Download Medical procedure choice and social interactions

Medical procedure choice and social
interactions- the role of patient
heterogeneity
(working paper do not circulate)
Emily TANIMURA∗
Centre d’Economie de la Sorbonne, Université Paris 1 Panthéon-la Sorbonne-CNRS,
106-112, Bld de l’hôpital, 75013, Paris
e-mail: [email protected]
and
Jean-Pierre NADAL
Laboratoire de physique statistique, ENS and
Centre d’analyse et mathématiques sociales, EHESS
54, bld Raspail, 75006, Paris
e-mail: [email protected]
Abstract: To identify possible explanations for high regional variation in medical procedure choices, we consider a dynamic model where
physicians base treatment choices on two factors: the characteristic of
the patient, drawn according to a random variable describing the patient population and on a variable that depends on the past choices
of the physicians’ network neighbors, the physician’s own past choices
or a combination of these. We characterize the long run treatment frequencies in some particular cases. This characterization shows that two
behavioral assumptions, social interactions or personal reinforcement or
a combination can reinforce differences between regions in a similar way.
This highlights a possible explanation for the observed variations other
than that of social interactions which has previously been suggested in
the litterature. In the model, we then compare steady state frequencies of treatments to those that would prevail without social or personal
influence. We relate the direction and magnitude of the reinforcement
effects to the distribution of patient characteristics.
Keywords and phrases: social interactions, social multiplier, Markovian dynamics, heterogeneity of characteristics, binary choice
JEL classification: I00, C02, D7.
∗
Corresponding author. E. Tanimura has benefited from a doctoral fellowship from the
CNRS project ELICCIR. We thank members of the interaction group at GREQAM for
helpful comments on this work
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1. Introduction
Large geographical variations in the rates of certain medical treatments are
well documented in the United States and elsewhere (e.g. Wennberg et al,
1973, Phelps, 2000). Such variations can be observed in relation to various
medical conditions for which several treatments are available and their amplitude may be large. For example, the rates of coronary angioplasty has
been shown to vary by a factor of seven between different regions in the
United States. While the characteristics of the patient populations also vary
across regions, studies indicate that this is not in itself sufficient to explain
the large variations in treatment rates even after controling for various factors (Phelps and Moody, 1993).
Thus the high variation in medical procedure choices has come to be seen
as something of an unsolved puzzle and has been much discussed in the literature on health economics. One possible explanation for the phenomenon
brought forward in Burke et al (2003) is that social influence between physicians could generate the high regional variation that is observed. In other
contexts, empirical studies and theoretical models have already indicated
that social interactions could explain the variance of for example crime rates
(Glaeser and Scheinkman, 1996) or stock market prices (Bannerjee,1992) in
absence of other underlying explanatory factors. In the case of medical procedure choice, it seems plausible that physicians would take into account
the treatment choices of their peers, either due to pure conformity effects
or in order to reduce the risk of mal practice charges by adopting widely
accepted procedure choices. It is also possible that spillover effects could
make it beneficial to coordinate on treatments.
In this paper, we will take an approach similar to Burke et al and analyze
the impact of social influence on medical procedure choice. The empirical
motivations for our work are thus essentially the same as for Burke et al
(2003) to whom we refer for a more detailed discussion of the background to
the problem. Our main objective is to go further in the formal modeling of
the situation in order to answer some questions that could not be addressed
in the model of Burke et al and which we will discuss in what follows. In
particular, the framework we develop also allows us to explore alternative
explanations for the observed variations. The presence of social interactions
is supported by econometric analysis in Burke et al. However, there could
be other explanations, which could exist together with social interactions
and which deserve to be explored, notably that of personal reinforcement.
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When agents face similar choices, repeatedly over time, it seems plausible
that they would reinforce the choices they have made in the past. Indeed, the
behavioral motivations could be quite similar to those for social influence.
Instead of pure social conformity one could invoke the pure force of habit. If
co-ordination with colleagues is beneficial there could also be efficiency gains
associated with the repeated use of the same procedure. For this reason we
consider a model in which agents may be sensitive to social influence, to
their own prior choices or to a weighted combination of these.
It is also interesting to better understand which factors determine the
relative frequencies of treatments in different regions. This question can only
be partly answered in the framework of Burke et al. In their model, there
are two treatments and two patient types. The physician’s utility function
takes into account the characteristics of his patient and the average of his
network heighbors’ choices in the previous period. The infinite population
of physicians are located on a line where each physician is linked to his two
closest neighbors. The line is divided into two regions. One patient type
is more common in the region to the left of the origin and the other type
to the right of it. From a qualitative point of view, this model shows that
interaction between physicians can indeed reinforce regional differences. The
main theoretical result is that social influence generates a long run outcome
where most of the patients in a region are treated in the way most suited to
the typical patient in that region. In a given region, all but a finite number
of physicians on the infinite line always prescribe the treatment that is most
suited to the patient type which is most frequent.
The link between demographics and long run outcomes is thus fairly weak:
whenever one of the two types is more common, then all but a finite number of patients in the infinite region receive the treatment best suited to
this type. This is true whether the dominant type represents fifty-one or a
hundred percent of the patient population. In reality data shows that both
treatments typically coexist although one treatment may be significantly
more common in some regions. It seems reasonable that the relative frequencies of different treatments would depend on a number of underlying
factors such as the interaction network, precise patient demographics, and
strength of social influence. Here, we will focus mainly on a more realistic
description of the characteristics in the patient population. Many variables
such as, age, gender, or insurance status may affect the treatment choice.
