Optimizing voting order on sequential juries: a median voter theorem and beyond

This paper extends the Condorcet jury theory to juries that vote sequentially (knowing earlier votes) rather than simultaneously. Since we consider jurors who are heterogenous with respect to their ability to determine the true state of Nature (equiprobably A or B, Innocent or Guilty), it turns out that the order in which they vote affects the reliability of the majority verdict. By reliability we simply mean the probability that the majority verdict agrees with the actual state of Nature. We prove that for a jury of size three with any fixed abilities, the voting order with greatest reliability is median ability first, then highest ability, then lowest. Using simulation, we suggest that, for medium-sized juries, the median ability juror should still vote first. Then the abilities should increase, and after the highest ability juror votes, the rest should vote in decreasing order. An intuition for this is that if the highest-ability juror votes first, then herding may reduce the contribution of the others; if he votes last, then the majority verdict may be decided before he even gets to vote.

In a model where jurors have private information in the form of a binary signal (A or B), Condorcet showed that for jurors of common ability \(p>1/2\) (probability of a correct signal), or of varying abilities all at least p, the reliability approaches 1 as the jury size increases.

Our model is a little more sophisticated than that of Condorcet in two ways: signals and abilities. Condorcet’s voters get a binary signal, for A or for B. Our jurors get a signal s in the interval \([-1,+1]\), with higher signals giving a higher probability of A. The signal \(s=0\) is neutral and useless, as A and B remain equally likely after such a signal. In analogy with Condorcet’s probability p (that the binary signal is correct) our jurors have an ability in the interval [0, 1], which is proportional to the probability of A given a positive signal s, \(0<s\le 1\). An ability of 0 makes that probability 1/2, the same as the a priori probability and hence useless (no ability at all). Jurors of higher ability have, in general, more useful private information. When a juror of given ability comes to vote, he will vote for the alternative, A or B, that he views as more likely, given prior voting and his private signal s. This type of voting, called honest voting, models the jurors as experts who have known abilities by reputation. They wish to vote for the alternative that turns out to be true, assuming Nature is eventually revealed. This would be true if they are making short term predictions (say weather or the economy). This type of voting by experts was introduced by Ottaviani and Sørensen (2001), although they do not employ a majority verdict, which we will discuss in detail in the next section. Overall, the model of signals and abilities used here is the same linear signal model introduced in Alpern and Chen (2017b) in a simpler voting model and is similar in spirit to the discrete signal and ability model of Alpern and Chen (2017a), which is unique in some sense (Alpern et al. 2021). The way in which a juror’s signal and ability affect his judgement of the probability of A is described fully in Sect. 4.

Given the above model, we are able to address four questions about the comparative reliability of different voting schemes or voting orders. The main question addressed in this article is the optimal voting order for jurors of different abilities. Putting higher ability jurors first is more likely to have them vote before a majority verdict is established, but also may create negative herding effects. We also consider when sequential honest voting, where jurors can take into account previous voting, is better or worse than simultaneous voting (or secret ballot)as in the Condorcet model. We also answer the question of whether, for fixed average ability, a more heterogeneous or homogeneous jury has higher reliability when voting in the optimal sequential order. Finally, we ask when sequential honest voting is optimal in that it maximizes the reliability of the verdict. If not, the jurors might be incentivized to jointly modify their voting thresholds to produce a better verdict, which could mean possibly voting for the alternative that a juror views as less likely. This might be accomplished by giving all jurors a reward later, if majority verdict turns out to be correct, or perhaps by giving them stock in the company making the decision. Since we are dealing with majority verdicts and, for a single member jury, all the above distinctions are meaningless, we consider a jury of three experts, where we find sharp answers to all of our above questions.

We note that while electorates in preference voting are typically large, juries in information voting are typically small, and often of size three, such as refereeing team in tennis or boxing, appellate levels in legal decisions, doctors giving second or third opinions on the necessity of an operation. For juries of size three we are able to answer all of our above questions regarding comparative reliability.

