The Blackstone ratio,modified

Abstract

In his discussion of evidentiary policies, Blackstone famously noted that ‘it is better that ten guilty persons escape than that one innocent suffer’ (Blackstone 1769). The conventional wisdom among lawyers, judges as well as academics holds that accepting this statement as a maxim necessitates the adoption of pro-defendant evidentiary rules. It is also commonly believed that costs associated with false convictions being greater than those associated with failures to punish offenders due to the presence of punishment costs provides a utilitarian rationale for Blackstonian principles. After formalizing Blackstone ratios (either as marginal rates of substitution or, alternatively, as the ratio between quantities of errors), I show these conventional views are incorrect. I then propose a simple modification of the Blackstone ratio, which shifts the focus from aggregate outcomes to consequences for individuals within the criminal justice system. This modification better aligns commonly held views about the Blackstone ratio with its actual implications and justifications.

1. Introduction

‘[B]etter that ten guilty persons escape, than that one innocent suffer’. ¹ This is William Blackstone’s well-known statement which later provided the motivation for many commentaries on, and was also used to capture the rationale behind, many pro-defendant procedures and doctrines in criminal law. Moreover, similar statements in which different numbers have been substituted for $10$ in Blackstone’s statement have been made by others, some pre-dating Blackstone. ² Thus, today, Blackstone’s ratio is used to refer to a more general proposition about the asymmetry between the perceived costs, or social tolerability, of punishing innocent parties and failing to punish offenders, and embodies one of the first principles to which students of the law are exposed. It is therefore unsurprising that Blackstone’s ratio is far from being merely an academic curiosity, but it has also greatly influenced actual legal decision making, precedents, and therefore, the law. Given its great importance, many have attempted to construe the precise meaning of Blackstone’s ratio, ³ and others have interpreted it as providing the basis for a variety of criminal procedural safeguards which call for different types of protections for criminal defendants. ⁴ Yet others have suggested that the ratio’s prioritization of wrongful convictions over wrongful acquittals can be rationalized based on various characteristics of the criminal justice system. ⁵

Here, I question whether Blackstone ratios, by which I mean ratios of the type Blackstone has referred to more generally, as opposed to the specific ratio of ten to one, can serve the function that they have been understood to serve. To do so, I provide a simple formalization of Blackstone’s ratio, which follows, in my opinion, the most natural economic interpretation of his original statement. Namely, I conceptualize Blackstone’s ratio as a lower bound on the marginal rate of substitution between the number of wrongful convictions and the number of wrongful acquittals associated with social preferences. I find this to be the most natural interpretation of Blackstone’s statement, because the statement specifies how two things ought to be traded-off, and this is precisely what is conveyed through the marginal rate of substitution. In fact, this appears to be the definition which has been implicitly adopted in some prior studies attempting to empirically assess Blackstone ratios most consistent with people’s preferences, ⁶ but which has surprisingly evaded prior attempts at formalizing the Blackstone ratio (see, e.g., DeKay (1996)).

It is also worth noting that Blackstone made this statement when describing a principle that ought to constrain the nature of evidentiary rules. Thus, Blackstone’s ratio implies that we ought to implement procedures and policies in our criminal justice system such that the rate at which we actually trade-off the number of failures to convict guilty people and the number of wrongful convictions is no smaller than some ratio $B$ greater than $1$ (and, specifically, $10$ in Blackstone’s statement). This trade-off, in turn, is measured by the marginal rate of transformation, that is, the rate at which we can trade-off the two errors, given behavioral and technological constraints. I allow $B$ to be any number greater than $1$ , to incorporate the asymmetry inherent in Blackstone’s statement, and also to allow for the variation in views regarding the precise ratio noted previously.

I then question whether the Blackstone ratio thus defined can provide a meaningful criterion that can guide criminal justice policy. My analysis reveals two important findings contrary to conventional wisdom. First, an arbitrarily large Blackstone ratio can emerge even when convictions are made through a ‘more likely than not’ (MLTN) standard—a phrase often, but not always, used interchangeably with the ‘preponderance of the evidence’ standard ⁷ —which is a weaker standard than the ‘beyond a reasonable doubt’ standard used in criminal trials. In fact, this standard is typically associated with a Blackstone ratio that is greater than one within standard law enforcement models. Thus, a Blackstone ratio that places greater emphasis on acquitting the innocent rather than convicting the guilty does not necessarily imply the high standards of proof adopted in criminal justice systems. It is worth noting that this analysis is purely positive in its nature, that is, it does not relate to a normative claim. It merely assesses the validity of statements made in the literature that implementing a large Blackstone ratio requires heightened standards of proof.

Then, I turn to a related normative issue. There is an impression created in the literature that punishment costs imply large Blackstone ratios. This claim is based on the observation that, given any crime rate, an additional wrongful conviction generates punishment costs, while an additional wrongful acquittal reduces them. ⁸ Thus, wrongful convictions carry greater social costs than wrongful acquittals, and therefore optimal policies ought to reflect this asymmetry, which in turn provides a basis for a Blackstone ratio greater than $1$ . I show that this implication does not hold, either: the optimal conviction criterion that emerges in the standard enforcement model with costly punishments can require tolerating more than one additional false conviction to eliminate a failure to convict, that is, a Blackstone ratio smaller than one.

These results, taken together, reveal a lack of fit between what Blackstone’s ratio is believed to stand for and its implications when interpreted in the intuitive manner which I have described. Thus, although I believe that the most natural interpretation of the Blackstone ratio is that it is a constraint on the marginal rate of substitution, I next question whether an alternative definition of Blackstone’s ratio, one which has been suggested in the literature, may perform better. This alternative definition focuses on the ratio between the quantities of the two types of errors, as opposed to how they ought to be traded-off. ⁹ Unsurprisingly, this definition leads to similar problems.

I then consider a simple modification of Blackstone’s statement which might mitigate the disconnect between what the statement describes and its purported implications. For this purpose, I propose a modified ratio, which I call the $M-$ ratio. The $M-$ ratio focuses on the trade-offs between the probabilities of erroneous outcomes, conditional on the defendant’s actual guilt, as opposed to trade-offs between the numbers of erroneous outcomes, as in Blackstone’s original statement. This is formalized by the marginal rate of substitution, $M$ , between two simple probabilities: $P($ conviction $|$ innocence $)$ and $P($ non-conviction $|$ guilt $)$ . I show that the $M-$ ratio performs better with respect to both issues described above. Specifically, (i) when punishment is costly, the optimal standard of proof is associated with an $M-$ ratio greater than $1$ , and (ii) an $M-$ ratio generates an evidentiary standard that is stricter than the MLTN standard as long as $M$ is greater than the ratio between the truly innocent to truly guilty parties, that is, the ratio between the compliance rate and the offense rate. Thus, I conclude by suggesting a reformulation of Blackstone’s statement that describes the $M-$ ratio:

‘It is better that a guilty person more likely escape conviction, than an innocent more likely suffer’.

