J

Referen

A111D4 5S1ET^

NBSIR 79-1711

Resource Requirements and

Allocations in IRS' Audit

Division

Karla L. Hoffman

Lambert S. Joel

Martin H. Pearl

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

February 1979

Final

-QC

100

,1156

79-1711

u7j

Technical Report to: The Audit Division

Internal Revenue Service

1111 Constitution Avenue

Washington, DC 20224

HATiajMX. BxmsMJ

OF WAITOmW

library

NOV 2 4 1960

NBSIR 79-1711

RESOURCE REQUIREMENTS AND

ALLOCATIONS IN IRS' AUDIT

DIVISION

/ q

Karla L. Hoffman

Lambert S. Joel

Martin H. Pearl

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

February 1979

Final

Technical Report to: The Audit Division

Internal Revenue Service

1111 Constitution Avenue

Washington, DC 20224

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

List of Tables . . ii

List of Figures . iii

Abstract 1

1. Introduction . 1

1.1 Role of IRS Audit Division 1

2. The Data Base 3

3. Audit Classes and Their DIF Formulas 5

3.1 Evaluation of the DIF Process 6

3.2 Exploration of the Discriminant Function

Analysis Methodology 12

3.3 Audit Class Definitions 15

4. Compliance 19

5. The Long Range Plan (LRP) 23

5.1 Implementation of Plan 27

6 . Summary 31

References 3^

Appendix A 35

Appendix B 5^

Appendix C 67

1
LIST OF TABLES

Table 1 Spearman Rank Correlation Coefficients

(Adjusted for Ties) Measuring Relationship

of Tax Change to DIF Score 8

Table 2 Pearson Correlation Coeffieicnts Based on Aggregated Data Measuring Relationship

of Average Tax Change to Average DIF Score . . 9

Table 3 Spearman Rank Correlation Coefficients

(Adjusted for Ties) Measuring Relationship

of Average Tax Change to Average DIF Score . . 10

Table 4 Comparison of DIF, Random and Perfect

Discrimination 11

Table 5 Program and Class Coverages in FY 1982

and the FY 1977 24

Table 6 Data Used for Developing Compliance Coverage

Curves for 1974 and Subsequent Year Long

Range Plans 26

Talle 7 Summary of Research Topics 33

Table A1 Numerical Values for the Sample Game 45

ii
LIST OF FIGURES

Figure 1 Average Yield Versus Coverage Level

for the Medium Business Class 29

Figure A1 Histogram of Tax Changes 44

Figure Cl Average DIF Score versus Average Tax

Change for Low Business Class 68

Figure C2 Average DIF Score versus Average Tax

Change for Medium Business Class ... 69

Figure C3 Average DIF Score versus Average Tax

Change for High Business Class 70

Figure C4 Average DIF Score versus Average Tax

Change for 1040A Returns 71

Figure C5 Average DIF Score versus Average Tax

Change for $10,000-15,000 AGI

Returns 72

Figure C6 Average DIF Score versus Average Tax

Change for $15,000-50,000 AGI

Returns 73

Figure C7 Average DIF Score versus Average Tax

Change for Above $50,000 AGI

Returns 74

iii

Resource Requirements and Allocations in IRS’ Audit Division

Karla L. Hoffman

Lambert S. Joel

Martin H. Pearl

Abstract: The Applied Mathematics Division of the National Bureau of

Standards (NBS) was asked by the Internal Revenue Service under IRS

order #7T2867 to "assess the validity and effectiveness of the IRS Audit

Division's Long Range Plan's strategies and approaches to resource

requirements and allocations". This report, which documents that

assessment, (l) summarizes the examination and evaluation by NBS of

IRS's current audit practices and plans for the future, and (2) presents our major conclusions and recommendations. In many cases the

available information did not permit recommending specific methods with

which to solve current problems. In such situations we sought to

identify those areas in which we believe further research is needed and

is most likely to lead to improvements over present practices.

1. Introduction

The Applied Mathematics Division of the National Bureau of Standards

(NBS) was asked by the Internal Revenue Service under IRS order #7T-2867

to "assess the validity and effectiveness of the IRS Audit Division's

Long Range Plan's strategies and approaches to resource requirements and

allocations". This report, which documents that assessment, (l) sum- marizes the examination and evaluation by NBS of IRS's current audit

practices and plans for the future, and (2) presents our major con- clusions and recommendations. In many cases the available information

did not permit recommending specific methods with which to solve current

problems. In such situations we sought to identify those areas in which

we believe further research is needed and is most likely to lead to

improvements over present practices.

This review pertains to IRS policies and procedures. Pertinent IRS

records were reviewed, data were analyzed and discussions were held with

national office personnel. However, no contacts were made with "field"

personnel and problems specific to them were not considered. Divergences between practice and policy, though obviously of potential significance, were beyond the scope of this study.

1.1 Role of IRS Audit Division

* As stated in the IRS 1975 Long-Range Plan,

*An unsigned IRS internal draft document.

1
"The audit mission is to encourage and achieve the highest

possible degree of voluntary compliance with the requirements of the internal revenue laws for the correct reporting

of income, estate, gift, employment and certain excise taxes

through implementation of appropriate programs for the

measurement of the types and degrees of compliance in

reported tax liabilities, the determination and analysis of

the reasons for noncompliance, and reduction of noncom- pliance by maintaining optimum levels of examinations for

all classes of taxpayers."

The audit process begins when a return reaches one of ten IRS service

centers where it is checked manually for completeness and obvious

inaccuracies (e.g. unallowable deductions). The service center's

computer then checks the correctness of the taxpayer's computations.

Eventually, each return is scored by the computer using a mathematical

formula ( DIF ) -'-obtained from a Discriminant Function Analysis—which

indicates the likelihood of tax error. The returns having the highest

DIF scores are then reviewed manually, and those which are deemed to

have the highest error potential are assigned for audit.

The first two steps in the procedure just described are straightforward

and will receive little attention here. However, a major portion of

this report is concerned with how best to carry through the last step,

i.e., how to decide which tax returns are to be audited. The second

part of that decision (the manual review) is left to the judgment of

experienced personnel. It is the prior portion, the identification by

the computer of tax returns proposed for audit, which will concern us initially.

In order to evaluate, on a continuing basis, the effectiveness of the

procedures it uses to perform its duties, the Audit Division must

incorporate into its Long-Range Plan: (l) the maintenance of an adequate

data base from which to estimate current levels of compliance; (2) the

development of optimal methods for identifying those tax returns with

high likelihood of error; and (3) means for measuring the effect on voluntary compliance of the number and type of audits conducted. Sections 2, 3 and ^ of this report will discuss respectively the present

data base, the methodology used to identify noncomplying taxpayers,

and the influence of auditing on compliance.

In the discussions we will address how each of these major areas of

concern relates to the Audit Division's goals, as stated in the IRS

1976 Long Range Plan, of: (l) promoting a high degree of voluntary

compliance; (2) satisfying considerations of equity; and (3) maximizing the net direct yield from Audit examinations. Section 5 will

then evaluate IRS' Long Range Plan, and Section 6 will summarize our findings.

2
2. The Data Base

As set down in its mission statement, a function of "Audit" is to

measure degree of compliance with the law and types of deficiencies in

reporting tax liability. IRS derives this information largely through

the Taxpayer Compliance Measurement Program (TCMP) surveys. Randomly

selected income tax returns from individuals, small corporations (assets

less than $1,000,000), estate exempt organizations, and fiduciaries

undergo intensive audits. Whenever an individual income tax return

selected in the TCMP process has a partnership schedule attached, and

that partnership has fewer than 11 partners, the partnership itself, and

all partners undergo an intensive audit. (These partnership returns

actually increase the number of returns in the final sample). The

results of these in-depth audits (which are reviewed for completeness,

fairness and accuracy) are used to determine the present levels and

trends of compliance by the tax return population.

Voluntary compliance level (VCL) is measured by comparing the tax

liabilities voluntarily reported (TR) by taxpayers, with the "correct"

total tax liabilities (TL) as determined by Audit in the TCMP survey.

Specifically,

VCL (in %) = X 100

and TL = TR + TC

where TC = Total disclosable tax change on all returns filed by

class (projected from the TCMP sample findings) as if all returns

filed were audited.*

The first TCMP Survey sampled individual returns filed in 1964, and

subsequent surveys have sampled returns filed in 1966, 1970, 1972, 1974,

and 1977 * Comparison of TCMP data for different years can reveal both

shifts in general compliance levels and specific changes in tax reporting behavior. TCMP data are also used to develop and test DIF formulae.

Moreover, they provide a means of estimating audit yields (average tax

changes and no-change rates) at various levels of examination coverage.

Without this information, IRS would have virtually no empirical basis on

*Until recently, the total disclosable tax change on all returns was defined to be the sum of the absolute values of all tax underpayments

and tax overpayments. In the revised computations now in use, "total

disclosable tax change" includes only tax underpayments. The change was apparently made for nontechnical reasons associated with bewilderment

arising from identification of overpayment as "deficiencies".

3
which to evaluate its programs or to plan an optimal audit strategy. We

applaud IRS's investment in this research.* It should be noted that

the research effort itself affects compliance behavior; the Taxpayer

Compliance Measurement Program is well known to sophisticated taxpayers,

who are therefore aware that they have a nonzero probability of being

audited no matter how successful they have been in maintaining low DIF

profiles. It is also known that TCMP audits will be especially intense,

and will not merely examine the most glaring line items on each return.

This awareness, that a random process is part of the Audit Procedure,

enhances the IRS's audit strategy in the sense that no one can ever be

certain of avoidance of audit. Such uncertainty should improve the

"voluntary" compliance of some taxpayers.

One of the most important uses of the TCMP data is to generate DIF

formulae for the various audit classes. The next section will analyze

the appropriateness of the present stratification into audit classes and

evaluate the effectiveness of the current DIF formulae for these classes.

*This statement, which may appear patronizing, is made to emphasize

our strong belief in the (frequently ignored) importance of basing

decisions on appropriate data.

h
3. Audit Classes and Their DIF Formulas

The present audit classes, shown in Table 1 following, are described in

the July 1976 draft document "Audit Planning and Resource Allocation"

prepared by the IRS Audit Division:

"For planning and program management purposes, both individual and corporation returns are stratified into classes

according to certain common characteristics, i.e., amount of

adjusted gross income (AGl) (plus the characteristics of

standard or itemized deductions in low AGI classes and the

presence or absence of business schedules) for individual

returns, and asset size for corporation returns. Presently,

eight classes are used in the individual return area and

eleven classes in the corporation return area. The detailed

class breakdown is shown on Chart 13, Appendix 10—Audit

Strategy For the Long-Range Plan.

Tax returns as a whole constitute a greatly diverse population, and are stratified to form relatively less heterogeneous groups to facilitate planning and program management.

For example, simple types of returns require relatively low

level of audit skills and lower average time of examination

as compared with the complex ones. Also, it has been found

that there is a reasonably good correlation between the

average size of tax error and the income level (or asset

level in the case of corporation returns), type of deductions and complexity of tax issues involved. Thus an appropriate stratification helps to enhance the effectiveness of

planning and program management by taking advantage of the

cost and benefit differentials among different classes of

returns

Five years ago, a study was conducted to test a return

selection procedure for individual returns without any class

stratification at all. Using TCMP data, a so-called one- class DIF formula was developed for all sample individual

returns. As expected, the formula would select primarily

high-income returns for audit. For a given level of audit

resources, it would mean a lower level of overall audit

coverage since examinations of high-income, complex returns

take more examination time per return. Moreover, with such

a workload Audit would need only technical personnel with

high-level accounting skills, who are in relatively short

supply. Furthermore, the highly one-sided concentration on high-income taxpayers and the corollary absence of coverage

elsewhere certainly causes concern about adverse impact on voluntary compliance of the great numbers of taxpayers.

These observations also indicate that a very gross scheme of

class stratification would tend to reduce planning and

program management effectiveness.

5
For pragmatic reasons, however, we cannot afford to have too

many audit classes, even though theoretically the greater

the number the more homogeneous are the elements within each

class for fuller revelation of the co-benefit differentials

between classes. This is because too detailed a stratification scheme would require a similarly detailed programming

of operations of all field units, not to mention the corollary

reporting and data processing requirements. Additionally,

increasing the number of classes significantly will cause a further fragmentation of the data base or a very "thin" or limited data base for each planning unit, which would tend

to result in highly unreliable planning estimates or projections .

The preceding discussion cannot be construed as an argument

that the current audit class stratification scheme is the

optimal one. It should be reexamined from time to time.

Limited studies of the reclassification problem have been

conducted in the past, e.g., classifying business returns on the basis of gross receipts instead of AGI, but these studies

have not produced definitive findings except that a splitting

of the medium non-business class into two income subclasses

would be helpful. However, in view of the recent inflation

phenomena, reclassification efforts should be resurrected to

test the idea of "indexing" the income class breaks (or

putting the class breaks on a constant dollar basis).

Similarly, perhaps it is worthwhile to split the mediumincome non-business class (MNB) into standard and itemized

ded .ctions subclasses to overcome the apparent anomaly of

having a planned coverage for low-income nonbusiness (itemized

deduction) class higher than the current MNB class (with

both standard and itemized deduction returns)."

We concur with the Audit Division's general philosophy of stratifying

returns into audit classes to form less heterogeneous groups, and

agree that a very large number of classes would cause many administra

tive problems and require a large expenditure being directed toward

TCMP surveys. We also agree that the existing stratification should

be reexamined regularly. Section 3.3 suggests a number of topics for

inclusion in such a reappraisal.

