J. Biomedical Science and Engineering, 2010, 3, 799-806 JBiSE
doi:10.4236/jbise.2010.38106 Published Online August 2010 (http://www.SciRP.org/journal/jbise/).
Published Online August 2010 in SciRes. http:// www. scirp. org/journal/jbise
Uncovering preferences from patient list data using benefit
efficient models
Jan Ubøe, Jostein Lillestøl
The Norwegian School of Economics and Business Administration, Bergen, Norway.
Email: jan.uboe@nhh.no, jostein.lillestol@nhh.no
Received 11 September 2009; revised 1 June 2010; accepted 5 June 2010.
ABSTRACT
In this paper it is shown how the benefit efficient
patient list model of Ubøe and Lillestøl [1] can be
used to infer strength of preferences from patient list
data. It is proved that the model allows the cons-
truction of unique sets of preferences replicating the
observed allocations. To illustrate how this theory can
be applied in practice, preferences are uncovered
from a small data set, obtained from the Norwegian
patient list system.
Keywords: Patient Lists; Efficient Welfare; Statistical
Distributions
1. INTRODUCTION
In the Norwegian patient list system in general practice
the patients can be assigned to a general practitio ner that
agrees to have the main responsibility for his or hers
patients. As there are limited numbers of doctors of
specific types, e.g. gender, it may happen that a signi-
ficant number of patients are assigned to doctors of the
wrong type, i.e. a type of doctor that they really do not
want. This gives rise to several interesting research
questions, among them: How can we characterize as sign -
ments that conform to reasonable criteria for total bene-
fits to the patient-doctor community, when patients are
individually competing for the scarce resource? Which
changes in allocation can be expected when the avail-
ability of the scarce resource changes, e.g. by increased
availability of female doctors? The latter is known to
happen in many countries, and was precisely the
question asked by the investigators on general practice
and community medicine that posed the problem in the
first place.
In [1] Ubøe and Lillestøl suggested a new statistical
framework for this context, enable to answer questions
of this kind, based on the concept of benefit efficiency.
The next research challenge was then to see if the model
allowed inferences, i.e., to say something about the
preference structure based on an observed allocation.
This turned out to involve some delicate theoretical
problems, among them non-uniqueness, and the purpose
of this paper is to present our solution to these problems.
As an illustration of the theory we will con sider a special
case, using patient list data from the Norwegian patient
list system in general practice. These data report the
registered allocation of male and female patients to male
and female doctors. While this is mainly a theoretical
paper, the paper also offers some guidelines for
practitioners that want to apply this kind of theory to real
world data.
The theory of this paper follows from the idea of a
benefit efficient allocation, which can be described as
follows: Let A1, A2, , AT denote the sets of actions that
agents of type 1, 2, , T can choose, let Ut: At R be
the real number utilities of each choice, and let Q1,
Q2, …, QT be a sequence of probability measures on A1,
A2, …, AT. Then Q = Q1 × Q2 ×…× QT is called a benefit
efficient probability measure if it satisfies the two
conditions:
Larger aggregated utility, i.e. sum of utilities of all
agents, of an allo cation implies larger probability of that
allocation.
As the number of agents of each type pass to
infinity, the numbers of agents making each choice must
satisfy a specified set of linear allocation constraints.
Quite surprisingly, there are extremely few probability
measures of this type. In fact when utilities and cons-
traints are given, these measures form a one parameter
family. In the inv erse problem we consider in this paper,
the parameter can be set to 1 without loss of generality.
Then the resulting allocation will be as given by For-
mula (1) in the theory section below, where we briefly
recall the model construction in [1], and then show how
we can obtain unique representations of preferences. To
enhance readability, proofs and technical arguments are
given in appendices.
The Norwegian patient list system in general practice
800 J. Ubøe et al. / J. Biomedical Science and Engineering 3 (2010) 799-806
Copyright © 2010 SciRes. JBiSE
is described in some detail in [1], and we refer to this
paper for a review of the system. Note that our modeling
framework extends beyond the Norwegian patient list
system and it can be understood without any particular
knowledge of that system. Note also that our model is
completely specified by Formula (1), and no further
knowledge of the model will be needed to understand
the issues we address in this paper.
In the application section we use our model to infer
structure and strength of preferences from observed real
world data. These data were collected from an official
panel survey of Norwegian living conditions (Levekårs-
undersøkelse n 20 0 3) .
We believe that our basic approach to this problem is
novel. It is fundamentally different from the theory of
revealed preferences in consumer theory, see e.g. [2],
and is not in any way related to the extensive economic
literature on the design of matching system in health
care, see e.g. [3]. Hence we will not enter into a
discussion of other models in this area with a different
scope.
2. METHODS
The model in [1] can be described briefly as follows:
Assume that there are S groups of patients, T types of
doctors, and let Pts denote the number of patients in
group s that has a doctor of type t, s = 1, 2, ... , S, t = 1,
2, , T.
Patients: We assume that there is a total of Es
patients belonging to group s, s = 1, 2, ..., S. A patient
belonging to group s is assumed to have a utility Uts of
having a doctor of type t, t = 1, 2, , T. It may some-
times happen that a patient prefer to wait for a vacancy
of a suitable doctor rather than being assigned to a
doctor of a type that the patient dislikes. We let Pt(s+S)
denote the number of patients of type s waiting for a
doctor of type t (not being assigned to any doctor), and
let Ut(s+S) denote the utility of these patients.
Doctors: Every doctor working within the system
is assumed to have a certain list length, i.e., a max imum
number of patients that he or she can serve. We assume
that there are Dt doctors of type t, and that these doctors
can serve a total of Lt patients, i.e., Lt is the sum of the
list lengths of all doctors of type t. Some doctors may
have vacancies, and we let Ut(2S + 1) denote the utility per
vacancy incurred by a doctor of type t.
Thus we have defined Pts and Uts for s = 1, 2, , 2S +
1, t = 1, 2, , T, which can then be represented by Tx(2S
+ 1) matrices P and U.
Utilities for vacancies and for being assigned to the
wrong type of doctor may of course be negative, in
which case we refer to these numbers as disutilities.
Note that the word utility is used in a broad sense as a
quantification of strength of preferences. Utilities are
hence not necessarily utilities in the von Neumann-
Morgenstern sense.
Clearly the (E1, , ES) patients can be allocated to the
(D1, , DT) doctors in many different ways. The basic
hypothesis in [1], however, is to assume that the system
is benefit efficient in the sense that states with large
aggregate utility (sum of the utility of all patients and
doctors) are more probable than states with smaller total
utility. If the system is benefit efficient with a large
number of patients in every group, it is possible to prove,
see [1], that the allocation will settle at a statistical
equilibrium given by the following explicit formula:
() 1,,
() 1,,2
() 21
ts ts
tsts Sts
tts
A
BexpUs S
P
D BexpUsSS
AexpUsS



