1 John Fortescue, De Laudibus Legum Angliæ 62 (London: Companie of Stationers, 1616).Google Scholar

2 2 Matthew Hale, The History of the Pleas of the Crown 289 (In the Savoy: E. & R. Nutt, & R. Gosling, 1736).Google Scholar

3 4 William Blackstone, Commentaries on the Laws of England 352 (Oxford, Eng.: Clarendon Press, 1769).CrossRefGoogle Scholar

4 May, J. Wilder, “Some Rules of Evidence,” 10 Am. L. Rev. 642, 651–64 (1876).Google Scholar

5 Coffin v. United States, 156 U. S. 432, 455–56 (1895).Google Scholar

6 Birmingham, Robert, “Remarks on ‘Probability’ in Law: Mostly, a Casenote and a Book Review,” 12 Ga. L. Rev. 535, 536–37 (1978); Alan D. Cullison, “Probability Analysis of Judicial Fact-Finding: A Preliminary Outline of the Subjective Approach,” 1969 Toledo L. Rev. 538, 566–67; id., “The Model of Rules and the Logic of Decision,” in Stuart S. Nagel, ed., Modeling the Criminal Justice System 225, 238, 244 n. 13 (Beverly Hills, Cal.: Sage Publications, 1971) (“Cullison, ‘Model of Rules' “); Ward Edwards, “Influence Diagrams, Bayesian Imperialism, and the Collins Case: An Appeal to Reason,” 13 Cardozo L. Rev. 1025, 1063–64 & n. 80 (1991); Bernard Grofman, “Mathematical Models of Juror and Jury Decision-Making: The State of the Art,” in Bruce D. Sales, ed., The Trial Process 305, 314–15 (New York: Plenum Press, 1981) (“Grofman, ‘Mathematical Models’“); Richard O. Lempert, “Modeling Relevance,” 75 Mich. L. Rev. 1021, 1038 (1977); Stuart Nagel, “Bringing the Values of Jurors in Line with the Law,” 63 Judicature 189, 191–92 (1979); Stuart Nagel, David Lamm, & Miriam Neef, “Decision Theory and Juror Decision-Making,” in Sales, The Trial Process 353, 355–57, 359–60 n. 4, 366, 376, 378, 381 (“Nagel et al., ‘Decision Theory’“) (suggesting that Blackstone's conviction is perhaps the best quantitative statement of what the Constitutional framers meant by due process); Stuart S. Nagel & Miriam Neef, “Deductive Modeling to Determine an Optimum Jury Size and Fraction Required to Convict,” 1975 Wash. U. L. Q. 933, 943 n. 17, 945–46, 959, 963, 965 (noting the influence of Blackstone on the authors of the Constitution).Google Scholar

In Ballew v. Georgia, 435 U. S. 223, 234 (1978), the Supreme Court referred to Nagel & Neef's, supra, use of a 10:1 utility ratio as “perhaps not an unreasonable assumption” but made no mention of the fact that this ratio was based on Blackstone's statement. Nevertheless, Richard S. Bell, “Decision Theory and Due Process: A Critique of the Supreme Court's Lawmaking for Burdens of Proof,” 78 J. Crim. L. & Criminology 557, 562 (1987), noted the Court's statement and suggested that “perhaps the Court has evaluated the relative disutilities of errors in criminal trials in light of Blackstone's principle.” However, the Court has never explicitly paraphrased Blackstone in terms of utilities, nor has it endorsed a particular ratio of utilities. On the other hand, Justice Douglas once elevated Blackstone's ratio of error frequencies to the level of a onstitutional guarantee when writing outside the Court: “Constitutional guarantees which ensure a fair trial are not slick lawyers' tricks by which known criminals are set free. In a democratic society, the means are all important. We are a civilized people…. We believe that it is better for ten guilty people to be set free than for one innocent man to be unjustly imprisoned.” William O. Douglas, Foreword to Jerome Frank & Barbara Frank, Not Guilty 11, 11 (Garden City, N. Y.: Doubleday, 1957). Although this last sentence was quoted in Furman v. Georgia, 408 U. S. 238, 367 n. 158 (Marshall, J., concurring), the Court has never committed itself to such a ratio. In summary, the Court has entertained both ratios of error utilities and ratios of error frequencies, but it has never equated the two. See infra notes 15 and 78 and accompanying text.Google Scholar