This paper is organized as follows. After a literature review in the next section, we present in Sect. 3 some simple results and intuitions under some simplified settings, before formally introducing our model in Sects. 4 and 5. We present our main results in Sects. 6–9, which are summarized as follows: Note that the proofs of all the aforementioned eight theorems are completely independent except that Theorem 2 is based on Theorem 1. The order in which these later results are presented could have been different. In Sect. 10, we consider majority voting of medium-sized juries and propose an analog of the optimal rank ordering of abilities for three-member juries. We complete our paper by making some concluding remarks in Sect. 11.

To see how these results aid an organization that has to make a binary decision, consider that the organization has hired three experts of distinct abilities and wishes to determine the optimal voting mechanism. First, suppose it has no additional funds to incentivize strategic voting, so voting is honest. If the abilities of the fixed set of experts are sufficiently homogeneous, it should keep the voting silent (or simultaneous) by not allowing later voters to know the results of earlier voting (Theorem 5). (This is done for preference voting in US elections, but for different reasons.) If the ability set is sufficiently heterogeneous, then later voters should be told the outcome of earlier votes (Theorem 5) and they should vote in the order given by Theorem 2. We note that other variations are possible, for example, later voters could be told the votes of only some of the earlier voters. If the jury is not fixed, then Theorem 6 suggests that, assuming the cost of an expert juror is increasing in ability, heterogeneous juries might be preferable. Finally, we consider whether it might be advantageous to pay a bonus to each expert for a correct jury (majority) decision, assuming the correctness of the verdict becomes common knowledge in the near future, as with weather predictions. In general, this will be a cost-benefit problem, but Theorem 7 shows that such payments are wasteful if the ability set is sufficiently heterogeneous.

For larger juries of odd size \(n=2m+1\), we propose the Ascending–Descending Order (ADO), where the most able \(m+1\) jurors vote first in increasing ability order and then the least able m jurors vote in descending ability order. We already know from Theorem 2 that ADO is optimal for \(n=3\). Using simulation techniques, we establish that the ADO is an excellent heuristic (and possibly optimal) for \(n=5\) and \(n=7\).

To conclude the section, we remark that our problem can be viewed as a simple optimization problem faced by a decision maker, who has an unordered set of people to form a jury to help him make a binary decision. He wants to arrange their voting order so as to make the majority decision of the jury as reliable as possible. On the other hand, we can also view our problem more game theoretically as a Stackelberg game or principal agent problem. First, we are given an unordered set of jurors. The Principal, who moves first, selects an order in which they must vote. He wins the game (payoff 1; otherwise payoff 0) if the majority verdict is the true state of Nature. Then the jurors vote openly in the order selected by the Principal. In this sub-game, a juror wins (payoff 1; otherwise payoff 0) if his vote is the same as the true state of Nature. That is, the Principal cares only about the correctness of the verdict, while the jurors care only about the correctness of their own votes. In the sub-game, honest voting is exactly the unique Nash equilibrium. In fact, no juror can improve his expected payoff (probability he votes correctly) by changing from honesty, if the prior voters stick to honest voting. Thus it is stronger than the usual definition of Nash equilibrium.

The celebrated Condorcet jury theorem (Condorcet 1785) is concerned with a model with an odd number n of jurors who receive independent binary signals for one of two states of Nature. Each juror receives the correct signal with the same probability p. Condorcet’s Jury theorem (CJT) states that if each juror votes his signal, the probability that the majority vote is correct goes to 1 as n approaches to infinity if \(p>1/2\). There is a large body of literature on extensions and discussions of this result. In this paper we are interested in extensions to sequential voting of heterogeneous ability jurors, where the jurors vote in order, with knowledge of all previous votes. In particularly we are interested in work related to voting order.