(1)
(2 1)
1
()
1
S
tst St
s
T
tst sSs
t
PP L
PP E

See Appendix 3 on how to compute the balancing
factors A1, A2, , AT, B1, B2, , BS. Note that these
allocations must not be confused with the allocatio n with
maximum total utility. In fact, the allocation with maxi-
mum total utility can be obtained as special case if one
multiplies the utilities in (1) with a constant and let that
constant pass to infinity.
The basic problem we want to address in this paper
can be formulated as follows: Assume that the system is
benefit efficient and that we observe
The total number of patients in each group, i.e. Es, s
= 1, , S.
The total number of doctors of each type, i.e. Dt, t
= 1, , T.
The total list length of doctors of each typ e, i.e. Lt, t
= 1, , T.
The final allocation of patients to doctors/waiting
lists, i.e. Pts, s = 1, , 2S + 1, t = 1, , T.
To what extent do these observations reveal the
strength of the preferences
Uts, s = 1, , 2S + 1, t = 1, , T ?
It is easy to observe that there are always an infinite
number of utility matrices leading to the same final
allocation. To obtain uniqueness we hence have to
impose some additional restrictions on specific utilities
and/or the relationships between them. Such restrictions
are typically based on prior knowledge of the context
and on known empirics, and modeling issu es of this kind
are discussed in detail in Section 3. In Theorem 2.1
below, we single out one of the infinitely many repre-
sentations, and refer to the representation in (2) as the
canonical choice. This is useful for two main reasons: To
J. Ubøe et al. / J. Biomedical Science and Engineering 3 (2010) 799-806 801
Copyright © 2010 SciRes. JBiSE
get insight to the degrees of freedom in modeling, and to
provide a basis for numerical calculations.
2.1. Theor e m 2.1
Assume that an observed patient list distribution P can
be replicated by a model that satisfies (1). Then we can
find a unique utility matrix U that replicates P on the
form (later referred to as (2)):

