We are interested in examining two considerations: breadth and alignment. Consider Judge Posner’s opinion in *Jutzi-Johnson*. Judge Posner disagreed with the prior cases identified by both the plaintiff and the defendant. He explained:

The plaintiff cites three cases in which damages for pain and suffering ranging from $600,000 to $1 million were awarded, but in each one the pain and suffering continued for hours, not minutes. The defendant confined its search for comparable cases to other prison suicide cases, implying that prisoners experience pain and suffering differently from other persons, so that it makes more sense to compare Johnson’s pain and suffering to that of a prisoner who suffered a toothache than to that of a free person who was strangled, and concluding absurdly that any award for pain and suffering in this case that exceeded $5,000 would be excessive.

*Jutzi-Johnson*, 263 F.3d at 760. Judge Posner ultimately concluded that “[t]he parties should have looked at awards in other cases involving asphyxiation, for example cases of drowning, which are numerous.” *Id*.

In the language of the current section, Judge Posner disagreed with the alignment of the plaintiff’s cases, implying that awards corresponding to cases involving hours, rather than minutes, of pain and suffering would be inappropriately high. He disagreed with the breadth (and the alignment) of the defendant’s cases, suggesting that a set of cases involving the pain and suffering of inmates, rather than the general population, is too narrow, and that the defendant’s focus on inmates led to alignment issues that resulted in “absurd” conclusions. Additionally, Judge Posner seems to suggest that the sets of cases identified by the parties suffered from small sample sizes as well, indicating that broadening the prior cases to include other cases involving asphyxiation in the general population would have led to “numerous” cases. *Id*. (Bavli 2017)

Thus, consider again a claim that receives an award $Y$ “drawn from” a distribution centered at the correct award ${\mathrm{\mu}}_{y}$ with judgement variability ${\mathrm{\sigma}}_{y}^{2}$ (known), where ${\mathrm{\mu}}_{y}$ is centered at ${\mathrm{\lambda}}_{0}$ with variance ${\mathrm{\eta}}_{0}^{2}$. That is,
$Y\phantom{\rule{thinmathspace}{0ex}}\sim \phantom{\rule{thinmathspace}{0ex}}({\mathrm{\mu}}_{y},{\mathrm{\sigma}}_{y}^{2}),\phantom{\rule{thickmathspace}{0ex}}{\mathrm{\mu}}_{y}\phantom{\rule{thinmathspace}{0ex}}\sim \phantom{\rule{thinmathspace}{0ex}}({\mathrm{\lambda}}_{0},{\mathrm{\eta}}_{0}^{2}).$

Assume that the court identifies a set of prior cases involving awards ${X}_{1},\dots ,{X}_{N}$ with correct outcomes ${\mathrm{\mu}}_{1},\dots ,{\mathrm{\mu}}_{N}$ centered at ${\mathrm{\mu}}_{0}$ with variance ${\mathrm{\tau}}^{2}$. Thus,
${X}_{i}\phantom{\rule{thinmathspace}{0ex}}\sim \phantom{\rule{thinmathspace}{0ex}}({\mathrm{\mu}}_{i},{\mathrm{\sigma}}^{2}),\phantom{\rule{thickmathspace}{0ex}}{\mathrm{\mu}}_{i}\phantom{\rule{thinmathspace}{0ex}}\sim \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}({\mathrm{\mu}}_{0},{\mathrm{\tau}}^{2});\phantom{\rule{thickmathspace}{0ex}}i=1,\dots ,N.$

Essentially, this means ${X}_{i}\phantom{\rule{thinmathspace}{0ex}}\sim \phantom{\rule{thinmathspace}{0ex}}({\mathrm{\mu}}_{0},{\mathrm{\psi}}^{2})$ for $1\le i\le N$, where ${\mathrm{\psi}}^{2}={\mathrm{\sigma}}^{2}+{\mathrm{\tau}}^{2}$. Now, consider an estimator of the form
${\stackrel{\u02c6}{\mathrm{\mu}}}_{y}^{s}=\frac{\frac{Y}{{\mathrm{\sigma}}_{y}^{2}}+\frac{{\mathrm{\mu}}_{0}}{{\mathrm{\psi}}^{2}}}{\frac{1}{{\mathrm{\sigma}}_{y}^{2}}+\frac{1}{{\mathrm{\psi}}^{2}}},$[8]

