Attributes for the Multiattribute Utility Function

Now, there are several different performance measures that one might consider in making the decision corresponding to the situation described here. We want to make sure that we consider the various stakeholders (two in this case) and their objectives. One approach would involve employing the following set of attributes (or performance measures):

CGOV = Cost to the government associated with the decision made regarding bail,

CINB = Cost associated with inconvenience (as a result of an imposed pretrial restriction) + the cost associated with bail to the defendant,

DFREE = Number of days of non-incarceration for the defendant as a result of release of the defendant prior to trial.

CGOV is of course associated with the stakeholder that we identified as the government. This attribute consists of four types of costs: the cost for incarceration of the defendant, the cost for imposition of pretrial restrictions, the cost associated with nonappearance of the defendant at trial given that he/she does not appear, and the cost associated with crimes committed by the defendant while out on bail, prior to his/her trial. In addition, if the defendant does not show for trial, any bail that the defendant paid could be thought of as a negative cost (i.e., income to the government).

CINB and DFREE are attributes mainly of interest to the defendant. CINB is the sum of two costs, CDEFPTj and CBAIL. which were discussed above. In valuing DFREE, the judge, as the “super decision maker” must consider the comparison to incarceration and the direct and social costs associated with that incarceration.

For any specific decision made by the judge (xR = 1, or xPri =1, for one value of i = 1,2, ..., NPT), the values associated with CGOV, CINB, and DFREE can be uncertain in nature as a result of the uncertain behavior of the defendant (as represented by the probabilities: PNA, PC, and PACCi for i = 1,2, NPT) and the uncertain values of some of the other model parameters as a result of their representation as random variables.

Criterion, Alternatives, Parameters, and Multiattribute Utility Function for the Example

One criterion that could be employed in the evaluation of the judge’s decision would be the maximization of the expected utility, where the utility is a function of the three attributes: CGOV, CINB, and DFREE:


In this illustrative example given below, we will employ a Monte Carlo simulation to estimate the expected utility of each alternative decision. Since, with the simulation, we will only be obtaining estimates of expected utility, we will employ one out of several available approaches for ranking the alternatives: the approach which involves “all pairwise comparisons” (pages 387-391, Evans, 2017). This approach is appropriate because of the relatively small number of alternatives under consideration.

Let’s consider a situation involving an arraignment for an alleged nonviolent felony. The judge is considering remand of the defendant, along with three other alternatives (NPT = 3) with respect to pretrial conditions:

  • 1. Release of the defendant on their own recognizance,
  • 2. Release of the defendant with the restriction of weekly checkins with the probation department, and
  • 3. Release of the defendant on bail of $10,000, with no other conditions.

Now, a typical amount of time from the arraignment to the start of a trial is 30-45 days (Schwartzbach, n.d.), and a felony trial will sometimes require several months to complete (Chapman, n.d.). So, let us assume that the time from arraignment to disposition of the case through trial (DAYSF, as defined above) is uniformly distributed with a minimum value of 60 days and a maximum value of 120 days.

Estimates of costs associated with incarceration of prisoners vary widely. For example, a Bureau of Prisons estimate (2018) of the cost for keeping a prisoner in a residential re-entry facility in FY 2017 was $88.52 per day, while the same estimate for a federal prison was $99.42 per day. Estimates of the cost per day for keeping a prisoner in a local jail in 2015 vary according to $48 in Cherokee County, Georgia; $49 in Dallas County, Texas; and $83 in Douglas County, Nebraska (Henrichson, et al, 2015). In our example here, we will assume a cost per day of $90 to keep a prisoner incarcerated, i.e., CINCG = $90.

The costs for the three pretrial conditions specified are assumed to be CPT, = 0. CPT, = $390, and CPT, = 0. Since the first and third pretrial conditions place no restrictions on the defendant and require no resources, these costs are just 0. The second pretrial condition requires a weekly check-in by the defendant; assuming 13 “checkins”, with each check-in requiring one hour of time for a court official who is paid $30 per hour, including overhead, we arrive at a cost of 13*$30 = $390.