This statement, too, involves a trade-off. It sacrifices clarity for brevity. A more precise and lengthier statement that describes the $M-$ ratio, and which departs from Blackstone’s original statement as little as possible, is the following:

‘It is better that a guilty person’s likelihood of conviction be reduced by $Mx\mathrm{\%}$ , than an innocent person’s likelihood of conviction be increased by $x\mathrm{\%}$ ’

Quite interestingly, what is responsible for the better performance of the $M-$ ratio is its focus on individuals rather than aggregates. The classic Blackstone ratio focuses on the numbers of guilty and innocent convicted, whereas the $M-$ ratio focuses on individual consequences (the probabilities of conviction, conditional on guilt and innocence). As a result, the trade-off that defines the $M-$ ratio is unresponsive to changes in the aggregate offense rate, while the classic Blackstone ratio is. Because evidentiary standards like the MLTN and optimal standard of proof also have similar properties, the $M-$ ratio can more naturally be related to them.

Before formalizing the points described above, it is worth describing how the this article builds on prior scholarship. While most prior scholarship focusing on the relationship between Blackstone ratios and evidentiary rules appears to be informal, in the mathematical sense, a notable exception is DeKay (1996).

As here, DeKay (1996) considers various mathematical definitions of the Blackstone ratio and questions how it relates to standards of proof. Unlike here, DeKay (1996) does not consider a definition of the Blackstone ratio which focuses on the marginal rate of substitution between the number of errors. Instead, it considers the ratio of error frequencies definition I consider in section 3, as well as the ratio of (relative) utilities from errors. However, DeKay’s utility based analysis is conceptually problematic, because it does not define utility functions in their ordinary economic sense, that is, as representations of rankings of possible outcomes, which in this case, corresponds to error pairs. ¹⁰ Also, as noted in Kaplow (2011a), DeKay’s analysis implicitly takes behavior to be exogenous, and is thus incapable of incorporating deterrence effects, which are at the heart of most theoretical economic analyses of crime.

Another important difference between DeKay (1996) and this article is their focus. The former is concerned with highlighting the fact that standards of proof and Blackstone-like ratios are different concepts, which are affected by different considerations. Thus, it concludes that ‘many authors have incorrectly assumed that the standard of proof completely determines the ratio of erroneous acquittals to erroneous convictions’ (DeKay, 1996, p. 131). This difference between concepts is presumably apparent to most theoretical social scientists today. Indeed, the question posed here is whether high Blackstone ratios necessarily imply pro-defendant biases, and whether optimal standards of proof in the presence of costly sanctions imply large Blackstone ratios. Thus, it takes the type of difference that DeKay notes, as given, and conducts a systematic analysis of the relationship between the two concepts to dispel conventional wisdom regarding the same. It shows that no Blackstone ratio, regardless of how large it may be, can on its own justify pro-defendant evidentiary rules, and that costly sanctions are insufficient to provide a utilitarian rationale for large Blackstone ratios. Moreover, unlike any prior work that I am aware of, including DeKay (1996), this systematic analysis allows me to propose a simple modification to the Blackstone ratio to restore its usefulness.

My analysis is also closely related to the economics literature on standards of proof, and in particular Kaplow (2011a). The pioneering work on optimal standards of proof, namely Lando (2002), and Demougin and Fluet (2005, 2006, and 2008), conceive of standards of proof as a threshold evidence to which the evidence observed in court must be compared to render decisions, and investigate the properties of optimal standards of proof. Subsequently, many economics articles that incorporate additional functions and effects of evidentiary requirements adopted similar conceptions of the standard of proof. ¹¹ Notably, none of these articles explicitly questioned the relationship between Blackstone ratios and (optimal) standards of proof, as I do here.

The closest in this regard is Kaplow (2011a), which focuses on a trade-off introduced in Kaplow (2011a, 2011b, and 2011c) and Mungan (2011). In these articles, evidentiary standards may affect not only people’s decisions to refrain from crime, but also innocent parties’ decisions to engage in socially benign (or even desirable) behavior. Kaplow (2011a) notes that when these types of ‘chilling behavior’ are accounted for, the optimal standard of proof balances deterrence benefits and chilling costs. The article then explains the mismatches between optimal standards of proof and standards considered in the literature, including the MLTN standard, as well as objectives like achieving a targeted ratio of judicial errors. Kaplow also notes that the presence of costly sanctions may have an ambiguous effect on the optimal standard of proof. ¹² However, Kaplow (2011a) does not focus on how the optimal standard of proof or the MLTN standard compares to Blackstone ratios. Moreover, in its comparisons it focuses on mismatches between the standards analyzed (e.g., the optimal standard of proof vs. MLTN) as opposed to questioning whether principles reflected by one concept can appropriately rule out other principles or practices as I do in this article.

This latter exercise is useful for ascertaining whether conventional and doctrinal approaches can act as useful constraints on the rules or standards that can be adopted. This is because principles that guide legal practices are often not precise, and thus do note define, for instance, the specific evidentiary standard to be applied to achieve optimality. The latter task is, in fact, impracticable as noted in various criticisms of Kaplow (2011a). ¹³ This is perhaps why legal decision makers often prefer to focus on more workable concepts that are easier to describe. The Blackstone ratio is perceived as providing this type of constraint. My analysis here highlights that its focus on the number of errors, as opposed to the probabilities of errors conditional on behavior, makes it ill-fit to serve the function it has been assumed to serve in the literature. Thus, slightly modifying it, as described above, may restore its function, when correctly interpreted, as a helpful and intuitive tool in discussing criminal justice policies.

Another point worth emphasizing is that the analysis here builds on prior studies (see, e.g., the survey in Polinsky and Shavell, 2007) which follow Becker (1968) in studying the deterrence effects associated with sanctions and other law enforcement policies. More specifically, following the prior literature on the optimal standard of proof noted above, deterrence effects are incorporated by allowing the offense rate to be responsive to the evidentiary standards adopted by the government. These effects play a central role in Section 2.2 in demonstrating that large Blackstone ratios need not imply strong pro-defendant protections; and they play a similar role in showing that punishment costs need not provide a rationale for large Blackstone ratios in Section 2.3.

Given the impact and prevalence of Blackstone’s ratio, it is important to clarify and formalize the points described above. This is done in the next three sections. I provide brief concluding remarks in the fifth section. An appendix in the end contains formalizations of some properties of evidentiary rules.

2. Blackstone’s ratio and implications

To formalize social preferences consistent with Blackstone’s statement, consider a social objective, $\mathcal{B}(N_G,N_I)$ , which is a function of the two objects of interest, namely the number of guilty persons who escape conviction, $N_G$ , and the number of innocent people who suffer, $N_I$ . Implicit in Blackstone’s statement is that ${\mathcal{B}}_{N_G},{\mathcal{B}}_{N_I}<0$ , that is, both types of erroneous outcomes are socially undesirable. Thus, conceiving of $\mathcal{B}$ as a social utility function, one can express the set of combinations of $N_G$ and $N_I$ which give rise to the same (attainable) social utility, $U$ , through a social indifference curve

$$N_G^U(N_I)suchthat\mathcal{B}(N_G^U(N_I),N_I)=U$$

(1)

Since, ${\mathcal{B}}_{N_G},{\mathcal{B}}_{N_I}<0$ , it follows that the marginal rate of substitution ( $MRS$ ) between $N_G$ and $N_I$ is:

$$S\equiv MR{S}_{N_I,N_G}=-\frac{d{N}_G^U(N_I)}{d{N}_I}=\frac{{\mathcal{B}}_{N_I}}{{\mathcal{B}}_{N_G}}>0$$

(2)

If, for instance, this marginal rate of substitution is constant and equals $10$ , then this suggests that reducing $N_I$ by one and increasing $N_G$ by $10$ would leave social utility unchanged. On the other hand, the same change in the number of erroneous outcomes would imply an increase in social utility if $S(N_I,N_G)>10$ , which corresponds to Blackstone’s statement: ‘[B]etter that ten guilty persons escape, than that one innocent suffer’.