In Ottaviani and Sørensen (2001) jurors have heterogeneous abilities (as in this paper), who care only about being right (they are “experts”, with reputations to uphold), rather than about obtaining a correct verdict. We call them honest voters. The main difference between our model and theirs is that our model stays within the Condorcet jury context, where the verdict is simply the majority vote, while their model has a “decision maker” who determines the group decision after all votes have been cast. They talk about the problem of groupthink, herding and conformism, and the last is also discussed in terms of committee decisions regarding the “secretary problem” in Alpern and Baston (2016).

Dekel and Piccione (2000) mainly consider the equilibria of simultaneous versus sequential voting, but they do make an important comment on voting order of heterogeneous jurors” in their Conclusions (p. 48):

“...if voters are endowed ex ante with differential information (some voters can be better informed than others), knowing which voters voted in favor and which against can affect the choice of a later voter. It can be shown that, in a common-value and two signal environment (as in Sec. IIIC above), if the player’s signals are completely ordered (in the sense of Blackwell), then it is optimal to have the better informed vote earlier. This provides an interesting contrast to the findings of Ottaviani and Sørensen (2001). They obtain the opposite optimal order in an environment in which information providers care not about the outcome but about appearing to be well informed. It is not difficult, however, to construct examples in which having the best-informed voter vote first is not optimal. Hence it seems unlikely that general insights into this question can be obtained.”

The point of this paper is that we are indeed able to obtain general insights into this question.

Ali et al. (2008) discuss the optimal voting order of experts and cite examples in which courts of judges follow either anti-seniority (increasing order of ability, in our terminology) or seniority (decreasing order) orders, respectively, in the ancient Sanhedrin court and in the contemporary American Supreme Court. Voting order in selection committees is analyzed in Alpern, Gal and Solan (2010), but there voting is by veto.

A sequential voting model, with discrete but non binary signals, was introduced by Alpern and Chen (2017a). Ability levels were also discrete, so propositions about which ability orderings are best could be obtained by exhaustive search among the finite number of possibilities. Most of that paper is about strategic voting where voters wish to obtain the correct majority verdict, even if this means voting for an alternative which a juror believes to be less likely. For the question of the optimal ability ordering of three jurors with abilities \(a<b<c\), that paper shows by exhaustive search that b, a, c and b, c, a are the optimal orderings (the order of the last two voters in strategic voting does not matter). For honest voting, it is shown for the finite number of cases considered, that the ordering b, c, a is optimal. This is the observation that led us to attempt the general algebraic proof, where a, b and c are real numbers, that is given here. The first algebraic proof of this kind was obtained recently by Alpern and Chen (2017b) for a simpler voting model, where first two jurors vote simultaneously and then the remaining (third) juror casts the deciding vote if there is a tie. This is a much simpler model algebraically as there are only two voting thresholds to consider (for the casting voter), depending on whether the the more able juror voted A or B. Alpern and Chen (2017a) also consider larger juries, up to size 13. It uses simulation to compare randomly generated juries, which vote honestly in increasing, decreasing or random ability order. For all these sizes, decreasing order has higher reliability than random, and random has higher reliability than increasing order.

For a simpler voting model described above with n-1 jurors voting simultaneously, followed by a tie breaking vote (if necessary) of the remaining voter, Alpern and Chen (2017b) showed that for \(n=3\) the voter of median ability should have the casting vote, with honest voting.

A related literature compares the efficacy of sequential versus simultaneous voting. A potential problem with the former is the possibility of herding and information cascades, especially when more able voters are at the beginning of the sequence. Here, voters ignore their own private information to vote instead based on a consensus achieved by earlier voters. On the other hand, with sequential voting each juror has more information to go on. Bikhchandani, Hirshleifer and Welch (1992) explore the probability of informational cascades and also the probability that they converge on an incorrect outcome. They explain many recognized phenomena based on this convergence. In this respect, Ottaviani and Sorensen find that “increasing the quality of some experts on the committee can exacerbate herd behavior and hence decrease the amount of information collected by the decision maker”. A more recent approach to this problem, by computer scientists exploring sequential voting in the context of what are known as social recommendation systems (such as Amazon product valuations, written with knowledge of previous customer’s evaluations) is given by Celis, Krafft and Kobe (2016). They analyze student-reported learning and find that “sequential voting systems can surface better content than systems that elicit independent votes”. A related paper, which deals with sequential evaluations, is Besbes and Scarsini (2018) and considers the possibility of only revealing the mean of prior evaluations. This brings the suggestion of varying our model so that jurors know the prior unascribed vote (such as three votes for A and one for B) without revealing how each prior juror individually voted. This model will be for our future work.