If the defendant does not show for their court date, then typically the bail will be revoked, and a warrant will be issued for arrest. Bailjumping is a crime in and of itself. The costs involved will include the administrative time for court officials as well as the time of police in searching for and re-arrest of the defendant. We will assume that this cost (denoted as CNA) is given by a random variable with a uniform distribution having a minimum value of $4,000 and a maximum value of $8,000.

There is also the possibility that the defendant will commit additional crimes if free on bail. We denote this cost associated with these additional crimes as CAC and assign its value as a random variable with a uniform distribution having a minimum value of $15,000 and a maximum value of $20,000.

The first step in developing a multiattribute utility function for this situation would be to determine the minimum and maximum values to allow for each attribute. These values would not have to be the exact minimum and maximum values, but they would need to just bound the actual minimum and maximum values.

Consider CGOV first. Without going into a lot of detail, the minimum value for CGOV would occur if the defendant is not incarcerated prior to trial, does not require any government resources prior to trial (i.e., is assigned pretrial release condition 1 or 3 above), does not commit any crimes while free on bail, and shows for his/her court date. This minimum value would be $0. The reader could verify this by assuming different alternative decisions and outcomes for the uncertain events and performing the relevant computations, relative to Figure 3.1.

The corresponding maximum value for CGOV would occur if the defendant were allowed to be free prior to trial by being released on his/her own recognizance, committed crimes at the highest level of cost according to the uniform distribution specified for this input ($20,000), and also did not show for their trial, also at the highest level of cost ($8,000). Hence, this maximum cost would be $28,000.

The minimum value for CINB would occur if the judge chose the first pretrial condition (to release the defendant on their own recognizance). This value would be $0.

The maximum value for CINB would be equal to CDEFPT, plus CBAIL, or $1000 + $1000 = $2000.

The minimum value for DFREE would be 0 days (this would be the worst outcome for the defendant for this attribute as DFREE represents the number of days of “non-incarceration”), while the maximum value for DFREE would be 120 days. This value of 120 days represents the maximum possible value for the number of days from arraignment to the end of the trial (i.e., the maximum possible value for U (60 days, 120 days)).

In order to keep this illustrative example relatively simple, we will assume that the multiattribute utility function is a scaled additive function (from 0 to 1), with the individual attribute utility functions being linear. Representing the attributes as X„ X2, and X3 we have:

X, = CGOV,

X, = CINB, and


We will assume a scaled, additive utility function:


u(x|,x2,x3) = y^kiui(xi)

where u,, u2, and u, are the individual attribute utility functions for Xj, x2, and x3, respectively, and k,, k2, and k3 are the corresponding scaling constants.

Given the minimum and maximum values for each of the attributes and the fact that the individual attribute utility functions are linear, we can express these individual attribute utility functions as:

u1(x,) =-.000035714x, + 1, u2(x2) = - OOO5x2 + 1, and u,(x3) = .OO83333x3

At the maximum values of attributes X( and X2, the corresponding respective utility values are each 0, and at their corresponding minimum values the respective utility values are each 1. At the maximum value for the third attribute, X3, the corresponding utility value is 1; when X, has a value of 0, its minimum value, the corresponding utility value is 0. These utility values correspond to the fact that we want to minimize both X, and X2 and maximize X3.

We assume values for the scaling constants as follow:

k, = .4, k2 = .3, and k3 = .3.

Note that the form of the multiattribute utility function (additive, instead of say multilinear or multiplicative), the form and specifications of the individual attribute utility functions: u,, u2, and u, (linear in this example), and the values for the scaling constants ( k|; k2, and k3), would all be determined through an assessment process, as alluded to in Chapter 2 of this book, and in other books (for examples, see pages 236-263 of Evans (2017), and pages 261-271 of Keeney and Raiffa (1993)).

