1 See, e.g., Ellman, Ira M. & Kaye, David, Probabilities and Proof: Can hla and Blood Group Testing Prove Paternity? 54 N.Y.U. L. Rev. 1131 (1979);Kaye, David, The Laws of Probability and the Law of the Land, 47 U. Chi. L. Rev. 34 n.1 (1979).Google Scholar

2 Letter from Richard Lempert to David Kaye, Apr. 2, 1980, p. 1.Google Scholar

3 Kaye, David, Naked Statistical Evidence (Book Review), 89 Yale L.J. 601 (1980) (reviewing Finkelstein), infra note 6.Google Scholar

4 E.g., Kaminsky v. Hertz Corp., 94 Mich. App. 356 (1979);Kaye, David, Paradoxes, Gedanken Experiments and the Burden of Proof: A Response to Dr. Cohen's Reply, 1981 Ariz. St. L.J. 635.Google Scholar

5 L. Jonathan Cohen, The Probable and the Provable (London: Oxford University Press, Clarendon Press, 1977); Brilmayer, Lea & Kornhauser, Lewis, Review: Quantitative Methods and Legal Decisions, 46 U. Chi. L. Rev. 116, 135–48 (1978);Cohen, L. Jonathan, Subjective Probability and the Paradox of the Gatecrasher, 1981 Ariz. St. L.J. 627.Google Scholar

6 Finkelstein, Michael O., Quantitative Methods in Law: Studies in the Application of Mathematical Probability and Statistics to Legal Problems 69 (New York: Free Press, 1978).Google Scholar

7 Saks, Michael J. & Kidd, Robert F., Human Information Processing and Adjudication: Trial by Heuristics, 15 Law & Soc'y Rev. 123, 151 (198081) (decrying “the myth of particularistic proof”); Williams, Glanville, The Mathematics of Proof I, 1979 Crim. L. Rev. 297, 305.Google Scholar

8 See Kaye, , supra note 4, at 635 n.1.Google Scholar

9 A less procrustean rule might serve this same function. Where a party inexplicably fails to produce evidence under circumstances in which he would be expected to have favorable evidence, an inference that the evidence is in fact unfavorable could be drawn. See authorities cited, Kaye, David, Probability Theory Meets Res Ipsa Loquitur, 77 Mich. L. Rev. 1456, 1475 n. 59 (1979).Google Scholar

In a few situations a rule disfavoring naked statistical evidence can be defended on another ground as well. Cohen's “paradox of the gatecrasher” (see note 5 supra) is a good illustration of a case in which the plaintiff has no evidence with which to distinguish any specific defendant from many other possible defendants who are equally likely to be liable. It can be argued that notwithstanding whatever probability theory may teach us, assuring the appearance of fairness precludes imposing liability in the absence of some evidence singling out particular defendants.Google Scholar

10 26 Cal. 3d 588, 163 Cal. Rptr. 132, 607 P.2d 924 (1980).Google Scholar

11 The Supreme Court left open the possibility that a defendant could somehow demonstrate at trial that it did not market the quantity of DES that had caused plaintiff's cancer. 26 Cal. 3d at 612. It also indicated that plaintiff must join the manufacturers of a “substantial share” of the DES marketed for the prevention of miscarriages. Id. Commentary on these and other aspects of Sindell includes Robinson, Glen O., Multiple Causation in Tort Law: Reflections on the des Cases, 68 Va. L. Rev. 713 (1982);Note, Sindell v. Abbott Laboratories: A Market Share Approach to des Causation, 69 Calif. L. Rev. 1179 (1981) [hereinafter cited as A Market Share Approach]; Note, Market Share Liability: An Answer to the des Causation Problem, 94 Harv. L. Rev. 668 (1981) [hereinafter cited as Market Share Liability]; Case Comment, Refining Market Share Liability: Sindell v. Abbott Laboratories, 33 Stan. L. Rev. 937 (1981) [hereinafter cited as Refining Market Share Liability].Google Scholar

12 26 Cal. 3d at 611, 163 Cal. Rptr. at 145, 607 P.2d at 937.Google Scholar

13 Id. at 616, 163 Cal. Rptr. at 147, 607 P.2d at 939.Google Scholar

14 It has been said that the particular form of cancer linked with DES, clear-cell adenocarcinoma, used to be rare. Comment, des and a Proposed Theory of Enterprise Liability, 46 Fordham L. Rev. 963, 965 n.8 (1978) (citing Ulfelder, The Stilbestrol-Adenosis-Carcinoma Syndrome, 38 Cancer 426, 428 (1976)) [hereinafter cited as Comment, DES and a Proposed Theory].Google Scholar

