Aggregating probabilities across cases: Criminal responsibility for unspecified offenses

Minnesota Law Review

Volumn 94, Issue 2, 2010, Pages 261-310

  Harel, Alon ^a   Poratt, Ariel ^a  

^a NONE

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EID: 79551697643     PISSN: 00265535     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (187)

1. Cf. Saul Levmore, Conjunction and Aggregation, 99 MICH. L. REV. 723, 729 (2001) (noting the absence of an aggregate-probabilities approach in criminal law, according to which probabilities are aggregated across the different elements of the same offense). Levmore does not raise, however, the question of aggregating probabilities across different cases, which this Article addresses. Cf. id.
2. This Article uses numerical probabilities for the sake of exposition. See infra notes 6, 17 and accompanying text.
3. The probability that the defendant committed each one of the offenses is .9, and therefore the probability, for each one, that he did not commit the offense is 1 - .9 = .1. Consequently, the probability that he did not commit any offense is (.I)2 = .01, and the probability that he committed at least one of the offenses is 1-.01 = .99.
4. If the defendant had been charged with four offenses instead of two, this would yield a probability of .9999 that he had committed at least one offense. Applying the APP would guarantee conviction for two offenses since the probability that two offenses had been committed would be higher than the threshold required for conviction. This is the outcome of a binomial distribution. There are four events and in each one the defendant either committed the offense or did not (thus he either committed zero, one, two, three, or four offenses and the probability that one of these scenarios transpired is one). To calculate the probability that the defendant committed at least two of the four offenses we subtract from one the probability that he committed zero offenses or one offense. Since the probability that the defendant did not commit any offense is (.I)4 - .0001, and the probability that he committed exactly one offense is (.9) * (J)3 z.ast; 4 = .0036 (.9 is the probability that he committed one specific offense; (.l)3 is the probability that he did not commit any of the other three offenses; we multiply by four because the specific offense committed by the defendant could be any of the four offenses), the probability that he com-mitted at least two offenses is i - .0001 - .0036 = .9963. To calculate the probability that the defendant committed at least three offenses we add the probability that he committed four offenses to the probability that he committed three offenses. Since the probability that the defendant committed four offenses is (.9)4 and the probability that he committed three offenses is (.9)3 * .1 * 4, the probability that he committed at least three offenses comes to .6561 + .2916 = .9477.
5. See, e.g., McCullough v. State, 657 P.2d 1157, 1159 (Nev. 1983) ("The concept of reasonable doubt is inherently qualitative.").
6. The .95 threshold has traditionally been used as an illustration in texts that interpret the principle of beyond a reasonable doubt in probabilistic terms. See David Kaye, Laws of Probability and the Law of the Land, 47 U. CHI. L. REV. 34, 40 (1979) ("Surely it is not some defect in probability theory that restrains us from instructing jurors that they should convict so long as they are, say, at least ninety-five percent certain that the defendant is guilty."). 7 See discussion infra Part II.B.2. 8 (.95)2 = .9025.
7. In other fields, however, the APP has been considered and discussed at length. For instance, legal theorists have proposed aggregating probabilities in civil cases, and, as such, that discussion will not be explored in this Article. See, e.g., Levmore, supra note 1, at 724 (discussing aggregating probabilities mainly in tort cases). Furthermore, Frederick Schauer and Richard Zeckhaus-er proposed aggregating probabilities across cases outside the judicial context. Frederick Schauer & Richard Zeckhauser, On the Degree of Confidence for Adverse Decisions, 25 J. LEGAL STUD. 27, 41-51 (1996). Schauer & Zeckhauser argue that it would make sense for a school to dismiss a teacher against whom several complaints of sexual harassment had been made in the past, even if each complaint, considered separately, would not constitute sufficient reason for dismissal. Schauer & Zeckhauser maintain, however, that such an argument is inapplicable to criminal proceedings. Id. "Of course, the practice of noncumulation of charges in the criminal law serves important goals⋯ . Obviously there are costs associated with these goals ⋯ but weighing the costs and benefits of the refusal to cumulate in the criminal process is not our goal." Id. at 45-16.
8. See, e.g., Wicks v. Lockhart, 569 F. Supp. 549, 565 n.18 (E.D. Ark. 1983) (noting that the prosecution must prove beyond a reasonable doubt each offense charged).
9. See, e.g., Estelle v. Williams, 425 U.S. 501, 517 (1976) ("One of the essential due process safeguards that attends the accused at his trial is the benefit of the presumption of innocence ⋯ .").
10. See, e.g., "Wicks, 569 F. Supp. at 565 n.18 ("It is axiomatic that the government bears the burden of proving beyond a reasonable doubt as to each offense charged ⋯.").
11. The APP could also be applied across civil cases. The considerations for and against such application differ from those relevant to criminal cases, and we leave that question to future consideration.
12. In re Winship, 397 U.S. 358, 364 (1970) ("[W]e explicitly hold that the Due Process Clause protects the accused against conviction except upon proof beyond a reasonable doubt of every fact necessary to constitute the crime with which he is charged.").