3.1 Evaluation of the DIF Process

To test the quality of IRS's stratification of returns into audit

classes, we must examine how well the corresponding DIF formulations

for each class succeed in differentiating those returns with a high

probability of significant tax change from those with low probability

To start, we can examine how well the DIF scores for a set of tax

returns (chosen randomly by a TCMP survey and presumably covering a broad range in degree of compliance), correspond to the disclosable

tax changes for the same set of returns. Disclosable tax change is

6
defined to tie the difference between tax liability voluntarily reported

on tax forms and the "correct" total tax liability as determined by

Audit. "Tax change" in this report will always refer to the absolute

value of the changes, i.e. to overstatement as well as understatement

of tax liability. Although, in principle, discriminant analysis is

meant merely to separate a population into a fixed number of classes

(in this case two: compilers and noncompliers ) the mechanism used for

this separation should yield "good" correlations with tax-change sizes

when the DIF is effective. It has been standard practice by IRS to use the DIF score for ordering the returns.

The Spearman rank correlation coefficient and the Pearson correlation

coefficient are two measures of the degree of relationship of two sets

of data. Since the Spearman test deals only with ranking, it is less

likely to distort the sense of the DIF (which attempts to discriminate

compilers from noncompliers rather than predict the degree of non- compliance) than is a test based on explicit numerical values, such as the Pearson test.

At NBS, the DIF scores and the tax changes within each audit class

were both ranked from highest to lowest. The Spearman correlation

coefficient produces an index of the agreement between these two sets

of rankings. Table 1 lists the correlation coefficients for some audit

classes. They were lower than had been expected. A ranking procedure

correlating poorly with a target ranking over an entire set might still

be quite useful if the correlation at the high end of the target ranking

were good. We therefore proceeded to check whether the discriminant

function might perform better for those returns within each class which

have high DIF scores. The Spearman rank correlation coefficients for

the actual coverages planned for 1977 were calculated, and are also

displayed in Table 1 below. Contrary to our hopes, the correlation

coefficients for the actual coverages planned for 1977 were lower than

those for the entire audit class.

7
TABLE 1

SPEARMAN RANK CORRELATION COEFFICIENTS (ADJUSTED FOR TIES)

MEASURING RELATIONSHIP OF TAX CHANGE TO DIF SCORE

TCMP SAMPLE OF RETURNS FILED IN 1970

For Coverage Levels

Entire TCMP Planned for 1977

Sample (Unweighted) Based on TCMP Data

INDIVIDUAL : Under $10,000 Standard

Itemized

$10 , 000-Under $15,000

$15 ,000-Under $50,000

$50,000 and Over

NONFARM BUSINESS : Under $10,000

$10, 000-Under $30,000

$30,000 and Over

.54655 .43327 ( .6$)

.36451 .28973 ( 4.2$)

.37254 .23364 ( 2.0$)

.27953 .14385 ( 2.2$)

.23924 .18944 O

1

—

1

1

—

1

.19102 .12733 ( 3.6$)

.34770 . 32190 ( 2.5$)

.31871 .13131 (10.2$)

* The numbers in parentheses indicate the coverage levels for each audit

class. For example, the coverage of individual returns with AGI or $50,000 or more would consist of the 11.0$ with highest DIF scores. Thi

table excludes taxpayers with income from farms, because those data were not supplied to NBS.

These results were discussed with IRS personnel, who then indicated that

they too had calculated correlation coefficients for TCMP data based on returns filed in 1966. They first aggregated the data by dividing each

audit class into between 15 and 31 intervals, depending on the class.

For each subdivision the mean (average) tax change and median DIF score were calculated. The Pearson (linear) correlation coefficient obtained

by comparing these interval measurements yielded significantly higher

results, as is shown in Table 2.
TABLE 2

PEARSON CORRELATION COEFFICIENTS BASED ON AGGREGATED

MEASURING RELATIONSHIP OF AVERAGE TAX CHANGE TO AVERAGE

TCMP SAMPLE OF RETURNS FILED IN 1966

i DATA

DIF SCORE

Aggregated into

the Following Pearson

No. of Intervals Coefficient

INDIVIDUAL:

Under $10,000 Standard 31 .8553

Under $10,000 Itemized 23 .9449

$10 , 000-$ 50,000 29 .8504

Over $50,000 25 .8207

NONFARM BUSINESS:

Under $10,000 21 .8194

$10 , 000-$ 30,000 28 .8908

Over $30,000 27 .8497

FARM BUSINESS:

Under $10,000 23 .7601

$10 , 000-$30,000 15 .8198

Over $30,000 26 .5991

The results in Tables 1 and 2 cannot meaningfully be compared. The two

tables use different data, different correlation coefficients, and one uses aggregated data whereas the other does not. For this reason we reexamined the data used in compiling Table 1, and for each audit class

aggregated the data so that approximately 4% of the population was contained in each of 25 intervals. A second aggregation was also

computed which divided the class into 100 intervals each containing

approximately 1% of the population. For each subdivision the average

tax change and median DIF score was calculated and a Spearman Rank

correlation coefficient obtained. The results, presented in Table 3, are discussed in the next paragraphs.

The correlations using aggregated data are seen to be higher than when

unaggregated data are used. This is to be expected with either the

Pearson or Spearman correlation coefficients. The aggregation process

smoothes the data, thereby diminishing the effect of outlying points and

increasing the correlation.

The Long Range Planning process determines the percentage level (to one decimal place) of audit coverage for each class, based on expected

direct yield from audit. The relationship between expected yield and

audit coverage is defined in terms of a discriminant function whose

9
values are used to separate the levels of coverage. If the process of

ranking returns is to merit the adjective "efficient", then although

some deviations from the ranking by tax change would he tolerable for

the unaggregated population, certainly the ranking for the population

aggregated into h% chunks (thus reducing many thousands of members

to just 25 ) should be perfect within the feasible audit coverage levels

(i.e. the average DIF score for the highest h% DIF scores should be

greater than the average DIF score for the next b%)

.

Plots of the data

exhibited in Appendix C highlight the ranking deficiencies which can be

deduced from Table 3 below.

TABLE 3

SPEARMAN RANK CORRELATION COEFFICIENTS (ADJUSTED FOR TIES)

MEASURING RELATIONSHIP OF AVERAGE TAX CHANGE TO AVERAGE DIF SCORE

TCMP SAMPLE OF RETURNS FILED IN 1970

Aggregated into

25 intervals

w in each interval)

Aggregated into

100 intervals

( 11 in each interval)

INDIVIDUAL

Under $10,000 Standard •95712 .83684

Itemized .84911 .84253

$10 ,000-Under $15,000 .83385 .69865

$15 ,000-Under $50,000 .93769 .75154

$ 50,000 and over •95077 .80961

NONFARM BUSINESS

Under $10,000 .84231 .43885

$10,000-Under $30,000 .96385 .77699

$30,000 and Over .95846 .80000

It is quite possible for two returns to be identical "on face" (so that

any automated classification scheme would fail to separate them), and

yet disclose widely differing tax changes after audit. Often a return

which looks suspect if only its line items are 'examined, has supporting

documentation attached which establishes the admissibility of the

questionable items. Such distinctions are made after the DIF process,

by classifiers who screen manually all returns having high DIF scores and choose among those returns the ones they believe have the greatest

potential for tax change. It will be convenient for us to refer to the

phenomenon of returns which cannot be separarated by any automated

classification scheme as "noise".

We must conclude from the correlation coefficients presented in Tables

1, 2 and 3 that the present DIF process does not produce a good ordering

of those returns with high tax change. In the presence of substantial

10
noise, even the "best" DIF formula would not he able to produce "good"

correlations. Thus it is quite possible that high correlations on unaggregated data may not be obtainable. At this stage of our study we

are not able to determine how much the low correlations can be attributed to large random error, how much to inappropriate stratification

into audit classes and how much to poor DIF formulae.

Insight as to how far from perfect the present methodology is can be

obtained by comparing DIF results to those from random and from perfect

selection. Table 4 shows the results of two comparisons by IRS of the

average tax change expected under the DIF system (at a specified audit

coverage level) with the average tax change under random selection

(represented by the total TCMP sample), and also with that obtainable

from auditing those returns with the highest tax change (representing a perfect selection scheme).

TABLE 4

COMPARISON OF DIF, RANDOM AND PERFECT DISCRIMINATION

Percent

of audit Average tax change per return

Audit coverage Random Perfect

class (note a) DIF selection selection

Nonbusiness

Low 1.3 $ 230 $ 36 $ 745

Medium 1.5 802 93 1,800

High 11. 7 9,460 2,178 15,312

Business

Low 2.0 940 230 3,265

Medium 2.4 2,755 345 5,512

High 12.6 5,781 1,662 10,590

a/—This is the approximate percentage of returns by class that

IRS audits in a fiscal year.

Clearly, although the DIF system is superior to a totally random se- lection process, the comparison between the DIF results and perfect

selection indicates considerable room for improvement. That is, even though it is intrinsically impossible to create perfect DIF selection

formulae, improvements in the present system may be possible. The

weakness of the present system may not be due entirely to random error. Making available to the discriminant- function analysis process a greater

body of information with which to distinguish compliers from noncompliers, creating more homogeneous audit classes and/or improving the

discriminant-function methodology, might permit a scoring system that

ranks tax returns more accurately.

11
Cursory analysis identifies three potential sources of supplementary

information for discriminant analysis:

1. There exists the possibility of requesting more information about

a taxpayer than is currently available from the tax return (e.g.,

age, education, home ownership, etc.) and incorporating some or all

of this information into the DIF formulae. However caution is

necessary when moving in this direction, since "privacy" questions

may arise.

2. Veteran tax-return classifiers could be consulted to determine on what basis they conclude that a tax return should be audited.

Although these judgments may be largely intuitive, it may very well

be possible to formalize some of the criteria used by these classifiers and to incorporate some of this information into the DIF

formulae.*

3. Tables of supplementary information should be available as input

into the Discriminant Function Analysis as a basis for making

comparisons with the tax returns. For example, the expected rate

of return or the ratio of inventory to gross receipts for particular broad categories of business (the first two digits of the

SIC code), as given by Department of Commerce figures, can be

compared with the corresponding figure for an individual business

of that type as reported on its tax form. Thus a 10% rate of

return might be completely within reason for one type of business

and yet completely anomalous for another type.

3.2 Exploration of the Discriminant Function Analysis Methodology

The methodology of the discriminant function analysis underlying the DIF scores was examined in order to determine whether deficiencies in the

method could account for the observed poor performances and whether

improvements in the process could be effected by a change in the audit

class definitions.

The methodology employed by IRS entails the use of two discriminant

functions, one developed in-house and the other developed independently

by an outside contractor. Both functions are applied to TCMP sample

returns in an audit class, and the one yielding the better results

(higher average tax change and lower no change rate, for the same audit

coverage level) is the one chosen which will be applied to all returns for

that class in subsequent years. The methodology of the outside contractor was not fully documented for IRS because of proprietary considerations . * We have been informed that IRS is currently making a study in which

TCMP returns are subjected (before the audit) to the customary manual

screening process by classifiers. This will provide some measure of

the discriminating power of the classifier and could serve to begin

the formalization of some of the classification procedures.

12
The methodology of the in-house development was described to us in

detail and, in theory, seems appropriate. It consists of choosing

approximately 200 income tax variables (data available from income tax

returns) to be used to differentiate those returns with significant tax

change potential from those with little or no potential. The tax

returns in the data base (TCMP tax returns) were grouped into three

intervals:

Subset 1: No tax change or less than $X,

Subset 2: Tax change between $X and $Y,

Subset 3: Tax change greater than $Y.

The range of each variable is divided into intervals (usually between

six and ten). Then the frequency distribution of the variable over the

returns in subset 1 is compared with its frequency distribution over subset 3. Those variables for which there is little difference in

distribution between subsets 1 and 3 are eliminated, as are variables

for which there are insufficient data. Subsets 1 and 3 are chosen in

the analysis in order to differentiate the "no change" population from

that with significant tax changes (the value Y varied among classes, but

X was always set at $25).

After this preliminary screening, approximately 100 variables remain.

Likelihood ratios for each interval (the ratio of relative frequency of

subset 1 to relative frequency of subset 3) for each variable are calculated and used as preliminary weights in the discriminant function.

The original likehood functions (preliminary weights) are then modified

using the covariance matrix to obtain the final weight or contribution

of that variable.

A linear discriminant function was developed. The methodology is

commonly justified in terms of assumptions of normality and identical

covariance matrices. Since the latter assumption was not met, the data

of Subset 1 and 3 were pooled, (a standard approach), and a single

covariance matrix calculated. It was hoped that this matrix obtained by

pooling the data was a good approximation to the underlying population

covariance matrix.

All 100 of the screen-surviving variables are considered in the discriminant function analysis. A weight for each variable is assigned in

the usual way by this process; the three variables found to contribute

the least to separating the two populations are then removed and another

discriminant function analysis performed. The three variables found to

contribute the least are again dropped and the process continues until a discriminant function formula with 25 variables has been calculated.

Everything in the analysis up to this point is totally automated, and

only the final 25 variables (and their correlation matrix) appear in the

computer print-out.

IRS staff examine this correlation matrix, deciding which (if either) of

pairs of variables having high correlation (greater than 0.7) will be

13
removed. At this point, they try a number of alternative modifications

by forcing variables either into or out of the discriminant function.

The formula having the "best" results in terms of no-change rate and

average tax change for the same coverage level, is the one chosen.

The contractor-developed DIF development involved an iterative procedure

in which the contribution of each return characteristic is measured in

terms of the incremental information it provides in discriminating

between the two subpopulations. This system, unlike the in-house

system, uses all the tax change information including the "gray area" of

Subset 2. It considers the little-or-no-change area to include those

returns with tax change less than or equal to Y, and the profitable-toaudit population to be those returns with tax changes greater than Y.

The methodology of this discriminant approach allows unequal variances

in the two subpopulations. Therefore two covariance matrices (rather

than the single pooled matrix used in the in-house analysis) are provided.