1
11 12
1( 1)
11 12121 221
2( 1)
21 222
1( 1)
(1)1 (1)212 1
0
00 0
0
0
0
0
S
SS
S
TS
TT TS
TT T
v
vv
u
uu v
vv w
u
uu w
u
uu v
vv w














  

(2)
PROOF: See Appendix 1.
The zeros in (2) can formally be interpreted as
reference points and the corresponding groups as
reference groups. Uniqueness is obtained when we
specify how much more/less utility the other groups
have in comparison to these reference groups. While the
representation given by (2) has several favorable
properties, results given on this form are quite hard to
interpret. Hence it might be profitable to look for other
representations offering more transparent interpretations.
In general the position of the S + T reference groups can
be chosen in several different ways, and an important
modeling issue is to specify natural reference groups for
the given context.
We can also obtain alternative unique representations
by assuming a utility structure with sufficient identities
and/or symmetries. Nevertheless, it is convenient to use
(2) as a canonical form, both for algorithmic program-
ming and for resolving theoretical issues. One important
issue is that of identification, i.e., recovering the para-
meters of an assumed utility structure from its estab-
lished canonical form. Equivalent structures are obtained
by transformations of U that leave P invariant. These are:
Add/subtract a fixed T-dimensional column vec-
tor a to all columns of U labeled s = 1, …, S and s =
2S + 1 (i.e. except s = S + 1, , 2S).
Add/subtract a fixed 2S + 1-dimensional row vector
of form (b,b,0 ), with b S-dim ensional , to all rows of U.
Add/subtract a constant c to column s = 2S + 1 of
U and at the same time subtract/add the same constant
from all columns s = S + 1, …, 2S.
However, the easiest way to check identifiab ility may be
to use the transform given by Formula (11) in Appendix 1
and check the uniqueness of the parameter recovery.
Note that the canonical form, as well as alternative
models with the same number of (linear) restrictions,
provides perfect fit to the available data. Hence any
inference does not lend itself to the usual statistical
standard error computations.
3. RESULTS AND DISCUSSION
The Norwegian patient list system was introduced in year
2001 and is monitored by the authorities. Data on availa-
bility of doctors are made readily available to the public,
and some aggregated data on list composition and vacan-
cies are also av ailable for research purposes. Reliable data
on doctor preference are not readily available. However,
some questions on the combination (gender of respondent,
gender of assigned doctor, preferred gender of doctor)
were included the official panel survey of Norwegian liv-
ing conditions (Levekårsundersøkelsen 2003). The ques-
tions asked relevant to our study were:
Q1: Do you want to relate to an assigned general
practitioner, or do you want to use several general practi-
tioners?
Q2: Do you mind whether your assigned general
practitioner is male or female?
Q3: For those who answered “yes” on Q2: Do you
want to have a male or a female assigned general practi-
tioner, or do you want to use both a male and a female?
Unfortunately the response rate to the preference
questions were low, and more so for males than females.
Hence this part of the paper must be considered more as
an illustration of the potential offered by the theory, and
not so much as an empirical survey in its own right.
With the way of questioning above we were left with
the problem of what to do with respondents who wanted
both gender of doctors available. Since the system of a
single assigned general practitioner was already firmly
established with no opportun ity of multiple assignments,
we decided to split these relatively few respondents
equally between the two preferences. The observed counts
are shown in Table 1.
In this case T = 2, with types denoted M (male doctor)
and F (female doctor), and S = 4 with groups denoted
mm, mf, fm and ff, where the first letter is the gender of
the patient and the second letter is the preferred gender
of doctor.
A total of 3489 persons, 1736 men and 1753 women,
were interviewed in the panel survey. As we can see
from Table 1, the responses to the preference questions
were very low, and more so for males than females.
That partly explains the strong bias we observe in the
data, e.g. 22% men, and we therefore scaled the data to
adjust for this. Two issues are taken into account: First
Table 1. Observed counts in each group-original data.
Patient group mm mf fm ff
M-docto
r
782 32 77
F-docto
r
54 2 200
802 J. Ubøe et al. / J. Biomedical Science and Engineering 3 (2010) 799-806
Copyright © 2010 SciRes. JBiSE
we have scaled the data so that there is an equal num-
ber of respondents of each sex, second we have scaled
the data to get the marginal frequencies in accordance
with the approximately known distribution of doctors at
the time, namely 70% male doctors and 30% female
doctors.
The results after these scalings are given in Tab le 2
per 1000 respondents.
3.1. Missing Data and Partial Information
Officially there are no waiting lists, and data on this are
hard to get. In Appendix 2, however, we prove that one
can infer the correct preferences for the groups in Table
2 even in the case where data on waiting lists and
vacancies are missing. The crucial result, Theorem 5.4,
states that these preferences are independent of the data
for waiting lists and vacancies.
A survey made by the Norwegian Ministry of Health
and Care Services (2004) r eports a total of 2 026 doctors
with vacancies, the average number of vacancies being
223. With the reported 4 563 751 patients served, this
gives 99 vacancies per 1 000 patients. For illustrative
purposes we round this in Table 3 to 100 patients per
1000 served. We have no information on how this is
distributed among the gender of doctors. If they are
distributed evenly among the genders, the number will
be as given in the parentheses in Ta ble 3 . Note that the
number of patients who want a doctor of the same
gender is higher for males than for females.
3.2. Inferring Canonical Utilities from
Observed Data
The utility matrix corresponding to the types and
groups in Table 3 is
1518 191114
2528 29
21 24
UUU
UU
UUU
UU