7 Nagel, 63 Judicature at 191.Google Scholar

8 See generally Leonard J. Savage, The Foundations of Statistics (2d rev. ed., New York: Dover, 1972 [1954]).Google Scholar

9 Kaplan, John, “Decision Theory and the Factfinding Process,” 20 Stan. L. Rev. 1065, 1071–77 (1968); Cullison, 1969 Toledo L. Rev. at 564–71, and id., “Model of Rules” at 237–39; among the others, see, e. g., Edwards, 13 Cardozo L. Rev. at 1062–65 & n. 80; Michael Fried, Kalman J. Kaplan, & Katherine W. Klein, “Jurot Selection: An Analysis of Voir Dire,” in Rita J. Simon, ed., The Jury System in America: A Critical Overview 49, 58–64 (Beverly Hills, Cal.: Sage Publications, 1975) (“Fried et al., ‘Juror Selection’“); Grofman, “Mathematical Models” at 309–15; Nagel, 63 Judicature at 191–92 & n. 5; Nagel et al., “Decision Theory” at 354–60; Lawrence H. Tribe, “Trial by Mathematics: Process and Ritual in the Legal Process,” 84 Harv. L. Rev. 1329, 1378–86 (1971).Google Scholar

10 C. M. A. McCauliff, “Burdens of Proof: Degrees of Belief, Quanta of Evidence, or Constitutional Guarantees?” 35 Vand. L. Rev. 1293, 1322 (1982).Google Scholar

11 Birmingham, 12 Ga. L. Rev. at 536–37; Cullison, 1969 Toledo L. Rev. at 566–67, and id., “Model of Rules” at 238; Grofman, “Mathematical Models” at 314–15; Lempert, 75 Mich. L. Rev. at 1038; Nagel, 63 Judicature at 191–92; Nagel et al., “Decision Theory” at 355–57, 359–60 n. 4, 366, 376, 378, 381; Nagel & Neef, 1975 Wash. U. L. Q. at 943 n. 17, 945–46, 959, 963, 965.Google Scholar

12 United States v. Fatico, 458 F. Supp. 388, 411 (1978).Google Scholar

13 In re Winship, 397 U. S. 358 (1970) (Harlan, J., concurring).Google Scholar

14 “The Supreme Court, 1969 Term,” 84 Harv. L. Rev. 1, 158 n. 13 (1970) (citation omitted); see Tribe, 84 Harv. L. Rev. at 1381 n. 162.Google Scholar

15 In fairness to Justice Harlan, I believe “The Supreme Court, 1969 Term,” 84 Harv. L. Rev. at 158 n. 13, and Tribe, 84 Harv. L. Rev. at 1381 n. 162, misrepresented the Justice's expressed views on this point. After citing Kaplan, 20 Stan. L. Rev. at 1071–77, who clearly based his analysis on error utilities rather than error frequencies, Justice Harlan wrote:.Google Scholar The standard of proof influences the relative frequency of these two types of erroneous outcomes. If, for example, the standard of proof for a criminal trial were a preponderance of the evidence rather than proof beyond a reasonable doubt, there would be a smaller risk of factual errors that result in freeing guilty persons, but a far greater risk of factual errors that result in convicting the innocent. Because the standard of proof affects the comparative frequency of these two types of erroneous outcomes, the choice of the standard to be applied in a particular kind of litigation should, in a rational world, reflect an assessment of the comparative social disutility of each.Google Scholar In re Winship, 397 U. S. 358, 371 (1970) (Harlan, J., concurring). Justice Harlan's assertion regarding the effects of the standard of proof on error frequencies was less bold than that credited to him. The Justice argued only that lowering the standard of proof will lower the ratio of erroneous acquittals to erroneous convictions, which is correct. He did not suggest that either the clear and convincing evidence standard or the reasonable doubt standard will lead to a ratio greater than one. Nor did Justice Harlan suggest, as his critics claimed, that the objective of the reasonable doubt standard is to ensure that there be many more erroneous acquittals than erroneous convictions. Although the source of misunderstanding seems to be the initial clause of Justice Harlan's final sentence, the Justice did not condition the rationality of utility-based standards in this manner elsewhere in his opinion. Id. at 370, 372; quotation infra note 78. In subsequent decisions, the Court has maintained Justice Harlan's emphasis, arguing for a balance of utilities rather than a balance of error frequencies. E. g., Mullaney v. Wilbur, 421 U. S. 684, 703–4 (1975); id. at 706 (Rehnquist, J., concurring); Patterson v. New York, 432 U. S. 197, 208 (1977); Santosky v. Kramer, 455 U. S. 744, 788 n. 13 (1982) (Rehnquist, J., dissenting).Google Scholar