The Simulation Model and Results

The Monte Carlo simulation model employs a process-oriented simulation software package. Each replication of the model involves the generation of a single entity which flows through a network of modules. The branching from one module to the next is accomplished in either a deterministic fashion (depending on the value of a control variable which represents one of the four policies regarding bail listed above) or in a probabilistic way, modeling (1) whether the defendant accepts the pretrial release alternative offered by the judge, (2) whether the defendant commits additional crimes while free on bail, and (3) whether the defendant shows for their trial if released prior to trial. Random variates are generated for (1) the three probabilistic branching situations listed above (i.e., according to the probabilities given by PACC,. PC,, and PNA,. for i = 1,.... NPT); and (2) the uncertain quantities associated with DAYSF, CAN, and CAC.

The main output of the simulation is an estimate of the expected utility associated with one of the four policies modeled with the current simulation run. In addition, the 95% confidence interval associated with the expected utility is output by the model. There is a large body of literature involving the choice of a best alternative using a simulation (see pages 371 through 412 of Evans (2017) or pages 548 through 574 of Law (2007)); a reason for the difficulty here is the fact that we obtain only an estimate of the criterion value (i.e., value for expected utility) used for making a choice. This subject area is beyond the scope of this book; suffice it to say, however, that since there are only four policies in this example, and that 100 independent replications of the simulation are made for each policy, we can easily distinguish between the expected utility estimates for the four policies.

In addition to expected utility, the model also outputs an estimate for the expected values of each of the attributes: CGOV, CINB, and DFREE.


Inputs to the Simulation Model



Value (for this example)


Number of alternatives under consideration for pretrial release



Number of days that defendant will be released under a pretrial release alternative

U (60 days, 120 days)


Cost per day to incarcerate the defendant to the government


CPT„ for i =


Cost to the government to implement pretrial release condition i

$0, $390, $0


Cost to the government associated with nonappearance of the defendant at trial

I) ($4000, $8000)


Cost to the government associated with additional crimes committed by defendant while on bail, given that additional crimes are committed

I) ($15000, $20000)

CDEFPT,, for i = 1,2,3

Cost to the defendant associated with “inconvenience'’ due to the adherence to pretrial restriction i

$0, $1000, $0

CBAIL,, for i = 1,2,3

Monetary cost to defendant of bail paid under pretrial release alternative i.

$0, $0, $1000

PNA„ for i =


Probability of nonappearance of the defendant for trial, given that he/she is released under the alternative of pretrial release condition i

.2, .1,.05

PC,, for i =


Probability that the defendant will commit additional crimes under the alternative of pretrial release condition i

.2, .1, .2

PACC, for i = 1,2,3

Probability that the defendant will accept pretrial release condition i, if offered; if not accepted the defendant will remain incarcerated until trial

1., .9, .6


Decision variable set to 1 if remand is chosen, 0 otherwise

Set according to policy chosen

XP„, for 1 =

Decision variable set to 1 if pretrial release

Set according to


alternative i is chosen, 0 otherwise

policy chosen

A summary of the inputs to the simulation model is shown in Table 3.3.

For access to the simulation model developed for this example, contact the author.

The four policies: remand (policy 1), release on own recognizance (policy 2), release on condition of weekly check-ins (policy 3), and release on bail money (policy 4) were simulated under two different sets of preferences (as represented by the three scaling constants of the utility function) and under two different sets of probabilities associated with the defendant committing additional crimes while free prior to the trial (i.e., the PC, values) and with the defendant not appearing for trial (i.e., the PNA, values). The combinations of scaling constants and parameter values yield four different sets of results, as shown in Table 3.4.