U (3)
where the row order is M, F and the column order is
mm, mf, fm, ff, mm-w, mf-w, fm-w, ff-w, vacancy,
where w indicates a waiting list state. According to
Theorem 5.4 in Appendix 2, the missing data in Table
3 can be chosen arbitrarily. Zero entries can be handled,
but unless special care is taken such entries lead to
serious numerical problems. For simplicity we have
carried out all the calculations using the numbers
reported in Table 4. We remark, however, that the
numbers reported on waiting lists are fictitious but to
some extent realistic.
To find replicating utilities, we use the construction
described in Proposition 5.3 in Appendix 1. The result is
shown in Table 5.
The numbers in parenthesis are based on the fictitious
waiting list numbers. Hence the strength of preferences
we can infer from the data in Table 2 is given by Table 6.
If we take the canonical structure as our model, we
may infer that that the patients with preference for the
scarce gender (female doctors), have high utilities for
correct matching compared with the reference groups,
and that female patients wanting a male doctor while
being assigned to a female doctor tend to have utilities
slightly below the reference group zero.
Be aware, however, that this inference may be due to
technical artifacts linked to the implicit assumptions
caused by a special choice of reference groups. The
canonical choice corresponds to an assumption where
male patients wanting a male doctor are in effect
indifferent to the gender of their doctor, and that may
very well be an artificial assumption.
3.3. Modeling and Further Inference
We will now look into the modeling aspects of alter-
native representations. While the utilities reported in
Table 5 are the unique utilities on the form (2) that
replicates the counts in Ta ble 4, there are several other
utility matrices that offer perfect replication. Also,
observations of this type are subject to considerable
amounts of randomness. Perfect replication is hence
relatively unimportant, and models offering less than
perfect fit may be superior if they carry more transparent
information.
A main issue in this context is to quantify disutilities
of incorrect patient/doctor matching. It is straight-
forward to verify that the solution of (1) is fixed if we
add the same constant to all utilities within the same
group, i.e., our model is sensitive to utility differences
but does not depend on the general level of utility.
Without loss of generality we can hence assume that all
utilities for correct patient/doctor matching are equal to
zero. These assumptions lead us to consider utility
matrices on the form:
21314141112
1323 24 2
21 22
00
00
abbc
ab b
aa bbc
bb
U (4)
Table 2. Adjusted data from Table 1.
Patient group mm mf fm ff
M-doctor 455 12 69 164
F-doctor 19 14 2 265
Total 474 26 71 429
Table 3. Scaled counts in each group per 1000 patients served.
Patient groupmm mffmffmm-w mf-w fm-wff-wvac
M-doctor 4551269164 - - - -(70)
F-doctor 19142265 - - - -(30)
Total 4742671429 - - - -100
J. Ubøe et al. / J. Biomedical Science and Engineering 3 (2010) 799-806 803
Copyright © 2010 SciRes. JBiSE
Table 4. Scaled counts with artificial waiting list data.
Patient group mm mf fm ff mm-w mf-w fm-wff-wvac
M-doctor 455 12 69 164 (8) (4) (9) (1)(70)
F-doctor 19 14 2 265 (2) (6) (4) (6)(30)
Total 474 26 71 429 (10) (10) (13) (7)100
Table 5. Canonical utilities acording to (2) with Table 4 data.
P mm Mf fm ff mm-wmf-w fm-w ff-wvac
M 0 0 0 0 (-7.7) (-4.7) (-5.7) (-8.7)0
F 0 3.3 -0.4 3.7 (-8.2) (-3.7) (-5.6) (-6.1)(2.3)
Table 6. Canonical utilities (2) inferred from Table 2.
Patient group mm mf fm ff mm
-w mf
-w fm
-w ff-w vac
M-doctor 0 0 0 0 - - - - 0
F-doctor 0 3.33 -0.37 3.66 - - - - -
Here a1, a2, a
3, a
4 are the disutilities of incorrect
patient/doctor matching. The question is now if it is
possible to find utility matrices of the form (4) rep lic at in g
the counts in Table 4. The transformation defined by
Formula (11) in Appendix 1 transforms any utility
matrix U to an equivalent matrix U on the form (2).
Equivalent means that the two matrices produce
exactly the same counts when they are used in (1). Using
(11) on the matrix in (4) we obtain Formula (5). We see
that we do not have identifiability, unless we add
restrictions. Nevertheless it is possible to infer some
non-trivial relationships. If the utility matrix in (5)
equals the matrix specified in Ta b le 5 or 6, we get the
equations
21
31
41
3.33
0.37
3.66
aa
aa
aa
 