If patients are described by a number of such characteristics, each of which
may take several or even continuous values, classifying patient according to
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two types is clearly a simplifying assumption. In our model we can describe
the patient population with an arbitrary continuous or discrete distribution
over types.
The model we propose for analyzing social influence in medical procedure
choice is close to the model of binary choice in Brock and Durlauf (2001) and
to those analyzed in Glaeser and Scheinkman (2002), or Glaeser et al (2003),
although these consider static settings. Ioannides (2005) studies a dynamic
model of binary choice with social interactions, focusing on the influence of
network topology. However, in our case, a different context of application
motivates different modeling choices. In the aforementioned studies as well
as in ours, agents are sensitive to social influence and to a second factor
which is in our context the patient type, drawn from the population of patients and which is in other social interaction models interpreted either as
an unobservable personal characteristic of the decision maker himself or as
a random perturbation of his best response (”noisy choice”). In a dynamic
context, only the second interpretation seems convincing since the random
variables are supposed independent across periods which is unnatural if their
realization is interpreted as a characteristic of the agent. Interpreted as a
random perturbation of the best response the second factor of influence is
naturally modeled by a random variable with a centered and symmetric
distribution. Moreover, in the literature it has been common to choose a
specification of the random variable and the utility function that leads to a
logit choice probability, which allows for analogies with well known models
of interacting particle systems in physics (see Liggett, 1985 or Blume, 1993).
This choice probability depends on a parameter which can be interpreted
as a exploration/exploitation trade off (see Nadal et al., 1998). More randomness implies exploration of the choices whereas less randomness implies
exploitation since the agent chooses his best response with high probability.
Analysis is often carried out letting the noise go to zero.
In our setting, the relevant variable represents the distribution of characteristics in the group of patients. In this context, it is not relevant to let
the randomness go to zero. Moreover, it seems interesting to explore a wider
range of distributions, in particular non symmetric ones, than in the ”noisy
choice” setting. Relating the properties of the distribution of characteristics
to the long run frequencies of treatments is one of the main objectives of
this study.
We may also note that the case of medical procedure choices is a particularly interesting setting in which to study social interactions due to the availimsart ver.
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ability of data concerning patient characteristics. In other contexts choices
are assumed to depend on idiosyncratic preferences of the decision makers which cannot be observed. It should thus be easier to test predictions
from theoretical models against data in the medical procedure choice context than in many other ones where the influence of social interactions has
been analyzed.
In the paper, we have chosen to initially adopt a fairly open and flexible
framework in which we may consider different assumptions about the nature
of the interactions, the distribution of patient characteristics, the interaction
network etc. We then consider some particular cases in which we are able
to characterize the long run treatment frequencies.
1.1. Organization of the material
In section two, we present general framework. We define the markovian dynamics of procedure choices and discuss how to model patient characteristics.
In section three, we characterize the average long run treatment frequencies
for particular choices of the utility functions and the distribution of patient
characteristics. In section four, we compare the effects of social influence
and personal reinforcement and characterize the long run behavior of the
treatment frequencies for uniform interactions and personal reinforcement.
In section five, we analyze the treatment dynamics when the population of
physicians is large and relate it to fixed points of the choice probability.
The final section explores two particular cases of distributions of patient
characteristics.
2. A general framework for medical procedure choice
2.1. Utility and choice of treatment
There are N physicians. Physician i = 1, ..., N has a utility function Ui .
There are two possible treatments, 0 and 1. Each patient is described by
a θ ∈ R, with the convention that higher values of θ make the patient
more suited for treatment 1. We adopt a dynamic model in discrete time.
At t ≥ 1, each physician i receives a patient and prescribes a treatment
ati ∈ {0, 1} so as to maximize his utility function. The utility derived from
each treatment choice also depends on the patient type θ ∈ R and on an
influence variable hi ∈ [0, 1] whose exact significance will be described in a
later section: Ui = Ui (ai , θ, hi ). We make the following assumptions about
the utility function:
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(1) Ui (1, hi , θ) − Ui (0, hi , θ) is strictly increasing in hi and in θ.
(2) Presence of dominant patient types: there is a θh such that for θ > θh ,
Ui (1, h, θ) > Ui (0, h, θ) for h ∈ [0, 1] and there is a θl such that for
θ < θl U (0, h, θ) > U (0, h, θ) for h ∈ [0, 1].
(3) Let θn be the ”neutral” patient type that makes the physician indifferent between choices when the environment is neutral; Ui (1, 12 , θn ) =
Ui (0, 21 , θn ). The physician has no personal preference over treatments
in the sense that for all hi ∈ [0, 1] and all α > 0, Ui (1, hi , θn + α) =
Ui (0, 1 − hi , θn − α).
At time t, the physician chooses 1 if and only if
Ui (1, hti , θ) ≥ Ui (0, hti , θ)
(1)
Since θ is drawn from the patient characteristics distribution, ex ante, at
each time of choice, the probability of choosing 1 is given by
P (Ui (1, hi , θ) − Ui (0, hi , θ) ≥ 0) =: P (1|hi ).
(2)
Thus, the probability P (1|h) contains all the information about utility and
characteristics that is relevant to choice. Often it is convenient to work
directly with the choice probability. The hypotheses (1) and (2) about the
utility imply that P (1|h) is increasing in h and that P (1|1) < 1, P (0|0) < 1.