13. See Jessica N. Cohen, The Reasonable Doubt Jury Instruction: Giving Meaning to a Critical Concept, 22 AM. J. CRIM. L. 677, 678 (1995) ("[B]ecause reasonable doubt is a term of art it should be defined for the jury.");
14. Thomas V. Mulrine, Reasonable Doubt: How in the World Is It Defined?, 12 AM. U. J. INT'L L, & POL'Y 195, 197-98, 210-25 (1997) (explaining various approaches to and definitions of reasonable doubt);
15. Lawrence M. Solan, Refocusing the Burden of Proof in Criminal Cases: Some Doubt About Reasonable Doubt, 78 TEX. L. KEV. 105, 105 (1999) ("Most debate in judicial opinions and in the scholarly literature has focused on whether reasonable doubt should be defined for the jury, and, if so, how it should be defined.").
16. See ALEX STEIN, FOUNDATIONS OF EVIDENCE LAW 65 (2005) ("Adjudi- cative fact-finding rests on probabilistic reasoning that derives from experience."); id. at 66 ("Any finding that fact-finders make can only be probable, rather than certain.").
17. For those readers who are skeptical about mathematical calculations in the legal context, it is possible to consider the same problem without resorting to probabilities: should a court convict a defendant when there is no reasonable doubt that he committed at least one of several charged offenses, but it cannot be established which one he specifically committed? See L. Jonathan Cohen, The Role of Evidential Weight in Criminal Proof, 66 B.U. L. EEV. 635, 635 (1986) (stating that by trying to give an account of the standard of criminal proof in Pascalian terms, one reserves the crucial place in reasoning for the assignment of a high value non-Pascalian function for the assessment of evidential weight);
18. Laurence H. Tribe, Trial by Mathematics: Precision and Ritual in the Legal Process, 84 HAEV. L. REV. 1329, 1372̂75 (1971) ("Both callousness and insecurity ⋯ might be increased by the explicit quantification of jury doubts in criminal trials-whether or not it would be factually accurate to describe the trial system as imposing criminal sanctions in the face of quantitatively measured uncertainty in particular cases.").
19. See, e.g., Wicks v. Lockhart, 569 F. Supp. 549, 549 (E.D. Ark. 1983).
20. Id. at 565 n. 18.
21. See supra Example 1.
22. See supra note 4.
23. This is not to say, however, that the APP is neutral overall with respect to defendants. There are not an identical number of expected convictions and expected acquittals resulting from an application of the APP. The transition from the DPP to the APP can be expected to bring about more convictions than acquittals, based on the observation that the APP, in taking into account all probabilities from .01 to .94 (assuming .95 is the threshold for conviction), increases the number of convictions, and only in taking into account probabilities from .95 to .99 does it reduce the number of convictions. See supra Example 2.
24. See discussion infra Part V.
25. Id.
26. As we elucidate later, interdependence could sometimes be a significant obstacle in employing the APP. See discussion infra Part 1I.B.2.
27. See Charles J. Walsh & Alissa Pyrich, Corporate Compliance Programs as a Defense to Criminal Liability: Can a Corporation Save Its Soul?, 47 RUTGERS L. REV. 605, 633 n.89 (1995).
28. See infra Part V.
29. See supra Example 1.
30. If the defendant was acquitted in the first trial because the probability of his guilt was only .9, and in the second trial the probability of his guilt was again .9, under the APP he should be convicted at the second trial. If, instead, that defendant was convicted at the first trial because the probability of his guilt was .95, under the APP he should not be convicted at the second trial. Moreover, if in the latter case the probability of guilt in the second trial was also .95 (and not .9), as illustrated by Example 2, applying the APP should also lead to acquittal at the second trial.
31. First, the information obstacles in applying the APP across trials are more serious than those that would arise across charges in the same trial. See discussion infra Part IV.D. Second, taking into account prior acquittals as a consideration for convicting the same defendant in a subsequent trial could violate the Double Jeopardy Clause. See U.S. CONST, amend. V. Third, applying the APP across different trials in cases represented by Example 2 (when the APP generates fewer, rather than more, convictions) could reduce deterrence of future crimes: a defendant who was convicted in a trial for one offense at a probability of .95 will not be punished for a subsequent crime as long as the prosecution cannot establish his guilt at a probability of 1. See discussion infra Part III.B.
32. LEO KATZ, ILL-GOTTEN GAINS: EVASION, BLACKMAIL, FRAUD, AND KINDRED PUZZLES OF THE LAW 67-69 (1996).
33. See, e.g., People v. Collins, 438 P.2d 33, 33, 40 (Cal. 1968); ROBERT COOTER & THOMAS ULEN, LAW & ECONOMICS 460̂66 (5th ed. 2008) (discussing the aggregation of probabilities in tort law).
34. See Robert Cooter, Adapt or Optimize? The Psychology and Economics of Rules of Evidence, in HEURISTICS AND THE LAW 379, 380 (G. Gigerenzer & C. Engels eds., 2006).
35. Id. at 384̂87.
36. See Levmore, supra note 1, at 729-30 n.ll (labeling this alternate method "reverse conjunction").
37. See RESTATEMENT (SECOND) OF TORTS § 281(b)-(c) (1965) (listing these two elements as required for negligence liability).
38. See Alex Stein, Of Two Wrongs That Make a Right: Two Paradoxes of the Evidence Law and Their Combined Economic Justification, 79 TEX. L. REV. 1199, 1205 (2001) (explaining the rationale of aggregating probabilities in the civil-cumulative case).