16 See generally David M. Green & John A. Swets, Signal Detection Theory and Psychophysics (Los Altos, Cal.: Peninsula, 1988 [1966]) (“Green & Swets, Signal Detection Theory”); Neil A. Macmillan & C. Douglas Creelman, Detection Theory: A User's Guide (New York: Cambridge University Press, 1991) (“Macmillan & Creelman, Detection Theory”); Clyde C. Coombs, Robyn M. Dawes, & Amos Tversky, Mathematical Psychology: An Elementary Introduction 165–201 (Englewood Cliffs, N. J.: Prentice Hall, 1970) (“Coombs et al., Mathematical Psychology”).Google Scholar

17 E. g., Grofman, “Mathematical Models” at 307–17; Norbert Kerr, “Stochastic Models of Jury Decision Making,”in Reid Hastie, ed., Inside the Juror: The Psychology of Juror Decision Making 116 (New York: Cambridge University Press, 1993) (“Hastie, Inside the Juror”); Robert D. Sorkin & Huanping Dai, “Signal Detection Analysis of the Ideal Group,” 60 Organizational Behav. & Hum. Decision Processes 1 (1994); Ewart A. C. Thomas & Anthony Hogue, “Apparent Weight of Evidence, Decision Criteria, and Confidence Ratings in Juror Decision Making,” 83 Psychol. Rev. 442 (1976); Barbara D. Underwood, “The Thumb on the Scales of Justice: Burdens of Proof of Persuasion in Criminal Cases,” 86 Yale L. J. 1299, 1331 n. 93 (1977).Google Scholar

18 In SDT, it is standard practice to use conditional distributions as in fig. 1. Although other authors (e. g., Grofman, “Mathematical Models” at 312) have used distributions that do reflect the relative proportions of truly innocent and truly guilty defendants, some important quantities–most notably the likelihood ratio P(x|G)/P(x|I)–cannot be read directly from such a figure.Google Scholar

19 For reviews, see Swets, John A., “Indices of Discrimination or Diagnostic Accuracy: Their ROCs and Implied Models,” 99 Psychol. Bull. 100 (1986), and id., “Measuring the Accuracy of Diagnostic Systems,” 240 Science 1285 (1988).Google Scholar

20 Labels for these outcomes differ across disciplines and methodologies. For example, convictions of truly innocent defendants (CI) are equivalent to false positives in medicine, false alarms in SDT, and Type I errors in statistics. Likewise, acquittals of truly guilty defendants (AG) = false negatives = misses = Type II errors, convictions of truly guilty defendants (CG) = true positives = hits, and acquittals of truly innocent defendants (AI) = true negatives = correct rejections.Google Scholar

21 See, e. g., May, 10 Am. L. Rev. at 653–54 (cited in note 4).Google Scholar

22 Terry Connolly, “Decision Theory, Reasonable Doubt, and the Utility of Erroneous Acquittals,” 11 Law & Hum. Behav. 101, 104 (1987).Google Scholar

