What Do Judges Want? How to Model Judicial Preferences

A.2.1 A Two-Dimensional Case Space with a One-Parameter Rule

Recall our example from Appendix A in the prior chapter in which the case space X had two dimensions. In this example a given case is a vector (x ₁, x ₂) (subscripts denote dimensions). For concreteness, imagine the case space as the unit square, so the space is X = 0,1 × 0,1 . We restricted attention to the class of rules indexed by the parameter b as in the following

Assume that the judge’s ideal rule is the 45° line, i.e. sets b = 0. We assume all other doctrines simply alter the intercept b Employing the same style of notation as above, call judge i’s most-preferred partition b ¯ i .

The case space and two cutting lines are shown in Figure 8.

Figure 8:

Two dimensional case space with a one parameter rule. The case space is the unit square. The dark line (x ₂ = x ₁) represents the most-preferred rule of the judge. An alternative rule is x ₂ = max{x ₁ − b, 0}. The conflict zone is the space between the two cutting lines. In the figure, b = 1 4 .

We need to modify the dispositional utility function in Equation (1) for this more complex case space. This extension is immediate for the constant loss function h ( x ; b ¯ i ) = 0 and g ( x ; b ¯ i ) = − 1 . Under this dispositional utility function the judge receives the payoff 0 for a correct disposition and the payoff −1 for an incorrect one.

Suppose however we wish to use the linear loss function or a similar function. In that case, we must characterize not simply whether the case was wrongly decided but “how wrong” it was. In the one dimensional case, we took that measure to be the distance between the wrongly decided instant case and the doctrinal cut-point. The obvious extension here is the distance between the wrongly decided two-dimensional case and the cutting line. But there are many such distances – which one to use? Here is one answer. For a given wrongly decided case x 0 = x 1 0 , x 2 0 consider the closest point on the cutting line to the case using the standard Euclidean distance. Call this closest point x ′ = x 1 ′ , x 2 ′ . We can regard the distance between the two points as “how wrong” the case is since this distance is the distance from the instant case to the nearest case that would be correctly decided if it received the same disposition as the instant case.

The Euclidean distance between the two points is:

Because x′ must lie on the “correct” cutting line x ₂ = x ₁ we can re-write this distance as x 1 0 − x 1 ′ 2 + x 2 0 − x 1 ′ 2 . To find the closest case on the cutting line to the instant case, we seek the value of x 1 ′ that minimizes this distance. Some algebra shows that the closest case x ′ = x 1 0 + x 2 0 2 , x 1 0 + x 2 0 2 .^[14] See Figure 9.

Figure 9:

The distance to a wrongly decided case.

Substituting x ′ = x 1 0 + x 2 0 2 , x 1 0 + x 2 0 2 into the Euclidean distance gives us δ ( x 0 , x ′ ) = x 1 0 − x 2 0 2 . If we use this as the loss from an incorrect disposition of the case, the linear loss dispositional utility function becomes:

Cases may be distributed over X in many ways, according to some distribution F(x _i , x ₂) with density f(x _i , x ₂). An easy distribution is a uniform distribution over the entire space, the unit square. In that case f(x _i , x ₂) = 1. A slightly more general distribution is a uniform distribution centered on x ̂ 1 , x ̂ 2 with support x ̂ 1 − ε , x ̂ 1 + ε × x ̂ 2 − ε , x ̂ 2 + ε . Using this notation, the uniform distribution on the entire unit square is centered on 1 2 , 1 2 with ε = 1 2 . In order to consider distributions of cases that move with the rule (that is, that move as b shifts).

A.2.1.1 Expected Utility of a Rule with a Fixed Distribution of Cases

Let’s return to the expected utility of rules given a fixed distribution of cases. The simplest baseline uses the constant loss utility function and a uniform distribution over the entire case space. Here, the expected utility of a rule is simply (minus 1 times) the area of the conflict zone shown in Figure 8. This is:

When b = 0 (so the judge employs her most-preferred partition) expected utility is zero. When b = 1, all cases below the judge’s most = preferred cutting line must be decided incorrectly, yielding expected utility − 1 2 . The geometric interpretation of expected utility as (minus one times) the area of the conflict zone should be clear.

Suppose we employ the linear loss dispositional utility function instead. Then we have for positive b:

Now let’s consider a slightly different distribution of cases, centered on 1 2 , 1 2 and uniform on 1 2 − ε , 1 2 + ε × 1 2 − ε , 1 2 + ε . The support thus forms a box around 1 2 , 1 2 , with the most-preferred cutting line running from the lower left-hand corner of the box to the upper right-hand corner. The density of the distribution is 1 4 ε 2 . Again assuming b > 0 the possible enforced doctrines are cutting lines lying below the most-preferred doctrine and running through the box or lying entirely below it. One must take some care with the limits of integration:

The first result occurs when the enforced doctrine lies entirely below the box containing the cases so that one-half of the cases must be decided incorrectly. The second result reduces to the earlier result, − b 2 ( 3 − 2 b ) 6 2 , when ε = 1 2 .