This set-up does not require any specific functional form for $\mathcal{B}$ . However, the simplest case, where the $MRS$ is constant, is obtained when $\mathcal{B}(N_G,N_I)=-(N_G+B{N}_I)$ , where the constant $B>0$ can be interpreted as Blackstone’s ratio. Figure 1, below, depicts this case through social indifference curves ( $N_G^U(N_I)$ ) in $N_I,N_G$ space, along with a ‘minimum error frontier’ labeled $\epsilon$ , that reflects the attainable $N_I,N_G$ pairs given evidentiary and behavioral constraints described in the next sub-sections. The point at which this frontier is tangent to a social indifference curve (e.g., point $A$ marked in Figure 1) is where $\mathcal{B}$ is maximized, because ${\mathcal{B}}_{N_G},{\mathcal{B}}_{N_I}<0$ .

Figure 1.

Blackstonian indifference curves and the minimum error frontier.

It is worth noting that the marginal disutility from each erroneous outcome need not be constant, as depicted in Figure 1, and the set-up described above allows for this possibility. Thus, more generally, one can conceive of Blackstone’s ratio as placing a lower bound $B$ on the $MRS$ associated with $\mathcal{B}$ around its maximizer, that is, $S\ge B$ , given the constraints formalized below. This section focuses on this interpretation of Blackstone’s ratio, and an alternative interpretation proposed in the literature is considered in Section 3.

2.1. Evidentiary standards and conviction probabilities

Blackstone made his statement as one which ought to guide principles governing evidentiary rules. Thus, to study the relationship between evidentiary rules and Blackstone ratios, I first describe how $N_I$ and $N_G$ are affected by evidentiary standards, first without employing any specialized assumptions, and subsequently specifying a relationship between these standards and crime rates to allow more specific observations. Generally, the number of innocent people who suffer punishment is given by

$$N_I=N\times P(I)\times P(T|I)\times P(C|I,T)$$

(3)

where $N$ is the number of people in the population; $P(I)=\frac{\mathrm{\#}\text{ of innocent people}}{N}$ is the unconditional probability of innocence; $T$ denotes trial, such that $P(T|I)$ is the probability of an innocent person standing trial; and $C$ denotes a conviction such that $P(C|I,T)$ is the probability of conviction conditional on trial and innocence. Similarly,

$$N_G=N\times P(G)\times (1-P(T|G)\times P(C|G,T))$$

(4)

where $G$ denotes guilt. Here $1-P(T|G)\times P(C|G,T)$ is the probability a guilty person escapes punishment, either because they don’t stand trial or because they stand trial but are not convicted.

To abbreviate expressions, I adopt the following notation for these probabilities

$$\begin{array}{rl}\mu & =P(G)\\ p& =P(T|G)\\ q& =P(T|I)\\ \alpha & =P(C|I,T)\\ \beta & =P(C|G,T)\end{array}$$

(5)

In general, the evidentiary standards adopted in trials affect $\alpha$ and $\beta$ , which may, in turn, have deterrence effects. To capture these effects, I adopt a modeling approach introduced in the prior literature on optimal standards of proof (see, e.g., Demougin and Fluet (2006), Mungan (2020b), and Fluet and Mungan (2025)). In this approach, $\beta$ and $\mu$ are modeled as functions of $\alpha$ , that is, $\beta =\beta (\alpha )$ and $\mu =\mu (\alpha )$ , ¹⁴ as opposed to functions of the evidentiary standards chosen by the government. This approach can be adopted, since each evidentiary standard corresponds to a particular false conviction rate, $\alpha$ . This eliminates the need to keep track of the evidentiary standard as a separate policy tool, and is convenient for purposes of identifying the relationship between evidentiary rules and outcomes of interest, by expressing them as a function of $\alpha$ .

Intuitively, when the government adopts the strictest evidentiary standard, such that no convictions are ever obtained, it follows that $\alpha =\beta =0$ . Similarly, when the government adopts the most liberal evidentiary standard, every defendant is convicted, and thus $\alpha =\beta =1$ . Consistent with this intuition, as one relaxes the evidentiary standard, both $\alpha$ and $\beta$ are increased. Thus, one can express $\beta$ as an increasing function of $\alpha$ with

$$\beta (\alpha )>\alpha \text{ for all }\alpha \in (0,1)$$

(6)

to keep track of how $\beta$ and $\alpha$ co-move as one changes the evidentiary standard. A more detailed and mathematical explanation of this modeling approach can be found in the appendix, below, as well as in Section 3.1 of Fluet and Mungan (2025).

As the government changes evidentiary standards, tracked through changes in $\alpha$ as explained above, this can have impacts on the behavior of individuals. For instance, when potential offenders notice that they are less likely to be convicted upon committing offenses, due to increased evidentiary standards, they may be more inclined to commit offenses. The precise impact of changes in evidentiary standards on the offense rate naturally depends on the behavioral responses of individuals, which can be modeled in different ways as explained in Sections 2.1 and 2.2., below. However, for current purposes it is sufficient to note the general relationship between evidentiary standards and the offense rate by letting $\mu =\mu (\alpha )$ . To simplify the analysis, I assume that ${\mu }_l\equiv min\mu >0$ and ${\mu }_h\equiv max\mu <1$ , that is, that there are compliers and offenders under all feasible policies. As will become clear below, these assumed conditions hold in standard models of crime and deterrence when some people derive large benefits from crime and others dislike committing them.

Next, note that the possible choices of $\alpha$ imply a pair of $N_I$ and $N_G$ which can be compactly expressed using the notation in (5 ) as

$${\overline{N}}_I(\alpha )=N(1-\mu (\alpha ))q\alpha and{\overline{N}}_G(\alpha )=N\mu (\alpha )(1-p\beta (\alpha ))$$

(7)

Thus, the collection of ${\overline{N}}_I(\alpha )$ and ${\overline{N}}_G(\alpha )$ pairs form the set of erroneous outcomes achievable through the trial system under various evidentiary rules, for example, the minimum error frontier $\epsilon$ depicted in Figure 1. Therefore, one can potentially ‘transform’ one of the errors into some quantity of the other through changes in $\alpha$ . I call the rate at which this can be accomplished the marginal rate of transformation between $N_I$ and $N_G$ , which—assuming $d{\overline{N}}_I/d\alpha \ne 0$ —is given by the following:

$$t(\alpha )\equiv MR{T}_{N_I,N_G}=-\frac{d{\overline{N}}_G/d\alpha }{d{\overline{N}}_I/d\alpha }=\frac{\mu (\alpha ){\beta }^{\mathrm{\prime }}(\alpha )p-{\mu }^{\mathrm{\prime }}(\alpha )(1-\beta p)}{(1-\mu (\alpha ))q-{\mu }^{\mathrm{\prime }}(\alpha )\alpha q}$$

(8)

Thus, $t(\alpha )$ measures the slope of the minimum error frontier $\epsilon$ in $N_I,N_G$ space as depicted in Figure 1, achieved when an evidentiary standard such that $P(C|I,T)=$ $\alpha$ is used. Importantly, $t$ depends on deterrence effects, ${\mu }^{\mathrm{\prime }}$ , because it measures the trade-off between aggregate concepts, namely the numbers of innocent and guilty convicted, which are directly impacted by deterrence. This property, which is caused by the Blackstone ratio being defined through references to $N_I$ and $N_G$ , plays a key role in causing the Blackstone ratio to serve as an inadequate tool in constraining evidentiary policies.

