Download THE LAW OF LARGE NUMBERS AND DETERMINISTIC

THE LAW OF LARGE NUMBERS AND DETERMINISTIC
APPROXIMATION: A POTENTIAL PITFALL
ROBERT MOLZON†
Abstract. The Law of Large Numbers is commonly used in economics to pass
from stochastic models to deterministic models. It is widely recognized that
one must use caution with this approach when population sizes are moderately
large. However for extremely large populations, it is frequently assumed that
one can justify a deterministic model by appealing to an “Exact Law of Large
Numbers”. We show that under some natural circumstances a deterministic
model constructed by an application of a law of large numbers will not approximate the corresponding finite population model even if the population
is arbitrarily large. The fundamental question of when a random continuum
economy is a good abstraction for a large finte population economy was raised
by H. Uhlig in [19]. This paper provides an answer to that question, and it
also provides a general method for capturing information encoded in the finite
population model that is typically lost by passing to the continuum economy
by applying the law of large numbers.
1. Introduction
The Law of Large Numbers (LLN), in some form or other, is perhaps the tool
most widely used in economics to approximate stochastic models by deterministic
models. It is widely recognized that one must use caution when making such
deterministic approximations, and that for moderate values of population size, the
variance in the stochastic model may play an important role. Moderate population
size may, in some cases, mean quite large populations.
However, it also seems to be commonly believed that if one takes the population
size to be extremely large one will be on firm ground with a deterministic model
obtained through an approximation based on the law of large numbers. By extremely large, one might mean in the limit as the population size tends to infinity.
Because of the complexity of many economic models the task of actually proving
that the deterministic model is a limit of a finite agent stochastic model is frequently neglected. In fact, a significant amount of effort has gone into obtaining a
mathematically rigorous “exact law of large numbers” for continuum populations
in order to avoid the necessity of proving limit results. Examples can be found
in [6, 7, 1, 2, 19]. In this paper we give simple examples that show why it may
sometimes be inappropriate to approximate a finite population stochastic model by
a deterministic model even if the population size is arbitrarily large. In fact, the
examples show that behavior of a stochastic finite population model can remain far
from the behavior of the corresponding deterministic model even as the population
The author would like to thank Bill Sandholm for a very helpful discussion of the ideas presented
in the paper.
† Robert Molzon, Department of Mathematics, University of Kentucky, Lexington, KY 40506,
USA, [email protected].
1
of the stochastic model tends to infinity. On the positive side, we present a general
result that justifies use of the law of large numbers to construct continuum models that approximate finite population models if the right conditions hold. We also
present a general method that allows one to obtain an appropriate continuum model
in case the law of large numbers approach fails to provide a good approximation to
the large finite population model.
In our examples strategy selection will depend upon aggregate costs or rewards
for the entire population. Models in which individual behavior depends upon an
aggregate outcome are typical in continuum models that appeal to a law of large
numbers, and examples can be found in all sub-disciplines of economics. A short
list can be found in Uhlig’s paper, A law of large numbers for large economies [19].
The focus of that paper was on an interpretation and proof of the LLN that avoided
the measurability difficulties that had been pointed out in the context of economic
models by Judd [12]. However, Uhlig’s paper also raised the general question of
the extent to which a continuum economy can serve as an approximation to a large
finite population economy. This paper provides an answer to Uhlig’s question in a
broad setting. Very roughly, if costs, rewards, or payoffs to agents are discontinuous
functions of random shocks or strategies, then variance in these quantities can play
a decisive role in the model. The law of large numbers discards information about
variance. Hence the answer to the question raised in [19] hinges on a careful analysis
of the role of variance in strategy selection or modification.
It has been recognized that average behavior can be misleading, and many nice
examples are presented by J. Miller and S. Page in [16]. In their book, these
authors formulate an agenda for the study of complex systems and in particular
they ask what it takes for a system to exhibit complex behavior. Here we partially
answer that question in terms of the inapplicability of the law of large numbers as a
tool to average out randomness when there are discontinuities in payoff functions or
strategy selection rules. These discontinuities therefore contribute to the complexity
of a system in an important way.