which we can approximate by plugging in unbiased estimators
$\stackrel{\u02c9}{X}=\frac{{\sum}_{i=1}^{N}{X}_{i}}{N}\text{\hspace{0.17em}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thickmathspace}{0ex}}S=\frac{{\sum}_{i=1}^{N}{({X}_{i}-\stackrel{\u02c9}{X})}^{2}}{N-1}$

for the unknown parameters ${\mathrm{\mu}}_{0}$ and ${\mathrm{\psi}}^{2}={\mathrm{\sigma}}^{2}+{\mathrm{\tau}}^{2}$, respectively, to obtain:
${\stackrel{\u02c6}{\mathrm{\mu}}}_{y}^{a}=\frac{\frac{Y}{{\mathrm{\sigma}}_{y}^{2}}+\frac{\stackrel{\u02c9}{X}}{S}}{\frac{1}{{\mathrm{\sigma}}_{y}^{2}}+\frac{1}{S}}.$[9]

Again, as $N\to \mathrm{\infty}$, ${\stackrel{\u02c6}{\mathrm{\mu}}}_{y}^{a}\to {\stackrel{\u02c6}{\mathrm{\mu}}}_{y}^{s}$. And the risk of the shrinkage estimator ${\stackrel{\u02c6}{\mathrm{\mu}}}_{y}^{s}$ (which is roughly equal to the approximation, assuming a reasonable sample size) is
${R}_{y}^{s}=\mathbb{E}({\stackrel{\u02c6}{\mathrm{\mu}}}_{y}^{s}-{\mathrm{\mu}}_{y}{)}^{2}=\mathbb{E}{\left(\frac{\frac{Y}{{\mathrm{\sigma}}_{y}^{2}}+\frac{{\mathrm{\mu}}_{0}}{{\mathrm{\psi}}^{2}}}{\frac{1}{{\mathrm{\sigma}}_{y}^{2}}+\frac{1}{{\mathrm{\psi}}^{2}}}-{\mathrm{\mu}}_{y}\right)}^{2}=\mathbb{E}{\left(\frac{\frac{Y-{\mathrm{\mu}}_{y}}{{\mathrm{\sigma}}_{y}^{2}}+\frac{{\mathrm{\mu}}_{0}-{\mathrm{\mu}}_{y}}{{\mathrm{\psi}}^{2}}}{\frac{1}{{\mathrm{\sigma}}_{y}^{2}}+\frac{1}{{\mathrm{\psi}}^{2}}}\right)}^{2}$
$={\left(\frac{1}{{\sigma}_{y}^{2}}+\frac{1}{{\psi}^{2}}\right)}^{-2}\left[\frac{1}{{\sigma}_{y}^{2}}+\frac{{({\mu}_{0}-{\lambda}_{0})}^{2}+{\eta}_{0}^{2}}{{\psi}^{4}}\right],$

which is smaller than ${\mathrm{\sigma}}_{y}^{2}$, the risk of the classical estimator, when
$({\mathrm{\mu}}_{0}-{\mathrm{\lambda}}_{0}{)}^{2}+{\mathrm{\eta}}_{0}^{2}<2{\mathrm{\psi}}^{2}+{\mathrm{\sigma}}_{y}^{2}.$[10]

Let us consider the meaning of this condition and then examine a number of numerical examples to gain a deeper understanding of the circumstances necessary to improve accuracy. On the right side of eq. [10], ${\mathrm{\psi}}^{2}={\mathrm{\sigma}}^{2}+{\mathrm{\tau}}^{2}$ is the total variance (the sum of claim variability and judgment variability) of the prior awards; and ${\mathrm{\sigma}}_{y}^{2}$ is the judgment variability of the subject case. On the left side of eq. [10], $({\mathrm{\mu}}_{0}-{\mathrm{\lambda}}_{0}{)}^{2}$ is the square of the misalignment, the square difference between the expected correct award in the subject case and the mean of the correct awards in the prior cases; and ${\mathrm{\eta}}_{0}^{2}$ can be understood as the claim variability of the hypothetical population to which the correct award in the subject case belongs. For simplicity, we can set ${\mathrm{\eta}}_{0}^{2}=0$ and view the correct award in the subject case as its own population or as a realization of *μ*_{y}. Note that, although we employ this assumption throughout this section, our conclusions and illustrations herein are robust to reasonable alternatives and potentially substantial values of *η* such as *τ*. The condition is satisfied, for example, if (1) the judgment variability of the subject award is greater than the hypothetical claim variability (*e.g*., where ${\mathrm{\eta}}_{0}^{2}=0$), and (2) the breadth of the prior awards is greater than the misalignment of the prior awards. Of course, the effects of one factor can be offset by the effects of the other. For example, the effects of extreme misalignment can be offset by the effects of extreme judgment variability.