Specifically, implementing an evidentiary rule that leads to $t(\alpha )<B\le S$ would be inconsistent with principles embodied by a Blackstone ratio of $B$ . Conversely, a policy leading to $t(\alpha )\ge B$ cannot be ruled out as being socially undesirable solely based on a Blackstone ratio of $B$ . Next, for any given Blackstone ratio $B$ , I question whether the ‘more likely than not standard’ (MLTN), denoted ${\alpha }^M$ , is necessarily such that $t({\alpha }^M)<B$ , that is, such that it can be dismissed as being socially undesirable solely based on Blackstone’s ratio.

2.2. The ‘more likely than not’ standard

I focus on the MLTN standard, because my objective is to relate the current analysis to prior legal scholarship and conventional wisdom that the Blackstone ratio is sufficient reason for adopting heightened protections for defendants. The analysis in this sub-section, summarized by Proposition 1 and Corollary 1, below, reveals that the MLTN standard cannot be ruled out by any arbitrarily large Blackstone ratio. Thus, higher standards of proof like ‘beyond a reasonable doubt’ are not necessarily implied by large Blackstone ratios.

In general, evidentiary standards can be thought of as mappings from any body of evidence which may be observed in court into convictions and acquittals at trial. ¹⁵ The MLTN standard, in turn, would require a conviction only if the observed evidence $e$ is such that ¹⁶

$$p\mu P(e|G)>q(1-\mu )P(e|I)$$

(9)

since only then would it be more likely that the defendant at trial has more likely than not committed crime. This, in turn, corresponds to a likelihood ratio test of the form:

$$\frac{P(e|G)}{P(e|I)}>\frac{q(1-\mu )}{p\mu }$$

(10)

To relate this rule to the probabilities of conviction $\alpha$ and $\beta$ associated with it, a subtle but intuitive observation is needed. To describe this observation in a simple way, I assume that the likelihood ratios associated with the evidence is continuously distributed. Because this observation is relatively simple, I only intuitively describe it here, but the appendix, below, contains a formalization (in particular, see (29), below).

The observation is the following: When likelihood ratio tests are implemented, ${\beta }^{\mathrm{\prime }}(\alpha )$ corresponds to the threshold likelihood ratio used, that is, ${\beta }^{\mathrm{\prime }}(\alpha )=\overline{L}$ when convictions occur if and only if evidence $e$ such that $\frac{P(e|G)}{P(e|I)}>$ $\overline{L}$ for some $\overline{L}>0$ is observed. This is because ${\beta }^{\mathrm{\prime }}$ represents the increase in the probability of correct convictions as one increases the probability of false convictions. This is achieved by letting marginal pieces of evidence, which previously would not suffice for a conviction, be deemed sufficient for a conviction. Because these marginal pieces of evidence are all such that $\frac{P(e|G)}{P(e|I)}=$ $\overline{L}$ , the ratio between the increase in $\beta$ versus $\alpha$ is $\overline{L}$ , and hence ${\beta }^{\mathrm{\prime }}(\alpha )=\overline{L}$ . Thus, an $\alpha$ associated with the MLTN standard, denoted ${\alpha }^M$ , assuming it is interior, is such that

$${\beta }^{\mathrm{\prime }}({\alpha }^M)=\frac{q(1-\mu ({\alpha }^M))}{p\mu ({\alpha }^M)};\mathrm{or}$$

(11)

$$N{\beta }^{\mathrm{\prime }}({\alpha }^M)p\mu ({\alpha }^M)=Nq(1-\mu ({\alpha }^M))$$

(12)

As long as the evidence generation process is sufficiently informative, ¹⁷ a standard that satisfies (11) exists, and in many cases it is unique (see, e.g., the examples used in the proof of Corollary 1). Thus, in what follows I assume existence and also the uniqueness of ${\alpha }^M$ to simplify descriptions and derivations of results. ¹⁸

A key insight is that the MLTN standard (captured by (11)) is not a function of the (marginal) deterrence effects of evidentiary standards (i.e., ${\mu }^{\mathrm{\prime }}$ ), while the manner in which the quantity of the two errors can be traded-off is. More precisely, the MLTN standard only specifies (through (11)) how conditional probabilities of judgement errors, that is, $P(C|I,T)=\alpha$ and $1-P(C|G,T)=1-\beta$ , ought to be traded-off, given the offense rate that is observed when a trial is taking place. In fact, it requires that, holding the offense rate constant, making evidentiary standards slightly weaker result in a reduction in the number of guilty people who avoid punishment (i.e., the left hand side of (12)) that equals the increase in the number of innocent people convicted (i.e., the right hand side of (12)).

However, changes in evidentiary standards around MLTN affect not only the conditional probabilities $\alpha$ and $\beta$ , but also the offense rate, as long as ${\mu }^{\mathrm{\prime }}({\alpha }^M)\ne 0$ . In particular, when making convictions easier leads to deterrence effects, i.e., ${\mu }^{\mathrm{\prime }}<0$ , the reductions in ${\overline{N}}_G$ and the increase in ${\overline{N}}_I$ obtained through laxer evidentiary requirements are greater than the left hand and right hand sides of (12), respectively. This is reflected by the second terms which appear both in the numerator and the denominator of (8). These impacts through the crime rate naturally affect reductions in ${\overline{N}}_G$ more than the increase in ${\overline{N}}_I$ , as long as a guilty person is more likely to escape conviction than an innocent person is likely to be punished (i.e., as long as $1-p\beta ({\alpha }^M)>q{\alpha }^M$ ). A sufficient condition for this result is that $1-p>q$ , that is, the guilty are more likely to avoid trials than the innocent are likely to stand trials. Thus, when this condition holds, the MLTN standard is associated with a marginal rate of transformation that is greater than $1$ . These observations are reflected in the proof of the next proposition which summarizes the relationship between Blackstone ratios and the MLTN standard.

Proposition 1

The more likely than not standard may be associated with any arbitrarily large marginal rate of transformation. Thus, large Blackstone ratios, on their own, are insufficient to necessitate standards that are stronger than MLTN.

In particular, (I) $t({\alpha }^M)>1$ , as long as ${\mu }^{\mathrm{\prime }}({\alpha }^M)<0$ and the guilty are more likely to escape trials than the innocent are likely to stand trial, i.e., $1-p>q$ . (II) $t({\alpha }^M)>B$ , if $-{\mu }^{\mathrm{\prime }}({\alpha }^M)$ is sufficiently large and $1-p>Bq$ .