Condorcet theory has also been applied to particular problems: statistical estimation in Pivato (2013), forecasting US election results in Murr (2015) and for project evaluation in Koh (2008). The last of these considers both heterogeneous jurors and voting order.

As a warm-up, this section serves the purpose of giving some simple results and intuitions for optimally reliable voting orders on heterogeneous sequential juries, without resort to the full continuous signal model mentioned above and defined precisely later in Eq. (3) of Sect. 4. We first consider Condorcet jurors, who are defined as receiving binary signals (for the two possible states of Nature A or B) and then we consider what we call sophisticated jurors, who receive real number signals in the interval \([-1, +1]\).

In both the Condorcet model and in our model, there are two equiprobable states of Nature, called A and B. In the Condorcet model, private information of jurors is in the form of a binary signal \(\{A, B\}\). The ability of a juror to guess the true state is given as a probability, called p, that the signal received corresponds to the actual state of Nature. This probability p ranges between 1/2 (useless information) and 1 (definite information). In our model, private information of a juror comes in the form of a continuous signal s, which ranges between \(-1\) and \(+1\), with signal 0 neutral, negative signals indicating B is more likely and positive signals indicating that A is more likely. Our analog of Condorcet’s probability p is a number, called the ability a, which ranges from 0 to 1. The number a is linearly related to the conditional probability p of A, given a positive signal s. In particular, the conditional probability of A, given that a juror of ability a receives a signal s above a threshold \(\tau\), \(\tau \in [-1,+1]\), is given byThis generalizes Eq. (1). For a juror of ability \(a=0\), this means that p, or more generally \(p_{\tau }\), is 1/2, the same as the a priori probability of A. Thus a juror of ability 0 essentially has no ability, his signal (his private information) is useless. On the other hand, if a juror has maximum ability \(a=1\) and receives a signal above a threshold \(\tau\) that is very close to \(+1\), his conditional probability of A approaches certainty (\(p_{\tau } \approx 1\)). The fact that the signal is above the threshold \(\tau =-1\) says nothing about the signal (they are always at least \(-1\)) and indeed \(p_{-1}=1/2\), the same as the a priori probability of A. Thus in our model, the ability level a is a proxy for Condorcet’s probability p, but in a more general context of continuous rather than binary signals. We now describe the signal distribution which gives these outcomes (1) and (2). This continuous signal-ability model has been used before in the literature (e.g., Alpern and Chen 2017b).

The fact that the signal distribution that we use (see Sect. 4.1) is the unique one that makes our definition of ability a as linear function of Cordorcet’s p is shown by Alpern, Chen and Ostazewski (2021).

In this section we determine the thresholds (on his private signal) that an honest voter adopts, based on previous voting. Using these results, we determine the reliability (of the verdict) for a jury of honest voters of given abilities who vote in a given order. In this and the next section we signpost the main steps of the algebraic calculations, with details left in the appendix.