Furthermore, in general, the greater the dispersion of the prior awards, the more “tolerance” there is for misalignment. On the other hand, higher prior-award concentration requires greater alignment. Thus, it may be beneficial for the breadth of the prior awards to reflect the court’s confidence in their alignment with respect to the subject award.

Let us consider an example based on data obtained from Saks et al. (1997), which tested the effects, with respect to variability, of providing mock jurors with certain information regarding prior awards. In one set of control conditions in which mock jurors were provided with a fact pattern (based on actual personal injury cases) involving a “high-severity injury,” a broken back, the mean and standard deviation of the award amounts determined by participants were approximately $3 million and $4 million, respectively (Saks et al. 1997).

Note that these values are based on amounts determined by mock jurors rather than mock juries. Also, however, “[b]ecause the distribution for the raw dollar awards was highly variable and positively skewed, awards greater than two standard deviations above the mean were recoded to the amount at two standard deviations.” *Id*. The authors thereby limited the variability of the data.

Based on these data we construct Figure 2, which assumes a correct award (${\mathrm{\mu}}_{y}$) of $3 million and judgment variability (${\mathrm{\sigma}}_{y}$) of $4 million. It is intuitive to imagine an approximately “normal” distribution with almost all awards falling between 0 and $11 million (that is, the mean ±2 standard deviations). We assume also that ${\mathrm{\mu}}_{y}={\mathrm{\lambda}}_{0}=3$ million and ${\mathrm{\eta}}_{0}=0$. Thus, the figure illustrates risk as a function of the mean of the prior awards (indicating alignment), and displays different curves corresponding to different levels of prior-award variability and a shaded horizontal line corresponding to the risk of the classical estimator (which is not dependent on the mean or variability of the prior awards). We can see, for example, that if the prior awards are centered at $4 million with standard deviation equal to $2 million, we reduce risk by 92 % by using the shrinkage estimator rather than the classical estimator; and we can expect the award to fall within the interval $74,000 to $5.26 million (that is, the mean ±2 standard deviations), rather than $0 to $11 million. If the prior awards are centered at $5 million with standard deviation equal to $3 million, we reduce risk by 76.8 % relative to the classical estimator; and we can expect the award to fall within the interval $0 to $6.85 million. Although not shown in the figure, it can be shown that, for prior award mean and standard deviation equal to $8 million and $4 million, respectively, we reduce risk by 36 % relative to the classical estimator. Finally, if the distribution of prior awards has a mean and standard deviation equal to $3 million (the correct award) and $2 million, respectively, the variability (in terms of standard deviation) of the estimator is $0.8 million rather than $4 million, and we reduce risk by 96 % relative to the classical estimator.

Note that, although Saks et al. (1997) used mock jurors rather than juries, our choice of judgment variability – an important factor for whether prior-award information causes accuracy to increase or decrease – is likely conservative, since our choice ($4 million) reflects the methodology in that study whereby all award amounts above two standard deviations above the mean were reduced to the amount of two standard deviations above the mean. To be sure, however, let us illustrate an example in which we set judgment variability to *half* the standard deviation used above. Thus, Figure 3 assumes a correct award (${\mathrm{\mu}}_{y}$) of $3 million and judgment variability (${\mathrm{\sigma}}_{y}$) of $2 million. In this example, if prior awards are centered at $4 million with a standard deviation of $2 million, we reduce risk by 68.75 % relative to the classical estimator. If the distribution of prior awards is centered at $1 million with a standard deviation of $400,000, a distribution that is concentrated around a significantly incorrect award, we nevertheless reduce risk by 7.5 % relative to the classical estimator. Finally, if the distribution of prior awards has a mean and standard deviation equal to $3 million (the correct award) and $2 million, respectively, we reduce risk by 75 % relative to the classical estimator. Furthermore, if the variability in this final scenario were $200,000, the risk associated with the shrinkage estimator is even smaller – representing a risk reduction of 99.99 % (corresponding to a reduction in standard deviation from $4 million for the classical estimator to $20,000 for the shrinkage estimator).