Proof.

Plugging (11) into (8) reveals that

$$t({\alpha }^M)=\frac{(1-\mu ({\alpha }^M))q-{\mu }^{\mathrm{\prime }}({\alpha }^M)(1-\beta ({\alpha }^M)p)}{(1-\mu ({\alpha }^M))q-{\mu }^{\mathrm{\prime }}({\alpha }^M){\alpha }^{M}q}$$

(13)

When ${\mu }^{\mathrm{\prime }}({\alpha }^M)<0$ , it follows that $t({\alpha }^M)>1$ if $(1-\beta ({\alpha }^M)p)>{\alpha }^{M}q$ which holds when $1-p>q$ .

$\genfrac{}{}{0ex}{}{lim}{{\mu }^{\mathrm{\prime }}({\alpha }^M)\to -\mathrm{\infty }}t({\alpha }^M)=\frac{1-\beta ({\alpha }^M)p}{{\alpha }^{M}q}$ . Thus, for any $B$ , it follows that there exists large enough $-{\mu }^{\mathrm{\prime }}({\alpha }^M)$ such that $t({\alpha }^M)>B$ if $1-p>Bq$ .▪

The proposition formalizes the claim that Blackstone ratios being greater than some arbitrary number is insufficient to rule out the MLTN standard as an adequate evidentiary standard to achieve Blackstonian social objectives under two conditions. One condition is relatively intuitive: the probability with which a guilty person is not brought to trial is $B$ times greater than the probability with which an innocent person is brought to trial. Presumably, this condition is quite plausible when the proportion of innocent people standing trial is small for a large range of Blackstone ratios (e.g., the condition holds for $B=10$ even when $p=90\mathrm{\%}$ , if $q<1\mathrm{\%}$ ).

The second condition requires deterrence to be enhanced when evidentiary standards are made slightly weaker than MLTN to make convictions easier. This condition may also appear intuitive to many, since making convictions easier ought to deter people from committing crimes, especially when they are more likely to be brought to trial upon committing crime than when they do not (i.e., $p>q$ ). However, in the law enforcement literature it has been noted that easier convictions can have counter-deterrent effects by increasing the likelihood of being wrongfully convicted and thereby reducing the expected benefits associated with refraining from crime, that is, reducing the opportunity cost of crime. ¹⁹ Thus, I note below that even within such models, the impact of weakening evidentiary standards around MLTN is an increased level of deterrence, provided that crime commission is not the norm, that is, $\mu <1/2$ . Also, the assumptions imposed within these models allow me to identify more specific relationships between evidentiary standards and the crime rate, and thereby illustrate that arbitrarily large Blackstone ratios can be consistent with the MLTN standard.

Implications in models where wrongful convictions lower deterrence

In standard models where wrongful convictions lower deterrence, ²⁰ a person who commits crime faces an imprisonment sentence with probability $p\beta (\alpha )$ whereas a person who does not commit crime is imprisoned with probability $q\alpha$ . People who commit crime receive benefit $b$ , which differs from person to person, and $F(b)$ is the cumulative distribution of these benefits across people (who are assumed to form a continuum of measure $1$ for simplicity) with $F^{\mathrm{\prime }}(b)=f(b)$ . Normalizing the cost of being punished to $1$ , a person’s expected net-benefit from crime is $b-p\beta (\alpha )$ , whereas the person faces an expected cost of $q\alpha$ when they refrain from crime. Thus, people, who are assumed to be risk-neutral, commit crime if

$$b^{*}(\alpha )\equiv p\beta (\alpha )-q\alpha <b$$

(14)

and the crime rate is

$$\mu (\alpha )=1-F(b^{*}(\alpha ))$$

(15)

It is worth briefly noting a couple additional properties of $\beta$ , which captures the evidence generation process, that becomes relevant in these models. First, whenever likelihood ratio tests are used, it follows that

$${\beta }^{\mathrm{\prime }}(\alpha )>0>{\beta }^{\mathrm{″}}(\alpha )$$

(16)

Again, because these properties are intuitive, I relegate their formalization to the appendix, below, and only provide the intuition behind them. The first inequality follows from the fact that as more types of evidence are found sufficient for a conviction, both the guilty and the innocent face a greater risk of being convicted, and thus $\beta$ and $\alpha$ move in the same direction (see (29) in the appendix). The second inequality follows from the fact that conviction probabilities are increased by allowing convictions based on types of evidence which are less and less indicative of guilt (i.e., are associated with a smaller likelihood ratio $\frac{P(e|G)}{P(e|I)}$ ) and thus ${\beta }^{\mathrm{\prime }}$ is decreasing in $\alpha$ (see (30) in the appendix). Given these observations, it follows that crime is minimized by an interior $\alpha$ , as long as ${\beta }^{\mathrm{\prime }}(1)<\frac{q}{p}$ , that is, when the evidentiary process is sufficiently informative, ²¹ and I assume this condition holds to abbreviate the analysis.

Building on these preliminary observations, note that when the evidentiary standard for convictions is MLTN, it follows that

$${\mu }^{\mathrm{\prime }}({\alpha }^M)=-f(b^{*}({\alpha }^M))(p{\beta }^{\mathrm{\prime }}({\alpha }^M)-q)=-f(b^{*}({\alpha }^M))q\frac{1-2\mu ({\alpha }^M)}{\mu ({\alpha }^M)}$$

(17)

where the second equality is obtained by using (11). Thus, ${\mu }^{\mathrm{\prime }}({\alpha }^M)<0$ if $\mu ({\alpha }^M)<1/2$ and vice versa.

Combining this observation with proposition 1 leads to the following corollary.

Corollary 1

For any Blackstone ratio $B\ge 1$ such that $1-p>Bq$ , there exists a population (i.e., a cumulative distribution function $F$ ) such that the marginal rate of transformation associated with the more likely than not standard exceeds the Blackstone ratio (i.e., $t({\alpha }^M)>B$ ).

Proof.

The $MRT$ expressed in (13) can be re-written by using (17) as follows:

$$t({\alpha }^M)=\frac{F(b^{*}({\alpha }^M))+H(b^{*}({\alpha }^M))(2F(b^{*}({\alpha }^M))-1)(1-\beta ({\alpha }^M)p)}{F(b^{*}({\alpha }^M))+H(b^{*}({\alpha }^M))(2F(b^{*}({\alpha }^M))-1){\alpha }^{M}q}$$

where $H(b)\equiv \frac{f(b)}{1-F(b)}$ is the hazard rate associated with $F$ . Thus, if $F=1-e^{-\lambda (b+k)}$ is a shifted exponential distribution with support $[-k,\mathrm{\infty })$ , it follows that $F(b^{*}(\alpha ))>1-e^{-\lambda k}$ for all $\alpha$ , and in particular for ${\alpha }^M$ . Thus, if $\lambda k>\ln 2$ , it follows that $-{\mu }^{\mathrm{\prime }}({\alpha }^M)<0$ , since $F(b^{*}({\alpha }^M))>1/2$ . Next, note that when this condition holds and $1-p>Bq$ , it follows that

$$t({\alpha }^M)>\underset{\_}{t}(\lambda )\equiv \frac{1+\lambda (1-2e^{-\lambda k})(1-p)}{1+\lambda (1-2e^{-\lambda k})q}$$