 
(6)
If we eliminate a1 from the first and the third equation
and rearrange the terms, we see that:
31
42
0.37
0.33
aa
aa

 (7)
By assumption, the utilities for correct patient/ doctor
matching are all equal to zero, and hence (by context)
utilities for incorrect matching must all be negative (if
they are not, agents must have been allocated to wrong
groups). In (7) a1, a
2 are the disutilities of incorrect
matching for men, and a3, a
4 are the disutilities of
incorrect matching for women. From (7) we can draw
the conclusion that disutilities of incorrect matching are
larger for women.
If we go one step further, we can introduce the addi-
tional assumptions: a1 = a2 = am and a3 = a4 = af , i.e. th at
all men have the same disutility of incorrect matching,
and all women have the same disutility of incorrect
matching. In this case the system is overidentified, and
perfect fit cannot be obtained. As mentioned above,
perfect fit is relatively unimportant, however, and we
can instead search for the best possible fit using utility
matrices on the form
13 14111 12
23 24 2
21 22
00
00
mf
mf
abbc
abb
aa bbcbb
U (8)
The best replication, in the sense that we get an
approximate solution to (6), is then
am = 1.66 a
f = – 2.02 (9)
In this case all the relevant parameters are identified
by our assumptions, and we maintain the conclusion
from (7), i.e., that disutilities of incorrect matching are
larger for women. It is somewhat surprising that this
rather crude approach produces the near perfect fit
shown in Ta b l e 7 (to be compared with observations in
Table 2).
If we want to infer disutilities for vacancies or for
being on waiting lists, we would have needed data for
the allocation of these groups. Such data are missing.
Nevertheless we will look into some general issues
connected to these data.
If we compare the lower right corners in Ta ble 5 and
Eq.5, we obtain the equation c2 – c1 – a1= (2.33). This
equation could in principle offer an alternative line to
identification. If we assume that the disutilities for
vacancies are equal for both gender of doctors, i.e., that
c1 = c2, we could infer the value a1 = (– 2.33). This
value could then have been inserted in (6) to identify the
remaining disutilities in (6).
To proceed one step fu rther, assume that all disutilities
for being on waiting lists are equal, i.e., that bij = b for
all i = 1,2 and j = 1, …, 4, and that the disutilities for
vacancies are equal for both gender of doctors, i.e., that
c1=c2=c. Then we could try to find the best possible fit
using utility matrices on the form
00
00
mf
mf
aabbbbc
aabbbbc
U (10)
When modeling the utility structure with sufficient
assumptions to get excess degrees of freedom, we essen-
131 14 1411112 12
2131 412312414211
21 12212
000 0
0
0
bcbca
bcb ca
aaaaaab cb cacca
bcbca


 
 
 

