Collective Decision Making and Jury Theorems
by
Shmuel Nitzan and Jacob Paroush*
Department of Economics
Bar Ilan University
Ramat Gan 529000
Israel
Abstract
It is more than two hundred years that issues related to collective decision making and to Condorcet jury theorems are studied and publicly discussed. Recently, there is a burgeoning interest in the topic by academicians as well as practitioners in the fields of Law, Economics, Political Science and Psychology. Typical questions are: what is the optimal size of a panel of decision makers such as a jury, a political committee or a board of directors? which decision rule to utilize? who should be the members of the team, representatives or professionals? what is the effect of strategic behavior, group dynamics, conflict of interests, free riding, social interactions and personal interdependencies on the final collective decision?
The purpose of this article is to present the state of the art in the field, to suggest further research and to allude to possible future developments regarding public choice and collective decision making.
Keywords: collective decisions, Condorcet jury theorems, decision rules, strategic behavior, freeriding, personal interdependencies
*The authors are indebted to Assaf Basevich for his most useful assistance.
Content
1. Introduction
2. Heterogeneous competencies
3. Optimal decision rules
4. Order of decision rules
5. Competencies are not fixed
6. Diverse preferences
7. Asymmetric alternatives
8. On violation of independence
9. Strategic Behavior
10. Latent competencies
11. Miscellaneous
11.1 The hungjury problem
11.2 Multicriteria decisions
11.3 Applications
1. Introduction
The Marquis de Condorcet (17431794) is considered one of the pioneers of the social sciences. In the English literature, Baker (1976) and Black (1958) were among the first to turn the attention of the scientific community to the importance of Condorcet's writings [see Young (1995)]. No doubt that his seminal work from 1785 has formed the basis for the development of social choice and collective decision making as modern research fields. In the last forty years, hundreds of articles have been written in these fields and at least five new journals which are mainly devoted to these subjects have been published. This article focuses on Condorcet jury theorems and related collective decision issues which might be of interest to scholars in the areas of Law, Economics and Political Science.
In 1785 no jury existed in France. Condorcet applied probability theory to judicial questions and argued that the English demand for unanimity among jurors was unreasonable and suggested instead a jury of twelve members that can convict with a majority of at least ten. In 1815 the first French juries used Condorcet's rule but later adopted the simple majority rule. At that time the mathematician Laplace argued that simple majority is a dangerous decision rule for juries. Since 1837 juries had been established on several different plans, but the French law has never believed that one could count on twelve people agreeing (see Hacking (1990) ch.11). Since the seventies of the last century, several works analyzed the jury system applying probability theory as well as statistical data. Gelfand and Solomon (1973), (1975), Gerardi (2000), Klevorick and Rothschild (1979) and Lee, Broedersz and Bialek (2013) are a few such studies. The common English jury system is an extreme form of a qualified majority rule while the French version is less so.
The debate about the size and the composition of the jury and the kind of decision rule it should use has been going on for more than two centuries and will probably continue. This article presents the state of the art in collective decision making and jury theorems as well as some expectations about further developments in the field.
Condorcet (1785) makes the following threepart statement:
1) The probability that a team of decision makers would collectively make the correct decision is higher than the probability that any single member of the team makes such a decision.
2) This advantage of the team over the individual performance monotonically increases with the size of the team.
3) There is a complete certainty that the team's decision is right if the size of the team tends to infinity,i.e., the probability of this event tends to one with the team's size.
A "Condorcet Jury Theorem" (henceforth, CJT) is a formulation of sufficient conditions that validate the above statements. There are many CJT’s, but Laplace (1815) was the first to propose such a theorem. Some of the following conditions are explicitly stated by Laplace as sufficient conditions and the others are only implicitly assumed. A major concern of the modern collective decisionmaking literature is whether or not each of these conditions is also necessary for the validity of Condrcet’s statements. These conditions that are listed below will serve as a road map for the rest of the paper.
Specification of the problem faced by the jury.