Therefore, $\genfrac{}{}{0ex}{}{lim}{\lambda \to \mathrm{\infty }}\underset{\_}{t}(\lambda )=\frac{1-p}{q}$ , and therefore there exists large enough $\lambda$ such that $t({\alpha }^M)>B$ , when $1-p>Bq$ .▪

The corollary extends the results in Proposition 1. It shows that these results are not only possible in an abstract sense, but that they can easily emerge in settings in which the effect of evidentiary standards on deterrence is modeled through one of the most common approaches. It is also worth highlighting that both Proposition 1 and Corollary 1 focus on identifying intuitive sufficient conditions for $t({\alpha }^M)>B$ ; none of the conditions reported are necessary. In particular, because the relationship between $\beta ({\alpha }^M)$ and ${\alpha }^M$ can differ greatly based on the nature of the evidence generating process, results are derived by focusing on their upper bounds (of $1$ ) to produce a lower bound for $t({\alpha }^M)$ . Thus, in future work, conditions weaker than $1-p>Bq$ can be obtained at the expense of using specific functional forms for $\beta (\alpha )$ .

2.3. Optimal evidentiary standards with costly punishment and implied Blackstone ratios

My previous analysis reveals that relatively permissive evidentiary standards can generate quite large Blackstone ratios. Next, I investigate a relationship that goes in the opposite direction. I question whether optimal evidentiary standards that emerge in models which incorporate costly punishments result in $MRT$ s consistent with large Blackstone ratios. To do so, I describe the standard model of law enforcement with costly punishment (see, e.g., Polinsky and Shavell, 2007) which incorporates erroneous judgements (see, e.g., Rizzolli and Saraceno, 2013).

This model follows the same behavioral assumptions as described via (14) and the surrounding text. Thus,

$$\mu =1-F(b^{*})where{b}^{*}=p\beta (\alpha )-q\alpha$$

(18)

The welfare consequences of implementing $\alpha$ , on the other hand, are evaluated by the following social welfare function

$$W(\alpha )={\int }_{b>b^{*}}((1-r)b-h-p\beta c)f(b)db-F(b^{*})q\alpha c$$

(19)

Next, I describe the components of this welfare function which have not yet been introduced.

In (19), $c\ge 0$ denotes the per person social cost associated with punishment, which consists of the cost to the person being punished (normalized to $1$ ) plus the cost, or minus the net benefit, associated with punishment due to its effects on third parties or the government. When $c=0$ , the sanction is fully transferable, in the sense that the losses to the punished person generate benefits of equal magnitude elsewhere. The most well-known case of transferable sanctions is when the sanction is a monetary fine which is assumed to be collected from offenders without friction. The case of imprisonment is typically incorporated by assuming $c\ge 1$ ; there are costs to the person imprisoned and costs associated with the administration of the punishment which are not offset by any benefits that third parties may obtain from the punishment. Although cases where $c\in (0,1)$ are possible (e.g., a largely monetary sanction with a small imprisonment/community service component), it is my impression that they are not analyzed as frequently. Total punishment costs are $(1-F(b^{*}))p\beta c$ (included in the integral term in (19)) plus $F(b^{*})q\alpha c$ , which represent the punishment costs of the guilty and the innocent, respectively.

The (average) harm from each crime is $h$ , and thus total criminal harm is $(1-F(b^{*}))h$ . The final component is the benefit from crime, given by $(1-r)b$ , where $r$ determines whether or not the benefits from crime enter the social welfare function. The case where $r=0$ corresponds to the conventional utilitarian social where function where the criminal’s benefit fully enters the social welfare function. On the other hand, the case where $r=1$ can arise under different circumstances. First, criminal benefits can be outright excluded from the welfare function (see, e.g., Stigler, 1970). Second, the harm from the crime can be interpreted as equaling $h+b$ (as opposed to just $h$ ) consisting of some harm that equals the benefit to the criminal plus some additional amount (see, e.g., Rizzolli and Saraceno, 2013). Finally, Curry and Doyle (2016) provide a framework wherein the benefits from crime can also be achieved through voluntary market exchanges, and show that maximizing welfare is then equivalent to minimizing the costs of crime, that is, the case of $r=1$ . Here, I let $r\in [0,1]$ to capture these cases, and any other cases that may arise in between, to increase the generality of the analysis.

With this set-up, I question whether evidentiary standards that maximize the welfare function in (19) necessarily imply marginal rates of transformation greater than $1$ . The next proposition answers this question where the maximizer of (19), assumed to be a proper and interior maximum, is expressed as function of the harm from crime as ${\alpha }^{*}(h)$ .

Proposition 2

The optimal evidentiary standard may be consistent only with Blackstone ratios smaller than $1$ , even with costly punishments. In fact, when the minimum achievable crime rate is less than half, (i) $t(\arg min\mu (\alpha ))<1$ , and, as a result, (ii) holding all else equal, a large enough harm from crime implies $t({\alpha }^{*}(h))<1$ .

Proof.

It follows that ${\beta }^{\mathrm{\prime }}(\arg min\mu (\alpha ))=\frac{q}{p}$ and ${\mu }^{\mathrm{\prime }}(\arg min\mu (\alpha ))=0$ , and therefore $t(\arg min\mu (\alpha ))=\frac{\mu (\arg min\mu (\alpha ))}{(1-\mu (\arg min\mu (\alpha )))}$ per (8). Thus, $t(\arg min\mu (\alpha ))<1$ as long as $\mu (\arg min\mu (\alpha ))<1/2$ . Next, note that ${\alpha }^{*}(h)$ is characterized by

$${\beta }^{\mathrm{\prime }}({\alpha }^{*})=\frac{q}{p}\frac{[h+(p\beta -q\alpha )(c-(1-r))]f(b^{*})+(1-\mu )c}{[h+(p\beta -q\alpha )(c-(1-r))]f(b^{*})-\mu c}$$

(20)

Thus, it follows that, $\genfrac{}{}{0ex}{}{lim}{h\to \mathrm{\infty }}{\beta }^{\mathrm{\prime }}({\alpha }^{*})=\frac{q}{p}$ . Therefore, $\genfrac{}{}{0ex}{}{lim}{h\to \mathrm{\infty }}t({\alpha }^{*}(h))=$ $t(\arg min\mu (\alpha ))<1$ . Thus, $t({\alpha }^{*}(h))<1$ for large enough $h$ .▪

Proposition 2 reveals that costly punishments, alone, are insufficient to rationalize policies and procedures that are consistent with large Blackstone ratios. This is because the optimal evidentiary standard can require trading-off more than a single false conviction to reduce the number of criminals unpunished by one, contrary to preferences consistent with large Blackstone ratios.