39. To consider the applicability of the "product rule"-which is the rule that mandates the aggregation of probabilities-compare Maya Bar-Hillel, Probabalistic Analysis in Legal Factfinding, 56 ACTA PSYCHOLOGICA 267, 269 (1984) ("[T]he conjunction of a small number of weakly probative characteristics can be strongly probative."), and Bernard Robertson & G. A. Vignaux, Probability-The Logic of the Law, 13 OXFORD J. LEGAL STUD. 457, 478 (1993) ("Once one regards probability as a generalisation of logic and has freed one's mind from the shackles of frequentist examples and the Mind Projection Fallacy, these objections [to the use of probabilities] evaporate. The logical rules for thinking about facts in legal cases are those of probability."), with L. JONATHAN COHEN, THE PROBABLE AND THE PROVABLE 58-67 (1977) (discussing the problems of conjunction of facts or probabilities and claiming that mathematical probability is inadequate as a model for rational thinking). See also Stein, supra note 37, at 1203-05 (considering the effect of the "conjunction paradox" on the use of the product rule, but suggesting that, in light of another major distortion in fact-finding, the product rule leads to a second-best solution).
40. See Levmore, supra note 1, at 752 nn.58-60 (arguing that no jurisdiction explicitly recognizes the product rule and explaining that such non-recognition could be warranted mainly in those cases where decisions are made by either a jury or another multimember panel, either unanimously or by supermajority).
41. The probability that none of the events took place is .7 * .7 - .49. The probability that at least one of them took place is 1 - .49 - .51. If the scenarios exclude one another, then the probability that at least one took place is .3 + .3 = .6.
42. See Levmore, supra note 1, at 729 n.ll, 745̂46 (explaining the "alternative routes" scenario, which he labels "reverse conjunction"). Levmore uses the same reasoning for rejecting the product rule in the civil-alternative case as in the civil-cumulative case. See id. at 752 nn.58̂60.
43. Compare Tribe, supra note 17, at 1361 (claiming that hard statistical data lead decisionmakers into "[djwarfing [the] soft variables" by assuming that "[i]f you can't count it, it doesn't exist"), with Jonathan J. Koehler & Daniel N. Shaviro, Veridical Verdicts: Increasing Verdict Accuracy Through the Use of Overtly Probabilistic Evidence and Methods, 75 CORNELL L. REV. 247, 265 (1990) (stating that psychological research does not support Tribe's assumption, but rather "suggests that, in a wide range of situations, people generally undervalue base rate evidence and attach too much weight to case-specific evidence").
44. See Levmore, supra note 1, at 729 n.ll (stating that courts do not apply the product rule).
45. DAN DOBBS, THE LAW OF TORTS § 154, at 370̂73 (2001) (describing res ipsa loquitur cases where the jury is permitted to infer that the defendant was negligent in causing the harm in a specific scenario, even though evidence of any specific negligent act cannot be established).
46. See Bar-Hillel, supra note 38, at 268̂70, 282̂83 (analyzing the use of probabilities in cases and suggesting a "soft role ⋯ for probability in the factfinding process"). But see People v. Collins, 438 P.2d 33, 33, 40 (Cal. 1968) (rejecting the use of probabilities in determining guilt on the facts of the case).
47. See supra note 37 and accompanying text.
48. See Levmore, supra note 1, at 729 (suggesting that the product rule can equally apply to the civil-cumulative case and the criminal-cumulative case).
49. Compare id. at 733 n.19 (suggesting that the defense might benefit from a rule of aggregation when it reminds the jury of all the doubts that have been raised and implies that, combined, they create more than a reasonable doubt), with Jonathan Kemy Nash, A Context-Sensitive Voting Protocol Paradigm for Multimember Courts, 56 STAN. L. EEV. 75, 138 (2003) (discussing the rule of aggregation in the context of voting by judges in a panel or by jurors and observing that "[a]lthough a criminal defendant cannot be convicted unless a jury unanimously finds each element of the crime charged proven beyond a reasonable doubt, 'a federal jury need not always decide unanimously which of several possible sets of underlying brute facts make up a particular element, say, which of several possible means the defendant used to commit an element of the crime'" (citation omitted)).
50. See supra note 44 and accompanying text.
51. For this example and others, see DOBBS, supra note 44, § 154, at 372.
52. See Sindell v. Abbott Labs., 607 P.2d 924, 936-38 (Cal. 1980) (holding drug manufacturers liable even when proving which specific manufacturer produced the drug in question is not possible).
53. Id. at 925.
54. Id.
55. Id. at 925-26.
56. Id. at 926.
57. Id.
58. Id. at 937; see also Hymovitz v. Eli Lilly & Co., 539 N.E.2d 1069, 1078 (N.Y. 1989) (applying a modified version of the MSL doctrine); Martin v. Abbott Labs., 689 P.2d 368, 381 (Wash. 1984) (same); Collins v. Eli Lilly [&] Co., 342 N.W.2d 37, 49 (Wis. 1984) (same). But see Kurczi v. Eli Lilly & Co., 113 F.3d 1426, 1433 (6th Cir. 1997) (rejecting application of the MSL doctrine under Ohio law).