The results of applying each of these two systems to the data base are compared, and the one yielding "better results (higher average tax

change and lower no change rate, for the same audit coverage level) is

the one chosen. Having two methodologies available allows IRS the

flexibility of choosing an approach which best reflects tax-payer

behavior in a particular audit class. This ability to switch approaches

also makes it more difficult for someone outside IRS to be able to

"break the code" and accurately estimate what a taxpayer's DIF scores might be.

In principle, this automated process assures that each of the over 70

million individual tax returns and each of the over 1.3 million cor- porate returns with assets under one million dollars is screened. Of

those returns, only those believed to have the greatest change potential

are then selected for audit. The objective of equity is thus advanced

because the DIF methodology again, in principle, assures that scarce enforcement resources are directed toward those taxpayers with the

greatest expectation of noncompliance, while assuring that those taxpayers expected to be in substantial compliance will not be asked to

incur the additional burdens of being audited. The process should also

promote the confidence of taxpayers in the equitable and evenhanded

administration of the tax laws.

It is easily seen from Table 4 that the current DIF system yields

results well below those which would be obtained if the selection

process were perfect. While perfect selection cannot be expected and

the current DIF methodology seems quite reasonable, we nevertheless

offer a few alternative procedures which we believe are worth exploring

for potentially producing a better separation of tax returns according

to high probability of error.

In a very lucid paper [4] (whose details we do not repeat here),

Kendall describes a discrimination procedure which makes no assumptions

l4
about the probability distribution of the populations. This "distribution-free" methodology, unlike standard discriminant analysis, does

not separate the discriminated populations by a simple curve (e.g. line,

plane) but by an irregularly shaped boundary. It requires less computation than standard discriminant analysis but it is logically more complex and does not furnish a single numerical "score".

As an alternative to any discriminant techniques which separate rather

than order populations, we suggest that methods based on the notion of

regression be evaluated. These methods have the conceptual advantage

that a function is chosen which is a direct measure of the expected tax

change on a return and is therefore designed to rank returns.

There are two alternative methodologies to explore within this framework. The first approach requires a preprocessing of the data using the

mathematical procedure known as principal component analysis to remove collinearities in the data, followed by a standard regression analysis.

The second approach is based on heuristic pattern searching procedures.

The rationale for this approach is that multi-valued functional relationships will escape standard regression techniques whereas this one has the possibility of disclosing them. (For instance, a response

pattern in the form of a circle will show up as "noise" in classical

linear regression. ) These procedures allow the data analyst to graphically draw in two dimensions meaningful pictures of higher dimensional

data, and can be used for still another kind of separation (as in

discriminant analysis) as well as a substitute method for regression.

An exposition of these procedures can be found in a paper by Dixon and

Jennrich. [2

Since the methodology used to determine which returns have a high

probability of audit is the backbone of any audit strategy, we believe a thorough study of these and other approaches is relevant to IRS' Long

Range Plan.

3.3 Audit Class Definitions

As already discussed, IRS stratifies returns into classes for purposes

of planning and program management. All decisions are then made on a class by class basis. For each class, a discriminant function formula

is determined, audit yields and costs of audit are projected and the

voluntary compliance of each class is estimated. Comparisons are then

made among classes to determine the proportion of audit resources each

class will receive. The method used for this determination will be

discussed in Section 5- Different definitions of audit classes (i.e.,

class boundaries) will normally yield different results. Since the

limited studies performed previously have not confirmed the preferability of the present audit class structure, we urge continued investigation in this area.

15
IRS is presently initiating a new study to evaluate the consequences of

using alternative definitions of audit classes. We believe that this

study should be carried out on a scale allowing a broad spectrum of

alternative definitions to be tested. Going beyond an examination of the

consequences of altering the boundaries of the existing classes, or of

putting the classes on a constant dollar basis, it should consider a variety of possibilities with the objective of obtaining definitive

answers on the benefits and costs of each. The possibilities to be

addressed include defining classes on the basis of:

- geographical location

- specific line items or groups of line items on the return

- industry groups (e.g., production, service, financial, wholesale, retail)

- gross receipts (for business classes)

- assignment to the nonbusiness class of individuals with income

primarily from nonbusiness sources

- occupational groups (professional, businessman, white collar,

blue collar, etc.)

- personal traits (single or married, number of dependents, etc.)

- self employed vs. wage earners.

Consideration should be given to developing a separate DIF score for

each of the most important schedules on the tax return. Actually, a second classification will probably also be necessary. That is, a number of DIF formulae for schedule C might need to be created (e.g..

Schedule C with gross receipts under $10,000; Schedule C with gross

receipts $10,000-$100,000; and Schedule C with gross receipts over $100,000). By generating DIF scores for such specific classifications,

one is assured that the audit classes will be more homogeneous.

Once these scores are generated, two approaches can be taken. The first

of these would generate a function which integrates into a single score the results from the DIF scores on each schedule of the return, and

thereby determines the returns most promising for audit.

The second approach would treat the schedules of the return as separate

entities in the planning process. Audit resources would then be directed towards identifying those partnership returns, business schedules,

10U0 tax forms, etc. with the greatest probability for error. Under

this approach the number of entities in the audit process would be

substantially increased, so that the approach would be useful only if

the corresponding yield from audit were significantly greater. This apl6
proach would also alter manpower allocations, since it considers portions of a return as separate entities.

The idea of generating DIF formulae for each major schedule arose from

an examination of the compliance behavior of present audit classes. The

low business class (individual returns with under $10,000 AGI and filing

schedule C) exhibits an extremely low voluntary compliance level (63.5%).

We suspected that examining the distribution of errors, by line item,

might highlight approaches to improving voluntary compliance. Analysis

showed that a large proportion of returns in this class require fillingout a number of complicated schedules. Twenty percent of the returns in

this class showed income from rents, with a corresponding expected

change in taxable income of $133 million (based on TCMP Sample data

projected for the entire class as if all returns were audited); 12%

showed income from miscellaneous sources (with expected changes in

taxable income totalling $68 million) , k% from farms (with $4l million

in changes in taxable income) and k% from partnerships (with a cor- responding taxable income change of $109 million). These data indicate

that a formula for "potential for audit" for this class should refer at

least to Schedule E, the partnership form, and the farm schedules, in

addition to the Schedule C and the 10^0 form itself, to be able to

identify those returns with a high probability of error. It seems unlikely that a DIF formula with relatively few variables (at most 25)

could capture sufficient information on a population as nonhomogeneous

as the low business class. Moreover, one would expect that individuals

with business or farm income whose adjusted gross income is greater than

$10,000 would be filing returns at least as complicated as those discussed above. Thus, generating DIF formulae for each major business

schedule may highlight erroneous tax returns overlooked otherwise.

These suggestions should not be construed as recommendations for an immediate redefinition of audit classes, but rather as identifying

promising ideas to be analyzed and evaluated in a research project

adequate to determine the costs and benefits of various new approaches

to stratifying taxpayers into classes. The study will have to examine a significant amount of data over a period of years to be sure that any

change in strategy is justified, and which is best. Previous studies

(cf. page 5) have been too small-scale to provide sufficient evidence

for choosing one approach over another.

Generally speaking, stratification of tax returns into audit classes has

been accomplished to date by a methodology which is almost totally

intuitive. The suggestions above have been formulated in the context of

extending and refining these intuitive notions. But a basically different approach is also possible:

Although the intuitive stratification concepts appear fundamentally

sound, palpable difficulties have surfaced in attempts to specify and

measure the most appropriate boundaries for the classes. The source of

these difficulties is the lack of a definition of within-class homogeneity which can be related to the prediction of tax change. These

IT
difficulties suggest that some investigation of "formal" methods for

stratification might be profitable.

A large number of techniques for stratification, with supporting theory,

have appeared in the mathematical-statistical literature under the

heading "cluster analysis". This collection of procedures has been

developed largely in response to problems of classification in the life

sciences and behavioral sciences, for example in medical diagnosis

(classifying heart patients, for instance, on the basis of analysis of a body of past histories and current cardiograms). At least one area of

extensive application, namely market research (selection of target

markets for particular products and projection of sales within markets)

appears to be closely analogous to stratification into audit classes

followed by projection of tax changes; in fact, some of the same var- iables could conceivably be employed in both of these two-stage processes .

We note that the clustering or grouping problem of defining audit

classes is complementary to the discrimination problem within a class.

The mathematical descriptions of the two problems are similar and, in

fact, "discriminant analysis" is frequently subsumed under "cluster

analysis" in modern treatments of the theory of multivariate analysis.

(For a more detailed description of cluster analysis, see [l] and [ 3 ].

The book by Anderberg is lucid and contains many examples; its introduction in particular is an excellent exposition. The monograph by

Duran and Odell is short but fairly technical, having been written for a mathematical reader. We list it mainly because of its bibliography of

U07 titles.''

The last two sections have discussed two major aspects of the Audit

Division's mission, namely, measuring existing compliance and determining procedures to identify noncomplying taxpayers. In addition, the

Audit Division must also conduct a sufficient number of audits to assure high compliance. The next section will briefly discuss the topic of

compliance and the factors which affect it.

18
4. Compliance

It is the mission of the Service as a whole, and of the Audit Division

specifically, to manage their programs in a manner which fosters a high

level of voluntary compliance by the taxpayers. Until recently, the

Audit Division had postulated the use of the "Gompertz curve" (an Sshaped curve widely used in actuarial analysis) to represent the re- lationship between audit coverage and voluntary compliance.

This function embodies the assumption of a close relationship between

prior-year audit coverage (the percent of the population audited) and

the compliance levels of the following year. It depicts decreasing

marginal growth in compliance: as audit coverage increases, voluntary

compliance increases at a diminishing rate, and will, at some level of

coverage, level off at a maximum attainable level. The maximum attainable level will probably be below 100% since certain taxpayers will

be reluctant, or possibly unable, to change their behavior even if a high probability of audit exists.

IRS has discontinued using this model for a number of reasons:

1. The model entails the assumption that audit coverage is the

factor having the greatest effect on compliance. Although a limited number of studies have illustrated that audit does have a positive effect on compliance, many other factors also alter

taxpayer behavior, and the Gompertz curve does not reflect these

factors

2. The model needs, as input, a numerical value for the maximum

attainable level of compliance. For large corporations (whose

audit coverage is 100%) voluntary compliance levels are not greater

than 94%, so that there is empirical evidence that compliance will

converge to less than 100%. But for most audit classes, no data

exist from which to predict this asymptotic level.*

3. The number of data points' at IRS' disposal for estimating the

three parameters of this nonlinear curve is at most four, which is

insufficient given the uncertainties in the data. This is because

TCMP data which yield the necessary empirical information on voluntary compliance are taken only every few years, and the

program has not been in existence long enough to produce more data.

Having discarded use of the Gompertz curve which had been used to

predict audit's effect on voluntary compliance, IRS has no formulated

alternative model to employ in. its planning process. No theory exists

to provide the mathematical form of a more suitable functional relationship.

*For some audit classes, a small fraction of the entire TCMP sample

of 1971 individual taxpayers had also been audited in the prior TCMP

cycle. Although compliance did improve for those taxpayers "hit twice",

this sample is too small to use in determining the asymptotic level.

19
IRS has been aware of the need for better understanding of the factors

which affect compliance, and a number of studies have addressed this

question. In a 1966 study, researchers from the National Opinion

Research Center [5] queried individuals who filed personal income tax

returns on their perceptions of the audit process. They asked people to

estimate the possibility that they would be audited, and what changes in

the tax returns they would expect to increase or decrease their

chances of being audited. They also solicited taxpayers' opinions on the fairness of the audit process and what types of punishments they

believed to be appropriate to taxpayers who violated the law, either

knowingly or unknowingly. This type of survey gives IRS information

about how taxpayers perceive the audit process. In modelling taxpayer

behavior, an understanding of these perceptions is necessary.

Another study performed in 1966 and described in the Chicago Law Review

[ 6 ] examined the effects of legal sanctions on taxpayer behavior. It

questioned the degree to which legal sanctions and appeals to taxpayer

conscience affect taxpayer behavior. The authors concluded that

although legal sanctions increase willingness to comply for certain

classes of individuals, they have negative effects on others. Alternatively, appeals to conscience may have greater effects on certain

groups of individuals than do threats. These preliminary results

suggest that IRS should study further what positive approaches can be

taken to affect taxpayer behavior.

In a third study, performed in-house and entitled "A Cross Section

Regression Model of Audit and Non-Audit Factors Effecting Taxpayer

Complian e (RARC 7.2)", the authors disaggregated the IRS master file

into fiv>* -digit ZIP Code areas and attempted to regress IRS policy

variables (audit coverage, collection activity, return complexity) and

external variables (unemployment rate, average AGI, educational level,

population density, etc.) against voluntary compliance. Unfortunately,

the study was terminated when it was found that the data available at

that time did not permit accurate measurement of voluntary compliance at

the three and five-digit ZIP Code level (three-digit codes used in rural

areas, five-digit codes in urban areas). We commend the concept underlying this attempt and believe that, if in the future it is possible

either to aggregate more than one TCMP Sample in order to obtain estimates of compliance at regional levels or to design future TCMP Samples

in a manner allowing estimates of compliance by region, then a study

similar to the one described above should again be undertaken. As the

report states,

"the outcomes of such a study could develop information of the

following types:

.01 a. For the first time, a valid voluntary compliance audit

coverage relationship may be measured,

b. In a similar vein it may be possible to determine the

impact of Intelligence activity on voluntary compliance.

20
c. The effects of return complexity and education on voluntary compliance may he isolated.

This type of information not only increases our general

understanding of the tax administration problem but it is also

necessary to help secure the resources needed to design and

operate an optimal tax administration system.