 
U (5)
804 J. Ubøe et al. / J. Biomedical Science and Engineering 3 (2010) 799-806
Copyright © 2010 SciRes. JBiSE
tially have an estimation problem with an opportunity to
judge the sampling error. It may then be helpful to bring
the problem within the framework of conventional contin-
gency table theory. In our application this may be done by
lifting the 2 × 4-block of waiting list patients up as a
second layer on top of the 2 × 4-block of assigned patients.
On top of the block of vacancies we add a block of struc-
tural zeros. Thus we have a three-way table of size 2 × 5 ×
2 with the correct marginal features. By taking logarithms
in the representation (1), we get a linear expression for the
log-counts. Our modeling assumptions then give rise to a
log-linear mod el with a specific p arametric st ructure to be
estimated, and for which readily available and applicable
general theory exists. In the model (10), we have 4 para-
meters am, af, b and c, in addition to the 8 scaling cons-
tants Ai, Di, i = 1, 2 and Bj, j = 1, 2, 3, 4. These parameters
may then be estimated by maximum-likelihood principle,
for which asymptotic theory is available and provides
approximate standard errors of estimates, see, e.g. Bishop
et al (1975).
Remark. To estimate the parameters in (10 ) we would
have needed to make use of the artificial data in Table 4.
These data were not available and were included for
illustration only. To some extent the method suggested
for (10) is also relevant for the analysis of (8). As shown
in Ta b l e 7 , a simplistic approach already produces near
perfect fit, and alternative/more refined methods would
not change this. For these reasons we will not pursue this
further.
4. CONCLUSIONS
In [1] Ubøe and Lillestøl proposed a new type of statis-
tical model to study the allocation of groups of patients
to different types of doctors according to given pre-
ferences. This paper clarifies the inverse problem of how
to identify preferences based a given allocation.
As an illustration of the theory we have applied the
model to patient list data from the Norwegian patient list
system in general practice. However, this type of model
can be used to infer preferences from much more refined
systems than the one we have studied here. We only
Table 7. Counts using am = – 1.66, af = – 2.02.
Patient group mm mf fm ff
M-doctor 455 12 69 163
F-doctor 19 14 2 266
Total 474 26 71 429
made use of two types of doctors and four groups of
patients, while the mod el allows arbitrary many types of
doctors and arbitrary many groups of patients.
The revealed preferences from the Norwegian patient
list data turned out to be very reasonable, and mostly in
accordance with prior beliefs. The main empirical find-
ing that disutilities for incorrect matching are larger for
women than for men appears to be a relatively robust
conclusion that can be inferred from different model
formulations.
Despite the weakness of our data, they may give some
backing for the health authorities, e.g., when asking
questions like: What changes are likely to happen when
the fraction of female doctors is on the rise? This may be
answered by using the model in the forward manner, as
described in [1], and in more detail in [5]. Revealed
disutilities are then used as input, representing the
current preference status. It would clearly be of interest
to have periodic updates on patient allocations and
preferences to investigate the stability of disu tilities.
5. ACKNOWLEDGEMENTS
The authors wish to thank Yngve Rønsen MD who proposed the
patient list problem and Gry Henriksen at the Norwegian Social
Science Services (NSD) for making the patient list data available to us.
REFERENCES
[1] Ubøe, J. and Lillestøl, J. (2007) Benefit efficient
statistical distributions on patient lists. Journal of Health
Economics, 26(4), 800-820.
[2] Varian, H.R. (2006) Revealed preferences. in Szenberg,
M., ed., Samuelsonian Economics and the 21st Century ,
Oxford University Press, 99-115.
[3] Roth, A.E. and Peranson, E. (1999) The redesign of the
matching market for American physicians: Some
engineering aspects of economic design. The American
Economic Review, 89(4), 748-780.
[4] Bishop, Y.M.M. Fienberg, S.E. and Holland, P.W. (1975)
Discrete multivariate analysis: Theory and practice. The
MIT Press, Cambridge Mass.
[5] Lillestøl, J., Ubøe, J., Rønsen, Y. and Hjortdahl, P. (2007)
Patient allocations according to circumstances and
preferences. Discussion paper, Norwegian School of
Economics and Business Administration, Bergen.
[6] Bregman, L.M. (1967) The relaxation method of finding
the common point of convex sets and its application to
the solution of problems in convex programming. USSR
Computational Mathematics and Mathematical Physics,
7, 200-217.
J. Ubøe et al. / J. Biomedical Science and Engineering 3 (2010) 799-806 805
Copyright © 2010 SciRes. JBiSE
APPENDICES
Appendix 1: Proof of Theorem 2.1
Consider the following matrix transformation
11 11
1()1(21)
11(21)11
1, ,
1,, 2
21
ts ts
tstss SS
ts tS
UUUUs S
UUU UsSS
UUUUsS