59. Sindell, 607 P.2d at 937.
60. Id.
61. Id. It is not clear whether this decision should be interpreted as imposing liability on each defendant for all the plaintiffs' damage (with the proper allocation achieved through indemnification claims between the co-defendants) or as imposing liability on each defendant for only part of the damage. See Brown v. Superior Court, 751 P.2d 470, 485-87 (Cal. 1988) (adopting the second interpretation of imposing several liability on each defendant for only part of the damage); ARIEL PORAT & ALEX STEIN, TORT LIA-BILITY UNDER UNCERTAINTY 138, 148 (2001) (discussing the policies behind the two alternative rules).
62. Both principles differ from the alternative liability principle set by the California Supreme Court in Summers v. Tice, 199 P.2d 1 (Cal. 1948), which bears some superficial resemblance to the APP. In Summers, the defendants were two individuals who had participated in quail hunting. Id. at 2. The plaintiff had been shot in the eye by a stray bullet negligently fired by one of the defendants. Id. The defendants pulled their triggers simultaneously, so it could not be determined whose bullet had actually injured the plaintiff. Id. The court resolved the case by establishing the "alternative liability" principle, which shifts the burden of proof from the plaintiff to the defendant "to absolve himself if he can." Id. at 4. Thus, "[defendants unable to disassociate themselves evidentially from the damage are, therefore, held liable for the entire damage." PORAT & STEIN, supra note 60, at 61. This principle ultimately found its way into the BESTATEMENT (SECOND) OF TORTS § 433B, illus. 9 (1965), but has nothing in common with the aggregation of probabilities dealt with in this article. In Summers, there was a fifty percent probability for each of the defendants that he had hit the plaintiff, and this probability was not the result of any aggregation. 199 P.2d at 2. It seems that the only aggregation of prob-abilities that could be conducted would be on the side of the plaintiff rather than the defendant: the probability that the plaintiff suffered an injury from wrongful shooting would be the aggregate of the probabilities that each defendant had separately caused the injury. This would yield a probability of 1.
63. PORAT & STEIN, supra note 60, at 64-65 (discussing the limits of the MSL doctrine and citing cases).
64. But see id. at 65-67 (2001) (discussing cases in which the MSL doctrine was applied).
65. See id. at 60-62.
66. See generally OLIVER WENDELL HOLMES, JR., THE COMMON LAW 34- 43 (Transaction Publishers 2005) (1881) (comparing the different goals of the criminal law); PORAT & STEIN, supra note 60, at 1-15 (considering the evolution of Anglo-American tort doctrine).
67. See PORAT & STEIN, supra note 60, at 130-59 (discussing the justifications for the MSL doctrine);
68. Mark A. Geistfeld, The Doctrinal Unity of Alternative Liability and Market-Share Liability, 155 U. PA. L. REV. 447, 449 (2007) (discussing the deterrent effect of the MSL doctrine).
69. See HOLMES, supra note 65, at 36-46.
70. But see Ronen Perry, The Role of Retributive Justice in the Common Law of Torts: A Descriptive Theory, 73 TENN. L. REV. 177, 188-92 (2006) (arguing that retributive justice has an influence on the development of tort law doctrines).
71. See FED. R. EVID. 403, 413, 414.
72. Id.
73. See id. 404(b). See also United States v. Woods, 484 F.2d 127, 137 (4th Cir. 1973), where the court stated prior to the enactment of Rule 404(b): Unlike other cases where evidence of prior crimes is admissible for only limited purposes and where it is necessary or proper to give limiting instructions, evidence of the prior events was admissible here to prove both that Paul was the victim of infanticide and that defendant was the perpetrator of the crime.
74. FED. R. EVID. 404(b).
75. See People v. Quinn, 486 N.W.2d 139, 140 (Mich. Ct. App. 1992) ("Where, however, evidence of a defendant's other wrongful acts has been admitted for the limited purposes allowed under MRE 404(b), the prosecutor deprives the defendant of a fair trial in arguing that the jury should consider the evidence as substantive evidence of the defendant's guilt."); see also Huddle-ston v. United States, 485 U.S. 681, 689-92 (1988) (holding that the trial court is not required to make a preliminary finding that the petitioner proved commission of the similar acts by a preponderance of the evidence). Evidence of other crimes is usually submitted in criminal, not civil, procedures. Rule 404(b), however, contains no such limitation, and potential civil applications occasionally arise. See, e.g., Barnes v. City of Cincinnati, 401 F.3d 729, 741-42 (6th Cir. 2005) (ruling that a statement made by a high-ranking official regarding lesbians in the city's police department was admissible under Rule 404(b)).
76. See, e.g., Dowling v. United States, 493 U.S. 342, 348-49 (1990) (holding that testimony tending to prove that the defendant had committed a crime, which had been brought in a prior trial that ended in acquittal, was rightly admitted under Rule 404(b) by the court in a subsequent trial because it established the defendant's identity).
77. United States v. Jones, 455 F.3d 800, 810 (7th Cir. 2006) (Easterbrook, J., concurring).
78. FED. E. EVID. 413, 414.
79. Id. Under Rule 415 of the Federal Rules of Evidence this doctrine is also applicable to civil cases involving sexual assault and child molestation. See Louis M. Natali Jr. & R. Stephen Stigall, "Are You Going to Arraign His Whole Life?": How Sexual Propensity Evidence Violates the Due Process Clause, 28 LOY. U. CHI. L.J. 1, 29 (1997) ("By requiring the admission of propensity evidence, the rules prevent a fundamentally fair trial, and thus violate due process ⋯ .").