A.2.1.2 Expected Utility of a Rule when the Distribution of Cases is Sensitive to the Rule

We represent this situation with the stylized distribution, the “box of cases” centered on the middle of the enforced cutting line. So, this example is similar to the previous example but the “box of cases” moves within the case space depending on b, the parameter characterizing the enforced doctrine. The essential idea of the distribution is that “centrally located” cases are rather likely, while those in the far edges of the case space are quite unlikely. What counts as “central” to the case space depends on which doctrine is enforced. In addition, half the cases lie below and half above the enforced cutting line. If ɛ is large, the judge may face cases far from the enforced doctrinal cutting line. But if ɛ is small, she faces only cases rather close to the doctrinal cutting line.

The center of the enforced doctrine is 1 + b 2 , 1 − b 2 .^[15] There are limits on how large ɛ can be in order to keep the entire “box of cases” in the case space. In particular, we require 1 + b 2 + ε ≤ 1 and 1 − b 2 − ε ≥ 0 ; both imply ε ≤ 1 − b 2 . Again taking some care with the limits of integration we have:

The first of these results, the “small ɛ” case, occurs when b is sufficiently large and ɛ sufficiently small that the entire “box of cases” lies below the preferred doctrinal cutting line. This implies that one-half of the cases lie in the conflict zone. The second case is the “large ɛ” case in which a portion of the “box of cases” lies above the most-preferred cutting line and only a band of cases lie in the conflict zone. Note that the second result goes to 0 as b goes to zero, in other words, as the enforced doctrine approaches the most-preferred doctrine expected losses go to zero.

Figure 10 displays the expected utility of rules for various values of b between 0 and one-half, for three values of ɛ. Not surprisingly, when cases are concentrated in the conflict zone and distributed farther from the most-preferred cutting line, expected utility is lower.

Figure 10:

Expected utility of rules in the two-dimensional example, with the distribution of cases sensitive to enforced rule.

A.3.1 One-Shot Regime

The sequence of play in the One-shot regime is as follows. First, Nature draws a case x from a uniform distribution on the unit interval. Second, the trial judge processes this case using her preferred rule; we assume the resulting disposition is the correct one from the judge’s perspective. In addition, the judge exerts costly effort e (with 0 ≤ e ≤ 1) whose effect is to reduce the possibility of reversible procedural error in the trial. Reversible procedural error is denoted by the variable ɛ = {0, 1} where ɛ = 1 connotes a reversible procedural error. The cost of effort is c(e) = e ². Third, Nature determines whether a procedural error occurred in the trial. In the event of procedural error the court’s disposition is dismissed and in effect the incorrect disposition prevails. Nature draws ɛ using known distribution p(ɛ = 0|e) = e (so p(ɛ = 1|e) = 1 − e). Fourth, the judge receives her payoff and the game ends.

The dispositional utility function for the judge is:

with 0 < λ ≤ 1. The parameter λ can be seen as a judge-specific parameter denoting the judge’s scrupulousness. Or, it can be seen as a case specific parameter, related to case importance. Note that this is a constant, symmetric gain/loss function.

We may write the judge’s expected payoff as

Via calculus, the optimal procedural effort level is e o s * = λ . Thus, the total effort expended is λ, the probability of reversible error is λ, procedural effort is increasing in λ (case importance or judicial scrupulousness), and procedural effort is independent of case location x.

A.3.2 Do-Over Regime

The sequence of play in the Do-Over regime is exactly the same as in the One-Shot regime with one exception: in the event of reversible procedural error, the judge must re-try the case. And, he must keep doing so until the case terminates unmarred by reversible procedural error. Thus, the game has an infinite horizon and belongs to the class of models considered in more detail in Chapter 4. We assume the following per-period dispositional utility function:

Note that this unusual dispositional utility function arises because there are three possible case outcomes, two of which are final outcomes and one of which is an interim outcome. We assume discounting across time periods, which are discrete and begin with t = 0. The discount rate is δ.

We first derive the judge’s objective function. We exploit the stationarity of the problem and focus on a history-independent allocation of effort. So, in each period with probability e the judge receives λ and the game terminates. With probability 1 − e he receives the interim payoff 0 and the discounted continuation value of the game. Call the continuation value V. But in either case the judge must pay e ². So the per-period payoff is:

The accumulated expected per-period payoffs are:

But this is simply the net present value of a perpetuity of λ − e e discounted at δ(1 − e). From finance, this is:

Via calculus the optimal per-period effort expenditure is:

We omit proofs but the comparative statics of optimal effort are intuitive, e.g., greater λ leads to greater per-period effort while greater δ (more future orientation) leads to lower per = period effort. In essence, the availability of the future “do-over” reduces per-period effort. In addition, lim δ → 0 e * = λ 2 while lim _δ→1 e* = 0. From probability theory, the expected number of rounds until the first error-free adjudication is simply 1 e * .

A.3.3 Comparison of Incentive Effects

For a case, in the one-shot regime the total judicial effort exerted is just e o s * = λ .What about under the do-over regime? Here, the total expected effort is:

And, lim _n→∞ TEe = 1.

In words, the do-over regime induces the judge to work as hard, in expectation, as would only the most scrupulous possible judge λ = 1 in the one-shot regime. Or, equivalently, it induces the judge in the do-over regime to treat every case the same way that a one-shot judge would treat only the most-important cases. This finding, while quite striking, is not equivalent to saying that the do-over regime is unquestionably better than the one-shot regime. For that conclusion, we would also need to model the resources needed to identify reversible error (for a systemic analysis with that flavor, see Chapter 6). But this analysis of the incentive effects of institutional design displays one application of a “labor market” model set in case-space.