Trial Selection and Estimating Damages Equations

4.1 Rationality Model

The Rationality Model employs backward induction to analyze settlement decisions in multi-stage litigation. At the second stage (appeal), the parties proceed in litigation if and only if π p 2 > π d 2 , or (equivalently) v > c p 2 + c d 2 p p 2 − p d 2 . Assuming litigation in the second stage, the plaintiff’s expected reward in the first stage is p p 1 p p 2 v + 1 − p p 1 p p 2 v − c p 1 − c p 2 = p p 2 v − c p 1 + c p 2 . Note that this is independent of the plaintiff’s first stage trial outcome prediction, p p 1 . The defendant’s expected cost in the first stage is p d 2 v + c d 1 + c d 2 , also independent of his first-period trial outcome prediction. Therefore, the litigation condition in the first stage is v > c p 1 + c p 2 + c d 1 + c d 2 p p 2 − p d 2 .

Now consider the sequence where settlement would be rational in the second stage, that is, v < c p 2 + c d 2 p p 2 − p d 2 . Following Bebchuk, we assume the settlement amount in the second stage takes the average of plaintiff and defendant’s settlement bids: S 2 * = 1 2 p p 2 + p d 2 v + 1 2 c d 2 − c p 2 . Given this expectation, should the plaintiff choose to litigate in the first stage, he expects to receive S 2 * − c p 1 . The defendant, on the other hand, will pay no more than S 2 * + c d 1 to settle. Therefore, the parties will settle in the first stage for S 1 * = 1 2 S H + S L = S 2 * + 1 2 c d 1 − c p 1 > S 2 * − c p 1 . Thus, given the expectation of settlement in the second stage, the parties always settle in the first stage.

Summarizing: (1) Litigation occurs when v > c p 1 + c p 2 + c d 1 + c d 2 p p 2 − p d 2 . (2) Disputes either litigate in both stages or settle in the first stage – there are no cases of litigation to trial verdict followed by settlement before appellate verdict. (3) Only the win probabilities at the final stage matter in determining the decision to litigation.

4.2 Myopia Model

An alternative to the Rationality Model assumes that the parties are myopic, in the sense that their incentives to settle are based entirely on the payoffs relevant to the stage in which they find themselves.

In the Myopia Model, the plaintiff in Stage 1 examines only the payoff from Stage 1 litigation, ignoring the likelihood of appeal to the second stage. The following conclusions apply: (1) if c p 1 + c d 1 p p 1 − p d 1 > c p 2 + c d 2 p p 2 − p d 2 , then all settlements occur in the first stage and the probability of settlement is equal to the probability that v < c p 1 + c d 1 p p 1 − p d 1 . (2) On the other hand, if c p 1 + c d 1 p p 1 − p d 1 < c p 2 + c d 2 p p 2 − p d 2 , then it is possible (contrary to the Rationality Model) to have disputes that litigate in the first stage and settle in the second stage. The probability of settlement in the first stage is equal to the probability that v < c p 1 + c d 1 p p 1 − p d 1 , and the probability of settlement in the second stage is equal to the probability that c p 1 + c d 1 p p 1 − p d 1 ≤ v < c p 2 + c d 2 p p 2 − p d 2 .

Since the Rationality Model, (1) and (2) of Myopia Model result in different regression specifications, the three models could be put in competition with one another through specification tests on the associated regression equations.

5.1 Description of Problem

In this part, we discuss a structural econometric model for estimation using appellate court data. We assume that the analyst has a basic theoretical regression model to explain some particular dependent variable. For example, the legal analyst might develop a linear regression model that explains the amount of damages awarded in a tort lawsuit. The independent variables in the model are drawn from information provided in the case, such as the age or education level of the plaintiff. However, a simple linear regression of the dependent variable on the independent variables drawn from court reports is likely to be biased because of selection.

Consider an equation for damages estimation. The theoretical model that the analyst has designed aims to explain the expected damages award v _i for each case i. Specifically, the theoretical structural model is E v i | x i = β ′ x i , where x _i is the vector of independent variables of individual case i that the analyst posits should explain the damages awarded in a lawsuit, and β is the vector of coefficients for the related independent variables. The corresponding regression model is v _i = β′x _i + ɛ _i , where E(ε _i |x _i ) = 0 and Var(ε _i |x _i ) = σ ².

The results from the estimation procedure just described are likely to be biased by the selection of disputes for litigation. The theoretical structural model is based on the analyst’s belief that all realizations v _i are determined in expectation by the structural form β′x _i . But the analyst never sees all realizations v _i that occur in the population. He sees only the realizations that have not been screened out as a result of the settlement process. Because of the screening due to settlement, the direct estimation of the theoretical structural model is likely to result in biased estimates. Given the likely bias resulting from sample selection when using the theoretical structural model, we derive an alternative structural model that incorporates the selection process below.