23 Kenneth R. Hammond, Lewis O. Harvey, & Reid Hastie, “Making Better Use of Scientific Knowledge: Separating Truth from Justice,” 3 Psychol. Sci. 80, 84 (1992).Google Scholar

24 Connolly, 11 Law & Hum. Behav. at 104; Hammond et al., 3 Psychol. Sci. at 84.Google Scholar

25 Birmingham, 12 Ga. L. Rev. at 536–37; Grofman, “Mathematical Models” at 314–15; Lempert, 75 Mich. L. Rev. at 1038; Nagel, 63 Judicature at 191–92; Nagel et al., “Decision Theory” at 355–57, 359–60 n.4, 366, 376, 378, 381; Nagel & Neef, 1975 Wash. U. L. Q. at 943 n. 17, 945–46, 959, 963, 965 (all cited in note 6).Google Scholar

26 Of course, one may make meaningful statements about ratios of temperatures using the Kelvin scale. Because the Kelvin scale has a nonarbitrary zero point characterized by the complete absence of heat (O°K =−273.15°C =−459.67°F), it is a ratio scale. Needless to say, the prospects for discovering a similar “absolute zero” for utility are incredibly slim.Google Scholar

27 John von Neumann & Oskar Morgenstern, The Theory of Games and Economic Behavior 20–29, 617–28 (3d ed. New York: Wiley, 1947 [1953]) (“von Neumann & Morgenstern, Theory of Games”).Google Scholar

28 See discussion infra p. 117.Google Scholar

29 Cullison, 1969 Toledo L. Rev. at 566–67 & n.42 (cited in note 6). Cf. Cullison, “Model of Rules” at 238, 244 n.13 (cited in note 6) (using identical nomenclature, but not mentioning utility differences explicitly).Google Scholar

30 Edwards, 13 Cardozo L. Rev. at 1062–65 & n.80 (cited in note 6).Google Scholar

31 See discussion infra p. 117.Google Scholar

32 Although the present article is concerned with SEU (Savage, The Foundations of Statistics (cited in note 8)), and not its predecessor, expected utility theory (von Neumann & Morgenstem, Theory of Games), the term “expected utility” is often used in place of “subjective expected utility” for convenience. The distinguishing characteristic of SEU is that probabilities are estimated by the decision maker rather than being given.Google Scholar

33 von Neumann & Morgenstern, Theory of Games 24–29, 617–28.Google Scholar

34 For real-world tests of the relationship between expected utility strategies and longrun outcomes, see Larrick, Richard P., Nisbett, Richard E., & Morgan, James N., “Who Uses Cost-Benefit Rules of Choice? Implications for the Normative Status of Microeconomic Theory,” 56 Organizational Behav. & Hum. Decision Processes 331 (1993).Google Scholar

35 Cullison, 1969 Toledo L. Rev. at 564–66, and id., “Model of Rules” at 238; Fried et al., “Juror Selection” at 60–61 (cited in note 9).Google Scholar

36 Dorothy K. Kagehiro, “Defining the Standard of Proof in Jury Instructions,”I Psychol. Sci. 194 (1990); Kagehiro, Dorothy K. & Stanton, W. Clark, “Legal vs. Quantified Definitions of Standards of Proof,” 9 Law & Hum. Behav. 159 (1985); Nagel, 63 Judicature at 194–95, and Nagel et al., “Decision Theory” at 377–78 (both cited in note 6).Google Scholar

37 Cullison, 1969 Toledo L. Rev. at 564–66, and id., “Model of Rules” at 238; Grofman, “Mathematical Models” at 313 (cited in note 6); Edwards, 13 Cardozo L. Rev. at 1063 n.80.Google Scholar

38 The formulas for the threshold likelihood ratio presented in Grofman, “Mathematical Models” at 313, Green & Swets, Signal Detection Theory 23 (cited in note 16), and Coombs et al., Mathematical Psychology 170 (cited in note 16), have pluses rather than minuses in front of the utilities for false alarms (CI here) and misses (AG here) because the authors assume that error utilities are negative. This assumption is both confusing and limiting. Moreover, the assumption of an absolute zero point is inconsistent with the interval nature of utility scales. von Neumann & Morgenstern, Theory of Games 23–29, 617–28. Equations (17)-(19), on the other hand, correctly express decision thresholds as ratios of utility differences. In these equations, any of the four outcomes may be evaluated positively or negatively. The expression in Macmillan & Creelan, Detection Theory 49 (cited in note 16), is identical to eq. (19) in this respect.Google Scholar

