGULARITY ASSUMPTION

Let w= (WI, ... ,WA+B+J) be a vector such that wi=I or 2 for i=1, . . . , A +B +J; tV i = 1 means that the prosecution has to decide firstabout the ith prospective juror to be questioned, while Wi = 2 means thatthe defense must decide first. For any vector y with at least two elements,let 0(y) denote the vector that is obtained by deleting the first element ofy. Suppose now that at some stage the prosecution has a challenges re­maining, the defense has b challenges remaining, there are j jurors stillto be selected, and the vector (Pli, P2i) associated with the present can­didate is (PI, P2). In this situation, for j = 1 or 2, let p(j') be the productof the pi/S yet to be added to the jury, including the present candidate ifhe is accepted, and then let E[p(l/}] and E[p(2

/)] denote the expected

values of these quantities under the optimal procedure described in Sec­tion 2.

Let v= (VI, ... , Va+b+j) be the vector consisting of the last a+b+jelements of w, so that v specifies who decides first for each remaining po­tential juror. We write M*(a, b, j, PI, P2, v) =E[p(l/)] and M *(a, b, j,PI, P2, v) =E[p(2

f

)] to show the explicit dependence of these quantities onthe relevant parameters. Next, let J:l*(a, b, j, v) =E[M*(a, b, j, PI, P2, v)],where the joint distribution of (Pt, P2) over the unit square has the c.dJ.F, and let J:l*(a, b, j, v) be defined analogously. The quantities J:l*(a, b,i, v) and JL*(a, b,i, v) represent the expected products of the Pli'S and theP2i'S for the remainder of the process prior to the interviewing of thepresent candidate. Whenever a, b, i, and v are not ambiguous, we shallconserve space by letting a = [a-I, b, i, ¢(v)], (j [a, b-l, i, ¢(v)], and'Y=[a, b, i-I, cf>(v)]. Thus, a, {3, and'Y each represent possible stagesthat might be reached during the jury selection process.

Peremptory Challenges in Jury Trials 907

We can think of the stage a being transformed into the stage (j by thedefense giving one of its available challenges to the prosecution for itsown use. In this way the number of challenges available to the prosecu­tion increases from a-I to a, while the number available to the defensedecreases from b to b-l. It is, therefore, reasonable to suppose that forany fixed history of the process, the prosecution finds it at least as pref­erable to be at the stage (j as at the stage a. In symbols, this relation canbe expressed by the inequality

~*(a) ~p,*(ft). (3.1)

For the same reason, it is reasonable to suppose that for any fixed his­tory of the process, the defense finds it at least as preferable to be at thestage a as at the stage (j, i.e., that

(3.2)

Surprisingly, we have not been able to prove that the inequalities(3.1) and (3.2) hold in full generality for every c.d.f. F(Pl,P2) and allvalues of a, b, j, and v. Nor, on the other hand, have we succeeded inconstructing a counterexample in which at least one of these inequalitiesdoes not hold.

We can show, however, that (3.1) and (3.2) must always hold in anyproblem in which both sides always agree on the p-values of prospectivejurors. In a problem of this type, P<l) and p(2) must be equal at eachstage and the process is actually a zero-sum two-person game. The argu­ment that establishes (3.1) and (3.2) in this case is somewhat roundaboutand not directly related to the other developments in this paper, and isomitted.

If one of the inequalities (3.1) or (3.2) does not hold at some stage ofthe selection process, it means that the lawyers for one side would actuallyprefer to give one of their available challenges to the other side. It doesnot seem reasonable that such a situation could arise in any practicalproblem of jury selection. Hence, we shall herewith assume that in everyproblem considered in this paper, (3.1) and (3.2) are satisfied for all stagesthat might arise during the jury selection process.

4. THE FORM OF THE OPTIMAL STRATEGY WHEN BOTH SIDES HAVE AT

LEAST ONE CHALLENGE REMAINING

Case 1: Prosecution Makes the First Decision on the Next Candidate

When the prosecution makes the first decision on the next candidate,Vl = 1; Le., v is of the form v = 10(v). By considering the consequences ofthe two possible decisions, first for the prosecution and then, if the prose­cution accepts the juror, for the defense, we can write (for a~l, b~l,

j~1)

908 Roth, Kadane, and DeGroot

(4.1)

M*[a, b,j, PI, P2, 10(v)]

= {max [JL:(a), p~*('Y)] if P2JL*( 'Y) <JL*({1);max [JL (a), JL (11)] if P2JL* ('Y ) > JL*(11 ) .