2.2. Social or Personal Influence
Our assumption about the influence factor hti is that it potentially takes into
account the previous choices of the physicians network neighbors and/or his
own past choices. Thus, the variable that describes the influence on the
physician is a weighted sum of his neighbor variable vit and his personal
experience pti : hti = γvit + (1 − γ)pti . When γ = 1(or 0), we have the cases
where only social influence and only personal experience respectively matters.We also assume that the physician attributes more importance to recent
observations. Accordingly, the past will be discounted at a given rate. The
neighbor variable is assumed to depend on the proportion of the agent’s
neighbors who have chosen the action 1 in the past. If we denote by ati , the
action of agent i at time t and by V (i), i’s neighbors in the network Γ, then
the neighbor state variable is updated by adding the average of the most
recent actions of the agent’s neighbors. It is thus defined recursively by
vit+1 = λ2 vit + (1 − λ2 )
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1
at+1
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where λ2 is the peer discount factor. Similarly, the agents’ own past experiences are summarized by pt+1
= λ1 pti +(1−λ1 )at+1
i
i , where λ1 is the discount
factor for personal experience. A high value of λ2 and λ1 respectively means
that the agents have a long memory. Note that the physician does not necessarily update his observations of his peers’ treatments and those of his own
past treatments at the same rate.
2.3. Modeling patient characteristics
In reality, patient characteristics are likely to be multi-dimensional. However, we will model them by a scalar, which we motivate in the following
discussion. Let each patient p be characterized by a vector of characteristics
(θp1 , ...θpd ), where each θpl is a characteristic relevant to procedure choice. Let
us assume that physicians use this vector to obtain an allover evaluation of
how suited the patient is for procedure 1 so that
e : (θi1 , ...θid ) → θ ∈ R
(4)
with the convention that the higher the value of Θ, the more suited is the
patient for treatment 1. If all physicians use the same evaluation e, we can
simply model patient characteristics by the scalar θ. In the patient population, the characteristics have a distributed given by a random variable Θ.
We should note that for some model specifications, the choice probability,
and thus the outcome, does not depend on the exact distribution of Θ but
only on some statistic of it. For example: if the discount factor is 1 so that
only the last period choice counts, and if each agent has only two neighbors
in the network, then the possible states of the environment variable h is
0, 21 and 1. Since U (1, h, θ) is increasing in h and θ, we can consider the
smallest θ such that U (1, h, θ) > U (0, h, θ). For h = 0, h = 21 and h = 1
there are θ1 > θ2 > θ3 such that this is verified for the different levels of
h. Now clearly, as long as θ ∈ (θ1 , θ2 ) its exact value does not matter since
such a θ requires choice 1 for h = 1 but not for h = 21 The same is true for
the other intervals. Thus only the probability of each of the intervals and
not the exact distribution of Θ matters. When the influence variable takes
more values, the dependence on the patient distribution becomes finer. It is
only when the environment is described by a continuous variable that the
choices depend on the exact population characteristic. Otherwise there are
several distributions Θ 6= Θ̃ that give rise to the same choice probability.
We are interested in situations where the characteristics of the population
of patients is biased towards one end of the spectrum. When characteristics
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are binary, it is straigthforward to define what it means that the type 1, for
instance, is dominant in the population. With continuous patient types this
is not the case. We recall that θn is the neutral patient type. Two possible
definitions are:
(1) (weak dominance) E[Θ] > θn
(2) (strong dominance)for all α > 0, P ([−∞, θn − α]) < P ([θn + α, ∞])
Condition 2 is more restrictive: in terms of choice probability it implies
P (1|h) > 1 − P (1|1 − h) for all h ∈ [0, 1].
Remark 1. We obtain an identical model if the patient distribution is unbiased but all physicians share a common deterministic bias.
2.4. Dynamics and long run quantities of interest
In the framework we have established, we will consider synchronous updating at discrete times t = 1, 2... At these times all agents receive a patient
and make a treatment decision. Agent i takes his decision conditionally on
i=N , (pt )i=N )
and then updates vi and pi . Consequently ((vit )i=1
ht−1
i i=1 t≥1 is a
i
N
N
homogenous ergodic Markov chain on [0, 1] × [0, 1] . Thus there is an invariant measure µ on [0, 1]N × [0, 1]N that describes long run behavior of
the system.
3. Long run behavior a particular case
This general framework gives rise to a precise model by specifying the utility
function U , the distribution of patient characteristics given by Θ and the
interaction structure by a graph Γ. In our first example we will consider a
utility function which corresponds to a choice probability that is linear in h
and patient characteristics that are distributed uniformly on an interval of
length 1 and characterized by its mean. Our reason for this choice is mainly
that we can characterize explicitly the aggregate state variables in the long
run, in this simple case, which gives a benchmark result. We consider a
utility function of the form
1
1
1
1
Ui (ai , hi , θ) = β(hi − )(ai − ) + (θ − )(ai − )
2
2
2
2
(5)
For this utility function, the neutral patient type is θ = 12 . Ex ante, the
probability of choosing 1 for a given value hi of the state variable is:
1
P (ai = 1|hi ) = P (U (1) ≥ U (−1)|hi ) = P (θ ≥ (β + 1) − βhi )
2
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We let Θ follow a uniform on an intervall of length 1 centered in Pα .
If Pα > 21 high types are more common. For this distribution one obtains:
P (ai = 1|hi ) = Pα + 21 −( 12 (β +1)−βhi ) = Pα +β(hi − 21 ). Pα corresponds to
the probability of choosing 1 in absence of a private bias and social influence,
i.e. when choice depends only on patient type.