5.2 Information Structure

The information structure assumed in the litigation model influences the selection of disputes into the settlement (or litigation) process. In the previous part, we examined settlement incentives under the assumption that inconsistent beliefs could determine trial outcome predictions. An alternative to this approach, not explored here, would assume informational asymmetry generates litigation (Bebchuk 1984; Nalebuff 1987; Png 1987). Sieg (2000) estimates the parameters of an asymmetric information model of litigation using medical malpractice data.

In the parts below, we will assume inconsistent beliefs, in particular, mutual optimism as the basis for litigation. We assume p p k , p d k are the subjective beliefs of the parties in stage k (trial = 1, appeal = 2), where p p k > p d k . This is the optimism model proposed in Shavell (1982) and formalized in Hylton (2023).

We adhere to (2) of the Myopia Model of multi-stage litigation described previously because the model allows settlements at any stage, which is more representative of real-world cases. Under the model, litigation occurs when

That is, the appellate court data is left-truncated, and the truncation threshold depends on the plaintiff’s and defendant’s litigation costs and their predictions of the litigation outcome at the appellate court.

5.3 Truncated Regression Model with Stochastic and Unobserved v ̄

As we discuss above, when the analyst estimates damages using appellate court data, damages awards observations are left-truncated at v ̄ _i for each case i. However, each threshold v ̄ _i is unobserved in the data. To deal with this issue, we apply the truncated regression model with stochastic and unobserved thresholds as described in Maddala (1983, 257–91):

where v _i and v ̄ i each denotes the damages amount and truncation threshold for individual case i. x _1i and x _2i are the vectors of explanatory variables for v _i and v ̄ i , respectively. v _i , x _1i , and x _2i are observable in appellate court data if and only if v i > v ̄ i . v ̄ i is unobservable and stochastic. We assume that (u _1i , u _2i ) are distributed bivariate normal with mean vector zero and covariance matrix Σ = σ 1 2 σ 12 σ 12 σ 2 2 . The likelihood function for this model is

where N is the number of observations, Φ(⋅) is the cumulative distribution function of the standard normal, and f ⋅ , ⋅ is the joint density of u _1i and u _2i . After simplifying the above equation as in Maddala (1983),^[11] we can write down the log-likelihood function as

Then, we estimate the parameters of the model with the standard maximum likelihood estimation. As in Muthén and Jöreskog (1983), we assume that σ 1 2 = exp σ 1 ̃ and σ 2 2 − σ 12 2 σ 1 2 = 1 , and we estimate σ 1 ̃ and σ ₁₂ to ensure positive values for σ 1 2 and σ 2 2 .^[12]

As noted in Maddala (1983), we do not know whether the above equation is well-behaved so that it has a unique global maximum. Thus, starting from different initial values and using different optimization algorithms in the maximum likelihood estimation may help in empirical applications.

Another issue in the truncated regression model with stochastic and unobserved thresholds is the reliability of the estimated parameters β ₂ for x ₂. Monte-Carlo simulation results in Muthén and Jöreskog (1983) show that the estimated parameters of β ₂ could be unreliable while it can still correct for selectivity bias in β ₁.^[13] Thus, this model cannot be used to directly estimate the selection criterion parameters β ₂. The model, however, still corrects for selectivity bias in β ₁ even with the unreliable estimates of β ₂. Thus, we can use the model in estimating theoretical structural models in empirical legal research to correct for sample selection bias from appellate court opinions.

5.4 Explanatory Variables for v ̄

Here we consider the explanatory variables for v ̄ . As we discuss in Section 5.2, v ̄ depends on the total litigation costs for both parties and the difference in the parties’ predictions of the litigation outcome.

The data for total litigation costs are generally unavailable, and certainly not for any historical series of appellate cases. However, the number of attorneys involved are generally available in most court opinions, even in reports from more than one hundred years ago. Additionally, total litigation costs vary depending on the difficulty of each case. We assume that more difficult cases will result in longer appellate opinions. Thus, we use the number of pages in each appellate decision as a proxy for the case’s difficulty. We use those two types of data as the explanatory variables for the total cost of litigation in our empirical application in the following section.