It follows from (3.1) that (4.1) can be rewritten (for a~1, b~l, j~1) as

M*[a, b,j, PI, P2, 10(v)]

(JL*(a) if PI < [JL*(a)/JL*('Y)] and P2<[JL*(I1)/JL*('Y)],= ~ PIJL*('Y) if PI> [JL*(a) / JL*('Y )] and P2 < [JL* (11) / JL* ('Y )], (4.2)

lJL*(11) if P2> [JL*(I1) / JL*( 'Y)].

TABLE I

OPTIMAL DECISIONS

FIRST DECISION FINAL DECISION(Opponent may challenge if you accept) (Opponent has already accepted)

'" '"",P2 Small Large ",P2 Small Large

PI "" PI '"

Defense: Defense: Defense: Defense:Accept Accept Accept* Challenge

Small SmallProsecution: Prosecution: Prosecution : Prosecution :Challenge Accept Challenge Challenge

Defense: Defense: Defense: Defense:Accept Challenge Accept Challenge

Large LargeProsecution: Prosecution : Prosecution : Prosecution:Accept Accept Accept Accept*

* Hypothetical case, opponent has already challenged.

The optimal decisions for both sides regarding the present juror can bededuced from (4.2) and are summarized in Table I. Using (4.2), wedefine PI to be large if PI> [JL*(a) / JL*('Y )] and small if PI < [JL*(a) / JL*('Y )].Similarly, we define P2 to be large if P2>[JL*(I1)/JL*('Y)] and small if P2<[JL* (11)/JL* ('Y )]. These definitions depend on a, b, j, and v. If the marginaldistributions of PI and P2 are both continuous, then PI and P2 are eacheither large or small with probability one and Table I completely de­scribes the form of the optimal decisions. For present purposes, we assumethat the marginal distributions are continuous, so that the case PI =J1.*(a) / JL*('Y) or P2 = JL* (11) / JL* ('Y ), which will be discussed at the end ofSection 5, need not be considered here.

By examining Table I and considering the process from the standpointof the defense, we can see that (for a~l, b~l,j~l)

(4.5)

Peremptory Challenges in Jury Trials 909

(JL* (13) if P2 is large;M*[a, b, j, pl, P2, 10(v)] = ~ JL*(a) if Pl is small and P2 is small; (4.3)

lp2JL*(1') if Pl is large and P2 is small.

Case II: Defense Makes the First Decision on the Next Candidate

When the defense decides first on the next candidate, Vl = 2; i.e., v is ofthe form v=20(v). By almost identical arguments to those used in Case Iwe can write (for a~l, b~l,j~l)

_ fmin [JL* (13), P2JL* (1')] if Pl is large;M*[a, b, j, pl, P2, 20(v)] - 'lmin [JL*(,8), JL*(a)] if Pl is small, (4.4)

(JL*(13) if Pl is large and P2 is large;=~ P2JL*(1') if Pl is large and P2 is small;

lJL* (a) if Pl is small,

since it follows from (3.2) that JL*(a) ~JL*({j). The optimal strategiesnow follow from (4.4), and we summarize these strategies for bothCase I and Case II in Table I. Note that the strategies of both sides donot depend on whether they are making the first or final decision exceptwhen Pl is small and P2 is large, i.e., when both sides find the same jurorundesirable. In that case, whoever decides first will accept the juror,forcing his opponent to be the one to use a challenge. Table I gives thecomplete form of the optimal strategy when a~l and b~l. It does not,however, tell us exactly what this strategy is because the categories "large"and "small" depend on a, b, j and v through the functions JL* and JL* (withvarious sets of arguments), which can be evaluated (see Section 6) onlythrough the backward induction algorithm.

It follows from Table I that (for a~l, b~l,j~l)

{

JL*(13) if Pl is large and P2 is large;M*[a, b, j, pl, P2, 20(v)] = PlJL*(1') if Pl is large and P2 is small;

JL*(a) if Pl is small.

5. THE ADVANTAGE OF MAKING THE FIRST DECISION AND A

CHARACTERIZATION OF REVERSIBILITY

We shall now compare the expected value M*[a, b, j, pl, P2, 10(v)]that the prosecution can attain if it makes the first decision about thenext prospective juror, with the expected value M*[a, b, j, Pl, P2, 20(v)]that it can attain if the defense goes first for that prospective juror, andthere is an arbitrary fixed rule 0(v) for decisions about all future jurors.