We recall the parameters in the model:
• β regulates the influence of the variable h. In order to satisfy P (1|1) <
1 and P (1|0) > 0, we must have β < min[2Pα , 2(1 − Pα )]
• Pα is the probability that an agent receives the treatment 1 when
only characteristics matter. High Pα means that individuals suited to
procedure 1 are more common in the population.
• γ ∈ [0, 1] is a variable such that hi = γpi + (1 − γ)vi . A high γ implies
that personal experience is more important than social influence.
• λ2 is a discount factor of the social influence variable such that vit+1 =
P
1
t
λ2 vit + (1 − λ2 ) Cardv(i)
j∈V (i) aj
= λ1 pti +
• λ1 is a discount factor for personal experience such that pt+1
i
t
(1 − λ1 )ai
Under the above assumptions, we want to characterize the long run bePi=N
Pi=N
vi and P = N1 i=1
pi .
havior of the aggregate quantities V = N1 i=1
As we have seen, given the environment hi an agent i chooses 1 with the
conditional probability P (ai = 1|hi ) = Pα + β(hi − 12 ). There is a unique
solution to h = Pα + β(h − 21 ) which gives the unique fixed point of the
choice probability,
Pα − β2
h=
.
(7)
1−β
The following theorem characterizes the expected long run behavior of
the aggregate variables P t and V t :
Theorem 1. Assuming that
(1)
(2)
(3)
(4)
(5)
utility is given by (5)
the characteristic Θ follows a uniform distribution on [Pα − 21 , Pα + 12 ].
All agents have the same degree in the network.
social/personal influence is limited: 2Pα > β
1 dominates in the population in the sense that Pα > 21
Then for all choices of γ, λ2 and λ1 in (0, 1) we have
lim EV 0 [V t ] = lim EP 0 [P t ] =
t−→∞
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Pα − 21 β
,
1−β
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for any initial conditions V 0 and P 0 .
Proof. We define recursive equations for the expectations of the aggregate
state variables P t and V t . At each instant t, the agent i makes a choice
ati that depends on two state variables, one corresponding to the previous
actions of his neighbors, v t−1 and the other one corresponding to his own
previous decisions pt−1 These variables are defined recursively at time t as
P
t+1
1
t
vit+1 = λ2 vit + (1 − λ2 ) CardV
= λ1 pti + (1 − λ1 )at+1
j∈V (i) aj and pi
i
(i)
respectively. We note that since all agents have the same degree l,
X X
X
1
atj =
ati .
Card(V (i)) i j∈V (i)
i
(8)
At a given time t, we can define the aggregate variable V t = (1/N ) i vit
P
and P t = (1/N ) i pti . We can now consider the conditional expectation,
and conditionally on (vit )i and (pti )i , the variables (at+1
i )i are independent.
We have
P
E[V t+1 |(vit )i , (pti )i ] =
P
t+1
1
= 1)|(vit )i , (pti )i ) =
i=1 Card(V (i))
j∈V (i) P (aj
Pi=N
[(Pα − β2 ) + βγpti + (1 − γ)vit )] =
λ2 V t + (1 − λ2 ) i=1
[β(1 − λ2 )(1 − γ) + λ2 )]V t + [β(1 − λ2 )γ]P t + [1 − λ2 ](Pα − β2 ).
λ2 V t + (1 − λ2 )
Pi=N
This last expression is (V t , P t ) mesurable, and by properties of conditional
expectation equals E[V t+1 |V t , P t ]. Similarly we obtain
β
E[P t+1 |V t , P t ] = [β(1−λ1 )γ]V t +[β(1−λ1 )(1−γ)+λ1 )]P t +[1−λ1 ](Pα − )
2
Taking expectations on both sides of (9), we obtain a recursive relation:
β
E[V t+1 ] = [β(1−λ2 )(1−γ)+λ2 )]E[V t ]+[β(1−λ2 )γ]E[P t ]+[1−λ2 ](Pα − )
2
This relation holds for all t. We put a = β(1−λ2 )(1−σ)+λ2 , b = β(1−λ2 )σ,
c = β(1 − λ1 )(1 − γ) and d = β(1 − λ1 )σ + λ1 . Then we obtain the difference
equation
E[V t+1 ]
E[P t+1 ]
!
=
a b
c d
!
E[V t ]
E[P t ]
!
β
+ (Pα − )
2
1 − λ2
1 − λ1
!
We denote by
M=
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c d
!
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We remark that a, b, c and d are all positive and satisfy a + b < 1 and
c + d < 1 when β < 1. Thus it is possible to write
M=
a+b 0 0 c+d
T
where T is a stochastic matrix. We have
n
M =
(a + b)n
0
0
(c + d)n
!
Tn
n =0
Since T is a stochastic matrix and a + b < 1 and c + d < 1, limn→∞ Mi,j
for all i and j. Therefore, the solution of the homogenous equation associated
Pα − 1 β
with (10) goes to zero for n large. We can show that v = p = 1−β2 is a
particular solution to the inhomogenous equation since (v, p) satisfies the
equations
1
p = [β(1 − λ1 )γ]v + [β(1 − λ1 )(1 − γ) + λ1 )]p + [1 − λ1 ](Pα − β)
2
1
v = [β(1 − λ2 )(1 − γ) + λ2 )]v + [β(1 − λ2 )γ]p + [1 − λ2 ](Pα − β).