80. As Example 2 illustrates, sometimes the APP leads to acquittal rather than to conviction. See supra Example 2.
81. The Racketeer Influenced and Corrupt Organization Act (RICO), 18 U.S.C. §§ 1961-1968 (2006), can be interpreted as a tool for punishing individuals for unspecified offenses. Under RICO, a person who is a member of an enterprise that has committed any two specified crimes within a ten-year period can be charged with racketeering. Id. The racketeering offense can thus be seen as a mechanism for punishing individuals who are more likely to have committed serious unknown crimes. Arguably, one can infer from the type of criminal activity committed by those convicted under RICO their engagement in other activities-activities that have not been detected or proven. Yet, it is quite evident that this is not the central purpose of RICO: the Act targets not those who are more likely to have committed other crimes but people whose criminal activity is particularly harmful because it contributes to organized crime. Hence, RICO cannot be construed as serving goals similar to those of the APP. Note that under the prior-acts and similar-crimes doctrines, the fact that a person committed several similar offenses in the past increases the chances of conviction in the present case. Under the APP, in contrast, as illustrated by Example 2, the fact that a person was convicted of several offenses in the past decreases the probability of conviction in a later case. See supra Example 2. However, we do not suggest applying the APP across different trials. See infra Part IV.D.
82. See supra Example 1.
83. FED. R. EVID. 403, 413, 414.
84. See, e.g., Gastineau v. Fleet Mortgage Corp., 137 F.3d 490, 495 (7th Cir. 1998) (admitting evidence of plaintiff's prior lawsuits to show, inter alia, "Gastineau's modus operandi of creating fraudulent documents in anticipation of litigation against his employers").
85. See Jodi Leibowitz, Note, Criminal Statutes of Limitations: An Obstacle to the Prosecution and Punishment of Child Sexual Abuse, 25 CAEDOZO L. REV. 907, 939 n.127 (2003) ("For any sexual abuser, the likelihood that he has performed a similar abuse in the past-and that he will repeat it in the future-is extremely high.").
86. 1-(.3P = .91.
87. 1 - (2p = .96.
88. See supra Example 3.
89. See supra Example 5.
90. See supra Example 1 and modified Example 4 where the pattern-of-behavior doctrines are not applied.
91. See supra Example 4.
92. Jeremy Bentham argued that when there is evidence that the same convicted person escaped detection by the law in the past, the sanction to be inflicted in the present conviction should reflect this fact. JEREMY BENTHAM, Of the Proportion Between Punishments and Offences, in AN INTRODUCTION TO THE PRINCIPLES OF MORALS AND LEGISLATION 165, 170 (J. H. Burns & H. L. A. Hart eds., 1970). Bentham maintained that in setting the punishment, "it may be necessary, in some cases to take into account the profit not only of the individual offence to which the punishment is to be annexed, but also of such other offences of the same sort as the offender is likely to have already committed without detection." Id. We thank Avraham Tabbach for referring us to Bentham's thoughts on this issue. One way to interpret Bentham's argument is as the converse to our understanding of the prior-acts and similar-crimes doctrines: whereas in the latter doctrines, the court infers from past behavior forward to the present charge, Bentham urged courts to infer from the present charge backward to past behavior. The ramifications of this reading of Bentham's claim are that we can increase punishment in a present conviction in order to punish the convicted defendant for past behavior that, in light of the present conviction, can be more easily attributed to him now. Indeed, both the APP and Bentham's proposal are motivated by a concern for the underen-forcement of the law: the APP would be rendered completely meaningless if there were no under-enforcement and it had been always possible to fully and accurately detect all criminals. But, as already explained, the APP is based on the conjecture of independence of the relevant probabilities, whereas Bentham's proposal is founded on the opposite assumption, namely, that if the defendant committed one offense, it is more likely he had committed other offenses in the past. Id.
93. See A. Mitchell Polinsky & Steven Shavell, The Theory of Public Enforcement of Law, in 1 HANDBOOK OF LAW AND ECONOMICS 403, 427-29 (A. Mitchell Polinsky & Steven Shavell eds., 2007) (discussing different ways to optimize Type I and Type II errors in law enforcement); STEIN, supra note 16, at 141-71 (discussing the allocation of risks of error in the law of evidence); I.P.L. Png, Optimal Subsidies and Damages in the Presence of Judicial Error, 6 INT'L EEV. L. & ECON. 101, 102-04 (1986) (discussing different ways to optimize Type I and Type II errors in law enforcement).
94. See infra text accompanying note 99.
95. See supra note 91.
96. See supra note 4.
97. See supra Example 1.
98. See id.
99. See id.
100. C.Y. Cyrus Chu et al., Punishing Repeat Offenders More Severely, 20 INT'L EEV. L. & ECON. 127, 136 (2000).
101. Cf Henrik Lando, The Size of the Sanction Should Depend on the Weight of the Evidence, 1 REV. L. & ECON. 277, 278 (2005) (suggesting that sanctions be correlated with the weight of evidence and noting that this would result in less unfairness to the innocent who are wrongly convicted and less cost to society); Talia Fisher, Rethinking the Bipolar Structure of the Criminal Verdict 19-29 (Oct. 13, 2009) (unpublished manuscript), available at http://papers.ssrn.com/sol3/papers.cfm?abstractjd=1488345 (arguing that sanctions should be correlated with the probability of guilt and pointing out that, among other things, adopting such a rule could reduce the costs of convicting the in-nocent). Our argument is analogous to a different argument made by theorists, according to which it is justified to punish repeat offenders more severely than other offenders because the risk of wrongly convicting the innocent is lower with the former than with the latter. See RICHARD A. POSNEE, ECONOMIC ANALYSIS OF LAW 228 (7th ed. 2007) (increasing punishment for repeat offenders is justified because the risk of convicting the innocent is lower in their case); Chu et al., supra note 98, at 135 (arguing that increasing the punishment for repeat offenders and decreasing it for other offenders could achieve the same level of deterrence and, at the same time, would reduce the risks of convicting the innocent).