Another problem in choosing the explanatory variables for v ̄ is the part for the probability prediction (or expectations) differential. It requires the identification of variables that would tend to generate plaintiff optimism (plaintiff believing he has a strong case) and simultaneous defendant optimism (defendant believing plaintiff has a weak case). We use three dummy variables: one for whether the defendant was subject to vicarious or direct liability, another for whether the decedent was claimed to be at fault or not, and the third for whether the plaintiff won or lost at trial. Vicarious liability means that the defendant is liable for the fault of another party (e.g. an employer being liable for the negligence of an employee), and direct liability means, as the label suggests, the actor is liable for his own fault. An employer, for example, can be held directly liable if it negligently hired an incompetent employee, whose foreseeable incompetence led to the death of the decedent. Allegations of direct fault, whether against the defendant or against the decedent, might tend to generate or exacerbate mutual optimism, through hindsight bias or correspondence bias (fundamental attribution error) – and their associated influences on tastes for risk bearing. According to the correspondence bias theory, individuals tend to evaluate responsibility for harmful actions of others by referring to or invoking personal traits rather than situational factors, while doing the opposite in evaluating their own harmful actions (Flick and Schweitzer 2021). Such biases can generate divergent expectations on the probability of a finding of a fault.^[14] After the trial court decision, the plaintiff and defendant adjust their probability predictions; the probability prediction differential is larger or smaller if the plaintiff won or lost at trial, respectively.

6.1 Data

Our data consists of wrongful death and survival action appellate decisions in the state of Louisiana from the year 1901 to the year 1950. We attempted to get every informative (i.e. discussing damages and litigant characteristics) appellate decision on record.^[15] Wrongful death actions are lawsuits brought by the survivors of a decedent for the loss in financial support resulting from the death of the decedent due to the defendant’s tortious conduct. Many states, such as Louisiana during the period of our data set, permit recovery for the emotional suffering of survivors as well. Survival claims, by contrast, are for the losses the decedent could have brought for injuries personally suffered from the moment of injury until his death, which include lost wages and emotional suffering. Damages awards in Louisiana courts did not, as a general matter, separate these separate grounds for damages in the final award.^[16]

6.2 Estimation

We start with the simplest regression model in Table 1. The first column shows the most basic OLS result. The dependent variable is the amount awarded in damages by the appellate court. The second column shows the result of the model in Section 5.3. The estimated coefficients for the truncation threshold equation are not reported.

The most striking difference between the two columns is the enhanced race effect in the second column. In both columns, the race effect is highly statistically significant, with the result in the first column showing that the survivors of white decedents, after controlling for the income of the decedent, status of the plaintiff-survivor (a widow with children or not), and occupation of the decedent, received roughly $4006 more in compensation than did the plaintiff-survivors of black decedents. The second column finds that this race premium was about 53.6 percent higher, at $6152. In the second column, a one-dollar increase in the monthly wage leads to a $10.34 increase in the award to plaintiff-survivors. Wriggins’ (2005) finding, based on an examination of average awards, that Louisiana courts used race as a basis for discounting awards to survivors of black decedents receives considerably stronger support in the new regression.^[17]

The variables coding for cases where the decedent worked for a railroad and as a driver are significant and positive. The baseline of the variables is the cases where the decedent was a child, housewife, or retiree. The awards were on the order of $4126 and $3484 higher for survivors of decedents who worked for a railroad and as a driver, respectively. This is interesting given that we have already controlled for the wage at the time of death. The occupation control must therefore convey something other than the effect of the decedent’s compensation on the court’s award. The most plausible explanation is that railroad jobs were more stable and secure than other occupations. In light of this, a court was more likely to perceive the railroad-employed decedent as a greater source of support for family members than decedents of other occupations. Many of the decedents listed as laborers, for example, worked seasonally and moved from job to job, leading to substantial fluctuations in income over time.

Table 2 repeats the same comparison between the simple OLS and the model in Section 4.3 but includes regional controls. The regional controls separate the regions of the state of Louisiana where the appellate court that rendered the judgment sits. These regions also generally hold the trial courts from which the case was appealed. We have 29 separate parishes in the state of Louisiana over this period. We grouped these parishes into 5 separate regions: Acadiana (Cajun Country approximately), Florida Parishes, North Louisiana (Sportsmen’s Paradise), Greater New Orleans, and Central Louisiana (Crossroads).^[18] These regions are understood to be culturally different,^[19] and we posited that these cultural differences might influence the way courts view these cases. The results indicate that North Louisiana courts typically give greater awards. The reasons for these North Louisiana premium are not obvious; perhaps the lower prevalence of slavery in its history may have generated different perceptions of the value of a work-life. The results of the other variables in Table 2 largely replicate those in Table 1.

Table 3 provides a fuller regression equation, including age, age-squared, and the number of plaintiff-survivors. Regional controls were included in this regression, though their coefficient estimates are excluded from the table to reduce clutter. The results of the regional controls were consistent with those of Table 2. The award goes up about $576.5 for every additional plaintiff-survivor. Age and age-squared show that the award to survivors generally goes up with work experience (proxied by age) but at a declining rate. For every year of age on the decedent, the award rises by roughly $176 dollars, but the estimated coefficient of the age-squared variable is about −2.07. This implies that the maximum contribution on average was observed at age 42.5. This is consistent with the rapidity of declining health for workers during this time period.