910 Roth, Kadane, and DeGroot

(5.1 )

From (4.2) and (4.5) we see that (fora~l, b~l,.i~l)

ll;f*[a, b, j, PI, P2, 10(v)]-M*[a, b, j, PI, P2, 20(v)]

= {J.L* ((3) - J.L(a) if PI small, P2 large;ootherwise.

Since the prosecution wishes to maximize M*, it follows from (3.1) thatthe prosecution is at least as well off making the first decision on the nextcandidate as it is going second.

Similarly, we can compare the expected value M*[a, b, j, PI, P2, 20(v)]that the defense can attain when it makes the first decision with the ex­pected value M *[a, b, j, PI, P2, 10(v)] that it can attain when the prose­cution makes the first decision. From (4.3) and (4.4) we see that (fora>l, b>l,j>l)

M*[a, b,j, PI, P2, 10(v)]-M*[a, b,j, PI, P2, 20(v)]

= {J.L*({1) - J.L* (a) if PI small, P2 large; (5.2)ootherwise.

Since the defense wishes to minimize M *, it follows from (3.2) that thedefense is also at least as well off making the first decision on the nextcandidate as it is going second.

Thus, both sides are at least as well off going first for the next prospec­tive juror as they are going second; there is no difference unless PI is smalland P2 is large. (In fact, we have seen from Table I that the strategiesare independent of order except in this case.) This argument can be ex­tended by induction to other elements of the vector v, and we can con­clude that it is always desirable to decide first for any prospective juror.Since reversibility must necessarily hold if either a=O or b=O (Le., onlyone side has any choices remaining), (5.1) and (5.2) suggest the followingresult.

THEOREM 1. The optimal strategy is reversible if and only if for any reach­able a, b, j, v either (i) the probability is zero that PI is small and P2 islarge, or (ii) the probability is zero that PI and P2 are either both smallor both large not only for the present values of a, b, j, v but also for any a, b,j, v that are reachable from these present values.

Proof. The theorem would follow immediately from (5.1), (5.2), (3.1),and (3.2) if we could show that condition (ii) is equivalent to

(ii') J.L*(a) = J.L*({1) and J.L*(a) = J.L*({1).

But (ii') means that either side could give the other side one of its chal­lenges without affecting its expected value. Since the J.L* and J.L*-functionsrepresent expectations over the entire future of the selection process,

Peremptory Challenges in Jury Trials 911

(ii') is equivalent to the condition that (with probability one) it is notpresently, and will never in the future be, the case that one side wants tochallenge a candidate that the other side wants to accept. But this isprecisely condition (ii ) .

Theorem 1, unfortunately, is a characterization of reversibility that isas hard to verify as the original condition itself. Hence, the strength ofthe theorem lies in its theoretical insight rather than in its practical use.In the usual problems where the defense and the prosecution have es­sentially opposite goals, JL*(ex) <JL*({3) for all a, b, j, v, and condition (ii)fails. If condition (ii) is ignored, then reversibility is equivalent to theproperty that there is zero probability that both sides will find the sameprospective juror unacceptable.

An important special case of Theorem 1 occurs when both sides alwaysagree in their assessments of jurors, i.e., PI = P2• A slightly more generalproblem is considered in the next theorem.

THEOREM 2. Suppose P 2 =kPI for some k>O, i.e., the joint distribution of(PI, P2 ) lies entirely on a line through the origin. Then universal reversibilityholds.

Proof. We can ignore the degenerate case where PI and P 2 are constantsand assume that P2 is a nondegenerate strictly increasing function of PI.Hence, condition (ii) of Theorem 1 cannot possibly be met, i.e., JL*(ex) <JL*({3) for all a, b, j, v. For any values of the arguments, M*(a, b, j, pl,P2, v) = E[p(2

f)] = kjE[P(lf)] = kjM*(a, b, j, PI, P2, v). Taking expectations

with respect to PI and P 2, we obtain JL*(a, b, j, v) =kjJL*(a, b, j, v). Nowsuppose that Pi is not large and P2 is not small for some a, b, j, v. (In thediscrete case this may be a weaker assumption than PI small, P2 large.)Then

(i) PI~ JL*(ex) / JL*('Y), i.e., P2 = kpi ~ kJL*(ex) / JL*('Y), and

(ii ) P2~ JL*((3 ) / JL*('Y ) = k jJL*({3 )/ kj-

1JL*('Y )

=kJL*({3) / JL*('Y ) > kJL*(ex) / JL*('Y ),

which is a contradiction. The result now follows from Theorem 1.