15 It might be argued that DES is but one of several contributing causes. If such joint causation were present, the epidemiological data might be used to assess the relative magnitude of the contribution from DES. Whether it would then be appropriate to apportion damages in light of this figure is an interesting question. See, e.g., text and accompanying notes 70–73 infra. The apportionment issue, however, is distinct from the naked statistical evidence problem. It would arise even if the extent of relative contributions were quantified on the basis of “individualized” evidence.Google Scholar

16 See text accompanying notes 22–25 infra.Google Scholar

17 See generally, e.g., Easterbrook, Frank H., Landes, William M., & Posner, Richard A., Contribution Among Antitrust Defendants: A Legal and Economic Analysis, 23 J.L. & Econ. 331 (1980);Landes, William M. & Posner, Richard A., Joint and Multiple Tortfeasors: An Economic Analysis, 9 J. Legal Stud. 517 (1980); Polinsky, A. Mitchell & Shavell, Steven, Contribution and Claim Reduction Among Antitrust Defendants: An Economic Analysis, 33 Stan. L. Rev. 447 (1981);Rizzo, Mario J. & Arnold, Frank S., Causal Apportionment in the Law of Torts: An Economic Theory, 80 Colum. L. Rev. 1399 (1980); Robinson, supra note 11.Google Scholar

18 See note 15 supra and text accompanying notes 70–73 infra.Google Scholar

19 Compare 26 Cal. 3d at 612, 163 Cal. Rptr. at 145, 607 P.2d at 937 (“If plaintiff joins in the action the manufacturers of a substantial share of the DES which her mother might have taken, the injustice of shifting the burden of proof to defendants … is significantly diminished”) with id. at 612 n.28, 163 Cal. Rptr. at 145 n.28, 607 P.2d at 937 n.28 (“[i]f X [manufacturer sold one-fifth of all the DES prescribed for pregnancy and identification could be made in all cases, X would be the sole defendant in approximately one-fifth of all cases and liable for damages in those cases” (quoting Comment, DES and a Proposed Theory, supra note 14)).Google Scholar

20 33 Cal. 2d 80, 199 P.2d 1 (1948).Google Scholar

21 Although the probability figures that will be used here are merely illustrative, the possibility of some measurable discrepancy is real. See Defoliant, Cancer: Studies Show Link, 117 Sci. News 230 (1980).Google Scholar

22 In an important sense all empirical reasoning is statistical. The distinguishing feature of the naked statistical evidence cases is that reliance on a quantified probability is invited, to the exclusion of all nonquantified evidence. In justifiably naked statistical evidence cases this invitation seems appealing because the latter sort of evidence is impossible or impractical to obtain.Google Scholar

23 For simplicity, I am ignoring any latency period in carcinogenesis. The timing of disease onset in evaluating causation can be very important. See, e.g., Reyes v. Wyeth Laboratories, 498 F.2d 1264 (5th Cir.), cert, denied, 419 U.S. 1096 (1974).Google Scholar

24 See, e.g., Kaye, supra note 3; Eggleston, Richard, The Probability Debate, 1980 Crim. L. Rev. 678, 680.Google Scholar

25 A little algebra shows that λ >½ if and only if a/b<2. In other words, the “balance of the probabilities” (where λ completely captures the relevant probability) favors causation as long as the industry cancer rate is twice as large as the base rate.½+if+and+only+if+a/b<2.+In+other+words,+the+“balance+of+the+probabilities”+(where+λ+completely+captures+the+relevant+probability)+favors+causation+as+long+as+the+industry+cancer+rate+is+twice+as+large+as+the+base+rate.>Google Scholar

26 Alternatively, one could allow the worker to recover D with a probability λ=⅔ The expected value of this gamble is simply (⅔)D. A risk-averse employee would prefer the fixed sum (⅔)D to a ⅔ chance of D and a ⅔ chance of 0. A firm with the same degree of risk aversion would not be as concerned, however, because over a large number of cases it can be confident of paying an average figure very close to (⅔)D.Google Scholar

