Conjunction and aggregation.

• Author: Levmore, Saul

INTRODUCTION

This Article begins with the puzzle of why the law avoids the issue of conjunctive probability. Mathematically inclined observers might, for example, employ the "product rule," multiplying the probabilities associated with several events or requirements in order to assess a combined likelihood, but judges and lawyers seem otherwise inclined. Courts and statutes might be explicit about the manner in which multiple requirements should be combined, but they are not. Thus, it is often unclear whether a factfinder should assess if condition A was more likely than not to be present -- and then go on to see whether condition B satisfied this standard -- or whether the factfinder's task is to ascertain if both A and B can together, or at once, satisfy the standard. A mathematically inclined judge or jury that thought a tort defendant .6 likely to have been negligent and .7 likely to have caused plaintiff's harm might conclude that plaintiff had failed to satisfy the preponderance of the evidence standard because the chance of both requirements being met is surely less than either alone and, indeed, less than .5. Yet, the law often instructs the jury to find the defendant liable, or is strangely ambiguous in its instructions. Legal practice seems at odds with scientific logic, or at least with probabilistic reasoning. I will refer to this puzzle as the "math-law divide." Although this divide is encountered frequently in law, its puzzling character is unfamiliar to most lawyers and (even) legal scholars, and it is missed entirely by most litigants and judges.

This Article seeks to explain or rationalize law's suppression of the product rule, or indeed any explicit alternative strategy for dealing with the conjunction issue. Part I discusses in greater detail the nature of the math-law divide and a number of traditional reactions to the puzzle. The Article then advances the idea that the process of aggregating multiple jurors' assessments hides valuable information. First, Part II.B. posits that the Condorcet Jury Theorem indicates that agreement among multiple jurors might raise our level of confidence in a particular determination beyond what the jurors themselves individually report. Second, Part II.C. urges that a supermajority's mean or median voter is likely to have a different assessment from that gained from the marginal juror. As such, a supermajority (or unanimity) rule may take the place of the product rule where there are multiple requirements for liability or guilt. An attempt to extract this inframarginal information more directly would likely generate strategic behavior problems in juries. Part III extends this analysis to panels of judges, for whom "outcome voting" may (somewhat similarly) substitute for the product rule.

1. THE CONJUNCTION PROBLEM

   1. Law's Suppression of the Conjunction Problem and the Product Rule

      Consider the straightforward problem of combining two judgments concerning two or more elements of a legal claim. If, for example, the law holds A liable to B when A is negligent and when this negligence has (proximately) caused B's injury, a factfinder must evaluate the likelihood of A's negligence and the likelihood of the causal link between this negligent behavior and B's injury. An illustrative jury instruction is as follows:

      In order to prove the essential elements of plaintiff's claim, the burden is on the plaintiff to establish, by a preponderance of the evidence in the case, the following facts: First, that the defendant was negligent in one or more of the particulars alleged; and Second, that the defendant's negligence was a proximate cause of some injury and consequent damage sustained by the plaintiff.(1) Quite generally, juries are told that the "essential question is whether the evidence taken as a whole, both direct and circumstantial, establishes every element of the plaintiffs' case by a preponderance of the evidence."(2)

      Imagine now that the factfinder concludes that there is a .7 chance of negligence and a .6 chance of causation.(3) Doctrinally, the law seems to require that A pay if and only if A is negligent and causes B's harm. The question is whether this "and" is conjunctive. Most people who are experienced in probabilistic thinking hurry to say that A should be liable if A is both negligent and the causal agent, and that this combined probability is (.7)(.6) = .42. The product of the two probabilities, or likelihood of these two events, is thus less than the .5 hurdle established by the preponderance of the evidence ("POE") standard normally applied to civil claims.(4)

      In contrast, most lawyers who have thought about this subject regard the (representative) jury instructions as calling for holding the defendant liable in this case because plaintiff apparently satisfies the first requirement (inasmuch as .7 exceeds the .5 trigger established by the POE standard), and also satisfies the second requirement (again, inasmuch as .6 exceeds the .5 benchmark). At the risk of oversimplification, the problem is that the mathematics of the matter instructs us to multiply the two probabilities, following what is known as the "product rule" (for combining independent probabilistic assessments).(5) Law, however, appears not to abide by this rule. Hence the math-law divide.