COROLLARY. If PI =P2, then universal reversibility holds.

This corollary is equivalent to Theorem 1 of Brams and Davis [2]. Ineffect, they assume that decisions are made simultaneously by the two sidesand show that they could just as well be made alternately; we assume thatdecisions are made alternately and show that they could just as ,veIl bemade simultaneously.

Theorem 2 and its corollary are true even when the marginal distribution

912 Roth, Kadane, and DeGroot

of PI or P2 is discrete. The fact that Theorm 1 also holds in this case is adirect consequence of the following argument.

If PI=J.L*(a)/J.L*('Y) or P2 = J.L* ({3)/J.L* ('Y) , then at least one of these P­values is neither "large" nor "small." If the marginal distribution of PIor P2 is not continuous, this may occur with positive probability. When itoccurs, the lawyer whose p-value is neither large nor small will be indif­ferent between his two possible decisions. In that case, the decision thathis opponent would prefer that he make is called the benevolent decisionand the other decision is called the malevolent decision. A lawyer whoalways makes the benevolent decision when he is indifferent, regardlessof whether he is deciding first or whether his opponent has already ac­cepted the juror, is said to adopt the benevolent strategy. The malevolentstrategy is defined analogously. Of course, a lawyer may make some benevo­lent decisions and some malevolent ones.

A lawyer will choose a different strategy depending on whether his op­ponent has adopted a benevolent or a malevolent strategy. It can beshown, however, that reversibility is not affected by the types of strategiesadopted by the two sides, provided that each side knows whether its op­ponent is benevolent or malevolent. Thus, reversibility will be presentwhen one side uses a benevolent strategy if and only if it is present whenthat side uses a malevolent strategy. Similarly, the absence of reversibilityis not affected by the types of strategies adopted by the two sides. Theproofs of these statements are omitted.

6. AN ALGORITHM FOR DETERMINING THE OPTIMAL PROCEDURE

The form of the optimal procedure (as long as a~ 1, b~ 1) was foundin Section 4. To completely specify the procedure, it remains only toevaluate the functions J.L* and J.L* in order to quantify the notions of "large"and "small" values for PI and P2. For any two real numbers sand t, definethe set

S( s, t) = {(PI, P2) :PI> sand P2<t}

and define the two transformations

and

(6.1)

(6.2)

VF(s, t) = 11 P2dF(PI, P2).S(s,t)

Note that UF( s, t) = V F( s, t) = 0 if either s> 1 or t<O; UF( s, t) = UF(O, t)and VF(s, t)=VF(O, t) if s<O; UF(s, t)=UF(s, 1) and VF(s, t)=VF(s, 1)if t> 1.

We shall let F1 and F2denote the marginal c.dJ.'s of PI and P2, respec-

Peremptory Challenges in Jury Trials 913

tively and let

P(Pl' P2) = P(P1>Pl, P2>P2) = 1-F1(Pl) - F2(P2) +F(Pl, P2). (6.3)

Taking expectations on both sides of (4.2 )-(4.5), we now obtain (fora~ 1, b~ 1, j ~ 1) the following relationships:

J-L*[a, b, j, 10(v)] = J-L*(ex)F[J-L*(ex)/J-L*('Y) , J-L*(fj)/J-L*('Y)]

+ J-L*(fj){ 1-F2[J-L*(fj) / J-L*('Y)]}

+ J-L*('Y) UF[J-L*(ex) / J-L*('Y), J-L*(fj) / J-L*('Y)], (6.4)

J-L *[a, b, j, 10(v )]= J-L *(ex) F[J-L*(ex )/ J-L*('Y ), J-L* (fj ) / J-L*('Y ) ]

+ J-L*(fj){ 1-F2[J-L*(P) / J-L*('Y)]}

+ J-L *('Y ) V F[J-L*(ex )/ J-L*('Y ), J-L *(fj ) / J-L *('Y ) ], (6.5 )

J-L*[a, b, j, 20(v)]=J.L*(ex)F1[J-L*(ex)/J-L*('Y)]

+ p* (fj )F[J-L*(ex) / J-L*('Y), J-L* (fj) / J-L* ('Y)]

+ J-L* ('Y) V F[J-L*(ex) / J-L*('Y ), J-L* (fj) / J-L* ('Y)], (6~6)

J-L*[a, b, j, 20(v)] = J-L*(ex) F1[J-L*(ex) / J-L*('Y)]

+ p*({3 )F'[J-L*(ex) / J-L*('Y ), p* ({3) / J-L* ('Y )] (6.7)

+ P*('Y) UF[J-L*(ex)/J-L*('Y) , J-L*(fj)/J-L*('Y)].