2. The Continuum Model as a Limit
To understand why continuum models may not be good approximations to very
large but finite population models, it will be useful to first summarize the conditions
under which the continuum models are good approximations. Therefore, we review
the basic results from probability that justify the use of a continuum model as an
approximation to a large finite population model in the simplest possible setting.
We first briefly discuss what we mean by “continuum model” since there is more
than one way to make this idea precise.
Continuum models in economics generally take one of two different forms. One
approach is to take the idea of “agent” literally and construct models in which agents
are identified with points of an uncountable set - typically the unit interval [0, 1] in
R. This approach was introduced by Aumann [3] in 1964 and these models are often
described as having a “continuum of agents” (CA model). The second approach
is to forget about individual agents and describe the continuum model in terms of
the distribution of economically relevant traits of the population. For example, if a
model involves only one relevant economic trait described by a single real number,
then a distribution function, F (x), can be used to describe the population. If F
is a continuous distribution function then one could not realize such an economy
2
with finitely many agents. Models that take this approach to a continuum are
often referred to as “distribution economies” (DE model) and were introduced by
Hildenbrand [14]. We note that the “distribution economy” approach does not
preclude uncertainty in the model since the distribution function that describes the
economy certainly could be a random variable itself. In this case one would typically
work with the associated random measure defined by the random distribution.
A brief discussion of the two approaches to modeling large economies can be
found in [11], and there are a number of papers in the literature that describe how
one might pass back and forth from one point of view to the other. If the economic
trait or traits in question are deterministic at the individual level then going from
one approach to the other presents no problem.
Continuum models in economics are generally designed to allow individual risk
but no aggregate uncertainty. If relevant economic traits at the individual level are
assumed to be random variables then certain well know difficulties can arise with
the Aumann approach. These difficulties had already been noted by Doob [8] in
his book on stochastic process, and a number of ways to deal with the difficulties
have been presented in the economics literature [7, 10, 12, 19, 1].
Roughly speaking the transformation between the Aumann continuum model
and the Hildenbrand model takes the form of a “Law of Large Numbers” for the
distribution function. This idea can be made more precise by specifying a probability space of agents, (A, A, Q), together with a probability space, Ω, to model the
uncertainty. Suppose Xa (ω) denotes real valued random variables indexed by the
agent space with common distribution F (x) and the random variables satisfy some
form of independence condition. The “law of large numbers” as it is frequently used
in economics then means
Z
(2.1)
Q {a ∈ A |Xa (ω) ≤ x } = I(−∞,x] (Xa (ω)) dQ(a) = F (x)
for almost every ω ∈ Ω. Here I denotes the indicator function and the equation
says that for almost every realization of random shocks, the fraction of agents who
experience a shock less than or equal to x is equal to F (x). A clear discussion of
this idea with additional details can be found in the paper of Berti, Gori, and Rigo
[4]. As noted above, there are a number of issues related to measurability, and
these issues have been addressed by the authors cited above.
A corresponding result that obtains the distribution as a limit is provided by the
Glivenko-Cantelli theorem. If {Xk }1≤k<∞ is a countable family of i.i.d. random
variables with common distribution F , and
n
1X
I(−∞,x] (Xk ) ,
Fn (x) =
n
k=1
then,
(2.2)
lim Fn (x) = F (x) a.s.
n→∞
and the convergence is uniform in x.
A somewhat less general version of equation 2.1 only requires that the “average”
of the Xa over a suitably large set of agents equal the expected value of the Xa .
This statement takes the form
Z
1
Xa dQ(a) = µ
(2.3)
Q Ã a∈Ã
3
where the Xa are assumed to have the common finite mean µ, and the equality
holds almost surely on the underlying probability space. The set à is assumed to
be a subset of A of positive Q measure.