39 Grofman, “Mathematical Models” at 312. Apparently, Grofman's incorrect conclusion resulted from attempting to infer likelihood ratios from a graph that (unlike fig. 1) reflected the relative proportions of truly innocent and truly guilty defendants. See supra note 18.Google Scholar

40 E. g., Barza, Michael & Pauker, Stephen G., “The Decision to Biopsy, Treat, or Wait in Suspected Herpes Encephalitis,” 92 Ann. Internal Med., 614, 646 (1980); John C. Hershey, Randall D. Cebull, & Sankey V. Williams, “Clinical Guidelines for Using Two Dichotomous Tests,” 6 Med. Decision Making 68, 71–72 (1986).Google Scholar

41 A desription of such a figure does appear in Nagel et al., “Decision Theory” at 357 n. 2.Google Scholar

42 The opposite utility differences (UCI - UAI and (UAG - UCG) were considered by Cullison, 1969 Toledo L. Rev. at 566–67 & n.42, and Edwards, 13 Cardozo L. Rev. at 1062–65 & n.80 (cited in note 6). See discussion supa pp. 108–9. Cullison referred to these differences as “desirabilities” and noted that they should generally be negative. I find such an approach confusing. In medical decision making, positively valued utility differences analogous to CCI and CAG are associated with the costs (i. e., adverse effects) of treating nondiseased patients and the benefits of treating diseased patients. Pauker, Stephen G. &. Kassirer, Jerome P., “Therapeutic Decision Making: A Cost-Benefit Analysis,” 293 New Eng. J. Med. 229, 230 (1975); id., “The Threshold Approach to Clinical Decision Making,” 302 New Eng. J. Med. 1109, 1112, 1116 (1980); Hershey et al., 6 Med. Decision Making at 69.Google Scholar

43 See Hershey et al., 6 Med. Decsion Making at 71.Google Scholar

44 E. g., Kaplan, 20 Stan. L. Rev. at 1073–76 (cited in note 9). However, rules of evidence and instructions to the jury may moderate such effects to a substantial degree. Id. at 1074–77; Lempert, 75 Mich. L. Rev. at 1035–40 & nn.41 & 53 (cited in note 6).Google Scholar

45 In an extreme case, the jury might conclude that U _cl > U _Als, in which case the defendant should be convicted regardless of guilt. Kaplan, 20 Stan. L. Rev. at 1076. In terms of fig. 3, SEU(Convict) would exceed SEU(Acquit) for all probabilities, the lines would not cross, and the notion of a threshold probability would be meaningless. Similarly, acquittal may dominate conviction if the penalty is too severe even for guilty defendants (i. e., if U _AG > U _CG). Id. +U+Als,+in+which+case+the+defendant+should+be+convicted+regardless+of+guilt.+Kaplan,+20+Stan.+L.+Rev.+at+1076.+In+terms+of+fig.+3,+SEU(Convict)+would+exceed+SEU(Acquit)+for+all+probabilities,+the+lines+would+not+cross,+and+the+notion+of+a+threshold+probability+would+be+meaningless.+Similarly,+acquittal+may+dominate+conviction+if+the+penalty+is+too+severe+even+for+guilty+defendants+(i.+e.,+if+U+AG+>+U+CG).+Id.>Google Scholar