.02 The project also has potential for real, short-term pay-offs

by isolating geographic pockets of noncompliance. . . .03 Knowledge of this sort helps pinpoint not only tax administration problems, but also basic social, economic, and political

problems. This project, with its broad criminal-sociologicaleconomic approach, should throw some light on whether the

traditional enforcement approach or a more 'enlightened 1 taxpayer service and education approach would be more effective in solving such basic problems as how to raise voluntary

compliance in ghetto areas."

The studies discussed above are designed to determine the factors, both

internal to IRS and external socio-economic, which effect compliance

behavior. We believe that more studies of this nature should be undertaken, and a list of other possible approaches follows:

1. Surveying taxpayers to monitor their perceptions of

IRS activities, their ability to compute their taxes accurately, and their actions taken in attempts to comply with the

law (e.g., withholding more than is prescribed).

2. Controlled studies to evaluate the effects on voluntary

compliance of media campaigns, of the distribution of

educational materials, of IRS' telephone-answering tax

advice service, and of other similar IRS activities.

3. A study which attempts to model taxpayer behavior by

using a game-theoretic approach. A detailed description

of some of the data necessary to perform such modeling,

and of the answers which this approach might supply, is

given in Appendix A.

The above recommendations do not necessarily pertain to the Audit

Division's responsibility directly, but rather to the entire IRS

mission. We include these recommendations here because actions based on the output of such research efforts will have direct and essential

effects upon the Audit Strategy.

Research efforts which relate more directly to the Audit function will

now be discussed. In a study recently completed by IRS, the effect of

auditing a taxpayer's return on his future compliance was examined.

Researchers analyzed 2,l69 returns examined in both Cycle 3 (1969

returns filed in 1970 ) and Cycle 4 (1971 returns filed in 1972) TCMP

samples

21
Three classes were examined: low nonhusiness (less than $10,000 AGI

with itemized deductions), medium nonhusiness ($10,000-$50, 000 AGI with

itemized deductions) and low business (less than $10,000 AGI). In all

three classes examined, compliance increased after audit. A similar

study, which examined effects on taxpayers who had previously been

subjected to a non TCMP sample audit, also found increased compliance in

the returns of previously audited taxpayers.

No studies have been made to test the "ripple" effect of audit, that is,

the effect that an audit will have on persons who have some contact with

the audited taxpayer. Even though studies to determine "ripple" effects

are difficult to design, and unlikely to produce definitive results,

such research efforts can produce useful insights and should be pursued.

When a district or region initiates audits of a certain class of individuals, (e.g., those in a specific profession or business), the

"ripple" effect on others in that profession or industry can be evaluated by subsequent auditing or by examining changes in the reporting

characteristics of tax returns for that class. Another topic calling

for investigation is the rate of decay of the increased compliance in

the second and subsequent years following an audit.

The IRS Audit Division has often asserted, among other justifications

for its manpower requirements, that audits do have a positive "ripple"

effect and that therefore an increase of any significant size in

coverage will have a manifold impact on voluntary compliance. The

assumption certainly appears reasonable, but studies to test this

hypothesis should be undertaken.

Examination of the previous studies initiated by IRS has shown that very

little ii .as yet known about the factors which affect compliance and how

these factors interact. Our recommendations are aimed at improving this

situation. Since the mission of the IRS is "to encourage and achieve

the highest possible degree of voluntary compliance with the tax laws

and regulations," substantial research addressing how both audit and

nonaudit factors influence taxpayer behavior is justified.

We have, in the past few sections, discussed the major components of the

audit process: the Taxpayer Compliance Measurement Program, the Discriminant Function Analysis methodology, the stratification of taxpayers

into Audit Classes, and Compliance. The next section will briefly

summarize the 1976 and 1977 Audit Long-Range Plans, explain how each of

these components fits into those plans, and give an overall evaluation.

22
5. The Long Range Plan (LRP)

The first part of this section will address the income and estate tax

program areas, where the bulk of the analysis and planning effort is

concentrated and for which the largest body of pertinent data exists.

Some general comments will then be made about the other tax areas

fiduciary, gift, excise and employment tax returns.

Since 1970, IRS has been using a so-called "balanced strategy" whose

objective is "maximizing total net direct yield subject to the con- straint of maintaining certain minimum levels of voluntary compliance in

the known low compliance classes." Under the strategy, sufficient

resources are placed in low complying classes to assure that compliance

within five years will be raised to a specified level ( 85 $, in the 1976

LRP). The remaining resources are then allocated among all classes to

maximize net yield, constrained again to assure that if sufficient

resources are available, they are placed in each class so that voluntary

compliance in the complying classes does not fall below a specified

level {90%, in the 1976 LRP).

It is a well-known theorem of mathematical economics that allocating

resources so as to equalize the marginal-yield to marginal-cost ratios

of all classes under consideration will achieve the same allocation as that of maximizing direct net yield. Although, given current cost and

yield data, pursuing this policy will result in the allocation of some resources to every audit class, some low complying classes will receive

very small allocations of resources. The strategy discussed above is

therefore modified to one of constrained maximization which forces

additional resources into the low complying classes. Under both strategies, taxpayers in the higher income (asset) classes receive high

coverage because they have the greatest possibility for significant tax

errors. The constrained approach promotes equity additionally in the sense that taxpayers in low complying classes are not overlooked simply

because the cost of audit is high relative to the likely yield. Table 5 (pg. 2b) presents the 1982 proposed coverage levels based on the

balanced strategy used in the 1976 Long Range Plan.

This balanced-strategy approach requires a methodology for estimating

the examination coverage needed to reach or maintain a specified level

of voluntary compliance. The so-called Gompertz curve is the mathematical expression which had been previously used to model the com- pliance-coverage relationship. Section H discusses the reasons for IRS'

abandonment of this procedure. Unfortunately, at present no better

model exists with which to represent this relationship.

Because of this lack of information, the new plan being initiated this

year takes the approach of improving compliance in low complying classes

and maximizing net yield (by equalizing the ratio of marginal yield to

marginal cost in each audit class) in all classes. This strategy re- quires that sufficient funds be allocated to low complying classes so that the gap between that which was voluntarily reported and that which

23
TABLE 5

PROGRAM. AND CLASS COVERAGES IN FY 1982 AND THE FY 1977 TENTATIVE EXAMINATION PLAN

FY 1977 ,, FY 1982 , (Tentative Plan)-7 (1976 LRP )—

INCOME INDIVIDUALS (Audit Class)

NONBUSINESS

UNDER $10,000

Standard 0 . 6% 1 . 2% Itemized 4.2 1,5

4.0

$10,000 - $15,000 2.0 2.8

$15,000 - $50,000 2.2 4.1 $50,000 & Over 11.0 32.3

BUSINESS

Under $10,000 3.6 5.1 $10,000 - $30,000 2.5 1.0

$30,000 & Over 10.2 15-8

TOTAL 2.1 4.1

CORPORATIONS (Asset Class) NO BALANCE SHEET 3-5 6.8 UNDER $50,000 3.4 5-7 $50,000 - $100,000 6.4 12.6

$100,000 - $250,000 7-5 8.9

$250,000 - $500,000 14.3 8.3

$500,000 - $1,000,000 19.2 19.6

$1,000,000 - $5,000,000 36.9 30.2

$5,000,000 - $10,000,000 49.2 24.1

$10,000,000 - $50,000,000 45-3 31.0

$50,000,000 - $100,000,000 73.8 100.0

$100,000,000 & Over 100.0 100.0

TOTAL 9.4 10.6

FIDUCIARY 0.9 1.0

ESTATE (Audit Class) UNDER $300,000 14.

0

8.1

$300,000 & Over 52.8 23.0

TOTAL 19-2 10.1 GIFT 3.8 3.9

TOTAL INCOME, ESTATE AND GIFT 2.4 - 4.2

EXCISE 11.1 11.1

EMPLOYMENT 0.6 1.6

TOTAL PROGRAM 2.1 3.7

PARTNERSHIPS (memo) 1.7 2.0

1/ Current services (not optimized)

2/ Plan optimized for equal marginal yield-to-cost ratios, except for classes

with low VCL. Coverages include Service Centers DIFF CORR. examinations.

These coverages are predicated on the realization of anticipated

productivity improvements from the implementation of TAS.

3/ Includes DIF Corr. examinations at Service Centers not shown in individual class coverages.

24
should have been reported will be reduced by a specified amount. (Overpayments are counted as negative addends in the "gap".) Once these

resources are allocated, the remainder of the manpower available is

distributed among classes to maximize net yield. The methodology for

this maximization will be discussed in the next section. Table 6 shows

the "target" voluntary compliance levels of each class.

The tax-gap for each audit class is estimated based on the total disclosable increases and decreases on all returns filed (projected from

the TCMP Sample findings as if all returns filed were audited). To

obtain the dollar amount of the tax gap for some future year, the tax

gap of the TCMP year is adjusted to reflect the projected increase in

the reporting population. Thus to project how much resources will be

allocated to low complying classes in 1983, the following computation is

made

A o_ m _ # o f Returns filed in 1982

1983 Tax Gap - # of Returns filed in 19lb

X TG

where TG = Tax Gap calculated for returns filed in 197^ (based on TCMP Sample of 1973 Returns)

Projections of 1981 returns filed in 1982 are used in the 1983 Tax Gap

calculation because audits usually take place in the year following that

in which returns are filed. For classes in which the voluntary com- pliance level is less than 90%, additional resources are allocated based

on the following rules:

If Voluntary Amount of

Compliance is Between Gap to Recover Resulting Compliance

85- 90% 5% 86-91%

80- 85% 0vs. 11— 82- 86%

75- 80% 15% 79-82%

70- 75% 20% 7^-76%

These allocations are constrained by the requirement that ratios of

marginal yield to marginal cost do not fall below 1-1.

It is not possible to assess the impact that the implementation of this

plan will have on coverage levels, no-change rates, manpower allocations, or compliance, because at the present time the necessary data are not available. We can, therefore, evaluate this approach only on a conceptual basis.

25
TABLE 6

DATA USED FOR DEVELOPING COMPLIANCE COVERAGE CURVES FOR

197** MD SUBSEQUENT YEAR LONG RANGE PLANS INDIVIDUAL RETURNS * PHASE 3 TCMP POINTS

AUDIT

CYCLE 1 (1963) 2 (1965) 3 (1969) 1* (1971)

Voluntary

CLASS Audit Compliance

Coverage Level

Audit

Coverage

Voluntary

Compliance

Level i Audit

Coverage

Voluntary

Compliance

Level

Audit

Coverage

Voluntary

Compliance

Level

AGI (000)

NBU-10 standard 3.1* 95-6 2.3% 95.W 1.3% 95-2?

itemized 6.2 92.5 5.1* 91-9 2.2 88.5 2.5 86.0

10-50 8.6 96.1 7-1* 96.6 3.3 96.1 1.6 95.9

50 & over 28.5 95.6 27.2 95-8 15.1* 91*. 1 - - BU-10 2.7 78.0 3.2 78.0 2.6 68.7 2.2 63.5

10-30 6.8 89.1* 6.6 90.7 1+.0 87.8 - -

30 & over 17.8 91-5 19.1 93.3 12.9 91.2 - -

CORPORATION RETURNS

PHASE 1*. CYCLE 1 ~TCMP

(Returns filed in 1969

)

Asset Class COV. VCL

Under $50,000 2.9% 59.9%

$ 50 , 000-$ 100,000 5.6 73.8

$100 , 000-$2 50 , 000 9-7 81*.

3

$250 , 000-$ 500 , 000 16.8 83.1*

$ 500 , 000-$l, 000,000 23.9 87.9

ESTATE TAX RETURNS FILED IN 1971

Gross Estate COV. VCL

Under $300,000 lh.1% 81.5?

$ 300,000 & over 56.0? 86.9%

* The coverage compliance relationship for a particular year's compliance

to prior year examination coverage.

26
On face value, this program of action seems to be conservative, and

workable as an interim measure. It places resources in low complying

classes on the basis of how much direct compliance can be obtained, and

makes no assumptions about what effects the application of this scheme

may have indirectly on future compliance. The availability of better

data and behavioral theories in the future are, however, required to

provide a solid basis for reliance on this approach, improvements in the

settings of its parameters, or (perhaps most likely) the directions for

a better long-term approach.

The only Audit Planning goal which does not seem to be directly addressed by this plan is that of promoting a high degree of voluntary

compliance. What is accomplished is the assurance that after-audit

compliance is at a satisfactory level. Since little information

presently exists which would allow the formulation of any program

promising the promotion of high voluntary compliance, this strategy

seems reasonable. However, it should be considered an interim improvisation to be replaced as soon as sufficient information exists by a strategy that does directly address the voluntary compliance issue.

The discussion in this section has, so far, addressed the overall

strategy of the Audit Plan. We now discuss the procedures to be used in

implementing this plan.

5.1 Implementation of Plan

The first step of the planning process consists of determining the

maximum attainable level of proficient manpower. Account must be taken

of the experienced manpower presently available, and of estimated attrition. Then plans must be made to recruit personnel with the proper

education or experience and to train them in audit techniques. The

number of new auditors is limited by (l) availability of qualified

candidates, (2) budget, and (3) capacity of IRS training facilities.

With the maximum level of trained available manpower determined, audit

personnel are first assigned to the low complying classes. The definition of the Amount of Gap to Recover insures that not all the availablemanpower is allocated to the low complying classes. The manpower

remaining is then allocated among all classes on the basis of maximization of net-yield. To determine the optimal mix among classes,

coverage-to-yield curves for each audit class must be constructed.

Returns are ranked by their DIF scores. Thus, given an efficient DIF,

the higher the score on a return, the greater the likelihood that the

return will offer a potential tax change. It is therefore anticipated

that as the number of returns examined within a class increases, the

average yield per return decreases. The TCMP sample returns are used to

determine this functional relationship as described next.