Table 3.4 illustrates that Policy 2 (release on own recognizance) is ranked first for two of the four parameter settings shown, while Policy 1 (remand) and Policy 3 (release on condition of weekly check-in) are ranked first on one of the parameter settings each. Note that the various rankings shown are a result of not only the parameters that were varied, but also of values for the other parameters, such as the cost associated with the committing of additional crimes (CAC) and the cost associated with non-appearance of the defendant at trial. For example, as the cost associated with committing additional crimes and the cost associated with non-appearance at trial increase, Policy


Estimates of Expected Utilities and Associated Policy Rankings Under Two Sets of Scaling Constants and Two Sets of Probabilities. (Policies are listed from most to least preferred, with expected utilities shown in parentheses following the policies.)

Parameter Settings

k, = .4, k2 = .3, k3 = .3

k, = .6, k2 = .2, k3 = .2

PC, = .6, PC2 = .3,

Policy 2 (.71)

Policy 3 (.74)


Policy 3 (.69)

Policy 4 (.72)

PNA, = .6, PNA2 = .3,

Policy 4 (.67)

Policy 2 and Policy 1 (.63)

PNA3 = .3

Policy 1 (.58)

PC, =.8, PC2 = .5,

Policy 2 (.64)

Policy 1 (.63)

PC, =.5

Policy 3 and Policy 4 (.61)

Policy 4 (.62)

PNA, = .8, PNA2 = .5,

Policy 1 (.59)

Policy 3 (.60)

PNA3 = .5

Policy 2 (.52)

1 (remand) becomes more attractive. The overall ranking of all policies is a function of all the probabilities, costs, and other parameters input for the model.

Final Discussion of the Model

The purpose of this example is to illustrate how simulation and multiple objective methodology (as embodied by the use of a multiattribute utility function) can be used to determine a policy with respect to bail reform. The modeling approach allows for explicit consideration of all relevant costs and probabilities, as well as the trade-offs between the various objectives of interest to the two main stakeholders: the “government” (or the “people”) and the “defendant” (including those associated with the defendant).

There are many variations on the model presented here. One group of variations would be concerned with making the model more accurate. Examples of these variations would be (1) a consideration of the dependence between the events of the defendant committing additional crimes while free on bail and the defendant not appearing for trial, (2) a delay between the decision of the judge with respect to a pretrial alternative and the decision of the defendant in whether to accept the judge’s decision, and (3) the modeling of a possible plea agreement by the defendant prior to the start of trial. These additional features of the model would not be difficult to represent through appropriate modifications of the simulation model.

Another variation on the model would be the consideration of additional inputs to the model. An example of such an input would be the probability of defendant being guilty of the alleged crime, the conditional probabilities of the defendant being found guilty/innocent given that he/she is guilty/innocent. Although it may not be appropriate to do so, many judges would consider the probability of guilt at least implicitly in making their pretrial decisions.

At least some of the input data used in the simulation runs described in this section may very well be inaccurate or even unrealistic. Hence, prior to using a model such as this for decision-making, additional research should be done to firmly establish these data values, even if they are input as random variables. Of interest here would be the inputs associated with the costs and probabilities of additional crimes committed and non-appearance for trial (CAC, CNA, PC, and PNA) since these values would have a large effect on whether the defendant would be released prior to trial. Another input of primary importance would be the multiattribute utility function used to represent the trade-offs and risk preferences that the stakeholders would be willing to make or accept. Assessment procedures, involving questions of and answers from a decision maker, are available to determine an accurate multiattribute utility function.

Related to the accuracy of the input data for the model would be the concept of additional experimentation. The experimentation reported in this section of the book was extremely limited in nature. Given the relatively large number of difficult-to-estimate input parameters, the use of experimental design methodology may be appropriate (Chapter 12 of Law (2007)).

Finally, and this may very well be the most important point of this section, the model along with the variations described in this section could easily be extended to consider new policies or laws with respect to bail reform. Instead of generation of one entity as input to the process model for each replication, many entities, representing the new defendants over some period to be affected by the new policy, would be input for each replication. Further, groups of entities, representing groups of defendants with approximately the same characteristics could be input for each replication, and in fact an optimization could be performed which would address the assignment of each particular defendant group to a particular pretrial alternative.

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