These equations define recursive relations for determining the values ofJ-L* and J-L* at (a, b, j) from the values of J-L* and J-L* at (a-I, b,j), (a, b-l,j), and (a, b, j -1). Hence, the algorithm defined by (6.4) to (6.7) merelyrequires a set of boundary conditions to completely determine J-L* and J-L*for all possible arguments. The boundary conditions, for arbitrary v, are

p.*(a, b, 0, v) J-L*(a, b, 0, v) = 1 for any a, b, (6.8)

J-L*(a, 0, j, v) = u*(a, j) for a~ 1, 1, (6.9)

J-L*(a, 0, j, v) = u*(a, j) for a~ 1, j ~ 1, (6.10)

J-L*(O, b, j, v) = v*(b, j) for b~ 1, j~ 1, (6.11)

JL*(O, b, j, v) = v*(b, j) for b~ 1, j~ 1, (6.12)

where u* and v* represent the expected values that can be attained inthe one-sided versions of this problem when ,)nly one side has any chal­lenges left, and where u* and v* are the expected values of the other sidein these one-sided versions as they helplessly watch their opponents carryout their strategy. Separate algorithms for evaluating these four functionsare given below.

Before generating the algorithms for u*, u*, v*, and v*, we note that u*depends only on F1(Pl) and v* depends only on F2(P2), but u* and v*depend on the joint c.dJ. F(Pl, P2). In preparation for dealing with the two

914 Roth, Kadane, and DeGroot

marginal univariate c.dJ.'s, we define for any univariate c.dJ. A on [0, 1]the transformation

TA(S) =f (x-s)dA(x). (6.13)

The properties of this transformation are given in Section 11.8 of De­Groot [6]. Suppose that X is a random variable with c.dJ. G, on [0, 1],that Y=DX for some constant D>O, and that H is the c.d.f. of Y. Thenfor any constant K, it is easily shown that

E [max(K, X)]=K+To(K), (6.14)

E[min(K, X)]=E(X)-To(K), (6.15)

TH(s) =DTo(sjD). (6.16)

Algorithms for u* and v* are now easily obtained using (6.14)-(6.16):

u*(a, j) =E {max[u*(a-l,j), P1u*(a, j-l)]}(6.17)

=u*(a-l,j) +u*(a, j-l) TF1 [u*(a-l,j) ju*(a, j-l)],

v*(b, j) = E{min[v*(b-l, j), P2v*(b, j-l)]}

= E(P2)v*(b, j-l)

- v*(b, j-l)TF2[v*(b-l, j)jv*(b, j-l)]. (6.18)

The appropriate boundary conditions for (6.17) and (6.18), as well asfor the functions u* and v*, are

u*(a,O) = u*(a, 0) =v*(b, 0) =v*(b, 0) = 1 for any a or b, (6.19)

u*(O,j) = v*(O,j)=[E(PI)]iforanyj~l, (6.20)

u*(O,j) =v*(0,j)=[E(P2)]jforanyj~l. (6.21)

Algorithms for u* and v* are more difficult to obtain. To computeu* (a, j), for example, we note from (6.17) that the prosecution will chal­lenge the next prospective juror if and only if PI<u*(a-l,j)ju*(a, j -1) =Q, say. Then u*(a, j) =E(W), where

W = {u*(a-I,j) if P1<Q, (6.22)P2u*(a, j-l) if P1>Q.

We now obtain

u*(a,j) = u*(a-l,j)F1(Q)

+u*(a, j-I) ~l E(P2IP1 =Pl)dF1(Pl) (6.23)

= u*(a-l, j)F1(Q)+u*(a, j-l) VF(Q, 1).

By similar methods, we can obtain

v*(b, j) =v*(b-l, j)[I- F2(R) l+v*(b, j-l) UF(O, R) (6.24)

where R=v*(b-l, j)jv*(b,j-l).