IRS divides DIF scores into 25-unit intervals in descending order of DIF

scores. Then the audit coverages relating to these intervals and the

resulting average yields for the intervals are computed. These data are

27
fed into a least-squares computer program to determine the test parammeter fit of coverage-yield curves of pre-assigned form. F^r individual

returns the curve has the form Y = a + b(log X) + c (log X) + d(log X) , whereas for corporate returns the curve has the form log Y = a + b(log X) + c(log X) ; here Y = average yield in dollars and X = coverage {%)

.

These curves are then adjusted to include an assumption of 3.3% productivity growth per year (direct revenue/return) .

In 1976, IRS asked the National Bureau of Standards' Statistical Engineering Laboratory to explore the possibility that other functional

forms might better fit the data. Two classes, the medium business class

and the low nonbusiness class (standard deduction) were examined.

The report, which examined two other models, stated that "An analysis of

the residuals showed both models to be statistically unsatisfactory for

both sets of data." The report states further that "obvious discontinuities in the data make fitting any model difficult." Figure 1

shows a plot of the data for the medium business class and one of the curves which were fitted to those data. These data lead to an un- acceptable conclusion, namely, that if k% coverage were applied, the

yield per return would be significantly less than if 5% coverage were

applied. This discontinuity occurs within the most critical portion of

the curve, i.e., in the feasible region of coverage. Additional plots

in Appendix C show similar discontinuities for the high nonbusiness,

medium farm business, and high nonfarm business classes. These data

further confirm that the current discriminant function analysis procedure is not ordering the returns efficiently.

IRS has decided to continue with its past formulation of yield/cost

curves, because within the region of greatest applicability these curves fit the data at least as well as the new curves suggested by NBS.

We suggest that if, at the conclusion of the audit-class definition

study, IRS implements new Audit Class definitions or new DIF formulae,

then IRS should reevaluate the functional forms used to model the

coverage-yield relationship. It should be noted that TCMP data were not

required nor developed for corporations with assets from $1 million to

$50 million because of the high level of coverage of this class;

coverage-yield curves for these classes are calculated from data obtained through examinations during previous fiscal years.

Once estimates of yield (as a function of coverage) are calculated, the

next step in the process requires the examination of audit costs. For

each audit class, a cost per return examined is calculated. This annual

estimate includes adjustments for fringe benefits, mix of technical

personnel, and types of examinations (field, office, correspondence) as well as a 25% add-on in cost per return to reflect the costs of Appellate, Technical, Intelligence, Collection, data processing and other

support functions.

*The 3.3% production level assumed in 1975 was reduced to a 2.0%

production level in 1976, and a 0% production level in the 1977 LRP.

28
PEDIUM

BUSINESS

CLASS

>-•—' LU —IQ Q_LUQ Cd |_U |— dD Cd

29

COVERAGE

LEVEL

FIGURE 1
As implied above, the cost estimation procedure employs an assumption

that cost per return is not a function of the level of coverage within

an audit class. IRS is initiating a study to test the validity of this

assumption, and if the assumption is found to he invalid, will develop

the appropriate functional relationships.

Besides the above-mentioned study, we believe a study should be made of

the effects of including in the cost calculations a cost to taxpayers

who are audited and found to have little or no tax change. This would

permit incorporation of the idea of "harrassment" into the audit strategy, thereby enhancing the "equity” goal of IRS.

Once yield/cost curves have been calculated, IRS uses an iterative

process to determine the optimal allocation of resources. It begins

with tentative coverage levels for the available manpower and tests the

related marginal yield-to-marginal cost ratios. Shifts in manpower are made in small amounts between classes until equality in these ratios has

been achieved. Computer programs have been developed to assist in these

calculations. Although this method is basically heuristic in its

approach, it seems to be performing satisfactorily and we believe that

efforts at improvement are not now indicated in the face of other

priorities

.

The preceding discussion has briefly summarized the methodology which

IRS uses to distribute manpower resources among the individual, corporate and estate tax returns. No data presently exist to allow similar

planning of manpower-resources in the fiduciary, gift, excise and

employment tax areas. Fiduciary returns were sampled for the first time

in the TCMP which began in the summer of 1975 » and these will supply

data necessary to allow this tax area to be evaluated on a comparable

basis in the future. Since total yield/total cost data do exist for

these other tax areas (those of gift, excise and employment returns),

manpower is allocated to them based on past operating results and

evaluation of trends projected from past operations.

30
6 . Summary

It has been our purpose in this report to present an overview of the IRS

Audit Division's current methodology and to assess its Long Range Plan.

To this end, we have discussed in detail the methods IRS currently uses to select returns for audit and some of the differences in the future

selection methods proposed in the 1977 long range plan. We have exam- ined the concept of voluntary compliance and what levels IRS has been

able to achieve. This has led to a consideration of random (TCMP)

audits as well as those mandated by DIF scores. We have suggested

several areas of investigation concerning the relation between these

audits and voluntary compliance.

We can characterize the Audit Division's response to the specific need

to filter likely candidates for audit from the population of tax

returns, and to general problems of planning resource allocation, as innovative, open minded and technically sound. We found, however, that

documentation was neither easily available nor well catalogued, and that

we had to rely too heavily on personal interviews. Departure of key

personnel would make it near- impossible to ascertain the origins of, or rationales for, a number of procedures.

The most frequent complaint in surveys of Federal modelling activities

is that while much description of supporting analysis and theory is made

available through reports and journal articles, user manuals and computer

program listings are hard to find. Ironically , in the IRS a complementary situation exists: the end result of studies is an operational

procedure to be used internally, which therefore must be (and is)

explicitly described so it can be put to use. However, documentation of

the background theory and experiments supporting adoption of these

procedures (rather than others) is very sketchy and vague. In either

situation, the main reasons for the deficiency is clear—inadequate

funding and priority to allow the documentation, and time pressures to

complete the research and to implement the resulting operating procedures .

A transcript of work in progress reduces duplication of effort and time

spent for orientation required by turnovers of the in-house project

staff, or by participation of outside people in the R & D effort. Even

in the absence of such circumstances, the transcript affords a review of

the progress of experimentation for the analysts themselves—sometimes a second look discloses new insights.

It is not at all necessary for such a working document to be "polished"

into suitability for publication, but it should be edited at least once with twin objectives of minimizing ambiguity and achieving conciseness

in the exposition. In addition, such a document should be as selfcontained as possible, so that a future reader need not be forced to

hunt for background rationale.

In this report, notice has been taken of a number of methodological

practices which, so far as we are able to see, are based on sound

31
principles (e.g., the general philosophy of stratifying returns into

audit classes and the allocation of resources based on a balanced

strategy). In addition, we have designated areas in which we believe

that research studies are likely to lead to significant improvements

(e.g., altering the audit class definitions, reformulating DIF formulae,

and understanding of taxpayer behavior). These suggestions are reca- pitulated in tabular format at the end of the text.

The Audit Division’s present Long Range Plan, has evolved from years of

working experience and experimentation. It is, therefore, unlikely that

off-the-cuff suggestions for specific changes will bestow immediate

improvements; with no solid evidence to support such suggestions, we refrain from making any. This evaluation has, however, highlighted areas

in which we believe there are plausible alternative approaches which may

in fact be preferable. Models or theories underlying these alternative

approaches often cannot be tested because of lack of data. If IRS is

committed to the goal of formulating an optimal or near-optimal audit

strategy for the long term, then the generation of such data is essential, and the in-depth research efforts to produce this information

should be initiated.

32
TABLE 7

SUMMARY OF RESEARCH TOPICS

1. Supplementary information input to the discriminant function

analysis

a. Additional variables from tax forms (and possible broadening

of tax form)

b. Formalization of manual classification methods

c. Tables of economic, socio-economic data

2. Modification of the DIF methodology:

a. Distribution Free Methods (e.g. Kendall)

b. Regression-like methods aimed at ranking tax returns.

3. Audit Class Redefinition Study:

a. Alternative class boundary criteria

e.g.. Geographical, line items other than AGI, gross receipts,

industry or occupational groups, primary source of income, etc.

b. Separate (additional) DIF scores for most important tax

schedules in 10^0.

c. Cluster Analysis Approach

h. Compliance related research:

a. Surveys of taxpayers' perceptions

b. Controlled studies to evaluate effects on voluntary compliance

levels of media campaigns, educational materials, etc.

c. Game theoretic analysis - see Appendix A

d. Double hits (effect of repeated audits)

e. Ripple effect (e.g., -within an occupational group)

5. Implementation of the Audit Plan:

a. Explore audit-yield relationship based on trial redefinitions

of audit classes.

b. Incorporate "harrassment" into model (e.g., add a "penalty"

cost in calculating yield/cost). See Appendix B.

c. Examine field implementation of plan.

33
REFERENCES

[1] Anderberg, Michael R. , Cluster Analysis for Applications , 1,

Academic Press, 1971*

[2] Dixon, W.J. and Jennrich, R.I., "Computer Graphical Analysis and

Discrimination," Discriminant Analysis and Applications , Edited by T. Cacoullos, Academic Press, 1972, pp. l6l-172.

[3] Duran, Benjamin and Odell, Patrick, Cluster Analysis, A Survey , Lecture Notes in Economics and Mathematical Sciences #100,

Springer-Verlag, 1970.

[4] Kendall, M.G. "Discrimination and Classification," pp. 168-185 in

Multivariate Analysis , edited by P.R. Krishnaiah, Academic

Press, 1966.

[5] National Opinion Research Council, "Taxpayers* Attitudes Towards

Enforcement and Cheating on Personal Income Tax Returns:

September 1966 ," a report submitted to the IRS* Research

Division as partial fulfillment of contract TIR-25197.

[6] Schwarz, Richard D. and Orleans, Sonya, "On Legal Sanctions,"

The University of Chicago Law Review , V 34, pp. 274-300.

34
APPENDIX A

GAME THEORY AND AUDIT:

A PRELIMINARY EXAMINATION OF A MATHEMATICAL TOOL

FOR ANALYSES OF AUDIT STRATEGY AND TAXPAYER COMPLIANCE

35
Game theory is one of several mathematical models used to study decision

making. (Other such models include classical decision theory, utility

theory and simulation, i.e., gaming.) Game theory is particularly

applicable to situations in which there is a conflict among several

parties. It will not be necessary for us to consider anything other

than the simplest case, namely, a conflict between two parties. As we

shall consider it, a game consists of two players, together with a set

of rules of play and a scoring mechanism called a pay-off function.

Both players are aware of the rules of the game and the pay-off function. Each player makes a decision - that is, he selects one course of

action from among those available to him according to the rules. In the

language of game theory, each course of action which does not violate

the rules of the game is called a strategy. Of course, each player must

choose his strategy without the knowledge of what the other player's

strategy will be. The pay-off function, with the two chosen strategies,

then determines what "payoff" each player receives.

There are many well-known examples of two-person games that fall within

the broad outline given in the last paragraph. Perhaps the simplest of

these is the children's game of matching pennies. Each player shows a

coin with either the head face or the tail face visible. The rules of

the game call for one player to win if both players ' coins show the same

face while the other player wins when the faces on the coins are different. Here each player must decide how often (i.e., with what probability) to play heads and how often to play tails. The pay-off is +1

for a player when he wins and -1 when he loses.

36
Clearly, each player in a game should decide in such a way as to maximize

(in some sense) his pay-off. In fact, game theory assumes each player is

a "rational person" who will do so. (This usage of "rational" comes from

classical economics.) The philosophy adopted in game theory is, however,

also extremely conservative, namely, each player plays so as to maximize

the expectation that he is able to guarantee for himself, irrespective of

the other player's decision. Thus, as each player plays in such a way

as to guarantee a certain amount, he may have to forego the (non-guaranteed)

possibility of larger payoffs. For each player, a strategy which maximizes

the amount that he is able to guarantee for himself, is called an optimal

strategy. A game is said to be solved when each player's optimal strategies (and the corresponding pay-offs) have been found. The Fundamental

Theorem of Game Theory insures that a broad class of games do actually

have solutions. The games described here are simple representatives of

two-person games, and the existence of solutions for them is assured by

the Fundamental Theorem.

In matching pennies, each player can guarantee an expectation of at

least zero for himself by selecting heads and tails with equal probability, but in some random way (e.g., by tossing his coin in the air

and letting it fall as it' may). It can be shown that in this game no

strategy exists for either player which will guarantee an expectation

greater than zero. Thus, for each player, playing heads and tails each

with probability 1/2 is an optimal strategy and the corresponding payoff for each player is zero.

37
In a recent study [l], [2] conducted by the National Bureau of Standards, game theory was applied to "optimizing" the inspection strategy

of state agencies concerned with regulating the accuracy of measuring

devices used in retail trade. Three models were constructed reflecting

different possible goals of the regulating agency. The general type of

scenario examined in [l], [2] appears to be adaptable to the audit

situation within IRS. One player in the game is IRS, whose decision (i.e., choice of strategy) consists of selecting which tax returns to audit, up to the limit of its resources. The other player is

the taxpayers of the country. A. J. Goldman and D. R. Shier [3] have

shown that such a set of taxpayers can be considered as a single player

for the purposes of game-theoretic analysis. The second player's decision consists of selecting those tax returns on which to "cheat".*

In the Audit context, applying the game theoretic notions of "rational

behavior" and "maximization of expectation" does not necessarily signify

the presence of fraudulent behavior. Any entry in a tax return for

which the taxpayer recognizes the possibility of a dispute by IRS

(such as giving oneself the benefit of the doubt in interpreting tax

regulations, conscious carelessness, "not bothering" to check tax regulations or deliberate nonfeasance) may represent a rational decision in the

standard context of game theory.

*It was found convenient in [l] to use the ugly word "cheat" to denote

any malfunction of a measuring device, either by design or by inadvertence.