2
Therefore, the general solution of the difference equation is such that
lim E[V t ] = lim E[P t ] =
t→∞
t→∞
Pα − 21 β
1−β
(10)
Thus in the model described above, in the long run, the expected values
of the private and public state variables are the same and equal the fixed
point whatever are the values of the updating parameters and the weight
given to private and public experience respectively.
4. Social Influence versus Personal reinforcement
In the previous section, we characterized aggregate variables for fairly general interaction structures and parameters but under very restrictive hypotheses about utility and patient characteristics. In this section, we do
the contrary. We consider two particular interaction structures in which we
can consider arbitrary choices of utility functions and patient characteristics. For reasons of analytical convenience, we slightly modify the updating
mechanisms. The first case we consider is average interaction with one period memory. This is a particular case of the previous model with updating
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parameter λ1 = λ2 = 1 so that only the actions in the most recent period is
taken into account. No particular weight is given to personal experience, ie.
γ = N1 . We let the updating be asynchronous. At discrete times, an agent
i ∈ {1, 2, ..., N } is drawn uniformly at random and receives a patient. He
chooses a treatment depending on the patient type and on the average choice
P
t−1
in the last period, N1 i=N
i=1 ai . The second situation that we consider is
the one where each physician only cares about his own previous treatment
choices. The model is not completely within the previous framework, but
this is due only to a minor modification of the updating mechanism. The
agent remembers his own N most recent decisions, which are summarized
in a memory vector v ∈ {0, 1}N . At each time t the agent’s choice is affected by the average value of his memory vector. Thus at time t the agent
P
s
chooses 1 with probability P (1|v t−1 ) where v t−1 = N1 s=t−1
s=t−1−N a is the
average value of his memory vector. He updates this vector by adding his
choice at time t and by dropping the most ancient choice still in memory,
vt−N . In both cases, we have Markov chains on the state space {0, 1}N . In
order to compare the behavior of these systems for different N , we consider
P
the aggregate variables SN = N1 i=N
i=1 ai which is the average action of N
P
i which is a time average over N periods. These
agents and VN = N1 i=N
v
i=1
variables take the values ( Nk )k=N
k=0 ⊂ [0, 1].
Proposition 1. The variables SN and VN both have the same invariant
measure µ. If we denote by µ(k) (rather than µ( Nk ) for notational convenience)
the weight that the invariant measure µ places on Nk , then
N
k
µ(k) =
Pl=N
l=1
!
Qi=k−1 P (1| Ni )
i=0
N
l
!
P (0| i+1
)
N
.
Qi=l−1 P (1| Ni )
i=0
P (0| i+1
)
N
Proof. We prove that the invariant distribution of VN is µ. The proof for
SN can be found in the appendix. We will use the generic notation vk for
P
any v ∈ {0, 1}N such that N1 i=N
i=1 vi = k. We assume that there exists an
invariant distribution that puts the same probability on all elements with
the same mean. We write the detailed balance conditions for an element
vk . There are two possible cases: if the most recent element in the memory
vector vk is 1, then the most recent decision was 1. There are two possible
antecedents of this vector, one where the dropped element was 1 and one
where it was 0. If the dropped element was 1, the unique antecedent is
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13
an element vk , if the dropped element was 0 the unique antecedent is an
element vk−1 , since the updating replaced a 0 by a 1. Thus the detailed
balance condition gives:
µ(vk ) = µ(vk )P (1|
k
k−1
) + µ(vk−1 )P (1|
)
N
N
(11)
In the second case, the most recent entry of the vector vk is 0. In this case,
there is a unique antecedent vk , whose most ancient entry is 0 and a unique
antecedent vk+1 whose most ancient entry is 1. For the element vk such that
the last entry is 0, the detailed balance conditions give
µ(vk ) = µ(vk )P (0|
k
k+1
) + µ(vk+1 )P (0|
)
N
N
(12)
In both cases, the equations simplify to
µ(vk ) =
µ(vk−1 )P (1| k−1
N )
P (0| Nk )
.
(13)
Recursively we find that
µ(vk ) =
i=k−1
Y
i=0
P (1| Ni )
µ(v0 )
P (0| i+1
N )
Since µ is a probability, and since there are
is k,we must have µ(0) =
Pk=N
1+
k=1
N
k
!1
N
k
(14)
!
elements whose average
.
Qi=k−1
i=0
P (1| i )
N
P (0| i+1 )
N
5. The nature of the reinforcement effects
Having obtained the expression for the invariant distribution, we can now
give some properties of its behavior. The fixed points of the function P (1|x)
turn out to play an important role.
Proposition 2. Consider the function
g(x) =
imsart ver.
P (1|x)(1 − x)
.
(1 − P (1|x))x
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Let c > 0, then for any
g(
k
N
14
∈ [c, 1], we have
µ(k + 1)
k
k+1
) − R(N ) <
< g( ) + R̃(N )
N
µ(k)
N
(16)
where limN →∞ R(N ) = limN →∞ R̃(N ) = 0.
Since µ is increasing when g > 1 and decreasing when g < 1, proposition
2 is useful for studying on what intervals µ is increasing and decreasing
respectively. This is the contents of the following proposition.
Proposition 3. Let x0 = 0, let (x2k+1 )k=m
k=0 , be the fixed points of P (1|x)
and let xm+2 = 1. Then, for any > 0, there is N0 such that for every
N ≥ N0 , the function µ is increasing on [x2k + , x2k+1 − ] for k = 0, ..., m
and decreasing on [x2k+1 + , x2k+2 − ] for k = 0, ..., m.