102. See supra Part II.B.
103. See infra Part IV.B.
104. There are different views on the question as to whether, in order to achieve optimal deterrence, repeat offenders should be punished more severely than other offenders. See David A. Dana, Rethinking the Puzzle of Escalating Penalties for Repeat Offenders, 110 YALE L.J. 733, 737 (2001) (arguing that declining penalties for repeat offenders are optimal since the probability of detection escalates with offense history); Winand Emons, A Note on the Optimal Punishment for Repeat Offenders, 23 INT'L REV. L. & ECON. 253, 254 (2003) (arguing that when punishment is a fine, under certain conditions, the optimal sanction scheme decreases);*Winand Emons, Escalating Penalties for Repeat Offenders, 27 INT'L REV. L. & ECON. 170, 171 (2007) (arguing that under cer-tain conditions, increasing sanctions for repeat offenders is optimal and, under other conditions, the reverse holds true); Thomas J. Miceli & Catherine Bucci, A Simple Theory of Increasing Penalties for Repeat Offenders, 1 REV. L. & ECON. 71, 72 (2005) (claiming that repeat offenders should be punished more severely than other offenders, because of their diminished employment opportunities); A. Mitchell Polinsky & Daniel L. Rubinfeld, A Model of Optimal Fines for Repeat Offenders, 46 J. PUB. ECON. 291, 291 (1991) (claiming that when the penalty is a fine and when the ill-gotten gains of the offenders are not considered part of the social good, it is optimal to punish repeat offenders more severely than other offenders in one type of case, less severely in another type of case, and with the same severity in other types of cases);
105. A. Mitchell Polinsky & Steven Shavell, On Offense History and the Theory of Deterrence, 18 INT'L REV. L. & ECON. 305, 306ˆ07 (1998) (arguing that when the ill-gotten gains of the offenders are considered part of the social good, it is optimal to punish repeat offenders more severely than other offenders); Richard A. Posn-er, An Economic Theory of the Criminal Law, 85 COLUM. L. REV. 1193, 1215 (1985) ("[A] repeat offender is usually punished more severely than a first offender even if the repeat offender served in full whatever sentences were imposed for the earlier crimes &mellip; ."); Ariel Rubinstein, On an Anomaly of the Deterrent Effect of Punishment, 6 ECON. LETTERS 89, 90 (1980) (arguing that punishing repeat offenders more harshly increases deterrence of offenders).
106. See, for example, the "dangerous special offender" statute, 18 U.S.C. § 3575 (1988) (repealed 1984), which provided for an enhanced penalty of up to twenty-five years imprisonment for repeat offenders, professional criminals, and organized crime offenders.
107. See Lucian Arye Bebchuk & Louis Kaplow, Optimal Sanctions and Differences in Individuals' Likelihood of Avoiding Detection, 13 INT'L REV. L. & ECON. 217, 223 (1993) (discussing optimal enforcement when some individuals are more sophisticated than others).
108. A possible counterargument is that repeat sophisticated offenders may increase their avoidance efforts under the APP, which would be of greater benefit to them than under the DPP. Under certain conditions, this would result in more, rather than less, crime. See Jacob Nussim & Avraham Tabbach, Deterrence and Avoidance 11-16, 18-24 (Oct. 20, 2005) (unpublished manuscript), available at http://ssrn.com/abstract=844828 (showing that under certain conditions, higher sanctions encourage criminals to take more avoidance measures and reduce their expected sanctions); cf. Chris William Sanchirico, Character Evidence and the Object of Trial, 101 COLUM. L. REV. 1227, 1276 (2001) (arguing that if bad character evidence were admitted at the conviction stage, the disincentive for engaging in crime would be weakened, since character evidence enhances the probability of conviction, both for those who committed the prescribed acts and for those who refrained from such behavior, leading to a decrease in the marginal cost of engaging in the criminal activity ex ante; banning bad character evidence thus promotes deterrence).
109. See infra Part IV.D.
110. See infra note 108.
111. See, e.g., Talia Fisher, The Boundaries of Plea Bargaining: Negotiating the Standard of Proof, 97 J. CRIM. L. & CRIMINOLOGY 943, 950̂51 (2007) ("One can posit a situation where the task of proving the final X percent of the prosecution's case requires a vast investment in resources on its part⋯. The prosecution may regard this evidence as crucial for proving its case 'beyond a reasonable doubt'⋯ .");
112. Richard J. Gilbert & Michael L. Katz, When Good Value Chains Go Bad: The Economics of Indirect Liability for Copyright Infringement, 52 HASTINGS L.J. 961, 970 (2001) ("If it is relatively easy to detect some infringers, but not others, this pattern may lead to decreasing returns to scale (i.e., increasing marginal costs of enforcement at a given stage).");
113. Ehud Guttel & Alon Harel, Matching Probabilities: The Behavioral Law and Economics of Repeated Behavior, 72 U. CHI. L. REV. 1197, 1213 n.54 (2005) (stating that increasing enforcement can be achieved by either requiring enforcers to work more or recruiting additional personnel, under either approach "the marginal cost of enforcement is likely to increase");
114. Chris William Sanchirico, Evidence Tampering, 53 DUKE L.J. 1215, 1333 (2004) ("The phenomenon of increasing marginal costs corresponds to the exhaustion of economies of scale in enforcement.").