1. There are two alternatives.
2. The alternatives are symmetric.
Specification of the decision rule applied by the team:
3. The team applies the simple majority rule.
Specification of the properties of the jury members:
4. All members possess identical competencies.
5. The decisional competencies are fixed.
6. All members share identical preferences.
Specification of the behavior of the jury members:
7. Voting is independent.
8. Voting is sincere.
The core of Laplace’s proof is the calculation of the probability of making the right collective decision, (to be denoted by Π), where Π is calculated by using Bernoulli's theorem (1713). Laplace shows, first, that Π is larger than the probability of making the right decision P by any single member of the team, second, that Π is a monotone increasing function of the size of the team and, third, that Π tends to unity with the size of the team. Of course, besides the above conditions there is an additional trivial condition that the decisional capabilities of decision makers are not worse than that of tossing a fair coin. Namely, the probability P of making the correct decision is not less than one half. Other properties of Π are that it is monotone increasing and concave in the size of the team, n, and in the competence of the individuals, P. Thus, given the costs and benefits of the team members' identical competence, one can find the optimal size of a team and the optimal individual's competence by comparing marginal costs to marginal benefits.
The above eight conditions are quite strong and there is much doubt whether they exist in reality. The present article studies the consequences of the relaxation of these conditions which has been and apparently will continue to be a major topic in the literature of collective decision making related to CJT.
Given the limited space, it is possible to cover only some of the important subjects. Naturally, to some extent this article will be biased towards the interest of the authors.
2. Heterogeneous competencies
Let us start by studying the consequences of relaxing the assumption that the members of the team possess identical competencies. Suppose that there are n members in a team facing two symmetric alternatives with decisional competence that can be represented by their fixed probabilities of making the right decision. Denote these probabilities by p = (p_{1},…,p_{n} ), where the p_{i}’s are not necessarily identical. By simple numerical examples it is shown that the first two parts of Condorcet's statement are no longer necessarily valid.
Suppose p=(0.95,0.80,0.80). Assuming that the team utilizes a simple majority rule, it is possible to find that the team's competence in this case is Π=0.944. One can see that the competence of the first member of the team is larger. Thus, the first part of the statement is violated. Add now two more members to the team both with a competence of 0.6 to obtain the enlarged team where p=(0.95,0.80,0.80,0.60,0.60). Utilizing simple majority rule, the team's competence is now reduced to Π=0.910. The second part of the statement is therefore also violated [The accuracy of majority decisions in groups with added members is discussed in Feld and Grofman (1984)].
The numerical examples that are presented here are far from being extreme. For instance, Nitzan and Paroush (1985 p.68) find that if one draws a random sample of individual's competencies from a uniform distribution over the range [1∕2, 1], then within fivemember teams, in more than 20% of the cases the first part of Condorcet's statement is violated. To wit, the competence of a random fivemember team which utilizes a simple majority rule is smaller in at least 20% of the cases than the probability that the most qualified person of the team would decide correctly.
In contrast to the first two parts of the Condorcet's statement, the survival of the third part is somehow surprising. Many attempts have been made to prove the validity of the third part in case of heterogeneous teams [see Boland (1989), Fey (2003), Kanazawa(1998) and Owen, Grofman and Feld ,(1989)]. In fact, the following is a well known version of CJT: "If a team of decision makers utilizes a simple majority rule, the decision would be perfectly correct in the limit given that the size of the team tends to infinity, even if the individual competencies, the p_{i}’s, are different_{,} provided that p_{i}≥1∕2+ε, where the value of ε is a positive constant regardless of how small it is". The proof of the theorem relies on the proof of Laplace where P= 1∕2+ε combined with the fact that Π is an increasing function of the team members' competencies.