The rationale behind this result is that the optimal evidentiary standard (unlike the MLTN standard) seeks to balance two competing objectives, namely achieving deterrence gains and reducing punishment costs. The former objective is best served by implementing a standard that maximizes the opportunity cost of crime commission. This is achieved when the marginal impact of the evidentiary standard on the conviction probability of a guilty person and an innocent person is equalized, i.e., $p{\beta }^{\mathrm{\prime }}=q$ . Thus, holding the crime rate constant, the deterrence maximizing standard trades-off innocent people who are punished against guilty people who escape punishment proportional to the size of each group, i.e., $\frac{\mu p{\beta }^{\mathrm{\prime }}(\alpha )}{(1-\mu )q}=\frac{\mu }{1-\mu }$ . Moreover, when deterrence is maximized, by definition the impact on the crime rate is zero, that is, ${\mu }^{\mathrm{\prime }}=0$ . Thus, the deterrence maximizing evidentiary standard is associated with a marginal rate of transformation that is smaller than $1$ , as long as the minimum achievable crime rate is less than half. As a result, when deterrence becomes a relatively important goal, which occurs when harms from crime relative to other values are high, the marginal rate of transformation between $N_I$ and $N_G$ is less than $1$ .

3. An alternative Blackstone ratio definition

As noted in the introduction, I find the interpretation of Blackstone’s statement as specifying a lower bound for the $MRS$ between $N_I$ and $N_G$ to be the most natural one. However, prior commentary and scholarship suggests that some interpret Blackstone’s statement as providing a lower bound for $N_G/N_I$ , instead. (See, for example, the description of Blackstone’s ratio in Connolly (1987) reproduced in note 9, below.) Thus, in this section I briefly note that similar issues related to existing claims regarding the Blackstone ratio emerge under this alternative interpretation, as well. I follow the same approach as in the previous section, and first note that any requirement regarding the $N_G/N_I$ ratio can be satisfied by the MLTN standard.

Proposition 3

The more likely than not standard may generate any arbitrarily large wrongful failure to punish to wrongful punishment ratio (i.e., ${\overline{N}}_G/{\overline{N}}_I$ ).

Proof.

For any $B>0$ , a sufficient condition for $\frac{{\overline{N}}_G}{{\overline{N}}_I}=\frac{\mu (1-p\beta )}{(1-\mu )q\alpha }>B$ is that

$$\frac{1-p}{Bz}>{\alpha }^M$$

(21)

where $z=\frac{q(1-{\mu }_l)}{{\mu }_l}$ . Next, note that for any $B>0$ , there exists a sufficiently informative evidence generating process (operationalized by the function $\beta$ ) such that (21) holds. As an example, suppose $\beta ={\alpha }^{1-\gamma }$ where $\gamma \in (0,1)$ represents the informativeness of the evidence generating process. Then, for $\gamma$ close enough to $1$ , ²² one can express the MLTN standard as a function of $\gamma$ as ${\alpha }^M(\gamma )={\left(\frac{(1-\gamma )\mu p}{(1-\mu )q}\right)}^{\frac{1}{\gamma }}$ . It follows that $\genfrac{}{}{0ex}{}{lim}{\gamma \to 1}{\alpha }^M=0$ , and therefore (21) holds.▪

Proposition 3 reports results analogous to those summarized by Proposition 1 and Corollary 1. However, its proof focuses on a different aspect of the ratio being analyzed. Specifically, because ${\overline{N}}_G$ and ${\overline{N}}_I$ depend on the probabilities conviction, conditional on trial, an evidentiary standard that generates a very small probability of conviction for the innocent in trial can lead to a very large ratio between them. The MLTN standard, on the other hand, is determined not by the levels of these probabilities (i.e., $\alpha$ and $\beta$ ) directly, but by how they co-move (i.e., on ${\beta }^{\mathrm{\prime }}$ ). Thus, as the proof of the proposition illustrates, the MLTN standard can yield a very small probability of conviction conditional on innocence (i.e., $\alpha$ ) and therefore a very large ${\overline{N}}_G/{\overline{N}}_I$ ratio. This implies that any requirement regarding the size of ${\overline{N}}_G/{\overline{N}}_I$ , on its own, is insufficient for suggesting the need for standards of proof which are more demanding than MLTN.

This insight also explains why the Blackstone ratio, interpreted as a lower bound on ${\overline{N}}_G/{\overline{N}}_I$ , cannot be justified as a requirement based on a welfare analysis when there are costly sanctions within the model described in Section 2.3. This is because the optimal evidentiary standard is then also a function of how conditional conviction probabilities co-move as one adjusts evidentiary standards, as noted next.

Proposition 4

The optimal evidentiary standard with costly sanctions may generate a wrongful release to wrongful conviction ratio (i.e., ${\overline{N}}_G/{\overline{N}}_I$ ) smaller than $1$ .

Proof.

Note that $\frac{{\overline{N}}_G}{{\overline{N}}_I}<1$ if $\mu ({\alpha }^{*}(h))(1-p\beta ({\alpha }^{*}(h))+q{\alpha }^{*}(h))<q{\alpha }^{*}(h)$ . This condition holds if

$$\mu ({\alpha }^{*}(h))<\frac{q{\alpha }^{*}(h)}{1-p\beta ({\alpha }^{*}(h))+q{\alpha }^{*}(h)}$$

(22)

As noted in the proof of Proposition 2, $\genfrac{}{}{0ex}{}{lim}{h\to \mathrm{\infty }}{\beta }^{\mathrm{\prime }}({\alpha }^{*}(h))=\frac{q}{p}$ , which implies that ${\underset{\genfrac{}{}{0ex}{}{lim}{h\to \mathrm{\infty }}}{\alpha }}^{*}(h)$ is independent of the criminal benefit distribution $F$ . Thus, there exists benefit distributions such that ${\mu }_h<$ $\genfrac{}{}{0ex}{}{lim}{h\to \mathrm{\infty }}\frac{q{\alpha }^{*}}{1-p\beta ({\alpha }^{*})+q{\alpha }^{*}}$ , in which case (22) holds for sufficiently large $h$ .▪

Proposition 4 shows that this alternative interpretation of the Blackstone ratio also does not provide a basis for the purported relationships between the Blackstone ratio and optimal evidentiary standards when there are costly sanctions.

4. Modified Blackstone ratio

Given the issues associated with the conceptions of Blackstone ratios considered in Sections 2 and 3, here I consider a modified Blackstone ratio, which focuses on the trade-off between the probability with which a person faces a conviction upon refraining from crime (i.e., $P_I$ ) and the probability with which one avoids conviction upon committing crime ( $P_G$ ), respectively. Thus, the modified marginal rate of substitution is given by:

$$\hat{S}\equiv MR{S}_{P_I,P_G}=\frac{{\hat{\mathcal{B}}}_{P_I}}{{\hat{\mathcal{B}}}_{P_G}}>0$$

(23)

where $\hat{\mathcal{B}}\mathcal{(}{P}_I,P_G)$ represents a social objective function analogous to the function $\mathcal{B}$ explained in Section 2. The modified Blackstone ratio, or the $M-$ ratio, can then be thought of as a lower bound to $\hat{S}$ around the maximizers of $\hat{\mathcal{B}}$ .