The corresponding result in the form of a limit is the standard LLN from probability. Suppose the Xa are uncorrelated for a ∈ A, Ã is any infinite countable
subset of A, and the an denote elements of Ã. If Xan have common finite mean µ,
and the variances of the random variables Xan satisfy a mild growth condition as n
tends to infinity, a version of the LLN proved by Doob [8] says that almost surely
n
(2.4)
1X
Xak = µ.
lim
n→∞ n
k=1
The “continuum of agents” approach to continuum models is often justified by
appealing to the existence of a measure Q (under the appropriate assumptions) that
satisfies equation (2.1) or equation (2.3). If one were only interested in describing
the random shocks in a continuum model then these equations, (2.1), (2.2), (2.3),
and(2.4), tell us that the continuum model is a good approximation to the finite
agent model if the population size is large. On the other hand, if the costs or rewards
of agents are functions of the distribution of shocks or the mean of the distribution
of shocks, then these equations are not sufficient to justify the continuum model as
a good approximation to the large finite population model.
In this paper we are concerned with the question of when a continuum model is a
good approximation to a finite agent model when costs or rewards are functions of
random shocks. We shall take the distribution economy approach to the definition
of continuum model and show that a DE model obtained by an application of the
law of large numbers may not be a good approximation to a large finite population
economy regardless of the size of the finite population. The interested reader should
have no trouble reformulating the examples and results for CA models.
The distribution economy continuum model will be defined through some set of
distributions that characterize economically relevant traits of the population. Distributions may describe deterministic traits, stochastic traits, or both. In general
the distributions that are used to describe the model will be multivariate and joint.
We shall limit the type of models to those for which the relevant aggregate property is the sample mean of random shocks or endowments of agents in the finite
population case and the mean in the continuum model. We briefly describe general
features of a typical model to make these ideas clear. The exact nature of the
strategy selection mentioned in the example below is not important at this point
since we are only trying to outline the elements that define the finite population
model and the DE analog.
Example 1. A Random Endowment Model.
Consider a population consisting of individuals who receive a random endowment.
The endowments of individuals are modeled as i.i.d. random variables from a distribution F . After observing her individual endowment, an agent selects a strategy
from a set of possible strategies to optimize a payoff that will depend upon endowments and strategies. The payoff to each individual is then determined as a
function of the individual’s random endowment and strategy, and the endowments
and strategies of all other agents in the population.
4
A finite agent model of strategy selection will necessarily be a stochastic model.
In particular, the average initial endowment (the empirical distribution) of the population will be a random variable. However the corresponding continuum model
will be deterministic, and the DE continuum model will be described by the distribution of endowments, the distribution of strategies selected by agents, and a payoff
function that maps the distribution of endowments and distribution of strategies
to a distribution of payoffs.
Our example above very roughly follows an example presented in [19]. However
the examples we present below differ from the example presented in [19] in one
important way. Uhlig uses incentive compatibility to demonstrate that a continuum
model may not be a good approximation to a finite population model. In the
continuum case, the behavior of a single agent cannot change aggregate outcome.
Hence in his example, the continuum model is not a good approximation to the
finite agent model because incentive compatibility is lost. In particular, one agent
can change strategy and not impact payoffs in the continuum model. We take the
DE approach to continuum models in this paper, and therefore it does not make
sense to talk about the shock or strategy of an individual since the (continuous)
distribution will not change if one changes the values (shocks or strategies) assigned
to a single individual. The examples we give below show that the distribution of
payoffs may be quite different in the finite agent model and the continuum model
even if the population size of the finite agent model is arbitrarily large. Hence,
in our examples, even if one ignores the behavior of a countable set of agents in
the continuum model, the continuum model still does not approximate the finite
population model.
Our examples show that variance can play an important role in policy optimization, and a reasonable finite population model may not converge to the corresponding continuum model as population size tends to infinity. The primary difficulty
involves discontinuities in cost or reward functions. It is not surprising that these
functions, which define the essential features of a model, are often discontinuous.
For example, a government may complete a public goods project or impose a new
tax, and a firm may initiate a price war or create a new production facility. Each
of these scenarios will create a discontinuity in reward or cost. Discontinuities may
also be created by random events such as natural disasters that impact the population as a whole. Many of the examples presented in [16] exhibit discontinuities in
rewards or payoffs, and these discontinuities are often referred to as triggers.