One comment is in order. Paroush (1998) shows by means of an example that Π = 1∕2 in the limit even if all p_{i }>1∕2. This counter intuitive example occurs when the sequence of the p_{i}'s decreases rapidly from above towards 1∕2. Thus, it looks that regardless of how small it is, the presence of a fixed ε is not only a sufficient but also a necessary condition. But it turns out that existence of a fixed ε is inessential for CJT. Relying on the law of large numbers, Berend and Paroush (1998) found a necessary and sufficient condition for the validity of the third part of the statement, even in the case that competencies are not identical. The condition is: (A_{n}1∕2) √n →∞ , where A_{n}=∑p_{i }∕ n.
The meaning of this condition is that if all p_{i }>_{ }1∕2, the team's competence Π is still equal to one in the limit, even if the sequence of the p_{i} ‘s decreases towards 1∕2 but only in a "moderate pace".
The next section shows that the first part of the statement can be retained if the team utilizes not a simple but a weighted majority rule where the weights are adjusted to the individuals' competencies given that they are known.
3. Optimal decision rules
Condorcet who was a great fan of the wisdom of the crowd was among the first to lay down the philosophical foundations of Democracy. In particular, he strongly believed in the superiority of simple majority over other decision rules [see Grofman (1975)]. But, it is known today that if the team members' competencies are "public knowledge", the simple majority rule may lose its superiority. Many studies suggested alternative criteria for the optimality of a decision rule [see for instance Rae (1969), Straffin (1977) and Fishburn and Gehrlein (1977)]. However, in what follows, the assumption that the team's competence, Π, is the criterion for the optimality of the decision rule is preserved. Note that given that the team members share homogenous preferences, this criterion is also consistent with (equivalent to) expected utility.
Allowing heterogeneous decisional capabilities, Shapley and Grofman (1981, 1984) and Nitzan and Paroush (1982) find that the optimal decision rule is a weighted majority rule (WMR) rather than a simple majority rule (SMR). By maximization of the likelihood that the team makes the better of the two choices it confronts, they also establish that the optimal weights are proportional to the log of the odds of the individuals' competencies, i.e. the weights, w_{i} ,are proportional to log[p_{i} ∕(1p_{i})].
Let us apply this result to the case of a team of three members with known competencies of p_{1} > p_{2} > p_{3 }, where p_{i} is the probability that member i decides correctly. It is obvious that the simple majority is the superior decision rule if and only if {log [p_{2} ∕ (1p_{2})] +log [p_{3} ∕ (1p_{3})]} > log [p_{1 }∕_{ }(1p_{1})] or [p_{2} ∕ (1p_{2})]•[p_{3} ∕ (1p_{3})] >p_{1} ∕ (1p_{1}). Thus, the simple majority rule looses its superiority to the expert rule if either p_{1} is close enough to 1 or p_{3} is close enough to 1 ∕ 2.
For the general case of a team of n members, one can make the following statement: A sufficient condition for the violation of the first part of Condorcet's statement is the inequality: p_{1 }∕ (1p_{1}) > Π [p_{i }∕_{ }(1p_{i})] where p_{1} is the competence of the most competent person in the team and Π indicates here the multiplication of the odds of the rest of the members.
In the case of a team with more than three members, besides the simple majority rule (SMR) and the expert rule (ER), there are other efficient rules that are potentially optimal. For instance, in a fivemember team there exist seven different efficient potentially optimal rules. These rules include “the almost expert rule, " "the almost majority rule" and "the tiebreaking chairman rule". The number of efficient rules increases very rapidly with the team size. For instance, in a team of nine members the number of efficient rules is already 172,958 [see Isbell (1959)]. Now the following question is raised: is there a mathematical relation between the size of the team and the exact number of efficient rules? This simple question is still an open one.
Note that a choice of the optimal size of a team may be considered as a special case of a choice of the optimal decision rule. For instance, the optimal rule within a team of seven members could be a restricted majority rule where simple majority rule is utilized only by the three or by the five most qualified members where the rest of the members are inessential (their decisions are disregarded, that is, the weight assigned to an inessential member is zero).