We maintain here the convention that "cheating" covers any situation in

which the taxpayer makes any entry in his tax return for which he is

conscious of the possibility that it might be challenged by IRS.

38
True error, or inadvertence without any overtones of violation, could

be accommodated within game theoretic structures, but since its inclusion in a game would require some complications of probability analysis,

we will for simplicity ignore its existence in the present discussion.

The rules of the game are the IRS regulations, including the penalties

imposed when errors are discovered through a tax audit. The pay-off to

the tax-payer is the profit that he can expect to receive (including a

loss as a negative profit) by the occurrence of deficiencies in his tax

return.

There are several choices for how the pay-off function to the IRS

should be formulated to reflect that Service’s "mission." The simplest

one (and the first one considered in [ 1 ] ) is for the taxpayer's gain to

be IRS's loss. Thus, following this scenario, IRS's aim is to

minimize the taxpayer's gain due to cheating. A second scenario (and the

second one considered in [ 1 ] ) calls for IRS to minimize the taxpayer's undetected cheating - with the rationale that taxpayers once

forced to pay additional taxes, interest and penalties as a result of a

tax audit are less likely than other taxpayers to cheat in the future.

A third scenario treated in [l] leads to a game in which it is assumed

that the taxpayer has become familiar with IRS's strategy before he

chooses his own strategy - and that IRS knows this. Thus, IRS wishes to

elect that strategy which, the taxpayer being aware of it, yields to the

taxpayer the smallest gain due to cheating.

39
A reader who is frightened of mathematical notation may omit the material

set off by quadruple asterisks **** without disrupting the flow of

ideas, but at the risk of escaping knowledge of how game theory works in

the current context.

We will number the tax returns from 1 to n (at the moment, no particular rank criterion for the tax returns is implied in this ordering),

th

For the i tax return, we define the following terms:

- the dollar Value to the taxpayer of undetected cheating, i.e.

the amount of change in tax liability which would take place

were the tax return to be audited,

- the amount (Penalty, etc.) assessed against the taxpayer when

cheating is discovered on his tax return by audit,

c_^ - the probability of cheating,

p^ - the probability that his return will be audited by IRS and

any cheating discovered.

 

Also, we denote IRS audit resources by m; that is, m is the largest

number of tax returns that can be audited by IRS. If specific amounts

of audit resources are preassigned to the audit classes (or to any other

partition of the tax returns) then we denote by m. the largest number of

J til tax returns of the j class that can be audited by IRS. The sum of

the m.'s must be at most m.

J

Since c_^ and p^ are probabilities, we must have

0 <_ c £ 1 and 0 < p. < 1. — i —

The limitation on IRS's audit resources is expressed by the inequality

40
n£ pi- m -

i = 1

In other words, the expected number (in the technical sense) of returns to

be audited cannot exceed the resource limits. Any set of numbers c ,

c , •** » c n

> sup,j ect to the restriction 0 <_ c_^ <_ 1 constitutes a strategy

for the taxpayer. A strategy for IRS is any set of numbers p^, p^,

p , subject to the two conditions

n n

0 p^ <_ 1 and ^ p^ <_ m.

i = 1

When the taxpayers and IRS have chosen strategies (call them c and

p respectively), then the expected pay-off to the i^

h

taxpayer (and consequent loss to the Treasury) is (\A - Consequently, the expected total loss to the Treasury is

n L(p,c) = £ (V

i - P i

p

i

) c

i

-

i = 1

u For each specific strategy that IRS adopts (call it p ) , there is a

ff maximum expected total loss to the Treasury (which depends on p ) that

ff can occur, namely, the maximum value that L(p , c) takes on as c varies

over all possible strategies for the taxpayers. Application of game

ff theory in this context calls for finding p for which the maximum expected total loss is as small as possible.

We were told by IRS that the assessment against the taxpayer, when

cheating is detected on his return by an audit, is the sum of (l) the

kl
unpaid tax, (2) an interest charge, and (3) a penalty.* Since (2) and

(3) are fixed percentages of (l), it follows that the value of V_./P_^ is

the same for all detected cheating. Let us denote the value of that quotient

by q. Clearly 0 <_ q <_ 1.

In order to apply the model described above to the IRS audit situation,

it is necessary to have numerical values for m, n, VI and P Equivalently,

it is necessary to have values for m, n, V_^ and q. Each player will need

to know some (or all) of these values in order to construct his optimal

strategy. The values of m, n and q are known to IRS; n and q are (probably fairly accurately) perceived by the taxpayers. However, IRS

is not able to determine V very readily since, according to the definition

th

of VI, its value depends on the outcome of an audit of the i tax return. In order to estimate the value of V. for each tax return, a mechal nism is needed which, like the DIF, would determine a gain potential for

a return from its entries. Only after an audit takes place can IRS determine the accuracy of this estimate. Since the definition of cheating

that we have used requires that the taxpayer be aware of the possibility

of IRS disputing the entries on his tax return, the taxpayer has more

information than IRS about before an audit takes place.

Audit coverage typically varies widely over audit classes; in recent

years, aggregate coverage has averaged about 2% t i.e. m = .02 times n.

The results of [l] imply that (in the IRS audit context), whenever q

(i.e., V' /P.

)

is greater than m/n then the optimal strategy for the

taxpayer is to cheat in virtually all cases. In order for the taxpayer's

*In the game model, the penalty which we have called P. consists of the

sum of (l), (2) and (3). ^ b2
optimal strategy to call on him not to cheat, it is necessary that

either m he increased or that q be decreased (hy increasing the penalty,

P^), or hoth, until q is less than m/n.

Here is a sample "play" of a realization of the prototype game, whose

parameter values are derived from IRS data. One player is "the auditor".

The other player is "the taxpayer" with 1000 tax returns (i.e. n = 1000 ).

The values of the 1000 Vi's ("taxpayer’s potential gain" part of the payoff function) were chosen to follow fairly closely the national percentage

distribution of actual "tax change" amounts of one audit class: - "Under

$10,000 AGI, individual, itemized" in the Phase III TCMP survey. (See

histogram. Fig. AL ) The values of P^ were set hy multiplying hy 1.1 the

value of the corresponding V\ (i.e. penalty + interest = 10% of deficiency; q= 10/11) . The value of m was set at 40, or h% of n (the

FY 1981 audit coverage level designated for the specimen audit class

under the 1976 long-range audit plan). The numerical specification of

the game, optimal strategies of the players, and the value of the

games are listed in Table Al.

A solution* for this game calls for the auditor to distribute his reT sources uniformly over the m =m/ q (= 40 x 11/10 or 44) returns with

largest values of AA, that is, to audit each of these returns with relative

frequency q

= 10/ 11 or probability p^ of approximately .909? and not

t to audit the remaining n-m (=956) returns. The other player cheats,

that is, the taxpayer understates his liability by AA, for the 956 returns

which will not be audited under the auditor's solution strategy. With

*We have not determined that it is unique.

43
\\

TAX CHANGE IN DOLLARS

FIGURE A1

Percent Distribution of Returns According to Tax Change Audit Class:

Under $10,000 AGI-Itemized-Ron Business Derived from TCMP Sample 1966

(The first tax change interval, marked between -25 and 0, represents

the zero-change returns. The total on the chart is less than 100.0$

because of unplotted very small percent values for large tax changes.

The 5 largest tax changes actually recorded in the sample were $67,96*+,

$25,897, $25,291, $11,696 and $11,300.)
c

o

rO •rH ft CO

•rH •H

rH EH o /^N

•rH H CD On

ft PI P O

•H ctJ ft On

ft ft < At • CA> o O x—

A ft ft

ft O •H 1—

.-I ft 1— < Al 2 \

> o < o v—

X -** rH s ' o a H CJ o

Eh A •H X ^ X A X A x /

X Jo CL) CO LAN O o O o

ft 01 •H CO LAN t

O ft- O Cd ft o O o o 1—

i—

ca CD <£ CD

Ah prl! P V - X /• x— x—

ft •H o A o

o (D £ CD o o o O O

t> >5 o o o o o

•H ft cc! o o o LAN LAN -p o ft rH t— Lf\ CO CM X \ \ i—

ft o o o o O

CD ft LAN LAN LAN LTN Lf\ ft CO CO CO CO CO

ft

COoo At

CD

-ee1—

cd •H £

ft m ft X

K

ft

•H

ft A cd CJ1 o o o o o

CD ft o o o lr\ IAN ft ft •H o E- LAN CO D

o O > 1—E- LAN CO CM ft ft 1—

o -eec

•H

cd CO A

CD a %

ft %

A cd

0) E-t “Ho o

Ah ft

o o o o o

o o o o o O O O IAIA O t— LAN CO CM

H CXI COJ- LAN

CO O ON CO O

co ir\co co o

CM CO CO ft LAN

o o

o o o o o

LAN O O O O

i—I i—I On 0O t

O O O O O

LAN LTN LAN LAN LAN CO CO 00 CO (O

O O O O O

LAN ON CO tr— VO H ON CO Ei—I i—

o o o o o

o o o o o

LT\ O On co t

i—I rH

VO E-CO O CO

rH r—

I I On rH

i—

ON--' O O

On •

• O

O

rH

CO O 0O LTN-—' CO t— C— o

LTN t— t— CO •

• • • • rH

o o o o O O UNO

VO LANft ft

O O O O H

IA L(N 1A liN\

CO CO CO CO H

O O IAO IA

VO LAN ON CO VO LANft ft CO

o o o o o

O O LAN O LAN VO LAN ft ft CO

t— CO CM ft . H CM CO ft LAN

I I I I ft 0O -4" CO LAN rH i—I CM CO ft"

O O O O O

O IAO IAO O E- LAN CM O

CO CM CM CM CM

O On H LAN CO

C— C— ON O CM

I I I rH rH

LAN H O I I LAN t— CO CM VO

On O

O O O O O

LAX LAN LTN LTN lr\

• • • • CM LAN t— O C\J lt\ c— o C\J O E~ ft CM On vo 00 H CO

CO CM CM CM I

1 rH 1—

1 1

LPs O LTN o LAN t— Lf\ OJ o ErH rH rH i—

CO CM CM ft" LAN ft vo o ft o

H H CM CM CO

I I I I I ft ft CO CO LAN CM J-VO O J H H H CM CM

O O O

LAN t— O

LAN CM

O LAN O

LAN CM

E~ ON

On CO CO VO

I I VO CO O ON

CO CO ftVO

o

o

o

I—

I o

o

o

CO

*N CM

LAN

-cell o

o

ft

CN

1—1 o

1—

l—

1—1 ce- o ll

1—1 CO

CO

II o

1—1 ft A

A

co

cd o CD ft A

Eh

ii

i1 £ cd •H

ftA

CD ft o O

o ft o

rH i

1 cd

II ftO

a EH

^5

TABLE

Al:

NUMERICAL

VALUES

FOR

SAMPLE

GAME
reference to the game parameters, this means that c = 1 for these

returns. For each of the other 44 returns, the taxpayer assigns a

? frequency of cheating equal to the ratio between the 45th (m + 1st)

largest V. (or P^) and the VI (respectively, ) of the return in question, yielding c^'s which range from about .9 down to about .04. The

value of the game, L, "loss to the treasury" under the solution strategies, is the sum of the Vi’s for the 95^- returns which are destined to

be ignored by the auditor, or $52,800. Each of those returns contributes

V. to the sum

i n

(V. - p.P. )c.

. , i ill

i=l

5

being of the form (V - 0-P^)*l. For each of the other m (or 44)

returns the expected (average) contribution is zero because under the

auditor’s optimal strategy the term in parentheses becomes V. - q -p =

i i

The total potential taxpayer gain (the sum of all the V ) is $101,400,

and so the game value represents abatement of 47.9% of the potential deficiency, by the optimal audit strategy.

If the "penalty rate" in this game were doubled, i.e. P^ = 1.2 V

instead of P^ = 1.1 V., then, given the coverage rate of 4%, we would

T I have m = 48 instead of m =44; the number of unrestricted cheats

would drop by 4 and the remaining cheat frequencies reduced slightly;

the value, L, of the game would be reduced by $1400 and the expected

deficiency abatement would rise to slightly above 49%.

46
To restrict the propensity to cheat to those returns with no larger

than $100 would require a value of q

of about 1/5, or a penalty + interest

rate of about 400% of deficiency, producing $19,452 or 80.8% abatement.

A target of "no cheating", but with a "no change" criterion set at "$25

or less" would require about 900% penalty + interest (L = $6050; 9^%

abatement), while reducing the game value to literal zero would require

a penalty rate of 1500%.

In interpreting this example, bear in mind that the values of V were

determined by invoking recorded realized levels of post-audit tax deficiency to stand for potential deficiency levels, and that the rules of

the game imply that both players have full knowledge of the values of n,

m and all the V\ and P^. Notice, thus, that the solution strategy for

the taxpayer assigns the smallest probability of cheating to the returns

with the largest values of Vh, meaning that these will pose the least

temptation for the taxpayer. This possibly startling idea becomes

intuitively reasonable when one realizes that according to the auditor's

solution strategy these are the returns most likely to be audited. The

fact remains, however, that according to the data base all of those high

returns did "cheat" . These caveats aside, the sample game solution suggests

one possibly useful idea that may have been overlooked heretofore: that

the solution strategy for the auditor tells him not to audit all of the

m "highest ranking" returns but to extend his attention to a slightly

wider group and audit them all "almost" all the time. The "taxpayer's"

response in reducing his cheat frequencies implies that under some

actual circumstances IRS might obtain a "bigger bang for the buck" in
terms of inducing compliance by spreading the audit resources slightly

thinner but further at the high end of the DIF. This completes discussion of the example.