27 The phrase is taken from Steven Shavell, An Analysis of Causation and the Scope of Liability in the Law of Torts, 9 J. Legal Stud. 463, 465 (1980). Shavell's explanation of the term may be instructive:.Google Scholar

28 Assume, for simplicity, that every case of employee stomach cancer has the same cost D and that N such cases arise in an appropriate time period. On efficiency grounds, it can be aruged that the firm should pay λDN, since λ represents the increment in the stomach cancer rate due to the firm's activity. See id. The picture with regard to efficiency, however, is not as clear as Shavell seems to suggest. Whereas he treats the costs of production as fixed, it could be contended that even when the employer need not pay an explicit sum for the increment in stomach cancer due to employment, he pays this cost implicitly in the form of higher wages. If wage rates are subject to renegotiation in an efficient labor market, then they will drop once the accident costs are imposed on the employer. E.g., Demsetz, Harold, When Does the Rule of Liability Matter? 1 J. Legal Stud. 13 (1972). Another complication arises if employer contributions are not tailored to the accident costs experienced by each employer. For the purpose of this article, I shall assume that each firm pays premiums calculated to provide the proper incentive to take cost-effective precautions.Google Scholar

29 See, e.g., Kaye, supra note 1; Anne Martin & David A. Schum, Quantifying Burdens of Proof: A Likelihood Ratio Approach, Rice University Report No. 78–02 (Dec. 15, 1978).Google Scholar

30 E.g., King, Joseph H. Jr., Causation, Valuation, and Chance in Personal Injury Torts Involving Preexisting Conditions and Future Consequences, 90 Yale L.J. 1353, 1396 (1981).Google Scholar

31 Cf. Tribe, Laurence H., Trial by Mathematics: Precision and Ritual in the Legal Process, 84 Harv. L. Rev. 1329 (1971) (arguing that most jurors cannot make reasonable quantitative estimates of the probability that a defendant is guilty).Google Scholar

32 See, e.g., Alvis v. Ribar, 85 Ill. 2d 1, 421 N.E.2d 886 (1981); Carroll R. Heft & C. James Heft, Comparative Negligence Manual (Mundelein, Ill.: Callaghan & Co., 1978, 1981 Cum. Supp.); LanDES& Posner, supra note 17, at 551.Google Scholar

33 See, e.g., Garrison v. Funderburk, 262 Ark. 711, 561 S.W.2d 73 (1978); Downum v. Muskogee Stockyards & Livestock Auction, Inc., 565 P.2d 368 (Okla. 1977). Some fact-finders proffer such quantitative estimates even when not obliged to do so. See The Times (London) 60,948 (June 8, 1981, at 2, col. 8) (industrial tribunal finds employee 60 percent to blame for his dismissal).Google Scholar

34 To preclude this possibility, one might modify the expected value rule by insisting on some threshold level of p before permitting any recovery. The formal analysis of the expected value rule (to be developed shortly) applies, with obvious variations, to this modified expected value rule.Google Scholar

35 Plaintiffs may also be able to induce settlements by imposing heavy discovery and other trial preparation costs on defendants. A change in the evidentiary standard of the sort contemplated here should have no long-run effect on such matters.Google Scholar

36 For a discussion of the attitude of firms toward risk, see, e.g., Easterbrook, Landes, & Posner, supra note 17, at 351–53 n.50 (concluding that “the extent and intensity of risk aversion among firms is an unsettled empirical question”).Google Scholar

37 See, e.g., Polinsky & Shavell, supra note 17, at 457–62; authorities cited, id. at 460 n.43 (considering the additional factors of litigation costs and differences of opinion about winning). See generally Riddell, W. Craig, Bargaining Under Uncertainty, 71 Am. Econ. Rev. 579 (1981);Steven Shavell, Suit, Settlement, and Trial: A Theoretical Analysis Under Alternative Methods for the Allocation of Legal Costs, 11 J. Legal Stud. 55 (1982).Google Scholar

38 One can say that to the extent that the expected value rule reduces the “risk premium” paid by parties who settle, it lowers social costs, enhancing the attractiveness of the rule from the standpoint of economic efficiency.Google Scholar

