# Cost–Benefit Analysis

Cost-benefit analysis (CBA) is a methodology which evaluates each alternative in a decision situation by categorizing all of the effects of the alternative as either a cost or as a benefit, attaching a dollar value to each, and then seeing if the dollar value of the benefits is greater than the dollar value of the costs. Future costs and benefits are discounted with an appropriate interest rate.

Appropriate modeling techniques must often be used to accurately predict at least some of the costs and benefits, especially if they correspond to effects scheduled to happen far into the future. In addition, some of the costs and benefits might correspond to intangibles, which can be difficult to calculate. Examples of these intangibles with respect to criminal justice systems would be things such as the costs associated with the fear of crime, the death/injury of an individual through an assault, the detrimental effects on the family of someone sentenced to prison.

An approach for applying CBA in criminal justice in general is embodied in the Manning Cost-Benefit Tool (MCBT) (Manning et al., 2016). This approach basically involves having the user input data to spreadsheets; the data consists of a variety of inputs corresponding to some type of intervention in the criminal justice system. One of the major contributions of the MCBT is the enumeration of most of the categories of the various costs and benefits of an intervention of this type. For example, there are the major categories of (1) costs in anticipation of crime, (2) costs as a consequence of crime, and (3) costs in response to crime. These are divided into the sub-categories of homicide, serious wounding, sexual offenses, common assault, robbery, etc. The tool has as one of its inputs a discount rate, which allows for the computation of the net present value of costs and of benefits. In a follow-on article, Manning et al (2018) suggest the use of a conceptual tool called Smart MCBT that uses machine-learning techniques to aid the user in the input of the vast amount of data needed for the use of the tool.

The state of Washington has performed cost-benefit analyses of several of its programs and policies. (See the Washington State Institute for Public Policy (n.d.) and Aos and Drake (2010)). Their probabilistic approach to the analyses allowed for an estimate of the probability that the benefit would exceed the cost, as well as an estimate of the net present value of the benefits minus the costs, and a benefit to cost ratio. For example, their analysis of an employment counseling and job training program for transitioning from incarceration into the community yielded a net present value of $43,502, a benefit to cost ratio of 18.21, and a probability associated with the benefits exceeding the costs of .88.

Fass and Pi (2002) used simulation and CBA to investigate juvenile crime in Dallas County, Texas. Their investigation ascertained that more severe sentencing for juvenile crimes could avert some offenses, but that the cost for implementing the actions was much greater than the benefits.

Finally, Zarkin et al (2015) noted that 50% of those incarcerated in state prisons meet the standards for diagnosis of drug dependence. They developed a simulation model to look at the net benefits of diversion from reincarceration to community-based substance abuse treatment and discovered that the benefits far outweighed the costs.

# Multiattribute Value Functions and Multiattribute Utility Functions

Once a set of attributes for a decision situation has been properly identified, one can develop a method for ranking outcomes which correspond to a set of values assigned to these attributes. These outcomes would be associated with the respective alternatives under consideration. For example, one may be considering various policies corresponding to the hiring of additional police; each policy can correspond to an outcome corresponding to additional costs and higher levels of arrests.

In some cases, these outcomes may be deterministic in nature, and in other cases they may be probabilistic. One way to rank these outcomes if they are deterministic (probabilistic) in nature is through the use of a multiattribute value function (multiattribute utility function).

A **multiattribute value function **is one which maps from the outcome space to the space of real numbers. It is used to represent the preference structure of a decision maker over the outcome space of multiple attributes. Suppose that we use the following notation, corresponding to that used in Evans (2017):

n is the number of alternatives under consideration,

Aj represents alternative i for i = 1,..., n,

p is the number of attributes that we have under consideration,

X_{k} represents attribute k for k = 1, ..., p, and

x_{k}(Aj) is the attribute value associated with attribute X_{k} for alternative A_{k}

In most cases involving the use of a multiattribute value function, the function v will be what is termed a scaled, additive function, as shown below:

**p **v(x„x_{2},x_{3},...,x_{p}) = ^c_{k}v_{k}(x_{k}), **k=l**

where:

v_{k}(x_{k}) = l,and v_{k}(x_{k}) = 0fork = 1,..., p where x_{k} is the best possible value for the kth attribute and x_{k} is the worst possible value for the kth attribute,

0

**k=l**

Note that the v_{k} for k = 1, ...p, are called individual attribute value functions. These are not necessarily linear. In addition, the slope of an individual attribute value function will be positive if the attribute is one that should be maximized, and negative if the attribute is one that should be minimized.

Determining a multiattribute value function for a decision maker is called the assessment process. There are four basic sequential steps for this assessment process: (1) determine the best and worst values for each attribute, (2) determine whether the decision maker’s preferences correspond to a multiattribute value function with an additive form, as shown above, (3) determine the individual attribute value functions, v„ v , through answers obtained from the decision maker about his/her preferences over the outcome space, and (4) determine the scaling constants: c„ c_{2}, ..., c_{p}, again through the answers obtained from questions posed to the decision maker.