 
 
 
(11)
LEMMA 5.1
Let

ts
UUbe given, let {}
ts
UU

be defined by
(11) and let
P
and
P
denote the corresponding
distributions of patients in (1) when we use U and U
,
respectively. Then
P
P.
PROOF
Let A1, …, AT, B1, …, BS denote the balancing factors
solving (1) when we use U, and define




111
12 1
112 1
exp1, ,
exp1,,
ttt
S
sss
S
AUA
B
UUtT
BUUsS


(12)
If s = 1, …, S we get
exp( )exp( )
t stst sts
ABUAU B
 (13)
If s = S + 1, …, 2S we get
 

 
1121
1121
exp( )exp
exp( )
exp( )
tsStstsSsS S
ts sS S
tsSts
DDBUU
UU U
DB
BU
U






(14)
If s = 2S + 1 we get


111
12 1
111
12 1
exp( )exp()
exp( )
exp( )
ttstt S
ts tS
tts
AUU U
UUU
AU
U
AU


(15)
which proves the lemma.
PROPOSIT ION 5.2
Let (1)
U and (2)
U denote two utility matrices, and
assume that (1) (2)
P
P in (1). Using the transforma-
tion in (11) we have (1) (2)
UU

.
PROOF
We have to prove that the balancing factors must be
equal, and then it follows from (1) that all the utilities
must be equal as well. It follows from Lemma 2.1 that
(1) (2)

P
P, and observe from (11) that (1)
1(2 1)S
U
(2)
1(2 1)0
S
U
b y definition. Since
(1)(1)(2)(2)
11(21)11(2 1)
exp( )exp( )
SS
AU AU



(16)
it follows that (1) (2)
11
AA

. Now put t = 1 and s = 1, …,
S, and observe from (11) that (1) (2)
11
0
ss
UU
 by
definition. Hence from (1) we get
(1)(1) (1)(2)(2) (2)
1s11s1
exp( )exp()
ss
AB UABU

 
(17)
Since (1) (2)
11
AA

it follows that (1) (2)
s
s
B
B

for all s
= 1, …, S. We then put s = 1 and t = 1, …, T, and
observe from (11) that (1) (2)
11
0
tt
UU
 by definition.
From (1) again we get
(1)(1) (1)(2)(2) (2)
111 1
exp( )exp()
tttt
AB UBUA

 (18)
Since (1) (2)
11
B
B

it follows that (1) (2)
tt
AA

for all
t = 1, …, T. We have hence proved the proposition.
Proof of Theorem 2.1
By assumption we can find a matrix U that replicates
P
.
According to Lemma 5.1 U
also replicates
P
. By
construction U
is on the special format given by (2).
Hence there exists a matrix on the form (2) that
replicates
P
. Conversely if a matrix is of the form
given by (2), it does not change when we apply the
transformation given by (11). Uniqueness then follows
from Proposition 5.2.
PROPOSIT ION 5.3
Let{}
obs
ts
P be the observed numbers on the patient lists.
The replicating matrix in Theorem 2.1 can then be
constructed as follows. Put:
ln()1, ,
ln(/)1, ,2
ln()2 1
obs
ts
obs
tsts t
obs
ts
PifsS
UPDifsSS
PifsS



(19)
If we put 1
t
A
, 1
s
B
in (1), it is easy to verify that
the model in (1) replicates the observed pattern. The
unique replicating matrix in Theorem 2.1 can then be
found applying the transformation in (11) to the utilities
in (19).
Appendix 2: Inference under Partial
Information
Assume that we know the number of patients on the
patient lists and the number of vacancies, but do not
know how many patients that are waiting for a vacancy.
Is it then possible to infer the strength of preferences of
the patients on the patient lists? The answer is yes, and
this can be demonstrated as follows:
Assume that(0)
ts
P s = 1, …, S, t = 1, …, T is given,
and let (1)
ts
P and (2)
ts
P be arbitrary numbers for s = S +
1, …, 2S + 1, t = 1, …, T. Define the following aggre-
gated quantities
806 J. Ubøe et al. / J. Biomedical Science and Engineering 3 (2010) 799-806
Copyright © 2010 SciRes. JBiSE
(0)(0) (0)(0)
11
,,
ST
ttst ts
ST
LPE P