46 Tribe, 84 Harv. L. Rev. at 1383 n.168 (cited in note 9), argued that there may be no single probability threshold if the utility differences (U _AI - U _CJ) and (U _CG–U _AG) depend on P(G|x) in a linear fashion. For simplicity, let P(G|x) =P. If the social cost of convicting a truly innocent defendant (U _AI–U _CI) is a linear function of P, is greatest when P= 0, and is least when P= 1, then this cost may be expressed as C _CI–b _CI x P), where C_AG and b _AG are both positive and b _AG is the slope of the line. Similarly, if the social cost of acquitting a truly guilty defendant (U _CG–U _AG) is a linear function of P, is greatest when P= 1, and is least when P= 0, then this cost may be expressed as (C _AG - b _AG) +b _AG±P) where C _AG and b _AG are both positive and b _AG is the slope of the line. According to SEU, the jury should convier only if the positive and b _AG is the slope of the line. According to SEU, the jury should convier only if the expected cost of doing so is less than the expected cost of acquitting. In marhematical terms, they should convict only if (C_CI–b _CI± P) ± (1 - P) [(C_AG–b _AG) +b _AG±P] ± P. Rearranging terms yields the quadratic expression (b _AG - b _CI) ±P ²+ (C_AG+ C_CI - b _AG+b _CI) ±P - C_CI > O. The threshold value P_T is the value of P that makes the expression on the left equal to zero. Tribe argued that because such quadratic equations may have two roots between 0 and 1, more than one value of P^T may be implied. However, it is easily demonstrated that the quadratic in question has only one root between 0 and 1. Evaluated at P= 0, the expression is equal to -C_CI, which is clearly negative. Evaluated at P= 1, the expression is equal to C_AG, which is clearly positive. Because a quadratic expression has only one maximum or minimum value (i. e., a parobola has only one vertex), it is impossible for the curve to cross zero more than once between these two points. Hence, there is but one value of P_T between 0 and 1. The same result is obtained if the individual utilities (rather than the utility differences) are assumed to be linear functions of P. Moreover, if one makes the reasonable assumption that the defendant's probability of guilt affects the costs of the two errors to the same degree (i. e., b _CI=b _AG), then the quadratic equation simplifies to the standard expression for P _T (eq. (17) exactly. In summary, the potential confounding of utilities and probabilities causes less severe difficulties for the SEU decision rule than might be expected.+O.+The+threshold+value+PT+is+the+value+of+P+that+makes+the+expression+on+the+left+equal+to+zero.+Tribe+argued+that+because+such+quadratic+equations+may+have+two+roots+between+0+and+1,+more+than+one+value+of+PT+may+be+implied.+However,+it+is+easily+demonstrated+that+the+quadratic+in+question+has+only+one+root+between+0+and+1.+Evaluated+at+P=+0,+the+expression+is+equal+to+-CCI,+which+is+clearly+negative.+Evaluated+at+P=+1,+the+expression+is+equal+to+CAG,+which+is+clearly+positive.+Because+a+quadratic+expression+has+only+one+maximum+or+minimum+value+(i.+e.,+a+parobola+has+only+one+vertex),+it+is+impossible+for+the+curve+to+cross+zero+more+than+once+between+these+two+points.+Hence,+there+is+but+one+value+of+PT+between+0+and+1.+The+same+result+is+obtained+if+the+individual+utilities+(rather+than+the+utility+differences)+are+assumed+to+be+linear+functions+of+P.+Moreover,+if+one+makes+the+reasonable+assumption+that+the+defendant's+probability+of+guilt+affects+the+costs+of+the+two+errors+to+the+same+degree+(i.+e.,+b+CI=b+AG),+then+the+quadratic+equation+simplifies+to+the+standard+expression+for+P+T+(eq.+(17)+exactly.+In+summary,+the+potential+confounding+of+utilities+and+probabilities+causes+less+severe+difficulties+for+the+SEU+decision+rule+than+might+be+expected.>Google Scholar