   2. Reactions to the Math-Law Divide

      There are a number of conventional reactions to the math-law divide, including simple denial. In asking whether it is more likely than not that two coin tosses will yield two heads more than one in three times, everyone would agree that the problem solver should multiply .5 times .5 to see that the answer is no, for two heads are expected but one in four times. But when the questions are whether A was more likely than not to have been driving negligently and whether such driving caused B's neck injury (where there is some chance that B has no real injury or a preexisting condition), reasonable people are comfortable with the idea that if each answer is that it is .6 likely we should stop there and find A responsible. We are, perhaps, not looking for the conjoined probability that both things are true at the same time. The question is why intuitions about conjunction vary.

      1. Independence

         One reaction to the law's disinclination to use the product rule focuses on the likelihood that issues like negligence and causation are not likely to be perfectly independent of one another. The product rule applies only where the events or requirements in question are independent. That interdependence changes the way we ought to combine probabilities is obvious; the chance of flipping three heads in a row with a fair coin is (.5)(.5)(.5) = .125; it is expected to occur once in eight times. But if I know that the coin is weighted so that it will always come out the same, I need only flip it once to see whether it is weighted one way or the other. Now the tosses are completely interdependent and the probability of three consecutive heads (or tails) is .5.

         If fact finders are given multiple requirements for liability, and these requirements are highly interdependent, then a blanket instruction to apply the product rule would seriously underdeter defendants and undercompensate plaintiffs compared to the ideal set out by law. To take an extreme case, if a factfinder assesses the likelihood of defendant's negligence in a case by defendant's demeanor as a witness, then it is quite likely that a comparable assessment of causation amounts to drawing the same conclusion twice in a way that makes the estimates highly interdependent. Somewhat similarly, if B claims that pharmaceutical company A's nondisclosure caused B's allergic reaction, then it may well be that causation and negligence are virtually the same question. If A's drug has the side-effect that B asserts, then the factfinder might well conclude that A was negligent not to have warned of it, and the same inquiry is likely to drive the causation assessment. If events are completely interdependent and the factfinder thinks that each is still .6 likely, then .6 rather than .36 is our best assessment of the likelihood that A satisfies the two requirements for liability. Where multiple requirements are entirely interrelated, application of the product rule is unnecessary. Moreover, there are cases -- but only some -- in which a greater likelihood of one requirement (such as negligence) does imply a greater likelihood of the other (such as causation) if only because it becomes less likely that the plaintiff would have suffered the injury if defendant had avoided its failure as to the first requirement.(6)

         The concern for independence is most convincing if factfinders are incapable of following instructions regarding interdependent and independent events. It may be that where elements are independent, factfinders allow their view of one element to influence their assessment of another.(7) But even so, if there is so much interdependence between two requirements for a decision that we are better off not multiplying a factfinder's estimates, then it often follows that there is little need to have the second requirement in the first place. If the connection between two requirements is nearly perfect, so that whenever there is negligence there is also causation, for instance, then the law need only ask whether it is more likely than not that there has been negligence. Generally speaking, if it is worthwhile to ask multiple questions, then logic or math suggests that we do better applying the product rule -- though we need to be careful about independence and conditional probabilities.(8) But such generalizations may be beside the point because we can instruct the factfinder(s) not only about multiple requirements and the product rule, but also about the necessary modifications if the same factfinder deems the requirements to be somewhat interdependent.

      2. Misuse of Probabilities

         A second reaction to the math-law divide is that...