All the peaks of the measure coincide with fixed points of P (1|x). Depending on the number of fixed points of P (1|x), the general shape of µ would be
as in figure 1, 2 or 3. However, Proposition 3 does not provide any information about the relative heights of the peaks of µ. The following propositions
describe the invariant measure when there is a strong bias towards one type
(say type 1) in the sense of.
Proposition 4. If there is strong bias towards high types: for all x ∈ [0, 1],
P (1|x)
1
P (0|1−x) > 1, then for l ∈ N [0, 2 [
j=N
Y−l P (1| j )
µ(N − l)
N
=
>1
N −j
µ(l)
P
(0|
)
j=l
N
Proof. After simplifying the expression
µ(N −l)
µ(l)
(17)
we find the announced iden-
tity. The strong dominance of 1-types in the population implies that
1 for x ∈ [0, 1].
P (1|1−x)
P (0|x)
>
Assuming strong dominance of the high type, we also have the following
result that describes behavior when N is large.
Proposition 5. If there is strong bias towards high types: for all x ∈ [0, 1],
P (1|x)
1
P (0|1−x) > 1, and if x is the only fixed point in [ 2 , 1], then for every > 0
and every M if we define A = N (x − , x + ) there exists N (, M ) such that
µN (A)
µN (Ac ) > M
Proposition 5 is proved in the appendix.
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The results in this section suggest that social interactions and personal
reinforcement can give rise to effects that are similar and by consequence
difficult to distinguish. As we have seen, the direction and magnitude of the
effects that are generated by uniform one period memory interactions in a
large population are identical to those obtained by personal reinforcement
for agents with long memory. If we only want to explain regional variations
that are larger than what is motivated by underlying factors, then social
interaction or personal reinforcement are equally good candidates. The average treatment frequencies that would be observed over a period of time
would be identical. We note however some differences in what would be observed under these two assumptions. In the case of personal reinforcement
only, the invariant measure determines the distribution over time of each
agent’s decisions. At any given date, the decisions of the agents are independent and thus the average decision should coincide with the average action
under µ. The fluctuations around this value correspond to a sum of independent random variables. In the case of average interaction, at a given time,
the choices would be correlated. This also implies that we should see larger
fluctuations over time than if decisions were affected by personal reinforcement. High geographical variation could be explained by social interactions
or personal reinforcement. Large fluctuations of average treatment frequencies over time would suggest social interactions. This would be valid if we
study for example a large unit such as a hospital where all agents are assumed to interact. If on the other hand agents are linked to a small subset of
a larger population and the network is not known it is likely to be difficult
to distinguish between peer effects and effects of personal reinforcement.
6. The influence of patient characteristics on long run treatment
frequencies
In the previous section, we have, assuming either social interactions or personal reinforcement, characterized the long run behavior of the average decision of N agents or alternatively the time average over N periods of the decisions of an individual agent by an invariant measure µ. In these situations,
we can now explore how different assumptions about patient characteristics
affect the long run frequencies of treatments.
6.1. The influence of heterogeneity
In this section, we explore how the variance of characteristics in the patient
population affects the outcome. We want to compare the strenght of the reimsart ver.
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inforcement effects for different levels of heterogeneity. To do so, we will not
consider mean preserving spreads but rather a certain type of transformation
that increases the variance but conserves another property: the proportion
of treatment choices that will prevail in absence of interaction effects, i.e.
in a neutral environment. When these are indentical in absence of interactions, we may compare how the proportions are modified for different patient
characteristics distributions. There are classes of random variables that can
be obtained from each other by the transformation we consider, such as
gaussian variables or uniform distributions on different intervals. The reinforcement effects related to these distributions can be compared based on
heterogeneity. However, the theorem does not allow us to compare arbitrary
distributions with identical treatment frequencies in a neutral environment.
Moreover, the result cannot be improved since one can find distributions
that cannot be obtained from each other by the transformation such that
the one with higher variance has a larger fixed point.
Proposition 6. Let Θ̃ be a spread of Θ that conserves the proportion of
choices in a neutral environment:for a fixed a > 1, Θ̃ = aΘ + m(a) and
PΘ̃ (1| 21 ) = PΘ (1| 21 ). If PΘ̃ (1|h) and PΘ (1|h) both have unique fixed points in
( 21 , 1], denoted hΘ̃ and hΘ respectively, then hΘ̃ < hΘ .
Proof. Let a > 1 be fixed and define Θ̃ = aΘ + m(a) where m(a) is determined by the relation
1
1
1
1
P (U (1, aΘ + m(a), ) > U (0, aΘ + m(a), ) = P (U (1, Θ, ) > U (0, Θ, ).(18)
2
2
2
2
We note that for every h ∈ [0, 1] there is a c(h), increasing in h such that
P (1|h) = P (U (1, Θ, h) > U (0, Θ, h) = P (Θ > c(h)).
(19)
Thus, (18) is equivalement to
1
1
P (aΘ + m(a) > c( )) = P (Θ > c( ).
2
2
(20)
c( 1 )−m(a)
Assuming that the cdf of Θ is strictly increasing, this implies 2 a
=
1
c( 2 ).
For h > 12 , we have c(h) < c( 12 ) and we can write c(h) = c( 21 ) − k with
k > 0. Therefore we have
1
PΘ̃ (1|h) = P (aΘ + m(a) > c(h)) = P (aΘ + m(a) > c( ) − k) =
2
c( 1 ) − m(a) k
1
P (Θ > 2
− ) < P (Θ > c(h)) = P (Θ > c( ) − k) = PΘ (1|h)
a
a
2
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.