Following arguments similar to those presented in Section 2, let

$${\overline{P}}_I=q\alpha and{\overline{P}}_G=(1-p\beta (\alpha ))$$

(24)

denote the pairs of $P_I$ and $P_G$ that can be obtained through different evidentiary rules. The marginal rate of transformation between these probabilities is then given by

$$\hat{t}(\alpha )\equiv MR{T}_{N_I,N_G}=-\frac{d{\overline{P}}_G/d\alpha }{d{\overline{P}}_I/d\alpha }=\frac{p{\beta }^{\mathrm{\prime }}(\alpha )}{q}$$

(25)

Thus, for any modified Blackstone ratio of $M$ , it follows that evidentiary rules such that $\hat{t}(\alpha )<M\le \hat{S}$ would be inconsistent with the maximization of $\hat{\mathcal{B}}$ . Similarly, policies such that $\hat{t}(\alpha )\ge M$ cannot be ruled out as being socially undesirable solely based on the modified ratio of $M$ .

The expression in (25) and the analysis in the previous sections highlight the intuitive appeal of this modified ratio. The optimal evidentiary standard that emerges in the model presented in Section 2.3. As well as the MLTN standard previously described are both closely related to the manner in which probabilities of conviction, conditional on behavior, co-move, which corresponds to $\hat{t}(\alpha )$ . The optimal evidentiary standard depends on this relationship because it trades off gains from deterrence against punishment costs, which both depend on this relationship as is clearly illustrated through (20) which characterizes the optimal standard. The MLTN standard, on the other hand, is related to this ratio, because a defendant on trial has more likely than not committed the crime if the evidence produced at trial is a stronger indicator of guilt than priors are an indicator of innocence, that is, the evidence is associated with a large enough likelihood ratio which corresponds to ${\beta }^{\mathrm{\prime }}$ . This is captured by (11) which again clearly illustrates the relationship between $\hat{t}(\alpha )$ and the MLTN standard. The relationship between the $M-$ ratio and the evidentiary standards operationalized by ${\alpha }^{*}$ and ${\alpha }^M$ can thus be formalized as follows.

Proposition 5

The optimal evidentiary standard with costly sanctions is only consistent with $M-$ ratios greater than $1$ (i.e., $\hat{t}({\alpha }^{*})>1$ for all $c>0$ ).

The marginal rate of transformation associated with the MLTN standard is smaller than the $M-$ ratio whenever this ratio exceeds the compliance to offense ratio obtained under the MLTN standard (i.e., $\hat{t}({\alpha }^M)<M$ whenever $M>\frac{1-\mu ({\alpha }^M)}{\mu ({\alpha }^M)}$ ).

Proof.

Plugging (20) into (25) reveals that

$$\hat{t}({\alpha }^{*})=\frac{[h+(p\beta -q\alpha )(c-(1-r))]f(b^{*})+(1-\mu )c}{[h+(p\beta -q\alpha )(c-(1-r))]f(b^{*})-\mu c}>1$$

Plugging (11) into (25) reveals that $\hat{t}({\alpha }^M)=\frac{1-\mu ({\alpha }^M)}{\mu ({\alpha }^M)}$ .▪

Proposition 5 formalizes the clear relationship between the $M-$ ratio and the evidentiary standards previously discussed. First, it reveals that costly punishments, in fact, provide a rationale for principles that are conveyed through asymmetric $M-$ ratios. Given costly punishments, a simple utilitarian welfare function always produces $MRT$ s which are consistent with $M-$ ratios that are greater than $1$ . Second, it reveals that weak evidentiary requirements, like MLTN, are inconsistent with $M-$ ratios which are larger than the compliance to offense ratios which would emerge under such standards. Thus, this result suggests that the $M-$ ratio may produce a better fit between the principles that were presumably meant to be conveyed through Blackstone’s statement as well as the implications that are assumed to follow from it.

5. Conclusion

Blackstone’s simple statement has given rise to numerous commentaries and academic debates. From an economist’s perspective, his statement is most consistent with a description of how the quantity of the two types of judicial errors, wrongful convictions and failures to convict, ought to be traded-off. It has been taken for granted by many scholars that social preferences described by Blackstone are sufficient to warrant pro-defendant evidentiary rules. Others have suggested that the presence of punishment costs in the criminal justice system imply that the quantities of judicial errors ought to be traded-off asymmetrically, as suggested by Blackstone, from a simple cost-benefit perspective. Here, I have shown that neither claim is correct. This is because Blackstone’s statement focuses on a trade-off between the quantities of the two types of judicial error, which is not a direct determinant of evidentiary standards that are the subject of discussion. Instead, these standards are more directly related to how the probabilities with which the criminal justice system convicts individuals, conditional on their behavior, can be traded-off. Correcting this misunderstanding is important for focusing efforts and academic interest on the more relevant subjects of interest, namely individuals’ probabilities of conviction, as opposed to judicial errors in aggregate.

6. Appendix: Derivation of $\beta (\alpha )$ From Evidence Generation Processes

Evidence observed in court ( $e$ ) is assumed to be a random variable whose distribution depends on whether the defendant is truly guilty or innocent. The evidence may be multi-dimensional, but the likelihood ratio associated with the evidence, $L(e)=\frac{P(e|G)}{P(e|I)}$ is single dimensional. Thus, let ${\varphi }_i(L)=P(L|i)$ for $i\in \{G,I\}$ denote the probability density function associated with the production of any evidence which involves a likelihood ratio of $L$ conditional on the defendant’s type, and let ${\mathrm{\Phi }}_i(L)$ denote the associated cumulative distribution function. An implication of the Neyman–Pearson lemma (Neyman and Pearson, 1933) is that threshold rules wherein a person is found guilty only if the evidence is associated with a likelihood ratio exceeding a threshold $\overline{L}$ maximize the probability of a correct conviction, $\beta$ , given any targeted level of false conviction, $\alpha$ . When these evidentiary rules are implemented, it follows that the probability of conviction given the trial of an innocent person is

$$P(C|I,T)=1-{\mathrm{\Phi }}_I(\overline{L})$$

(26)

Thus, one can express the critical likelihood ratio, as a function of the targeted level of wrongful conviction probability as

$$\overline{L}(\alpha )={\mathrm{\Phi }}_I^{-1}(1-\alpha )$$

(27)

where ${\mathrm{\Phi }}_I^{-1}$ is the inverse of ${\mathrm{\Phi }}_I$ .

Next, noting the relation $P(C|G,T)=1-{\mathrm{\Phi }}_G(\overline{L})$ , one can express the probability of correct conviction, as a function of $\alpha$ , as:

$$\beta (\alpha )=1-{\mathrm{\Phi }}_G({\mathrm{\Phi }}_I^{-1}(1-\alpha ))$$

(28)

Thus, it follows that

$${\beta }^{\mathrm{\prime }}(\alpha )=\frac{{\varphi }_G(\overline{L}(\alpha ))}{{\varphi }_I(\overline{L}(\alpha ))}=\overline{L}(\alpha )>0$$

(29)

where the first equality is obtained by differentiating (28) and the second equality follows from the definition of ${\varphi }_i(L)$ . Thus, it follows that

$${\beta }^{\mathrm{″}}(\alpha )=\frac{d\overline{L}}{d\alpha }=-\frac{1}{{\varphi }_I(\overline{L}(\alpha ))}<0$$

(30)