47 This solution is not novel. Tribe, 84 Harv. L. Rev. at 1384–85, for example, wrote: “[O]ne will expect a lawmaker rather than the factfinder to use a model such as Kaplan and Cullison propose, and one will define the decision problem to be solved not as the one-shot problem of fixing a standard of proof for a particular trial with four possible outcomes, but as the much larger problem of establishing such standards for the trial system as a whole.” The Supreme Court argued similarly: “[T]his court never has approved case-by-case determination of the proper standard of proof for a given proceeding. Standards of proof, like other ‘procedural due process rules[,] are shaped by the risk of error inherent in the truth-finding process as applied to the generality of cases, not the rare exceptions.’ Since the litigants and the factfinder must know at the outset of a proceeding how the risk of error will be allocated, the standard of proof necessarily must be calibrated in advance.” Santosky v. Kramer, 455 U. S. 745, 757 (1982) (alterations in original) (citation omitted) (quoting Mathews v. Eldridge, 424 U. S. 319, 344 (1976)). Contra Santosky v. Kramer, 455 U. S. 745, 775, 787 (1982) (Rehnquist, J., dissenting) (arguing that it is necessary to consider the peculiarities of individual cases).Google Scholar

48 Kaplan, 20 Stan. L. Rev. at 1071, made the assumption implicitly, whereas Birmingham, “Remarks on Probability” at 536 (cited in note 6), Lempert, 75 Mich. L. Rev. at 1033, 1036 n.41, 1038, and Anne W. Martin & David A. Schum, “Quantifying Burdens of Proof: A Likelihood Ratio Approach,” 27 Jurimetrics J. 383, 398 (1987), made the assumption explicitly.Google Scholar

49 Lempert, 75 Mich. L. Rev. at 1036 n.41 (emphasis added).Google Scholar

50 Nagel, 63 Judicature at 192 n.5; Nagel et. al., “Decision Theory” at 355–57; Grofman, “Mathematical Models” at 314 (all cited in note 6).Google Scholar

51 E. g., Nagel, 63 Judicature at 191–92; Nagel et al., “Decision Theory” at 366; Lempert, 75 Mich. L. Rev. at 1038–39; Martin F. Kaplan, “Cognitive Process in the Individual Juror,”in Norbert L. Kerr & Robert M. Bray, eds., The Psychology of the Courtroom 197, 216 (New York: Academic Press, 1982).Google Scholar

52 Rita J. Simon & Linda Mahan, “Quantifying Burdens of Proof: A View from the Bench, the Jury, and the Classroom,” 5 Law & Soc'y Rev. 319 (1971).Google Scholar

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54 Id. at 1039.Google Scholar

55 Tribe, 84 Harv. L. Rev. at 1379–80.Google Scholar

56 Patricia G. Milanich, “Decision Theory and Standards of Proof,” 5 Law & Hum. Behav. 87, 93 (1981); id. at 90–91.Google Scholar

57 Connolly, 11 Law & Hum. Behav. at 109 (cited in note 22).Google Scholar

58 von Neumann & Morgenstem, Theory of Games 23–29, 617–28 (cited in note 27).Google Scholar

59 Ronald J. Allen, “The Restoration of In re Winship: A Comment on Burdens of Persuasion in Criminal Cases after Patterson v. New York,” 76 Mich. L. Rev. 30, 47 n.65 (1977).Google Scholar

60 Michael O. Finkelstein, Quantitative Methods in Law: Studies in the Application of Mathematical Probability and Statistics to Legal Problems 65–78 (New York: Free Press, 1978) (“Finkelstein, Quantitative Methods”).Google Scholar

61 Bell, 78 J. Crim. L. & Criminology at 570–72, 579 (cited in note 6).Google Scholar

62 Id. at 573–74, 579–83.Google Scholar

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64 Robert D. Sorkin, personal communication, 7 Sept. 1994.Google Scholar

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68 E. g., Ronald J. Allen, “A Reconceptualization of Civil Trials,” 66 B. U. L. Rev. 401, 410 n. 28 (1986).Google Scholar

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70 Id. at 105.Google Scholar

71 Id. Google Scholar

72 Alan E. Gelfand & Herbert Soloman, “A Study of Poisson's Models for Jury Verdicts in Criminal and Civil Trials,” 68 J. Am. Stat. Ass'n 271, 278 (1973); id., 69 J. Am. Stat. Ass'n at 35–36.Google Scholar