Peremptory Challenges in Jury Trials 915

Equations (6.17), (6.18), (6.23), and (6.24), together with the bound­ary conditions (6.19)-(6.21), form complete algorithms for evaluatingthe original boundary conditions (6.9)-(6.12). The functions p.* and Jl.*

can now be computed for any arguments, and the optimal procedure iscompletely specified for a~ 1 and b~ 1. When a = 0 and b~ 1, the defenseis playing a one-sided game and we see from (6.18) that the optimalstrategy is to challenge the next prospective juror if and only if P2>R.Similarly, when b=O and a~l, we have seen from (6.17) that the beststrategy for the prosecution is to challenge the next prospective jurorif and only if Pl<Q. When a=b=O, no strategy at all is involved. Theentire optimal strategy has now been specified.

7. EXAMPLES THAT ARE NOT UNIVERSALLY REVERSIBLE

Example 1. It follows directly from Theorem 1 that reversibility cannothold for any a, b, j at all if PI and P2 are independent and neither is aconstant.

Example 2. Since the problem is universally reversible if P2 = kPl ,

we might suspect that this is also the case when the distribution of (PI,P2 ) lies on the union of two such straight lines, i.e., either P2 =kl Pl (de­note this line L1 ) or P2=k2Pl (denote this line L2). However, we show thatsome F's that are not universally reversible satisfy this condition. Withoutloss of generality, assume O<kl <k2• The, three possible cases dependingon whether or not kl and k2 are larger than 1 are illustrated in Figure 1.

Let T(k l , k2 ) be a subset of the unit square with the following property:If (Xl, Yl)ELl nT(kl,k2) and (X2, Y2)EL2nT(kl , k2), then Xl~X2. Such a setcan be found for any O<k1 <k2-see, e.g., the shaded areas in Figure 1.Suppose that the distribution of (PI, P2) lies entirely on T(k1, k2) n(L1UL2).Suppose, furthermore, that very little of the probability lies on L 2 and,hence, most of the probability lies on L l • Then for relatively close valuesof a and b, it is clear that any prospective juror for whom (PI, P2) lies onL 2 will be unsatisfactory to both sides. Lack of universal reversibilityfollows from Theorem 1.

Comment. We cannot "fix" Example 2 by requiring that the support ofthe distribution of (PI, P2 ) be all of L l UL2• In that case we can constructessentially the same example by putting arbitrarily little probability onthose parts of the lines outside T(k l ,k2 ). We conjecture that if the jointdistribution of (PI, P2 ) is absolutely continuous with respect to theLebesgue measure on the plane, then the optimal strategy is not uni­versally reversible.

8. TWO NUMERICAL EXAMPLES WITH THE SAME MARGINAL DISTRIBUTIONS

Suppose that PI and P2 each have a marginal uniform distribution on[0,1] and that A=B=J=I; Le., one juror is to be selected and each side

916 Roth, Kadane, and DeGroot

has one challenge. We shall determine the optimal strategies for twospecial cases where (i) PI =P2 and (ii) PI and P 2 are independent.

Example 3: PI=P2• By Theorem 2 or its corollary we have universalreversibility. From the proof of Theorem 2 it follows that M* =M* andp,* =p,* for any possible common arguments. Furthermore, we can writeM*(a, b, j, p, v) since PI and P2 will always be the same. Thus

Case 1: k 2 ~ 1 Case 2: k 1 ~ 1 < k 2

1(l/k2,1)

1

(l,k2)

(l,kl )(1, k l )

0 1 0 1

Case 3: k l > 1

(1/k2,1) (l/k1 ,1)1

o 1

Figure 1

M*[l, 1, 1, p, le(v)]=M*[l, 1, 1, p, le(v)]

= M*[l, 1, 1, p, 20(v)]=M*[I, 1, 1, p, 20(v)]. (8.1)

If Fo denotes the c.d.f. of PI, then since PI = P2 with probability one,

{o if s~t,

UF(s, t)=VF(s, t)= t wdFo(w) if s<t.

We can now obtain the following results:

p,*[1, 0,1, e(v)]=u*(1, l)=u*(O, 1)

+u*(l, 0) TP1[u*(0, l)ju*(l, 0)]= 31+11

(x-~)dx=%, (8.2)1/2

Peremptory Challenges in Jury Trials 917

JL*[O, 1, 1, 0(v)]=v*(1, 1) =v*(O, 1){1-F2[v*(O, l)/v*(l, D)]}

+v*(l, O)UF[O, v*(O, l)/v*(l,O)]

11/2

=72[1-F2(72)]+UF(O,72)=7~+0 wdw=%.