Paroush and Karotkin (1989) investigate the robustness of optimal restricted majority rules over teams with changing size. One of the useful findings is that optimal restricted majority rules are robust over reduction of the size of the team. For instance, suppose that a SMR is the optimal rule in a team of seven members, then a SMR will stay optimal even in a team of five where the two least qualified members of the original team are discarded. The clarification of the conditions for the robustness of optimal decision rules when the team members' competencies are allowed to vary is an important topic for future research.
4. Order of decision rules
The existence of order relations among decision rules is firstly noted in Nitzan and Paroush (1985, p.35). Existence of such an order means, first, that the number of rankings of m efficient rules is significantly smaller than the theoretical number m! of all possible rankings of these rules and, secondly, that its existence is independent of the team's competence.
Beyond the theoretical interest in studying order among efficient decision rules, the information about the order has useful applications. Since the order relations are independent of the specific competencies of the decision makers, the knowledge about the order of the rules is important in cases where the competencies are unknown or only partially known. For instance, if for some reason(e.g., excessive costs) the optimal rule cannot be used, then even in the absence of knowledge about the decisional competencies, the team can identify by the known order of the decision rules the second best rule, the third best and so on.
Karotkin, Nitzan and Paroush (1988) show that six out of the seven efficient rules available to a team of five members generate a complete scale or an essential ranking. Thus, only less than 200 rankings out of 7! = 5,040 possibilities are feasible.
Paroush (1997) finds that the SMR and the ER are always polar rules in the sense that if one of them is optimal and thus the most efficient, the other one is the least efficient. He also identified second best rules and penultimate rules in cases that majority rules (simple or restricted) are optimal or the most inferior.
Karotkin (1998) exposes an interesting analogy between the network yielding the ranking of all weighted majority rules by efficiency and the weighted majority games in terms of winning and losing coalitions.
Paroush (1990) shows how to apply the knowledge of essential ranking of decision rules when the team of decision makers faces multichoice problems. Karotkin and Paroush (1994) apply the essential order in cases where the team members’ competencies may be subject to some fluctuations. For instance, if the optimal rule loses its optimality status due to fluctuations, then one of its neighbors on the ranking will become the second best.
However, the complete order relations among efficient decision rules as well as the coiled network among them are still to be discovered; the task of studying these issues is far from being exhausted.
5. Competencies are not fixed
Assume that the individual’s decisional competence p_{i} is not constant anymore but monotonically increasing (in a decreasing rate) with some invested resources, c_{i },_{ }which are intended to improve the individual's competence p_{i}. Such an investment in human capital might be a direct cost in the form of pecuniary outlay or an indirect cost in the form of money equivalent of the time and efforts necessary to collect data or evidence, to elaborate, process and transfer the relevant information into a decision. It also includes the costs necessary to shorten the delay of reaching a decision. Note that in most cases such an investment is very specific and depreciates completely immediately after the voting.
The question of optimal investment in human capital is important and still open for future research. Only few results are available and they have been obtained under restrictive assumptions.
Assume that all the team members possess an identical learning function. To wit, for each one of the members p_{i}(c_{i}) = p(c_{i}). It is worth noting that this assumption is consistent with a liberal viewpoint that the differences among individuals 'competencies are due to reasons such as unequal opportunities and are not due to genetic, gender ,race or origin differences.
Based on this model, Nitzan and Paroush (1980), Karotkin and Paroush (1995) and BenYashar and Paroush (2003) obtain the following results.
1) Given a standard learning function such as the logistic function, which is commonly used in psychology and education, the optimal social policy is first to invest in the least competent member of the team and only then in the more competent ones and only finally in the most competent member.
2) Denote by c* = (c*_{1},c*_{2},…,c*_{n}) the socially optimal investment in each member where the investment in each person is decided in a centralized system. Denote by c** = (c**_{1},c**_{2}, ..., c**_{n}) the optimal investment that is determined by each individual in a decentralized system. Assume that individual's reward is independent of her own vote but depends only on the success of the whole team. Under this assumption, the motivation to freeride emerges and the necessary result is that c* >c**.