115. FED. R. CRIM. P. 8(a).
116. Id. 14(a).
117. Prosecutorial misconduct was one of the main concerns expressed by Justice Brennan in his dissent in Bowling v. United States, 493 U.S. 342, 363 (1990) (Brennan, J., dissenting) ("The Court today adds a powerful new weapon to the Government's arsenal⋯ . Indeed there is no discernible limit to the Court's rule; the defendant could be forced to relitigate these facts in trial after trial."). The risk of prosecutorial misconduct is a consideration in shaping procedural and evidentiary doctrines.
118. See Charles P. Bubany & Frank F. Skil-lern, Taming the Dragon: An Administrative Law for Prosecutorial Decision Making, 13 AM. CRIM. L. REV. 473, 476-77 (1976) (noting the lack of controls over prosecutorial decision making);
119. Angela J. Davis, Prosecution and Race: The Power and Privilege of Discretion, 67 FORDHAM L. REV. 13, 20-25 (1998) (discussing prosecutors' vast discretion and power);
120. Angela J. Davis, The American Prosecutor: Independence, Power, and the Threat of Tyranny, 86 IOWA L. REV. 393, 410-15 (2001) (asserting that prosecutorial misconduct occurs at numerous stages of the criminal process-including the pretrial stage and during trial-and that only on rare occasions is the misconduct discovered);
121. Bruce A. Green, Policing Federal Prosecutors: Do Too Many Regulators Produce Too Little Enforcement?, 8 ST. THOMAS L. REV. 69, 70 (1995) ("[E]vidence of prosecutorial misconduct, particularly in federal cases, may be difficult to obtain ⋯ .");
122. Lesley E. Williams, The Civil Regulation of Prosecutors, 67 FORDHAM L. REV. 3441, 3442-47 (1999) (discussing how professional norms and statutory and constitutional law fail to regulate prosecutorial behavior in light of prosecutorial immunity).
123. See infra Part IV.B.
124. See infra Part IV.B.
125. STEIN, supra note 16, at 154.
126. Cf. Andrew James McFarland, Note, Lewis v. United States: A Requiem for Aggregation, 46 CATH. U. L. REV. 1057, 1077-79 (1997) (discussing cases in which defendants subjected to multiple charges for petty offenses were held to be entitled to the additional protections offered by jury trial).
127. See infra note 121 and accompanying text.
128. See Charles Nesson, The Evidence or the Event? On Judicial Proof and the Acceptability of Verdicts, 98 HAW. L. REV. 1357, 1366-67 (1985).
129. STEIN, supra note 16, at 206-07 (citation and footnotes omitted).
130. See, e.g., Eric S. Janus & Robert A. Prentky, Forensic Use of Actuarial Risk Assessment with Sex Offenders: Accuracy, Admissibility and Accountability, 40 AM. CRIM. L. REV. 1443, 1448-58 (2003) (discussing the use of statistical evidence in evaluating risks related to sexually violent predators).
131. STEIN, supra note 16, at 206-07.
132. Id. at 207. It is beyond the scope of this Article to discuss the pros and cons of using naked statistical evidence in criminal cases. Cf. Henry M. Hart, Jr. & John T. McNaughton, Evidence and Inference in the Law, in EVIDENCE & INFERENCE 48, 54 (Daniel Lerner ed., 1958) C'[T]he law refuses to honor its own formula when the evidence is coldly 'statistical.'");
133. Nesson, supra note 117, at 1379 (stating that cases based only on probabilistic evidence are unlikely to reach the jury because "the factfinder cannot reach a conclusion that the public will accept as a statement about what happened").
134. See supra Part I.
135. See Nesson, supra note 117, at 1366̂67.
136. Charles Nesson, Reasonable Doubt and Permissive Inferences: The Value of Complexity, 92 HAW. L. KEV. 1187, 1195 (1979).
137. Id. at 1199 (citation omitted).
138. Cf. supra Part III.A.
139. See supra Part III.C.
140. See supra Part III.B.
141. See supra Part II.B.2.
142. See infra text accompanying notes 132-33.
143. See supra Part II.B.2.
144. See supra Example 4.
145. See STEIN, supra note 16, at 64 ("In real life, evidence is constantly missing ⋯. Fact-finders have to settle for less.").
146. See supra text accompanying notes 16-17.
147. See STEIN, supra note 16, at 65 (noting that fact-finders are forced to make probability estimates based on inadequate evidence).