Although we are mainly concerned with the reliability of majority verdicts under sequential voting, the related question of unanimous verdicts will be useful to partly resolve here, as it has an impact on the analysis of majority verdict theory. So, we consider sequential voting of \(n\ge 2\) jurors and, between the two alternatives for verdict, there is one that is designated in advance as the favored, to which the verdict goes unless the other alternative gets all votes of the jury. Such voting rule, which we call unanimous voting rule, has been considered in the organizational studies literature, e.g., Romme (2004). In some courts of law a unanimous vote is required to convict. Such a requirement has been suggested for capital cases to reduce the number of false convictions. However, Feddersen and Pesendorfer (1998) have shown that, for strategic voting (to be discussed in more detail in Sect. 9) there is no such reduction when unanimity is required for conviction. Ben-Yashar and Danziger (2016) show how to allocate heterogeneous ability experts (with binary signals) to disjoint juries which adopt the unanimity rule. Most of the study of unanimous voting is done in a simultaneous rather than sequential manner discussed here.

Here is an example where sequential unanimous “voting” takes place. Suppose a patient considers having a major operation. It is important enough to get a second opinion. The patient will have the operation carried out only if both doctors recommend it. The second doctor to give an opinion will know the recommendation of the first doctor and will know his reputation as well. In what order should the patient query the two doctors (in terms of their abilities) to have the most reliable verdict?

In this section we first consider how reliability changes when two jurors switch their voting positions (Theorem 1, Propositions 3 and 4) and we then combine these results to determine the optimal voting order of three jurors (Theorem 2).

While little attention has been paid (until recently) on determining the optimal voting order, there has been much discussion on the relative performance (reliability) of seniority order (SO), in decreasing order of seniority or ability, with respect to anti-seniority order (AO), where voting is in increasing order. Ottaviani and Sørensen (2001) give an extensive survey of the practice and theory of SO vs. AO. They point out that the AO practiced in the ancient Sanhedrin and until recently for voting in the United States Supreme Court. In tennis and badminton, the referee, usually a more senior figure, votes after the lines-persons. It can be argued that Hawkeye, the computer arbiter of close calls, who is the final arbiter, is the most able. Sometimes, both SO and AO are used. Hall and Brace (1992) note that in courts where opinions are assigned at the discretion of the Chief Justice, conference discussion is SO and voting is AO. Appellate decisions in courts usually have more senior (equals more able?) judges in the higher courts, thus AO. Qualitatively, it would seem that AO is less susceptible to herding. On the other hand, since the most able jurors vote late, the majority decision may have already been confirmed before they are reached. So while conventional opinion may be on the side of AO, we find (at least for three-member juries) that it is dominated by SO.

To see this property of three-member juries, we look back at an earlier result. If the jury consists of three jurors of abilities \(a_1<a_2<a_3\), then seniority order \((a_3,a_2,a_1)\) can be obtained from anti-seniority order \((a_1,a_2,a_3)\) by simply transposing the first and last juror. We considered the effect of such a transposition in Proposition 4, where we showed that \(Q(b,c,a)>Q(a,c,b)\) for any c if \(a<b\). By taking \(a_3=b\), \(a_1=a\) and \(a_2=c\), we obtain the following dominance of seniority sequence over anti-seniority sequence.

For larger juries of odd size \(n>3\), we ask the following question: When a random jury is chosen, and assigned signals according to abilities, what is the probability that the seniority order is correct when the two orders (SO and AO) give different verdicts? Our results are given below.

For each odd-size jury \(n=5,7,9,11,13\), we generate one million random juries with abilities taken uniformly and independently from the interval [0, 1]. To each, we assign random signal vectors, generated according to the abilities and assuming state of Nature A. We calculate the majority verdicts X and Y by each, where X is based on AO and Y is based on SO. Thus A is the correct verdict. We record in Table 1 the number of verdict pairs (X, Y) (which we denote by \(\#(X,Y)\)), where neither ordering is correct (B, B), both are correct (A, A), only AO is correct (A, B) and only SO is correct (B, A). The final column \(\rho\) gives the fraction of the cases where SO is correct when exactly one of the two is correct:

As we can see from Table 1, When the verdicts under the two orderings differ, SO is correct about 58% of the time for juries of size 5, increasing to about 61% of the time for juries of size 11. This analysis refines earlier simulation work of Alpern and Chen (2017a).