Within each of the four steps, there are one or more approaches that can be used. The exact assessment process is beyond the scope of this book, but one can view any of several sources to obtain additional information about assessment (pages 94-112 of Evans (2017), or pages 82-125 of Keeney and Raiffa (1993)).

The reader can view an example of the use of a multiattribute value function in Chapter 3. This hypothetical example scores and ranks proposals from private companies for the operation of a prison.

A **multiattribute utility function **is also typically scaled to provide a value between 0 and 1 when acceptable values for attributes are input as independent variables for the function. The main difference between this function and a multiattribute value function however is that a multiattribute utility function will allow for the ranking of **probabilistic **outcomes, through the criterion of maximization of expected utility.

As with a multiattribute value function, a multiattribute utility function represents the preferences of a specific decision maker. In addition, typically, when applied in specific situations, these functions will assume any of several different specific forms, e.g., additive, multiplicative, or multilinear (see pages 236-263 of Evans (2017)).

The process of assessment of a multiattribute utility function, as with a multiattribute value function, also involves the decision maker(s) answering questions about their preferences over probabilistic outcomes defined over multiple attributes.

## An Illustrative Example: Using a Multiattribute Utility Function for the Decision of Whether to Use a Court-Appointed Attorney

Let us consider a hypothetical example where one could use a multiattribute utility function. This example illustrates how one can employ a multiattribute utility function.

Suppose a defendant has a decision to make. The defendant can use a court-appointed attorney to handle the felony case (decision 1), or the defendant can hire an attorney (decision 2), at a cost of $3500, to handle the case (How much are attorney fees?, n.d.). If the defendant uses the court-appointed attorney, then the probability of being convicted and serving two years in a state prison is .7, and the probability of being found innocent and therefore set free is .3. On the other hand, if the defendant hires the (expensive) attorney at a cost of $3500, the probability of being found guilty is .4, and if found guilty, there is a .5 probability of serving two years in prison, and a .5 probability of serving one year in jail.

From the description of the problem there are basically two attributes with which the defendant will be concerned:

X, = cost in thousands of dollars and

X, = number of months spent in jail/prison.

The outcome associated with each decision will be probabilistic in nature. For example, if the defendant decides to use the court-appointed attorney, the outcome will be:

X,: 0 with certainty,

X_{2}: 0 with .3 probability and 24 with .7 probability.

The outcome associated with the second alternative decision of hiring the attorney is:

X,: 3.5 with certainty,

X_{2}: 0 with .6 probability, 12 with .2 probability, and 24 with .2 probability.

Now, the best possible values for X, and X_{2}, denoted as xf and x_{2}, respectively, are 0 and 0. The worst possible values for X, and X_{2}, denoted as x" and x_{2}, respectively, are 3.5 and 24.

Let’s suppose that the defendant (who is the decision maker in this case) has a multiplicative utility function over the two attributes of concern. This multiplicative function, denoted as u(x,, x_{2}) could be written as follows:

u(X|,X_{2}) = W|U_{1}(X|) + W_{2}U_{2}(x_{2}) + WW_{1}W_{2}U_{1}(X|)u_{2}(x_{2}), where w, w_{h} and w_{2} are called scaling constants in this situation, and u/x,) and u_{2}(x_{2}) are called individual attribute utility functions. Since the function is scaled, we will have:

u,(3.5) = 0. Uj(O) = 1, u_{2}(24) = 0, and u_{2}(0) = 1.

It can be shown that w = (1 - w, - w_{2})/w_{(}w_{2} (see pages 244-245 of Evans (2017)). Let’s assume that w, = .2 and w_{2} = .5; then w will equal 3. Hence the utility function can be written as:

u(x_{p} x_{2}) = .2U|(X|) + .5u_{2}(x_{2}) + 3(.2)(.5)u_{l}(x_{l})u_{2}(x_{2}) = .2u,(x,) + .5u_{2}(x_{2}) + .3u_{l}(x_{1})u_{2}(x_{2})

We have the function values for u, and u_{2} for the best and worst values of X, and X_{2}, respectively, but not the functions themselves. Let us suppose that these functions are given by:

u,(x,) = -.2857Xj + 1. and

u_{2}(x_{2}) = -.04167x, + 1.

Note that both individual attribute utility functions are linear with a negative slope. The negative slope implies that these utility functions correspond to attributes that we want to minimize, as opposed to maximize.

We want to choose the decision that maximizes expected utility, or, in other words, that maximizes the expected value of a function (the utility function). For the first decision under consideration, use the court-appointed attorney, the outcome will be:

X, = 0, X_{2} = 0, with probability of .3,

X, = 0, X_{2} = 24, with probability = .7.

Substituting these attribute values into the utility function, we obtain:

u(0, 0) = 1 with probability of .3 and

u(0, 24) = .2 with probability of .7.

So the expected utility for this decision of using the court-appointed attorney is:

.3 (u(0, 0)) + .7 (u(0, 24)) = .3(1) + .7(.2) = .44.