148. See supra note 1 and accompanying text.
149. Id.
150. A similar argument to the one discussed here is sometimes raised against the application of the Hand formula in torts, which, arguably, requires courts to calculate expected damages and costs of precaution and then compare them with each other in order to determine whether the defendant was negligent or not. See COOTER & ULEN, supra note 32, at 351-52 ("The marginal Hand rule states that the injurer is negligent if the marginal cost of his or her precaution is less than the resulting marginal benefit ⋯. To apply the Hand rule, the decision-maker must know whether a little more precaution costs more or less than the resulting reduction in expected accident costs."). However, in order to implement the Hand formula, it is sufficient that the court determine whether the marginal expected damages are higher or lower than the marginal costs of precautions, and it need not make any accurate calculation of those figures. See id.; Ariel Porat, Offsetting Risks, 106 MICH. L. REV. 243, 272-73 (2007) (explaining how probabilistic rules can be applied with rough, rather than accurate, information about probabilities).
151. See, e.g., Andrew D. Leipold & Hossein A. Abbasi, The Impact of Joinder and Severance on Federal Criminal Cases: An Empirical Study, 59 VAND. L. REV. 349, 367 (2006) (showing empirically that criminal defendants who face multiple charges in a single trial have a harder time prevailing than those who face several trials of one count each).
152. Id.
153. The prior-acts and similar-crimes doctrines allow courts and jurors, under certain conditions, to consider the accumulation of the evidence of all charges. See supra Part Il.B.l. But as we explained previously, these two doctrines differ from the APP. Id.
154. See supra Part IV. Interestingly, those who oppose aggregating probabilities across different offenses, both explicitly and implicitly, could make use of exactly the opposite argument: different charges should not be brought at the same trial to avoid the risk of aggregation of this type.
155. See Nesson, supra note 124, at 1188 ("[D]ue process requires that the prosecution in a criminal case prove each and every material element of a criminal offense beyond a reasonable doubt.⋯").
156. Dana, supra note 102, at 742̂43 ("The question whether probabilities of detection escalate is ultimately an empirical matter, but not a matter easily subject to study. Because offenders are reluctant to provide candid information regarding their undetected violations, researchers face huge obstacles in developing any comparative assessments of the success of different groups of offenders in evading detection.").
157. See supra Part III.C.
158. See Oren Gazal-Ayal, Partial Ban on Plea Bargains, 27 CARDOZO L. REV. 2295, 2325 (2006) (noting that prosecutors incorporate rules which ex- elude reliable evidence in the plea bargaining stage because the bargaining "takes place in the shadow of the trial").
159. IMMANUEL KANT, THE METAPHYSICS OF MORALS 105 (Mary Gregor ed., Mary Gregor trans., Cambridge University Press 1996) (1797).
160. We do not argue, however, that Kant was committed to the versions of retributivism that we present below. Kant's theory of punishment has been interpreted by many theorists, and we do not purport to provide an interpretation of it here.
161. David Dolinko, Some Thoughts About Retributivism, 101 ETHICS 537, 541-42 (1991).
162. Id. at 542.
163. Id. at 550 (discussing Jean Hampton's characterization of retributivism in her essay The Retributive Idea, in FORGIVENESS AND MERCY 125-26 (1988)); cf. Thomas E. Hill, Kant on Wrongdoing, Desert, and Punishment, 18 LAW & PHIL. 407, 409 (1999) (illustrating retributivists' use of proportionality in sentencing).
164. Hill, supra note 151, at 425.
165. See Russell L. Christopher, Deterring Retributivism: The Injustice of "Just" Punishment, 96 Nw. U. L. REV. 843, 845 n.l (2002) ("[A] precise definition of retributivism has proven elusive ⋯.").
166. See infra notes 155-60 and accompanying text.
167. See, e.g., Joel Feinberg, The Expressive Function of Punishment, in DOING AND DESERVING 95, 98 (1970);
168. Jean Hampton, The Moral Education Theory of Punishment, 13 PHIL. & PUB. AFF. 208, 212 (1984).
169. See Dolinko, supra note 149, at 541-42 (comparing justifications for punishment).
170. ROBERT NOZICK, PHILOSOPHICAL EXPLANATIONS 370(1981).
171. Id. at 376̂77.
172. Feinberg, supra note 155, at 98.
173. Hampton, supra note 155, at 212.
174. It is possible, of course, to develop an expressivist theory that focuses on the condemnation of the character of the actor or his culpability rather than condemnation of the acts he has performed. This is not the route taken by traditional expressivist theories of punishment. See, e.g., id. at 225 (discussing that punishment should educate someone that a particular act is wrong and not concern itself with their character or moral duties).
175. See, e.g., id. (acknowledging that punishment is only one form of moral education).
176. See Feinberg, supra note 155, at 98̂99, 263; Hampton, supra note 155, at 225.
177. Hampton, supra note 155, at 216 ("[0]ur principal concern as we punish is to get the wrongdoer to stop doing the immoral action by communicating to her that her offense was immoral.").
178. See, e.g., id. at 225 (discussing how punishment should focus on educating that a specific act is wrong).
179. Id. at 227 (emphasis added).
180. Feinberg, supra note 155, at 100.
181. See, e.g., A. P. Simester & G. R. Sullivan, On the Nature and Rationale of Property Offences, in DEFINING CRIMES 168, 186̂87 (R. A. Duff & Stuart P. Green eds., 2005).
182. Id.
183. See supra Part V.A.
184. See supra Part IV.
185. See supra Part III.B.
186. See supra Part II.B.
187. See supra Part IV.A.