39 This equality is not intended to reflect the values held by jurors or judges in particular cases or to describe the costs to or utility functions of particular plaintiffs and defendants. It is a statement about institutional values, about the relative importance of these types of mistakes in the eyes of “the law.”. This conception of “blindfolded” justice is, I believe, widely shared. Whether it can be motivated solely by an efficiency argument is a nice question. One such argument goes something like this: It would be best to look to the opportunity costs (willingness to pay to avoid errors) to each party in each case if these costs could be cheaply measured. Because this administrative cost is very high, however, it is more efficient to use average figures, and on average the cost of each type of error is the same. This efficiency argument would not hold to the extent that there are easily identified classes of cases in which the costs of errors to each side diverge. If large businesses are risk neutral and individuals with small assets are risk averse, for example, the cost of a dollar erroneously “paid” by the latter exceeds that for the former. One suspects, however, that an instruction to consider the depth of a litigant's pocket would be superfluous. In cases where moral censure or other collateral effects would result from an award of damages, costs to plaintiffs and defendants clearly are not proportional to the dollars wrongfully paid, and the law requires plaintiffs to do more than adduce a preponderance of the evidence. See, e.g., John Kaplan, Decision Theory and the Fact finding Process, 20 Stan. L. Rev. 1065, 1072 (1968).Google Scholar

40 An essentially identical proposition was proved from a slightly different perspective in Kaye, supra note 3, at 605 n.19. The present development provides additional insights.Google Scholar

41 Figure 2 shows that when p₁= p₂ =½, all the decisions are equally effective in minimizing expected losses. To break these ties the p>½ rule awards the verdict to the defendant in those cases.½+rule+awards+the+verdict+to+the+defendant+in+those+cases.>Google Scholar

42 The proof is straightforward, but worth stating, paying attention to the way the choice of the function x(p₁) affects the rate of type I versus type II errors. The loss matrix is now given by figureGoogle Scholar

If the costs of each type of error are the same, the expected loss function is just the sum of the expected number of false positive and false negative dollars:Google Scholar

In a given case p₁, p₂ , and D are fixed, and f is therefore minimized by choosingGoogle Scholar

Since p₁p₂ is the same as p₁>½, we have again arrived at the more-probable-than-not rule.½,+we+have+again+arrived+at+the+more-probable-than-not+rule.>Google Scholar

43 Finkelstein has spoken of raising the threshold of the p>½ rule to some figure larger than ½ to make it an error equalizing rule. See Finkelstein, supra note 6. The expected value rule equalizes expected errors in the same sense as Finkelstein's modified p>½ rule, but it operates in a distinctive way.½+rule+to+some+figure+larger+than+½+to+make+it+an+error+equalizing+rule.+See+Finkelstein,+supra+note+6.+The+expected+value+rule+equalizes+expected+errors+in+the+same+sense+as+Finkelstein's+modified+p>½+rule,+but+it+operates+in+a+distinctive+way.>Google Scholar

44 Since p₁p₂ and p₁+p₂=1, it follows that 2p₁p₂D>p₂D.p2D.>Google Scholar

45 Statistical inference consists of using sample data to reach conclusions about the population being sampled. Suppose we wish to estimate some numerical characteristic of a large group on the basis of a limited number of randomly drawn observations. For any particular sample, the estimate will not necessarily correspond precisely to the population value. Sometimes it may be on the high side, sometimes on the low side. If the errors systematically fall in one direction, the estimator is said to be “biased.” If the errors are balanced, so that on average (in the limit) the estimator is accurate, it is said to be “unbiased.”. “Unbiased” estimators can be a mixed blessing, however. For any sample, an unbiased estimator could be very inaccurate. In some applications a more accurate, albeit biased, estimator may be preferred. Selecting the “best” estimator is often a subtle matter, not amenable to rigid rules.Google Scholar

46 One could also randomize the guesses in such a way as to announce that a marble is red in two-thirds of the selections. See note 25 supra.Google Scholar

47 See note 27 supra.Google Scholar

48 It should be clear that the analysis is confined to the expected value rule that weights damages by the probability of liability. Another expected value rule is appropriate for measuring damages themselves in situations involving future contingencies and losses of valuable chances. See, e.g., King, supra note 30.Google Scholar

49 It does this by solving the most elementary sort of problem in the branch of operations research known as linear programming. For a nontechnical introduction to the field, see Robert G. Bland, The Allocation of Resources by Linear Programming, 244 Sci. Am., June 1981, at 126.Google Scholar