73 Harry Kalven, Jr., & Hans Zeisel, The American Jury 488 (Boston: Little, Brown & Co., 1966).Google Scholar

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78 Bell, 78 J. Crim. L. & Criminology at 558 (cited in note 6), suggested that “[t]he intellectual origin of the Court's due process doctrine for burdens of proof is Kaplan's article, Decision Theory and the Factfinding Process.” Indeed, Justice Harlan's concurring opinion in Inre Winship appears to have been influenced by Kaplan's analysis:. [E]ven though the labels for the alternative standards of proof are vague and not a very sure guide to decisionmaking, the choice of the standard for a particular variety of adjudication does, I think, reflect a very fundamental assessment of the comparative social costs of erroneous factual determinations……. I view the requirement of proof beyond a reasonable doubt in a criminal case as bottomed on a fundamental value determination of our society that it is far worse to convict an innocent man than to let a guilty man go free.Google Scholar In re Winship, 397 U. S. 358, 369–70, 372 (1970) (Harlan, J., concurring) (citing Kaplan). Mullaney v. Wilbur, 421 U. S. 684, 703–4 (1975); id. at 706 (Rehnquist, J., concurring); Patterson v. New York, 432 U. S. 197, 208 (1977); Santosky v. Kramer, 455 U. S. 744, 788 n. 13 (1982) (Rehnquist, J., dissenting). See discussion supra note 15 and accompanying text.Google Scholar

79 Kagehiro, 1 Psychol. Sci. 194, and Kagehiro & Stanton, 9 Law & Hum. Behav. 159 (both cited in note 36); Nagel, 63 Judicature at 194–95, and Nagel et al., “Decision Theory” at 377–78 (both cited in note 6).Google Scholar

80 “[Blurdens of proof are founded on considerations of policy. These considerations are explicated by the policy for factfinders' erroneous decisions as a wanted ratio of errors.… A burden of proof is inconsistent with due process only if it apportions factfinding errors in an impermissible way.” Bell, 78 J. Crim. L. & Criminology at 583 (emphasis added) (footnote omitted); id. at 563, 573–74, 580–81; “If the burden of errors is deemed to be the same for both parties, the aim should be equal rates of errors for both parties.” Finkelstein, Quantitative Methods 67 (cited in note 60) (emphasis added); “[T]he reasonable doubt standard seeks to assure that erroneous acquittals of the guilty are far more common than the erroneous convictions of the innocent.”“The Supreme Court, 1969 Term,” 84 Harv. L. Rev. at 158 n. 13 (cited in note 15) (emphasis added); “A demanding burden of proof is imposed on the prosecution in order to assure that men are wrongly convicted much less often than they are wrongly acquitted…. [T]he objective [is to assure] that erroneous acquittals of the guilty occur with greater frequency than erroneous convictions of the innocent.” Tribe, 84 Harv. L. Rev. at 1381 n. 162 (cited in note 9) (emphasis added); see discussion supra pp. 97–98 and note 15. Cf. Finkelstein, Quantitative Methods 69 (suggesting that different goals might apply in different situations). But see Kaye, David, “Naked Statistical Evidence,” 89 Yale L. J. 601, 605–8 (1980) (book review) (criticizing Finkelstein's preference for standards based on ratios of errors).Google Scholar

81 Bell, 78 J. Crim. L. & Criminology at 580 (emphasis added). But cf. supra notes 6, 15, and 78 and accompanying text (arguing that the Court has favored a utility-based approach and has not adopted Blackstone's ratio).Google Scholar

82 Connolly, 11 Law & Hum. Behav. at 104–5 (cited in note 22).Google Scholar

83 Finkelstein, Quantitative Methods 73. Cf. quotation, supra note 80 (advocating an error-equalizing rule).Google Scholar

84 Connolly, 11 Law & Hum. Behav. at 104.Google Scholar

85 Frisch, Deborah & Clemen, Robert T., “Beyond Expected Utility Theory: Rethinking Behavioral Decision Research,” 116 Psychol. Bull. 46, 49–50 (1994).Google Scholar