Now let us consider some ideal situation for achieving maximum possible reliability, where the jurors jointly choose thresholds to maximize verdict reliability. We say such voting is strategic. Although strategic voting apparently cannot be realized easily in practice, we use the reliability under strategic voting as a benchmark. Another well-known voting scheme is simultaneous voting (or secret ballot), in which all voters vote independently.

Up to now we have considered the optimal voting order problem only for juries of size three, where we have the definitive result Theorem 2. For larger juries we only know the optimal ordering for the last two jurors (decreasing ability, see Theorem 1) and we have some numerical results comparing the seniority order (SO) with the anti-seniority order (AO) in Sect. 8, namely, the former is more likely to be correct when the verdicts differ. For larger juries the “curse of dimensionality” prevents us from using the earlier technique of deriving a formula for the reliability of a jury with a given order of abilities because the number of thresholds increases exponentially in the jury size. So in this section we will use simulation to estimate this reliability, by generating many random signal vectors for a given ability order jury, assuming one of the alternatives (say A) holds. For each, we compute the verdict. Finally, we estimate reliability as the fraction of verdicts which are correct (in this case, A).

In general, we cannot consider all possible orderings (permutations) of a set of abilities. In addition to SO and AO, we propose a new ordering which we call the Ascending-Descending Order (ADO). For a jury of odd size \(n=2m+1\), this ordering begins with the most able \(m+1\) jurors voting in increasing (anti-seniority) order and then has the m least able jurors voting in descending (seniority) order. In particular, the median ability juror votes first, the highest ability juror votes in the median position and the least able juror votes last.

Note that for \(n=3\) our main result, Theorem 2, says that the ADO is the unique optimal ordering. Furthermore, ADO has its last two jurors voting in decreasing order, which is a requirement of any optimal ordering according to Theorem 1. Our simulations, admittedly limited in number, have not yet produced any counter-examples to the following.

Of course we have already proved the conjecture for juries of size three in Theorem 2. Even if the ADO turns out to not always be optimal, our simulation results seem to show that it a good heuristic, producing high reliabilities.

Our first simulation is concerned with a jury of size five, which we take as the ability set \(\{0.1,0.3,0.5,0.7,0.9\}\), which is in some sense uniformly distributed. For simplicity, we denote them by \(\{1,3,5,7,9\}\) for the rest of this section. If two abilities are very close, then transposing the voting order of the two corresponding jurors would not greatly affect the reliability, and the difference might be less than the error in the simulation. There are \(5!=120\) orderings of such a jury, but only 60 for which the last two abilities are in decreasing order. We calculated 100,000 random signal vectors (generated from alternative A) for each such ordering and counted the number of (correct) verdicts for A, this number divided by 100,000 is an estimate of reliability. Of these 60, exactly 9 had an estimated reliability \({\hat{\rho }}\) above 76%. We then calculated 1,000,000 trials for these, listing the results in Table 2 below. The estimates have error less than 0.1% with a confidence of 90%. Note that ADO has the highest estimated reliability, at 77%.

The increasing order (1, 3, 5, 7, 9) (AO) had \({\hat{\rho }}=71.5\%\); the decreasing order (9, 7, 5, 3, 1) (SO) had \({\hat{\rho }}=75.5\%\); the ordering with the lowest estimated reliability was (5, 3, 1, 9, 7) with \({\hat{\rho }}=64.2\%\). It is worth noting that there is more than a 10% difference in reliability between ADO and the worst ordering.