Fitting the above analysis framework to the audit situation within

IRS will not be straightforward. For example, the conservative philosophy which underlies game theory may not be an accurate reflection of

the thinking of the American taxpayer and may be too conservative a

strategy for IRS as well. Also, it is clear that more is involved in

the proper evaluation for each taxpayer of (gain from understating

tax liability) and P_^ (loss through penalty) than the amount of money at

stake. If there were no considerations other than monies paid in penalties (P_^) or underpayments of taxes (Vt) and if the players were fully

aware of the rules of the game* (in particular, if the taxpayers were

aware of the amount of audit resources now available to IRS), then

an optimal strategy for the taxpayer today would call for virtually

universal cheating. Why does this not, in fact, take place? Clearly the

game model in the simple version described above does not adequately

mirror the income tax situation as it currently exists. We will suggest

several possible ways of improving the model, that is, making it reflect

more accurately the behavior of the taxpayer. Further study, however,

will then be needed to determine whether the new models (l) do indeed

reflect more accurately the behavior of the taxpayer, and (2) are still

mathematically tractable.

*It can be anticipated that, with the Freedom of Information Act in

effect, taxpayers will soon become more aware of the true rules of the

game.

48
One drawback of the model described above is that a single value of a,

the same for all taxpayers, does not exist. There are many often intangible, aspects of V\ and other than the numbers of dollars involved.

We present here several other possible constituents of the "true" values

of V. and P.

1) Satisfaction from gambling and risk-taking.

2) One-upmanship (i.e., keeping up with one's friends and neighbors).

3) Evidence of one's cleverness and sophistication (i.e., "Beating

the System").

k) "Revenge" on the government for following policies unpopular with

the taxpayer.

5) Benefit of cash available now compared to the future possibility

of loss.

P.

l

l) Financial losses in addition to the assessment P^ (e.g. legal

fees, time lost from work, etc.).

2) Loss of time in preparing for an audit and in being audited.

3) Conscience.

h) A generalized distaste for gambling and risk-taking.

5) Fear of exposure and publicity.

6) Fear of prosecution (even if the case should be decided in

the taxpayer ' s favor )
7) Fear of being convicted of tax evasion (criminal record, prison,

etc . )

.

8) Belief that the government is omnipresent and omnipotent (i.e.,

you can't fight City Hall).

We believe that these (and possibly additional) considerations play a

very important role in the way that the taxpayer perceives and plays the

game. Of course we have introduced a serious problem in requiring

realistic values for both and P.. The distribution of pertinent

attitudes and psychological traits among taxpayers must somehow be

reflected in efforts to assign reasonable dollar equivalents to each of

the items listed above. Clearly these values will differ from one

taxpayer to the next, and a realistic model must allow for this. Research along these lines has already begun [H] and promises to continue

in the future [ 5 ].

The problems and difficulties which we have just described indicate

that the simple game-theoretic model presented earlier is not by itself

an adequate tool with which to study the audit problem as related to

the individual taxpayer. In particular, the assumption made in our game

that the taxpayer's loss is the Treasury's gain (and visa versa) is not

really valid. Game models which allow for differing gains among players

(called non-zero sum games) have been studied extensively and this (somewhat more complicated) model might be more representative of the audit

situation.

50
The non- zero sum model mentioned above as well as the first model previously

described (and treated in [l] and [2]) call for the taxpayer to decide

whether or not to cheat a specified amount, V_^ . A second way in which

the game described above does not accurately model the income tax situation is that no provisions exist in the model for the taxpayer to decide

how much to cheat. A game which mirrors the IRS audit scenario more

closely would permit the taxpayer to cheat any amount he wishes (including

negative amounts), that is, the amount that a taxpayer will cheat is

described by a probability function and, at least in theory, many different amounts of cheating are possible. Of course the taxpayer understands that the larger the amount that he decides to cheat, the greater

may be the probability of audit. Incorporating this additional variable

into the taxpayer's strategy would significantly alter the rules of the

game and may require substantial further analysis in order to obtain a

solution. For example, it is not clear whether the set of taxpayers

could still be considered as a single player for the purpose of gametheoretic analysis. As before, IRS requires a mechanism (such as

DIF) with which to estimate the tax change potential of a tax return,

and an audit process to verify that the estimates given by this mechanism are reliable. Also, IRS must formulate a strategy, based on the

rules of the game and the estimate of the taxpayer's strategy, with

which to maximize its pay-off. Presumably, this strategy would require

that the probability of audit for a particular tax return reflect the

amount of cheating likely to have been selected in the taxpayer's strategy. Thus it is important that the tax-change potential of a return.

51
as estimated by IRS, correlate strongly with the amount of cheating

in the taxpayer's strategy. A more accurate model of the income tax

administration process would result from considering the probability

that an entry on the tax return which the taxpayer knows is subject to

dispute will eventually (either by the auditor or through litigation) be

upheld.

More substantial changes in the game model may be needed in order to

properly describe the income tax situation. It may be necessary to

introduce new formulations of the rules such as bluffing, games with

imperfect information, cooperative games, etc. These sophisticated

models have been treated extensively in the mathematical literature

and can be used to more accurately reflect the current income tax

picture.

It is unlikely that any mechanical formulation can furnish entirely reliable forecasts of taxpayer compliance. We believe, however, that game

theoretic analysis, which focuses on the divergence of objective which

separates the taxpayer from IRS—or if you will, the taxpayer as

economic creature from the taxpayer as citizen—is the formal technique

which offers the greatest independent hope of profitable application,

in addition to furnishing a yardstick for measuring representation of

simply motivated "extreme cases" of taxpayers' behavior, in conjunction

with other models.

52
REFERENCES

A. J. Goldman and Pearl, M. H., "The Dependence of InspectionSystem Performance on Levels of Penalties and Inspection Resources"

Journal of Research of the National Bureau of Standards-B. Mathematical Sciences , 80B, 1976, pp. 189-236.

M. H. Pearl and Goldman, A. J. "Policing the Market Place"

Mathematics Magazine V50, No. b , 1977, PP 179-189.

A. J. Goldman and Shier, D. R., "Player Aggregation in Noncooperative Games", -working paper. Center for Applied Mathematics, National

Bureau of Standards, Washington, D.C. 2023^.

Schwarz, Richard D. and Orleans, Sonya, "On Legal Sanctions" The

University of Chicago Law Review V3U, pp 274-300.

"A Study to Determine the Relative Impact of the Internal and

External Factors Affecting Compliance with Individual Tax Laws".

Request for Proposal IRS 78-35 January 10, 1978.
APPENDIX B

SOLUTIONS OF THE GAME-THEORETIC MODELS

This technical attachment presents (and proves the validity of) particular solutions in symbolic form for two of the three basic gametheoretic models alluded to in Appendix A. The first of these

games was described in some detail in that Appendix and generates the

numerical example of its text; a second one was mentioned briefly as

a variant. The notation of Appendix A is maintained. Model 1, in thi

treatment, is slightly more general than its version in the foregoing

material: the simplifying assumption of a constant ration V^/P^ is

discarded.
Model 1

For this model we use a zero-sum game, with objective function

n F(c,p) = Z ^ V

i"

P

iPi^ c

i i=l

for the taxpayer and its negative as objective function for the auditor,

Define q. , i=l,2,...,n by

q. = V./P.

1 11

Then

n

F ( c ,p ) = Z P

i

( qi-p.)c

i

. i=l

We assume that the returns are numbered so that

P-, > PG

> ... > P . 1 — 2 — — n Then the set T of "tempting" returns, whose size is denoted |t|, is defined by

T = { i I q_. >1} = {i|v.>P. }. 1 l 'll

The complementary set (of "untempting" returns) is denoted T.

Case 1 . The defining relation for Case 1 of Model 1, which describes

the inspection resources as being below a certain threshold, is n m < I T I + Z {q.lieT} = Z min{l,q.}.

l i=l

Let k be the largest integer for which

k

Z min{l,q. } < m. i=l 1 “

Then 0 < k < n. Set

a = m - Z min{l,q. }

.

i=l

Then 0 < a<min{l,q1 , n }

.

— k+1

Theorem : An optimal strategy for the auditor is given by

p.° = min{l,q.} i<k

i l —

Pk?l = a

oP

i = 0 i>k+l.

55
An optimal strategy for the taxpayer is

0

c . =

1 1 i<k. ieT,

0

c . =

1 1 i>k.

0

c . =

1 P /P k+1

7 1 i<k. ieT

.

The resultant payoff is

F° = 11 0

r ft 0

0

0

ft Z V. + IV. - Z P

ieT 1 i>k 1 ieT

i<k i<k

Proof: First we wish to show that

F(c,p°) £ F(c°,p° ) for all c

.

For any c,

F(c,p°) == Z (V.-P.

ieT 1 1 )c

i + ( Vi-pk+ i“>

Ck+1

+

.1

Z V.c.

>k+l

1 1

i k+1

i<k

since, for ieT, i<k, we have

V.-P.p? = V.-P.q. = V.-P.(V./P.) = 0 . 1 11 l ii l ill

Because all c.l—ii— < 1 , V.-P.X) for ieT, and a<q, , , it follows that k+1

’

F(o,p°) < E (V.-P.) (Vk+ 1

-P

k+1

a) + I V.

ieT i>k+l

i<k

= Z V. + IV. - Z P. - P, .-.a = F(c°,p°)

ieT

1 i>k 1 ieT 1 k+1

i<k i£k

for all c.

Next we wish to show that

F(c°,p°) <_ F(c°,p) for all p.

For any p,

F(c°,p) = E (V.-P.p.) + l (V.-P.p ) ieT i>k

i<k

- . E-(vi-pih )pk+i/p

i ieT

i<k

56
= I V. + E V. + P.^. E q. - 2 P.p.

ieT 1 i>k

1 k PieT 1 ieT 1 1 i<k i<k i<k

- .ww.yi x>k ieT

i<k

E V. + 2 V. + P_ 2 q. - 2 P.p.

ieT 1 i>k 1 k XieT 1 ieT 1 1 i<k i<k i<k

" Z P

i P

i " Pk+1 ( E-Pi + } * i>k+l k 1

ieT 1 k 1

i<k

Since E p. < m, it follows that

i=l l —

F(c ,p) > E V + E V. + P... E q. - E P.p.

. m i i k+1 m i^i ieT i>k - ieT

i<k— i<k i<k —

- E P.p.-P (m- E_ p. - E \ i>k+l 1 1 k 1

ieT

1 i>k+l P

i

i<k

= E V. + E V. + P E q. ieT

1 i>k 1 k P ieT 1 i<k i<k

-

. Z .

(PiP

k+l

)pi +

. E>v+ , (P^l-P

i )P i"

P

k+lm - ieT i>k+l

i<_k

The first three terms of the sum do not depend on p. Since the P. in descending order of magnitude,

F(c°,p) > E V + E V + P E_q. - E P -1

ieT i>k ieT ieT

i<k i<k i<k

" pk+1 ( m - 2 1

)

ieT

i<k

1 s are

57
= E V. + E V. - E P. ieT 1 i>k 1 ieT 1 i<k i<k

\ min{l,q. }

)

i=l 1

= E V. + E V. - E P.

ieT 1 i>k 1 ieT

1 i<k i<k

P OL k+l

T-,/ o O \ = F(c ,p )

for all permissible strategies p.

It now follows that (c°,p

0

,F°) is a solution of the game. This concludes

the discussion of Model 1, Case 1.

Case 2 . The defining relation for Case 2 of Model 1 is

i M + Uq.|icf} = z min{l,q.}.

i=l 1

We also define

Vm

= Z{V. | ieT}

.

T l

1

Theorem: Any strategy p for the auditor which satisfies the conditions

by

11— II o

•H ft ieT

o

p. > q.

l — l

ieT

An optimal strategy for

i

—

i

li o

•H O ieT

O i. o II o ieT

.

The resultant payoff is

F(c°,p°) = V - E P . ieT

Proof: First we show that

F(c,p°) <_ F(c°,p°)

n F(c,pu

) = Z (V

i -Pi p.

0

)c.

i=l

for all c.

= E (V.-P. )c. + E (V.-P.p. )c.

•m 111 . m 111 1 ieT ieT

< E (V.-P. ) = V- E P. — li T ieT ieT

58
since , for ieT, we have Vi-P^>0 and, for ieT, we have "Vl-P^p^ .1

It is also clear that

F(c°,p) = I (V -P p.) = V - E P p.

ieT 1 1 1 ieT

1

> v

T

- ^ P

± ieT

for all p. Hence (c°,p°,F°) is a solution of the game.

Example 1 . (Fixed penalty rate): Let qjL = q for all i. Recall that the

returns are numbered so that

P > P„ > . . . > P . 1 - 2 - - n

Case A : 0<q<l, i.e. 0<Y^/V

± <1 for all i.

In this case, |t| =0 and we are in Case 1 or Case 2 of Model 1 ac- cording as m<q|T| or m _> q | T | =qn.

Case Al : m<q|T|, i.e. m/n<q. Let k he the largest integer no greater

than m/q, and set a = m-kq. Then k<n and by the preceding analysis of

Case 1 of Model 1, a solution of the game is given by

o P

i = P i<k

P

k+1 = a

p

i°

= o

c.° = 1

i

Wh thO / o o \ n _ , Ov i F (c ,p ) = I P (q-p. ) c

i>k+l

i>k

i<k

i=l

= pk+1 ( q-a ) + q E P.

i>k+l 1

= 9

. E

. pi"“p

k+i' i>k

We can eliminate a by the substitution a = m-kq, yielding

F(c°,P°) = q( E Pi+kP^) - mPk+1

. i>k

59
Case A2 : m>q

|

2 of Model 1,

Case B : q>l.

In this case | k = m and a =

where V, , tot

]' e ' m/nAj‘ Then by the Preceding analysis of Case a solution of the game is given by

= m/n >_ q

c.° = 0

l

F(c°,p°) = 0 .

for all i

for all i

i.e. V./P.>1 for all i.

l i

T| =0 andjT|= n. Since m<n we are in Case 1, with

0. Thus a solution of the game is ° -t P, = 1

Pi

° - 0

C.° = 1

1

i<m

i>m

for all i.