3) An incentivescheme which compensates not only for the team's success but also for the individual's success is a possible remedy against free riding . However, such a remedy has two deficiencies as byproducts. First, it encourages individuals to vote insincerely (see Section 9). Second, it encourages individuals to violate the assumption of independence (see Section 8). For instance, even if the simple majority is the optimal decision rule and even if individuals share identical preferences, each of them will try in this case to copy the vote of the most competent member in order to increase the likelihood of her personal reward.
4) Modern technology offers ways to overcome the above deficiency. Secret voting together with automatic recording of votes will guarantee both the independence as well as the possibility to compensate individuals for correct voting.
It is worth noting that freeriding behavior in information acquisition affects not only the optimal investment in human capital, but also the optimal size of the team. Gradstein and Nitzan (1987) and Mukhopadhaya (2003) find, for instance, that in contrast to Condorcet's second statement, a larger jury may reach worse decisions.
In an extended setting of endogenous competencies, Ben Yashar and Nitzan (2001c) demonstrate that it is no longer the case that the SMR is superior to the ER, even if individuals share identical skills. Since now the objective of the team takes into account both the probability of making a correct collective decision as well as the aggregate costs associated with investment in human capital of the team members, it may very well be the case that it is better to invest in a single member and apply the ER rather than in all the members. Thus, Condorcet's statement may not be valid. Further discussion on variability of decisional skills and of information acquisition within teams appears in Gerling et al. (2005), Gradstein and Nitzan (1988) and Nitzan and Paroush (1984c).
In concluding this section, we wish to point out that the area of the economics of decision making that takes into account various elements of cost and benefit is only in its initial stage and further developments are expected.
6. Diverse Preferences
Consider a group of individuals with different preferences (note that such a group can no longer be called “a team”). A typical such group is a political committee, a parliament, a randomly selected jury or any representative decision panel whose members have different utilities or social values. In a binary setting, it is clear that the alternative which
is considered “correct” by one of the members might be considered “incorrect” by another. There is no agreed upon “truth” to seek; each member seeks her own subjective truth. Social choice theory struggles to find a decision rule that aggregates the diverse individual preferences and aims to come up with some “desirable” social preference relation or some common social objective. The classical normative approach establishes such objectives or social welfare functions on the basis of philosophical considerations or principles of justice, Rawls (1971). But the more modern approach in the social choice literature is axiomatic. The axiomatic approach assumes a set of “reasonable” axioms or “desirable” properties that are deemed as necessary and sufficient conditions for the derivation of an aggregation rule that enables society to take a collective action even in situations where conflicts of interests prevail among its members. We shall mention here a few studies that use the axiomatic approach in a binary setting. May (1952) derives SMR, Fishburn (1973) and Nitzan and Paroush (1981) characterize WMR and Houy (2007) and Quesada (2010) come up with qualified majority rules.
The following dilemma still remains. Which of the two juristic systems is socially preferable: a randomly selected group of jurors who possess diverse preferences as well as a variety of social norms or a team of professional judges who share a common goal of seeking the truth? Obviously, this dilemma has important applications, but as far as we know, there is neither an answer to this question nor even a theoretical framework within which the question can be analyzed.
In the context of Condorcet's setting, given the individuals' common objective and diverse information which yields their decisions, the optimal collective decision rule can be identified, as we have seen in Section 3. However, in a binary setting and diverse preferences, one can reach these same optimal collective decision rules by their unique axiomatic characterization. In a more general multialternative setting, however, the potential success of the axiomatic approach is clouded by the classical Impossibility Theorems of Arrow (1951) and his followers. As is well known, if few reasonable axioms have to be satisfied by the aggregation rule, a social welfare function does not exist. This type of finding, which has become one of the cornerstones of social choice theory, has raised a wave of works that attempted (mostly in vain) to disperse the pessimistic atmosphere implied by Arrow's ”Impossibility Theorem”.