To look at seven-member juries, it is no longer feasible to check the reliability of all \(7!/2=2520\) orderings even of a particular set of abilities. Instead, we generated 500 random juries \(\mathbf {a}\) (independently picking each juror with an ability uniformly in [0, 1]). For each, we calculated the estimated reliability \({\hat{\rho }}(\mathbf {a})\) and \({\hat{\rho }}^{*}(\mathbf {a}^{*})\), where \(\mathbf {a}^{*}\) is the reordering of \(\mathbf {a}\) to ADO. For example, if \(\mathbf {a}=(0.3,0.6,0.4,0.1,0.8,0.5,0.7)\) then \(\mathbf {a}^{*}= (0.5,0.6,0.7,0.8,0.4,0.3,0.1)\). We used 10,000 trials for each randomly generated jury \(\mathbf {a}\), and calculated the difference \({\hat{\Delta }}(\mathbf {a}) = {\hat{\rho }}^{*}(\mathbf {a})-{\hat{\rho }}(\mathbf {a})\) between reliability in ADO order and in the originally generated order. The mean value of \({\hat{\Delta }}\) was 0.036969 (roughly a \(4\%\) improvement in the reliability of the verdict), the maximum was 0.1609 (a \(16\%\) improvement in reliability) and the minimum was \(-0.0064\). The frequency distribution is given in Table 3 (no data at boundary points).

So from the heuristic point of view, it certainly seems a good idea to rearrange a jury (of size 7), which arrives in random order, into one in ADO order, if this is allowed. It nearly always gives an improvement in reliability. From the theoretical point of view, it at first may appear that we have found ten counter-examples to our Conjecture 1 regarding the optimality of ADO, in that the original order of the jurors that produced a negative \({\hat{\Delta }}\) is more reliable than ADO. However, the negative values were very close to zero. There were 11 juries \(\mathbf {a}\) where \({\hat{\Delta }}(\mathbf {a})<0.001\) and we recalculated \({\hat{\Delta }}\) for all of them with one million trials. All came out with a positive value of \({\hat{\Delta }}\). There was one additional jury \(\mathbf {a}\) for which \({\hat{\Delta }}(\mathbf {a})\) was not negative but was less than 0.001. We also recalculated this for one million trials and checked that it still came out positive.

Despite the aforementioned efforts, the search for a counter-examples to Conjecture 1 is far from exhaustive.

Another way of evaluating the ADO is to compare it to SO (Seniority, or Descending Order), in the same way we compared SO to AO in Sect. 8. For odd values of \(n=5, 7, 9, 11\) , we generate one million random juries with associated random signals generated, assuming Nature is A. We record in Table 4 (with the same notation as in Table 1) the number of verdict pairs for ADO and SO, respectively, and determine the probability that ADO is the correct verdict (that is, A) when the two verdicts differ.

From Table 4, we see that when ADO and SO give different verdicts, ADO is the correct one about 53% of the time for juries of size five, rising to about 55% of the time for juries of size 11.

When jurors of differing abilities vote sequentially, the reliability of the majority verdict depends on the voting order. We have shown that, regardless of the set of abilities on a three-member jury, if the two states are equally likely, then the optimal voting order is always median ability first, then highest ability, then lowest ability. The optimality of the ordering is robust for the a priori probability \(\theta\) around 1/2.

Our earlier paper, Alpern and Chen (2017a) showed that this was true for many ability sets, but this is the first paper to establish this result rigorously for all ability sets. Analogous results for larger juries appear to be beyond current methods, but presumably special cases could be studied, for example, when only one juror has a different ability than the others.

Since the optimal ordering is established, we can fix this ordering and then make various reliability comparisons. To do this, we have defined indices that measure homogeneity and heterogeneity of the set of jurors. We find that for sufficiently homogeneous juries, simultaneous voting is more reliable than sequential voting. On the other hand, when juries are sufficiently heterogeneous, sequential voting is more reliable. In a similar vein, we find that the reliability of a jury of fixed average ability is increasing in its heterogeneity. For sufficiently heterogeneous juries, the thresholds given by honest voting are in fact the optimal joint thresholds, so that honest voting optimizes the reliability of the majority verdict. That is, honest voting is also strategic.

For larger juries, we have introduced the Ascending–Descending Order of voting and shown that it is at least an excellent heuristic for obtaining high reliability, if not actually optimal. Further work in this direction for large juries would be useful, including asymptotic results analogous to those of Condorcet.