Figure 2: Comparison of the risk corresponding to the shrinkage estimator (black curves) and the risk corresponding to the classical estimator (gray horizontal line) plotted against the mean of the prior awards when the correct award is $\mathrm{\$}3$ million (vertical black line), and assuming ${\mathrm{\eta}}_{0}=0$ and the judgment variability of the subject case, ${\mathrm{\sigma}}_{y}$, is equal to $4 million. The black curves correspond to different values of prior-award variability, which ranges from $\mathrm{\$}0.2$ million to $\mathrm{\$}3$ million.

Figure 3: Comparison of the risk corresponding to the shrinkage estimator (black curves) and the risk corresponding to the classical estimator (gray horizontal line) plotted against the mean of the prior awards when the correct award is $\mathrm{\$}3$ million (vertical black line), and assuming ${\mathrm{\eta}}_{0}=0$ and the judgment variability of the subject case, ${\mathrm{\sigma}}_{y}$, is equal to $2 million. The black curves correspond to different values of prior-award variability, which ranges from $\mathrm{\$}0.2$ million to $\mathrm{\$}3$ million.

Lastly, we construct a third example using data from Bovbjerg et al. (1988), which examined real award data by severity of injury to analyze the variability of awards for pain and suffering. The data presented in this example are arguably conservative as well, since 1) the authors excluded the 5 % of award values farthest from the median; 2) the data include reported incidents of additur and remittitur; and 3) the data reflect the value of dollars in 1987. Thus, in Figure 4, we consider the example of severity level 7 (out of 9, representing severe, but not maximum-severity, injuries), with mean and standard deviation values of approximately $2 million and $2 million. The graph again displays different curves corresponding to different levels of prior-award variability and a shaded horizontal line representing the risk associated with the classical estimator at $4 million squared, which is equal to the standard deviation squared. First, we see that if prior awards are centered at $2 million (assumed to be the correct award) with a standard deviation of $500,000, we reduce risk by 99.65 % relative to the classical estimator. If the prior awards have a mean and standard deviation equal to $500,000 and $500,000, we reduce risk by 49.83 % – a milder reduction, due to an introduction of bias (but a reduction nevertheless). If the prior awards have a mean and standard deviation equal to $500,000 and $1.5 million, respectively, we reduce risk by 64 % – an improvement relative to the former scenario, due to the increase in breadth, which reduces the impact of the bias. If the prior awards have a mean and standard deviation equal to $4.2 million and $200,000, respectively, we *increase* risk by 18.63 %, since we have a tightly bound distribution centered at a significantly incorrect award value. On the other hand, if the prior awards have a mean and standard deviation equal to $4.2 million and $1.5 million, we reduce risk by 37.48 %, since, now, the introduction of bias is reduced due to high prior-award breadth, and the beneficial effect of the prior awards on award variability dominates.

Figure 4: Comparison of the risk corresponding to the shrinkage estimator (black curves) and the risk corresponding to the classical estimator (gray horizontal line) plotted against the mean of the prior awards when the correct award is $\mathrm{\$}2$ million (vertical black line), and assuming ${\mathrm{\eta}}_{0}=0$ and the judgment variability of the subject case, ${\mathrm{\sigma}}_{y}$, is equal to $2 million. The black curves correspond to different values of prior-award variability, which ranges from $\mathrm{\$}0.2$ million to $\mathrm{\$}4$ million.

Thus, using the derivations and illustrations above, we state the following conclusions regarding the alignment and breadth of prior awards:

Prior awards that are relatively aligned with the correct award in the subject case can lead to large accuracy benefits. These benefits are robust to changes in the alignment and breadth of the prior awards.

Prior awards that are misaligned – even significantly misaligned – but have relatively high breadth can lead to accuracy benefits, but benefits that are small relative to those that result from prior awards that are aligned and have lower breadth.

Increasing only the breadth of the prior awards (without affecting alignment) will generally not harm accuracy, but will reduce the influence of the prior awards, and therefore reduce their benefits with respect to accuracy. Considerations for determining an appropriate breadth include the sample size and the court’s confidence in the alignment of the prior awards.

Prior awards that are significantly misaligned and have low breadth can lead to harmful effects on accuracy. However, such effects generally require the unusual circumstance of tightly bound prior awards that are significantly misaligned.

In short, under relatively mild conditions, the shrinkage estimator outperforms the classical estimator, an adjudicated award.

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