If the defendant hires the attorney, the outcome will be:

X, = 3.5, X, = 0, with probability of .6,

X, = 3.5, X, = 12, with probability of .2, and

X, = 3.5, X, = 24, with probability of .2.

Substituting these attribute values into the utility function, we obtain:

u(3.5, 0) = .5 with probability of .6,

u(3.5, 12) = .25 with probability of .2, and

u(3.5, 24) =0. With probability of .2.

So the expected utility of the second alternative, hire the attorney, is given by:

,6(,5) + .2(.25) + .2(0) = .35.

Comparing the expected utilities of .44 and .35, the alternative of using the court-appointed attorney gives the larger expected utility and is therefore the preferred alternative.

The main purpose of this example is to illustrate the computation of expected utility. The defendant in this case probably could have made the correct choice just by looking at the probabilistic outcomes, that is, without the use of a utility function. In Chapter 3, we include a much more involved example where maximization of expected utility is used to determine the pre-trial disposition of a defendant.

It is important to keep in mind that both multiattribute value functions and multiattribute utility functions are personal in nature; that is, for a group of people faced with a specific decision situation, either deterministic or probabilistic in nature, every one of them will have either a value function or utility function that is at least slightly different from everyone else in the group. Note that this does not necessarily mean that they would not rank the alternatives in the same way. For example, with one decision maker, the expected utilities for three alternatives might be given as alternative one: .47, alternative two: .68, alternative three: .55, while for a second decision maker, the expected utilities might be given as: alternative one: ..64, alternative two: .78, alternative three: .68. In this case the two decision makers would rank these alternatives in the same way: alternative one: third; alternative two: first; and alternative three: second.

# Simulation Modeling and Analysis

*Simulation refers to the broad collection of methods and applications to mimic the behavior of real systems, usually on a computer with appropriate software* (page 1, Kelton, Sadowski, and Zupick (2015)). As such simulation allows one to re-create the operation of a complex system in a digital fashion on a computer. It is an especially useful technique when that system operation involves dynamic and probabilistic behavior over time.

With a simulation model of a system, one can experiment with different policies, procedures, and operations (via input to the simulation model) and examine the outputs from the model to determine the best policy or rank the various policies. The use of statistical analyses is important in the examination of the output because of the probabilistic nature of the simulation (which represents the probabilistic nature of the system). For example, in Chapter 3, we present a simulation model to examine various policies for pretrial conditions imposed by a judge on a defendant. Inputs to the model include the probability that the defendant will show for the trial, and the probability that the defendant will commit additional crimes while free on bail prior to the trial. This probabilistic representation of the defendant’s behavior requires multiple (e.g., 10s or 100s) replications (runs) of the model to perform a proper statistical analysis. In the first replication the defendant may show for the trial and in the second replication the defendant may not show, according to the appropriate input probabilities.

One of the advantages of a simulation model in policy analysis is the fact that many different policies can be evaluated (and ranked in order of preference), typically in just a few minutes of time. Each replication of the model in these cases may represent several months or even years of simulated time. Hence, experimenting with the actual system in these cases would be impossible, not only from the consideration of the time involved, but also from the consideration of disruption to the actual system.

Another advantage of simulation as a tool for policy analysis is the fact that these models can output estimates for attribute values, and corresponding estimated values for multiattribute value functions or multiattribute utility functions, for any specific policy. Using multiple replications of the model, an estimate of the expected utility of a policy can be made when expected utility is the criterion of interest.

Finally, when a policy can be defined using values assigned to several control variables, there may be too many policies to evaluate using the model, given the combinatorial nature of the inputs. An example of this situation would occur when one has six different alternative programs of rehabilitation to implement with ten different categories of prisoners, as defined by age, sex, crime background, etc. In this situation, an alternative would be defined by the assignment of a rehabilitation program to a prisoner category. Hence, the number of combinations (or the number of alternatives) would be given by 6^{10}, more than 60 million alternative combinations. In this situation one may want to use an optimization procedure to implicitly enumerate many of the policies. Most simulation software packages include an optimization package which allows the user to interface their simulation model with an optimization process in order to find an estimated optimal policy.

Although a simulation model can be constructed using a general-purpose programming language, most such models are built with a simulation software package. Examples of these packages include Simio®, Arena®, ProModel®, AnyLogic®, SIMUL8®, and Flexsim®. Many of these packages employ a process-oriented approach in which the simulation represents a process, composed of various activities, through which entities flow. The activities can be performed by various resources, and are represented by icons which the model builder places on the computer screen.

For example, if one used Figure 1.1 as a guide to build a simulation model of prosecution, pretrial services, and adjudication for a criminal justice system, the entities would be the defendants flowing through the process. These entities would have descriptors called attributes. (Note that the word attribute has two different meanings in this book: performance measures in reference to a multiattribute value/utility function, and descriptors of entities in reference to a simulation model.) The attributes for the entities would be things like: crime accused of, demographic information, etc.