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Now, suppose that h̄ is the largest value of h such that PΘ (1|h̄) = h̄.
By the assumption that PΘ (1| 1) < 1, we must have for every h > h̄ that
PΘ (1|h) < h. Since PΘ̃ (1|h) < PΘ (1|h) for every h ≥ h̄, necessarily the
largest fixed point of PΘ̃ (1|h) is inferior to h̄.
We illustrate proposition 6 by a numerical example. We use the utility
function Ui (ai , hi , θ) = β(hi − 12 )(ai − 21 )+(θ− 12 )(ai − 12 ), and we fix β = 12 and
we consider gaussian distributions of the patient characteristics. For every
value of the variance σ, we fix the mean of the characteristics distribution
in such a way that in absence of interaction or personal reinforcement, 60
percent of the patients in region 1 and 40 percent of the patients in region 2 receive treatment one. Thus, in absence of interactions, the relative
frequency of treatments between regions would be the same for these gaussian distributions with different variance. Table 1 shows how the fixed point
varies as a function of σ. With low variance these frequencies could be modified to 20 and 80 percent respectively, making treatment 1 four times more
common in the second region. Such a high variability would not occur when
patients are more heterogenous.
variance
σ=1
Table 1 σ = 12
σ = 13
σ = 14
m(σ)
0.76
0.63
0.59
0.56
fixed–point region 1
0.62
0.67
0.73
0.8
fixed point region 2
0.38
0.33
0.27
0.2
However, we should note that if we compare two arbitrary distributions,
Θ̃ and Θ, such that PΘ̃ (1| 12 ) = PΘ (1| 21 ) and V ar(Θ̃) > V ar(Θ), it is not
necessarily true that hΘ̃ < hΘ . With the same utility function as previously,
we can find a counter example, by comparing, for example, a gaussian distribution with a uniform distribution: the uniform
distribution on an interval
√
of length 1, centered in 0.6 has variance 0.08 < 13 and the associated fixed
point is 0.7 < 0.73
6.2. The influence of minorities with extreme characteristics
It results from proposition 3 in the previous section that whenever the condition P ([0, Θ]) > P ([1 − Θ, 1]) is satisfied, the average of the invariant
distribution favors action 1: Eµ [v] > 21 . We may ask if this remains true if
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the patient population verifies only a weak bias towards 1, E[Θ] > 12 . We
consider a simple example where it is easy to do explicit calculations. Let the
utility function be Ui (ai , hi , θ) = β(hi − 12 )(ai − 21 ) + (θ − 12 )(ai − 12 ), and we
fix β = 12 . We specify a distribution of characteristics such that E[Θ] > 12
is verified but P ([0, Θ]) > P ([1 − Θ, 1]) is not. We let the distribution of
characteristics be discrete and assume that there are only four types whose
probability is:

9

P (Θ = 0) = 50


 P (Θ = 1 ) = P
(Θ = 12 ) = 0
4
 P (Θ = 43 ) = 45



1
P (Θ = 1) =
(21)
50
We fix N = 2, so that h ∈ {0, 12 , 1}. Our specification of utility is then such
that if Θ = 0, the physician always chooses 0, if Θ = 41 , he chooses 1 if
h ≥ 34 , if Θ = 21 , he chooses 1 if h ≥ 12 , if Θ = 34 , he chooses 1 if h ≥ 14 ,
and if Θ = 1, he always chooses 1. Explicit computations of the invariant
distribution show that

81

 µ(0) = 140
µ(1) =
9
(22)
70

 µ(2) = 41
140
5
< 12 , and there are long periods
The average of h under this distribution is 14
where the action 0 is chosen repeatedly. This shows that the fact that that
the average patient type is more adapted to treatment 1 is not sufficient
to guarantee the prevalence of this treatment if there is a greater presence
of very low than very high types. Many models of social interactions show
that the action preferred by the majority is increased to a point where the
minority suffers. This example indicates that social interactions can also
produce situations where the majority falls victim to a minority.
7. Appendix: Proofs
7.1. proof of proposition 1 (average interaction case)
i=k and
By symmetry, the invariant measure places the same weight on (ai )i=1
i=k
(ãi )i=1 if the average decision is the same. In this proof, we will use the noPi=N
tation ak for any element a ∈ {0, 1}N such that N1 i=1
= k. The invariant
measure µ(ak ) is well defined. We use the detailed balance conditions for an
arbitrary fixed element represented by ak . This element can be reached from
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19
itself in two ways: by drawing one of the k agents who previously chose 1
when he chooses 1 again, or by drawing one of the N − k agents who chose 0
before and whose new decision is still 0. An element ak also has antecedents
of type ak−1 . There are k such antecedents: any element where one of the k
1 choices in ak is replaced by a 0 would be an antecedent. Similarly there
are N − k antecedents of type ak+1 . This gives the equation:
k
N −k
k
P (1|k) + µ(ak )
P (0|k) + µ(ak−1 ) P (1|k − 1) +
N
N
N
N −k
P (0|k + 1)
(23)
µ(ak+1 )
N
µ(ak ) = µ(ak )
We will use!the notation µ(k) =: µ({a ∈ {0, 1}N | N1
N
µ(k) =
µ(ak ), the relation (23)gives:
k
Pi=N
i=1
ai = k}). Since
k
N −k
P (1|k) + µ(k)
P (0|k) +
N
N
N −k+1
k+1
µ(k − 1)
P (1|k − 1) + µ(k + 1)
P (0|k + 1)
N
N
µ(k) = µ(k)
(24)
If we write these equations for µ(k)...µ(0) and add them, we obtain:
i=k
X
X
1 i=k−1
µ(i) =
µ(i)[iP (1|i) + (N − i)P (1|i) + iP (0|i) + (N − i)P (1|i)] +
N i=1
i=0
1
1
µ(k)[kP (1|k) + (N − k)P (0|k) + kP (0|k)] + µ(k + 1)(k + 1)P (0|k + 1) +
N
N
1
µ(0)[N P (1|O) + N P (0|0)]
N
Most of the terms cancel and we obtain the relation
k
N −k
P (1|k) + µ(k)
P (0|k) +
N
N
k+1
k
µ(k + 1)
P (0|k + 1) + µ(k) P (0|k)
N
N
µ(k) = µ(k)
(25)
Finally this gives
µ(k) =
imsart ver.