50 I use the term “two-defendant” (and “n-defendant”) rather loosely to indicate that more than one person might have independently caused the legally cognizable injury. Factual ambiguity prevents us from knowing which such person did so. Whether all such potential defendants are actually joined or impleaded is not crucial. Independence—that either defendant one or defendant two or defendant three, etc., caused the single injury—is critical. Persons acting in concert, concurrent causes producing indivisible injuries, and indemnity defendants all present “one-defendant” cases. See text accompanying notes 69–70 infra.Google Scholar

51 Including the possibility that neither defendant is truly liable is straightforward. One need merely introduce an additional probabilty p _o for the additional state s _o, (that plaintiff is liable). As far as the mathematics go, we have a three-defendant problem with the plaintiff playing the role of the third defendant.Google Scholar

52 The inequalities (1)–(3) constrain x ₁ and x ₂ to the triangular region depicted in figure B.Google Scholar

The function f, being a linear combination of x₁and x ₂, is a portion of a plane lying above this feasible region. To sketch this plane, we need a third axis perpendicular to the x ₁ and x ₂ axes. From equation 4, we can readily find the height off at the vertices of the feasible region. When x ₁=D and x ₂=0, then f =p ₂ D. When x ₁= 0 and x ₂=D, then f=p₁D. When x ₁=x ₂= 0, then f=D. Since three points determine a plane, we can now graph f. Suppose that p ₁=½. Then p₁D = p₂D = D/2, and the bottom edge of the portion of the plane projecting above the feasible region parallels the hypotenuse of that region, as shown in figure C.Google Scholar

We can see that f is at its lowest as long as plaintiff is fully compensated, regardless of how much each defendant contributes. In other words, when each defendant is equally likely to be the liable party, the minimization criterion gives no guidance as to how Δ₁ and Δ₂ should share in the payment of D to the plaintiff. The criterion is met as long as the plaintiff recovers. Either Δ₁and Δ₂may be treated as jointly and severally liable, or equitable principles of contribution may be applied. For discussion of the merits of these alternatives, see, e.g., Landes & Posner, supra note 17; Robinson, supra note 11. Of greater interest are the more prevalent situations in which p ₁ does not equalp ₂. If P ₁ P ₂, f drops to its lowest point when x₁= D and X ₂= 0. This is shown in figure D.Google Scholar

Finally, if pl<pl, the lowest value off lies above the point x= 0 and x= D, as revealed in figure E.Google Scholar

53 The expected loss function is now a portion of a plane extending over a feasible region in an n-dimensional space. Let x be a vector whose n components x ₁ through X_n represent the money paid by Δ₁ through Δ _n , respectively. Similarly, let p be a vector whose i th component p ₁ stands for the probability that Δ_i caused the damage. Then the expected loss is justGoogle Scholar

whereGoogle Scholar

The functionf is minimized by making x·p as large as possible for eachp. Suppose that max (p₁)=P₁ In light of the constraints on the components of x and the meaning of the scalar product, x·p then maximized by letting X ₁=D and x _i= 0 (where i≠j)—by having the single most probably liable defendant pay all the damages. In the event that no single component of p is larger than all the others, several defendants emerge as the equally likely and most probably liable parties, and the minimization principle does not enable us to choose among them.Google Scholar

54 Where there are only two defendants, one of whom must be liable (or one plaintiff and one defendant), this cannot happen because the larger of the probabilities associated with the parties must exceed ½ That is, the n-defendant solution reduces to thep>½ rule when n= 2, as indeed it must.½+rule+when+n=+2,+as+indeed+it+must.>Google Scholar

55 In deriving this result, we assumed that the only issue in dispute is the identity of the single, fully culpable party, so that the justifiably naked statistical evidence supplied the probabilities that each potential defendant is in fact liable. The same result also applies if the probablities p_o, P _i, …, P_n are subjective estimates of liability based on all the evidence in the case. How the probablities pertaining to each element of the cause of action should be combined to obtain the overall probability of liability is beyond the scope of this article. This “problem of conjunction” is discussed in Cohen, The Probable and the Provable, supra note 5; Carl G. Wagner, Book Review, 1979 Duke L.J. 1071 (reviewing Cohen, The Probable and the Provable).CrossRefGoogle Scholar