Hence, from (4.2) we obtain

(% if p<%;M*(l, 1, 1, p, v) = ip if %<p<%;

l% if p>%.It is found from (8.4) that

JL*(l, 1, 1, v) = %,

(8.3)

(8.4)

(8.5)

which merely verifies the fact that both sides start with expectation 72.The optimal strategy is as follows. The first candidate will be acceptedif his p value is between % and %; otherwise, he will be challenged byone side or the other. If either side challenges the first prospective juror,his opponent will challenge the second prospective juror if and only if hefinds % preferable to this second p value. If the defense uses the first chal­lenge, the expectation for both sides becomes %; if the prosecution usesthe first challenge, this common expectation becomes %. The mutualexpectation returns to % if both sides use their challenges.

Example 4: PI and P2 Independent. By Example 1 there is no reversi­bility. Therefore, we must determine four different values of 1l1* or M *and four different values of JL* or JL*. It can be shown that (for O<x<l,O<y<l)

UF(x,y)=%y(1-x2) and VF(x, y)=%y2(1-x). (8.6)

We can now obtain the following relations:

(8.8)

(8.7)

(8.9)

(8.10)

JL*[O, 1, 1, 0(v)]=v*(1, 1) =v*(O, 1){1-F2[v*(0, l)/v*(l, D)]}

+ v*(l, O)UF[O, v*(O, l)/v*(l, 0)]

=72[1-F2(%)] +UF(O, %)=%,JL*[l, 0,1, 0(v)]=u*(1, 1) =u*(O, 1)

+u*(l, 0)TF1[U*(0, 1)/u*(l, 0)] = 72+TFl (~) = %,

JL*[0,1,1,0(v)]=v*(1, 1) = 72v*(l, 0)

-v*(l, O)TF2 [v*(0, l)/v*(l, 0)]= 72-TF2(72) =%,JL*[l, 0,1, 0(v)]=u*(1, 1) =u*(O, 1)F1[u*(0, l)/u*(l, 0)]

+u*(l, O)VF[u*(O, l)/u*(l, 0),1]

= 72Fl(72) +VF(72, 1) = 72·

These equations allow us to obtain the optimal strategy and the M*

918 Roth, Kadane, and DeGroot

and M * values regarding the first prospective juror as presented in Table II.Note that each side has expectation 31 after it uses its challenge since theopponent's strategy is independent of its own perception of the p values.The optimal strategy after the first prospective juror is the same as inExample 3 since each side's strategy when the opponent is out of chal­lenges depends only on the appropriate marginal distribution.

To see" how much is gained by making the first decision on the first

TABLE IIOPTIMAL STRATEGIES AND THEIR VALUES FOR THE FIRST PROSPECTIVE JUROR IN

EXAMPLE 4

PI <)i PI> )i PI <)i PI > !1

.%*** PI* %*** P2*

%** %** .:Y2** !1**

M*[1, 1, 1, PI , P2 , 10(v)] M*[1, 1, 1, PI , P2 , 10(v)]

PI < !1 PI > !1 PI < ~ PI > ~

~*** PI* %*** P2*

~*** %** %*** ~**

M*[1, 1, 1, PI , P2 , 20(v)] M*[1, 1, 1, PI , P2 , 2~(v)]

* Both accept; ** defense challenges; *** prosecution challenges.

(8.11)

(8.13)

(8.14)

(8.12)

prospective juror, we use (6.3)-(6.6) to find

J.t*[I, 1, 1, 10(v)] = 31F(~, 72)+%[1-F2(~)]

+UF (72, ~)=%,

J.t*[l, 1, 1, 20(v)] = 72Fl(72)+%F(~, 72)

+Up (72, 72)=1%2=%-~2,

Jl*[l, 1, 1, 10(v)] = %F(72, ~)+72[I-F2(72)]

+VF(~, 72) = 1%2,

Jl*fI, 1, 1, 20(v)] = %F1(72)+72F(72, 72)

+VF (Y2, ~)=%=1%2-%2'Thus, each side can improve its expectation by >~2 by going first ratherthan going second.

ACKNOWLEDGMENT

This research was supported in part by the Office of Naval Researchunder Contract N00014-75-C-0516, Task NR 042-309 and the NationalScience Foundation under grant SOC 74-02071 A02.

Peremptory Challenges in Jury Trials

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919

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