(N − k − 1) P (1|k − 1)
µ(k − 1)
k
P (0|k)
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Recursively we obtain
µ(k) = CkN
i=k−1
Y
i=0
P (1| Ni )
µ(0)
P (0| i+1
N )
(27)
7.2. proof proposition 2
By the definition of the invariant distribution µ, we have
µ(k + 1)
=
µ(k)
N
k+1
!
N
k
!
N
P (1| Nk )Ck+1
(28)
P (0| k+1
N )
We have
P (1| Nk ) N − k
P (1| Nk )N − k
µ(k + 1)
N −k
=
(
>
−
)
k
k
µ(k)
k
k(k + 1)
P (0| N )k + 1
P (0| N )
(29)
P (1| k+1
P (1| k+1
1
µ(k + 1)
N )N − k
N ) N − (k + 1)
<
+
).
=
(
k+1
k+1
µ(k)
k+1
k+1
P (0| N )k + 1
P (0| N )
(30)
We can now write:
P (1| k+1
P (1| Nk )(N − k)
µ(k + 1)
k+1
k
N )
<
(31)
g(
)−
< g( ) +
N
µ(k)
N
P (0| k+1
P (0| Nk )k(k + 1)
N )(k + 1)
P (1|x)(1−x)
k
(1−P (1|x))x . We note that if we fix δ > 0, then for N ∈ [δ, 1],
k
P (1| k+1 )
P (1| N
)(N −k)
R(N ) = P (0| k+1 N
and
R̃(N
)
=
decrease to 0 as
k
P (0| N )k(k+1)
)(k+1)
N
with g(x) =
the terms
N increases since k > N δ.
7.3. proof
proposition 3 As we have seen, we have
g(
k+1
µ(k + 1)
k
) − R(N ) <
< g( ) + R̃(N ).
N
µ(k)
N
(32)
where R and R̃ are such that for any fixed δ > 0, if Nk ∈ [δ, 1], then
limN →∞ R(N ) = limN →∞ R̃(N ) = 0. Let > 0. Consider k and k + 1 such
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k
that k+1
N , N ∈ [x2j + , x2j+1 − ] for some j ≥ 0. The function g attains its
minimum m > 1 on the compact [x2j +, x2j+1 −]. Therefore, it is possible to
k+1
choose N0 such that for N ≥ N0 , µ(k+1)
µ(k) > g( N ) − R(N ) > m − R(N ) > 1.
Thus µ is increasing on this interval. Similar arguments show that µ is decreasing on the compacts [x2j+1 + , x2j − ]
7.4. proof proposition 5
Let xf denote the unique fixed point of P (1|x) in [ 12 , 1]. Fix > 0 and
define A = [xf − , xf + ]. We need to bound µ(l) for all l ∈ Ac . Let
x ∈ N be such that Nx ∈ A. (such an x exists if N is sufficiently large).
We define miny∈[ 1 ,xf − ] g(y) =: m1 > 1. Let us consider m ∈ N such that
m
N
2
2
∈ [ 21 , xf − ]. We have
Y P (1| l ) N − l
Y
µ(x)
(N − m)P (1|m) l=x−1
(N − m)P (1|m) l=x−1
l
N
=
=
g( )
l
µ(m)
xP (0|x)
l
xP (0|x)
N
l=m+1 P (0| N )
l=m+1
For
m
N
∈ [ 12 , x − ], the product contains
N
2
N
2
terms that are larger than m1 .
Thus µ(x) ≥ (m1 ) µ(m). Similar arguments show that for all l ∈ [xf +, 1],
N
there is m2 > 1 such that µ(x) ≥ (m2 ) 2 µ(l).
When
l
N
∈ [0, 21 ] we have µ(l) = µ(N − l)
Qi=N −l P (1| Ni )
i=l
P (1|x)
P (0|1−x)
. By hypothesis
P (0|1− Ni )
k
1
N ∈ [ 2 , xf −
≥ 1 for all x ∈ [0, 1]. If l = N − k with
] [xf + , 1],
then µ(l) ≤ µ(k), and we have already bounded µ(k). If l = N − k with
k ∈ [xf − , xf + ], then µ(l) = µ(N − l)
Qi=N −l P (1| Ni )
i=l
P (0|1− Ni )
S
. We have xf >
1
2
(1|x)
and we may assume xf − 21 ≥ 3. Define minx∈[xf −2,xf −] PP(0|1−x)
= m3 >
1. Thus the product contains at least terms greater than m3 . Finally,
N
µ(A)
1
2 . This quantity is greater than M for suffiµ(Ac ) > N (min(m1 , m2 , m3 ))
ciently large N .
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