F(c ,p ) = 2 (V.-P.p.°)c.° = 2 P.(q-p.°)

i=l

111 1 i=q 1 1

.11 j

n

Z V. . =i 1

= .E Pjtq-D^E 4Pi=V t

-_Z , Km i>m i<m

60
Example 2 (Two-tier penalty system). Here we assume that N = {l,2,...,n}

is partitioned into two subsets B and S, of respective sizes |b| and |s|,

corresponding respectively to "big" and "small" levels of V^/P_^:

q

= V /P = q

l li b

ieB

q. = V./P. = q

1 1 1 s ieS

and q > q . b s Case A : q > q > 1. In this case T = N and so | T j = n. Since m<n,

we are in^Case3 ! of Model 1 with k=m and a=0. A solution is given by

P

i° = 1

Pi° 0

C.° = 1

1

i<m

i>m

for all i

F(c°,p°) = 2 P

i^ qi"pi°^

c

i

C i=l

= § P. (q.-p.°) = 2 V.- ?P..

i=l

1 1 1 i=l 1 i=l 1

Case B:: \

• - = v - Sp..

tot . _ 1 1=1

> 1 > q . In this case T=B and T=S. There are two cases according as m < |b| + q

|s| or m>|B|+q |S

Case Bl: m < |b| + q |s|. This falls under Case 1 of Model 1. With

k and a as defined earlier, a solution is given by

° - 1 Pi - 1

o P i = *

s o

p, , = a ^k+1

Pi

° - 0

ieB, i<_k

ieS, i£k

i>k+l

,

1 ogether with

c? = 1

i

ieB, i<k,

6l
° - T c . = 1

l

i>k

< “ PK+l/

P

i

ieS, i<k.

The resultant payoff is

F(c°,p°) = EV. + E V. - EP

ieB

1 i>k 1 ieB

i<k i£k

’• “ P, . T a * i k+1

Case B2: m>|B| + q _| S | . This is Case 2 of

given by strategies satisfying

Model 1. A solution is

o _ , p. =1

l ieB

o ^ p. > q

i — s ieS

c . =1

1 ieB

o )• 0 II o ieS,

i

with payoff

F( c°,p°) = (q,-l) £ P . ieB

Case C: lxj^q^. Then T=N. There are two cases according as m<q

b |B|+qJs| or m>q

b

(B| + qjs|.

Case Cl; m<qb

|B| + q^ | S

| . Then we are in Case 1 of Model 1. A solution

is given by

H- o II q

i

i<k

p

°

pk+l = a

o P

i = 0 i>k+l

o

c . =

1 1 i>k

o

c . =

1 p /p

k+1' i

i<k

E V.

i>k 1 " P

k+1 a *

Case C2: m>q | B | + q_|s|. Then we are in Case 2 of Model 1 and

0 s a solution is given by the conditions

.

° > q. for all i

l — l

.

° = 0 for all i

l F(c°,p°) = 0.

62
Model 2

For this model we use a nonzero-sum game with the two objective functions

F(c,p) = 2 (V.-P.p.)c. = 2 P.(q.-p.)c.

. , 1 11 l . _ i i l l

1=1 i=l

for the taxpayer, and

G(c,p) = - 2 V.c . (l-p.

)

i=l

i i

for the auditor. (Thus the latter wants to minimize the extent of undetected cheating. ) It will be convenient to number the returns so that

V

1 i V2 ± . > V . — n The solution concept used in this nonzero-sum case is that a pair

(c ,p ) of strategies is an equilibrium point if c maximizes F(c,p ) over all choices c of the taxpayer's strategy, and p maximizes G(c ,p) over all choices p of the inspector's strategy. (Thus neither player

has an incentive to deviate unilaterally from "his" member of the pair

(c°,p°) .

)

Case 1 . This is defined by m<|T| + Z{q.|ieT}. As in Model 1, let k

be the largest integer such that

. E_,min{l ,q. }< m. i=l i —

Then 0<k<n. Set

a = m- £ min{l,q^};

i=l

it follows that 0<a<min{l,oi }; — k+1

Theorem: The pair of strategies (c ,p ), where

o

c . = 1 ieT,

1 o

c . =

1 V /V. k+1 7 i

i<k+l, ieT , o

c . = 1 i>k+l, ieT , 1 o p i = min{l,q^} i<k.

Pk+ 1 = a

*1° 0

is an equilibrium point,

i>k+l.

63
Proof: Clearly p. < m as required. For any strategy c for the

taxpayer we have

i=l

F(c,p°) = § P.(q.-p.°)c.

i=l 1 1 1 1

= E P.(q.-l)c. + E P.q.c. +p fq -a)c

ieT 1 1 1 i>k+l

11 1 °' C Wr

i<k

Since each P.(q.-l) > 0 for ieT and since q, , -a>0 and c° = 1 •} +

i l — k+1 — ^ k+1 ’ 1

0

follows that F(c°,p°) F(c,p°) for all c. On the other hand.

G(c°,p) = - § V.c.^l-p.)

i=l

= - E V (1-p ) - £ V

k+l (l-Pi>- z V (1-p ) ieT i< k+l k+1 1

i>k+i

1 1

i e T ieT

= z VV

k+l

z X-E V z V

ipi

+Vk+1 E P

i + Z V.p.

ieT k 1 i<k+l i>k+l ieT 1

jL<k+1 i>k+i

1 i ieT ieT ieT l e T

Note that the first three sums do not depend on p. Since the V.'s are

in decreasing order 'of magnitude, it follows that in order to maximize

G(c?p) we should set

<*) pi

1 for ieT, i<k

This is possible, since by the definition of k, we have ,L1 £ m. Thus, with

( * )

,

we have l£

i<k

G(c°,p) = - E V -V Z 1- E V. + E V.p.

ieT 1 i<k+l i>k+l 1 11

i>k ieT ieT

ieT

i> k+1

+ \+l E P

i + Z Vi Pi i<k+l_ i>k+l_

ieT ieT

" E V

i ’ V

k+l.

S 1 + E v, P,+v_ Pl i >k+l 'i<k+l_ i>k+l

i e T

i i k+1 k+1

+ V Z p . . k+1 *1

i<k

ieT

6k
Note that the coefficient of V , 1C“*" -L E p.). In order to maximize G(c°

i<k 1 ieT

apart from a constant, is (p +

-L ,p), this coefficient should he made

n as large as possible consistent with (*) and the condition E p, <_ m. Writing these restrictions as i=l

1

Pk+1

+ Z p . +

i <k x ieT

Z p. < m

i>k+l 1 Z 1

ieT

i<k

= m - Z min{l,q.

}

ieT -1 i<k

= Z q + a,

i<k

ieT

o we see that p satisfies this relation as an equality and makes the

coefficient as large as possible. Thus p° maximizes G(c°,p) and (c°,p°)

is an equilibrium point.

The resulting values of F and G are

F = EV. M V. - Z P.-ctR ,, , ieT 1 i>k+l 1 ieT 1 k -1 i<k i<k

G Z V + V (m-k).

i>k+l K -1

Case 2 . This is defined by m > |t| + Z{a. |ieT}.

Theorem: Any pair of strategies (c°,p°) where

c.° = 1

l

c.° = 0

l

p> ° > min{l,q.

}

l “ 1

ieT,

ieT',

for all i.

is an equilibrium point.

Proof : For any c, we have

F(c,p°) = l P

i (Gi -Pi

0

)c

i i=l - £ P.(q.-l)c + Z_ P (q -p °)c.°.

ieT 1 1 1

ieT 1 1 1 1

(

65
For ieT, since P.(q.-l)>0, it follows thal: in order tojnaximi ze F(c,p°)

we must have c^ 1!.

1 Also, since P

i (li “P i

) 1 0 for ieT’ it follows that

F(c,p°) £ F(c°,p°) for All c.

On the other hand,

G(c°,p) = - § V c °(l-p ) i=l

= - zv.d-p.).

ieT

Hence G(c°,p) <_ G(c°,p°) = 0 for all p and we have shown that (c°,p°)

is an equilibrium point. The resultant values of F and G are

Z V.- Z P. , ieT 1 ieT

1

0 .

66
APPENDIX C

This attachment presents graphical representations of the relationship

between discriminant function scores and tax change for each of the audit

classes. The correlation coefficients derived from this data are discussed on page 10 of the text.

67
AUG

DIF

SCORE

US

AUG

TAX

CHANGE

FOR

LOU

BUSINESS

CLASS


 <S> 



 <s» 



 CO CO ^r OJ

<cz>tdQ'<rohJ hex ox<xz:oyj




68

240

260

280

300

320

340

360

AUERAGE

DIF

SCORE

TCtIP

SAMPLE

OF

RETURNS

FILED

IN

1966

EACH

POINT

REPRESENTS

4*4

OF

POPULATION

(RANKED

BY

DIF

SCORE)
AUG

DIF

SCORE

US

AUG

TAX

CHANGE

FOR

MEDIUM

BUSINESS

CLASS

X

X

X

X

X _

X

X

X

X

X

X

X

TT I I T'l I T t

~

t

" r

~

i I I I'T T I I T

0 O 
 <S> 0

O 0 
 
 LO O LO 0 LO

03 03 tH

<ez>ujq:<eou h<EX oi<rzou

G>

69

100

150

200

250

300

350

400

AUERAGE

DIF

SCORE

TCMP

SAMPLE

OF

RETURNS

FILED

IN

1966

EACH

POINT

REPRESENTS

4%

OF

POPULATION

(RANKED

BY

DIF

SCORE)

FIGURE

C2
AUG

DIF

SCORE

US

AUG

TAX

CHANGE

FOR

HIGH

BUSINESS

CLASS

© © © © © ©

© © ©

CO CD T CU

©

<E Z> LU QC <r OSjJ h<EX OX<EZ(5U

-400

-300

-300

-100

0

100

200

300

AUERAGE

DIF

SCORE

TCNP

SAP1PLE

OF

RETURNS

FILED

IN

1966

EACH

POINT

REPRESENTS

4

%

OF

POPULATION

(RANKED

BY

DIF

SCORE)

FIGURE

C3
AUG

DIF

SCORE

US

AUG

TAX

CHANGE

FOR

1040A

RETURNS

oo (jd rr ru

<£ UJ <E OLU h<CX OICZCDLU

71

50

100

150

200

250

300

350

AUERAGE

DIF

SCORE

TCMP

SAMPLE

OF

RETURNS

FILED

IN

19S6

EACH

POINT

REPRESENTS

4%

OF

POPULATION

(RANKED

BY

DIF

SCORE)

FIGURE

CU
AUG

DIF

SCORE

US

AUG

TAX

CHANGE

FOR

ii@,

-§19^000

AGI

RETURNS

© © © © © n cu €3WQ£4XOLiJ 1-<xx c

©©

©

72

200

250

300

350

400

450

AVERAGE

DIF

SCORE

TCMP

SAMPLE

OF

RETURNS

FILED

IN

1966

EACH

POINT

REPRESENTS

454

OF

POPULATION

(RANKED

BV

DIF

SCORE)

FIGURE

C5
AUG

DIF

SCORE

US

AUG

TAX

CHANGE

FOR

$15,-50,000

AGI

RETURNS

<E I> LlI 0£ <L OLxi h<CX OKCZOUJ

73

-400

-300

-200

-100

0

100

200

300

400

AVERAGE

DIF

SCORE

TCMP

SAMPLE

OF

RETURNS

FILED

IN

1966

EACH

POINT

REPRESENTS

4*4

OF

POPULATION

(RANKED

BY

DIF

SCORE)

FIGURE

C6
ftUG

DIF

SCORE

US

AUG

TAX

CHANGE

FOR

ITEM*

RETURNS

>S50,000

© © © CD CD

<d CD © CD ©

LD <S> ID © LD

CU ©SSP8® r~ LD 03




<E I> LU Q£ <E OLU h€X OKEZOUJ

7 ^

-300

-200

-100

0

100

200

300

AVERAGE

DIF

SCORE

TCMP

SAMPLE

OF

RETURNS

FILED

IN

1966

EACH

POINT

REPRESENTS

AX

OF

POPULATION

(RANKED

BY

DIF

SCORE)

FIGURE

C7
NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. 2. Gov’t Accession

No.

3. Recipient’s Accession No.

4. TITLE AND SUBTITLE

Resource Requirements and Allocations in IRS' Audit Division ; A Systems Analysis

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S)

Karla L. Hoffman, Lambert S. Joel. Martin H. Pearl

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 7120^11

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Internal Revenue Service

1111 Constitution Avenue

Washington, D.C. 2022b

13. Type of Report & Period Covered

14. Sponsoring Agency Code

IBS

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) The Applied Mathematics Division of the National Bureau of Standards ( NBS ) was asked by the Internal Revenue Service under IRS order #7T2867 to "assess the

validity and effectiveness of the IRS Audit Division's Long Range Plan's strategies

and approaches to resource requirements and allocations". This report, which

documents that assessment, (l) summarizes the examination and evaluation by NBS

of IRS's current audit practices and plans for the future, and (2) presents our major conclusions and recommendations. In many cases the available information

did not permit recommending specific methods with which to solve current problems.

In such situations we sought to identify those areas in which we believe further

research is needed and is most likely to lead to improvements over present

practices

.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Audit; behavior; compliance; discriminant analysis; game theory; law enforcement;

taxation; utility

18. AVAILABILITY | | Unlimited

For Official Distribution. Do Not Release to NTIS

Order From Sap. of Doc., U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. Cl 3 Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

22. Price

USCOMM-DC 2904 2- P 7 4
:

.

'

W

.

1\II