56 See Rizzo & Arnold, supra note 17; Robinson, supra note 11.Google Scholar

57 26 Cal. 3d at 612, 163 Cal. Rptr. at 145, 607 P.2d at 937.Google Scholar

58 See text accompanying note 48 supra.Google Scholar

59 All this is a straightforward application of what was said in part II about the workers' compensation problem to the DES context. Yet, there is an intriguing distinction between the two cases. The point can be elucidated by recasting the DES situation slightly. Imagine that every DES victim joins in a class action against every DESproducer. Although no single victim can prove which producer distributed the quantity that harmed her, such individualized proof would seem unnecessary and wasteful. If the probabilities used in dividing the damages among the producers are accurate, then the expected value approach quickly accomplishes what the individualized method of proof laboriously strives for: compensating each DES victim and charging each injurer for the cost of the injuries it caused. The fact that company A may pay part of company B's victim's costs while company B does the same for company A's victim is hardly a cause for alarm. Seen in this light, the DES problem is better suited to the expected value rule than the phenoxy acids illustration. In the latter, only a fraction of the workers “deserve” compensation, but there is no way to match the employer only to these deserving workers, and we end up awarding every afflicted worker a reduced sum. In the DES situation, the causation problem amounts to matching the right firm with the right victim. See Refining Market Share Liability, supra note 11. There is no way to do this, but here the expected value approach does not reward any “undeserving” victims, and it awards the proper sum to each victim. For this reason, its use in Sindell is even more defensible than in the hypothesized phenoxy acids case. But see A Market Share Approach, supra note 11, at 1187–88 (administrative costs imposed on defendants may be excessive in cases involving many potential defendants).Google Scholar

60 If the expected value rule is used, however, it would seem that the “substantial share” requirement serves no meaningful function. See Robinson, supra note 11. But see A Market Share Approach, supra note 11, at 1197–99 (suggesting, among other things, that “[h]aving the major producers in court will facilitate the determination of the dimensions of the relevant market”). In addition, a more exacting measure of the probability than overall market share may be available. See Refining Market Share Liability, supra note 11. It should also be clear that the expected value rule is tantamount to what some commentators have called “pro rata” liability. E.g., A Market Share Approach, supra note 11, at 1196. It does not permit 100 percent of the liability to be apportioned among defendants who collectively marketed less than 100 percent of the relevant DES—a point that troubled the sole dissenter in Sindell. See 26 Cal. 3d at 617, 163 Cal. Rptr. at 148, 607 P.2d at 940.Google Scholar

61 33 Cal. 2d 80, 119 P.2d 1 (1948). For a perceptive comparison of the two cases developing this theme, see Robinson, supra note 11.Google Scholar

62 This much is required by the assumption that p(s_O)= 0, i.e., that the plaintiff would not be required to bear the cost of the accident if all the material facts were known with certainty. See note 51 supra. One might ask why the substantive law requires an injurer to compensate his innocent victims. See, e.g., George P. Fletcher, Fairness and Utility in Tort Theory, 85 Harv. L. Rev. 537 (1972); Richard A. Posner, The Concept of Corrective Justice in Recent Theories of Tort Law, 10 J. Legal Stud. 187 (1981); Robinson, supra note 11. The analytical tools constructed here can shed no light on such questions.Google Scholar

63 The court also relied on the more dubious proposition “[o]rdinarily defendants are in a far better position to offer evidence to determine which one caused the injury.” 33 Cal. 2d 80, 86 (1948). As the California court recognized in Sindell, however, the principal concern of Summers is that “if one [defendant] can escape the other may also and plaintiff is remediless.” 199 P.2d 1 (1948) at 4.Google Scholar

64 See, e.g., Market Share Liability, supra note 11, at 672.Google Scholar

65 See text accompanying note 51 supra.Google Scholar

66 495 F.2d 213 (6th Cir. 1974).Google Scholar

67 Id. at 216, –quoting Maddux v. Donaldson, 362 Mich. 425, 108 N.W.2d 33 (1961).Google Scholar

68 See text accompanying notes 76–77 infra.Google Scholar

69 This is not to say that the damage from pollution is necessarily a continuous function of the quantity of the pollutants. On the contrary, if a certain threshold amount is required before a type of injury occurs, the cost function will be discontinuous. See, e.g., Bruce A. Ackerman, Susan Rose Ackerman, & Dale W. Henderson, The Uncertain Search for Environmental Quality: The Costs and Benefits of Controlling Pollution Along the Delaware River, 121 U. Pa. L. Rev. 1225 (1973). Where discontinuities exist, it may be possible to find that some defendants are “but for” causes. Had they not contributed, the threshold would not have been reached and the damage would not have occurred. There is a problem, of course, in deciding which defendants come within this category. If some of the defendants begin polluting earlier and have a prior right to pollute so that their conduct is not tortious, then the polluter whose emissions pushed the total above the threshold could be held liable (in the amount the cost curve jumps). This attention to marginal costs in apportioning damages in some concurrent cause cases seems promising, but the concept plainly needs more development. See Wittman, Donald, Optimal Pricing of Sequential Inputs: Last Clear Chance, Mitigation of Damages, and Related Doctrines in the Law, 10 J. Legal Stud. 65 (1981).Google Scholar