21
()() ()()
11
,,1,2
ST
iiii
ttssts
SS T
LPEPi
 


THEOREM 5.4
For i = 1,2 put (0)( )(0)( )
,
ii
ttts ss
L
LLEEE and find a
unique matrix ()i
U of the form (2) such that the system
given by (1) replicates the numbers
(0)
()
1,, ,1,,
1,, 21
ts
ts i
ts
PifsSt T
PPifsS S
 
 
(20)
If (2) (1)
1(2 1)1(2 1)
/
SS
KP P

then the two utility matrices
(1)
Uand (2)
Uare connected through the formula (here
referred to as Eq.21):




(1)
2
(2)(1)
1
2
(1)
1
1, ,,1,,
lnln()1, ,2,1, ,
lnln( )21,1,,
ts
ts
ts ts
ts
ts
ts
ts
UsStT
P
UUKsSSt T
P
P
UKsStT
P
 

 




 



PROOF
Define a new utility matrix
U by the right hand side of
(21) and let (1)
t
A, t = 1, …, T and (1 )
s
B
s = 1, …, S
denote the balancing factors solving (1) using the
replicating utilities (1)
U Now put (2) (1)
tt
AAK
and
(2)
s
B(1) /
s
B
K.
If s = 1, …, S, t = 1, …, T w e get by (1)
 
 
11
(2) (2)(1)
111 (0)
exp( )/()
exp( )
ts tst sts
t ststs
ABAKBKexp U
AB U
U
P


If s = S + 1, …, 2S, t = 1, … , T we get by (1)


(2)
2
11
1
(2)
(1) (2)
(1)
exp( )
/(lnln( ))
ln()
tsSts
ts
tsSts
ts
ts
ts ts
ts
DB
P
DB KexpUK
P
P
U
PPK
P






If s = 2S + 1, t = 1, …, T we get by (1)
 

2
1
(2) (1)
1
exp()(lnln( ))
ts
ttst ts
ts
P
A
UAKexp UK
P





(2)
(1) (2)
(1)
ts
ts ts
ts
P
P
P
P
 
The marginal con strain ts are au tomatically sa tisfied when
the model re plicates each ent ry in the matrix. Note that





2
12 1
(1) (1)
1(21)11
12 1
lnln0
S
Ss ts
S
P
UU KU
P





and that if s = 1, …, S , then ts ts
UU
. This proves that
Uis of the form (2). Hence if we put (2)
UU this
matrix is the unique matrix on the form (2) that
replicates the system in (20) when i = 2.
As we can see from Theorem 6.1, the utilities s
= 1, …, S, t = 1, …, T do not depend on the values of
ts
P for s = S + 1, …, 2S + 1, t = 1, …, T. Hence we
have the following corollary:
COROLLARY
Assume that ts
P s = 1, …, S, t = 1,…,T are known,
while data on ts
P for s = S + 1, …, 2S + 1, t = 1, …, T
are missing. If we choose 0
ts
P for s = S + 1,…, 2S
+ 1, t = 1,…,T arbitrarily, we can still infer the correct
values on ts
Ufor s = 1, …, S, t = 1, …, T.
Remark. From the bottom line in Formula (21) we see
that we can also obtain strength of preferences for
vacancies in cases where information on the number of
patients waiting for vacancies is missing. Clearly,
however, it is impossible to infer strength of preferences
for groups of patients waiting for vacancies unless we
have data for these groups.
Appendix 3: Numerical Methods
How to find a numerical solution to (1) when utilities U
and marginal constraints L and E are given? We need to
find numerical values for the S + T balancing factors
A1, …, AT, B1, …, BS. This is done as follows:
Initially we put all the balancing factors equal to 1.
Then for t = 1, …, T we update At using

(2 1)
1exp
t
tS
ststS
s
L
ABUP
(22)
Once these are updated, then for s=1,…,S we update
Bs using


()
1(expexp)
s
sT
ttsttsS
t
E
BAUDU
(23)
We then repeat the updates in (22) and (23) until the
system settles. The algorithm is a variant of the Bregman
balancing algorithm, see Bregman (1967). Like the stan-
dard Bregman algorithm this algorithm is surprisingly
efficient, and solves large systems in a very short time.