70 See note 50 supra.Google Scholar

71 Rizzo & Arnold, supra note 17.Google Scholar

72 Refining Market Share Liability, supra note 11.Google Scholar

73 Robinson, supra note 11.Google Scholar

74 Consider, e.g., Rizzo and Arnold's proposal to apportion damages among “simultaneous causes” according to a particular function of certain probabilities. Rizzo & Arnold, supra note 17, at 1410–11, 1415. When the causes act “simultaneously,” they advocate dividing the aggregate damages in proportion to the probability that each defendant's conduct, acting alone, would have caused this amount of damage. When the causes interact, they offer a formula for measuring the “synergistic” contribution and dividing it equally among the defendants. There are serious difficulties with their formula (see David Kaye & Mikel Aickin, A Comment on a Proposed “Economic Method” of “Causal Apportionment” (1982) (unpublished article manuscript)), and there is reason to question the ability of “economic theory” to allocate jointly caused damages without resort to what are basically accounting conventions. See Armen Alchain & William R. Allen, Exchange and Production: Competition, Coordination, and Control 256 (2d ed. Belmont, Cal.: Wadsworth Publishing Co., 1977) (impossibility of apportioning the cost of a common input to two products).Google Scholar

75 If there are several equally likely causes, as in Summers v. Tice, the maximum likelihood rule leaves us indifferent as to how to allocate the damages among these causes.Google Scholar

76 The cases that seemed suitable candidates for the expected value rule involved the real possibility of repeated recovery from the same defendant. For example, under the maximum likelihood rule the pharmaceutical house that marketed the largest quantity of DES in California could be held liable for every DES -caused injury in California. The argument for the expected value approach also can apply when distinct defendants are implicated in different cases, yet certain probabilities remain fixed from one case to the next. Consider the scenario suggested by Glen Robinson: “Suppose [plaintiff] is exposed to carcinogen A that … creates a 20% probability of cancer. Carcinogen B enhances the risk of cancer by 10% (given a state of the world, which now includes A, the probability of cancer is now 30%)…. Is it not possible that carcinogen B might never (or hardly ever) be the ‘most likely’ cause of cancer, albeit still a ‘culpable’ (unreasonable) contribution to the risk of cancer? Should we not therefore be concerned about the lost deterrence?” Letter from Glen O. Robinson to David Kaye, Feb. 25, 1982. The analysis developed in this article will apply if either A or B, but not both, caused the plaintiff's cancer. (If a single tumor could have resulted from the simultaneous action of the two carcinogens, the mathematical proofs are inapposite, and substantive doctrines of contribution must be considered.) If the risk due to A is independent of the risk due to B, then the maximum likelihood rule does make A liable (since A had a greater chance than the next most likely contender, B, of producing the cancer). If this is a recurring situation—if reasonable victims never confront the risk from B in the absence of that created by A—so that producers of B never would be found liable, then the maximum likelihood rule would yield biased results and “lost deterrence.” This much follows from the fact that the probabilities associated with A and B are presumed to be fixed across a wide range of cases. At the same time, we might hesitate before moving to an expected value rule in cases like these. In light of the vast number of carcinogens, manufacturers, and activities in which people engage, the cost of trying to apply expected value liability through the system of private tort litigation could well be prohibitive. Further discussion of the broad-based use of expected value liability can be found in Robinson, supra note 11.Google Scholar

77 As to Sindell, we also questioned the need for individualized matching in the first place.Google Scholar

78 Again, in cases of “ties” among defendants, other doctrines or theories about contribution and joint and several liability must be consulted.Google Scholar

79 In the situations of potential “bias,” as I have defined them, one can say that the expected value approach reduces one sort of “error,” since it imposes correctly estimated aggregate costs on the culpable parties. In addition, in a few such situations, the expected value approach is error minimizing, even as that term is used here. See text accompanying notes 53 and 59 supra.Google Scholar