While it may be clear that discontinuities are a natural feature of economic
models, it is perhaps not so clear when and why these discontinuities may make
it difficult to approximate a very large finite population economy by a continuum
economy. The main point that our examples will demonstrate can be succinctly
summarized as follows.
If there are discontinuities in policies, then even small variance
matters. But, unfortunately the Law of Large Numbers discards
information about variance. Consequently, a continuum model
based on an assumption that random shocks “average out” may
not approximate a finite agent model even with arbitrarily large
population size.
The main objectives of this paper are:
5
(1) Provide simple natural examples to show that a continuum model obtained
through a LLN will not in general be a good approximation to the corresponding finite agent model if payoff (policy) functions are discontinuous,
(2) Provide sufficient conditions under which a continuum model, obtained
through a LLN will be a good approximation to the corresponding finite
agent model, and
(3) Provide a general method of obtaining a continuum model that preserves
the information about variance necessary to make the continuum model a
good approximation to the finite agent model.
3. A Tax Example
Consider a population of agents each of whom owes a tax of 1 dollar. Suppose
individuals decide to pay this tax with a certain probability p , which may be less
than one. If all agents pay the tax, then the state realizes the total tax owed.
However, some agents will not pay the tax, and so we suppose that the state will
audit individuals with probability q if the realized revenue falls below some fraction
λ of the total amount owed. If an audit occurs and an individual is found to have
not payed the 1 dollar tax, that individual will be fined an additional 1 dollar and
thus pay 2 dollars. If the individual has not paid the tax and is not audited, she
pays nothing. Consider the problem of determining a value of p such that the
expected tax paid by individuals is minimized.
We examine the optimization problem posed above in three distinct ways. First
we assume that the population is a continuum and that an “exact law of large
numbers” holds. The “exact law of large numbers” assumption more precisely
means that we assume that exactly the fraction p of the population initially pays
the tax.
We next examine the optimization problem as an n- person game with strategy
set for each individual equal to the unit interval, [0, 1] representing the probability
that the agent pays the tax. In this sense we consider a value p as a pure strategy
for the agent, and we look only for pure strategy equilibria. Alternatively, one
could could look for mixed strategies for the two strategy (pay or do not pay)
game. However, when we compare the n- person game approach with the continuum
economy approach it is a bit more convenient to work with the continuum of pure
strategies p in [0, 1] for agents.
In Section 5, we examine the optimization problem by approximating the number
of agents who pay the tax by a normal distribution. In other words, instead of
using an “exact law of large numbers” to obtain a deterministic approximation to
the stochastic optimization problem, we obtain a continuum model by using the
central limit theorem. The advantage of this approach is that the information
provided by the variance in the number of agents who pay the tax is not lost.
3.1. Continuum Economy Approach. In this section we construct and solve
a continuum economy model of the tax problem described above. We assume
the distribution of strategies is degenerate, that is, the entire population selects
the same value p as the probability that she pays the tax. Since the continuum
economy is motivated by the LLN, this implies that the total fraction of tax due
to the amount actually paid is also equal to p. Denote this initial payment fraction
by To (p) = p. Following the description of the example given above, if To (p) ≥ λ
the government will not conduct an audit and the final total fraction of tax paid to
6
tax owed, which we denote by Tf (p), will be p. One the other hand, if To (p) < λ,
then after the state conducts audits, a fraction of the population will pay a tax of
0, a fraction will pay a tax of 1, and a fraction will pay a tax of 2. These fractions
depend upon the strategy distribution, the aggregate outcome, and the government
policy. We assume q, the audit probability, satisfies q > 1/2. The three possible
payments and the fractions of the population that pay the corresponding amount
are given by
Amount
Fraction
0
(1 − p)(1 − q)
.
1
p
2
(1 − p)q
Finally we compute, as a function of p, the final aggregate fraction of tax paid
to tax owed. It is
2q − (2q − 1)p if 0 ≤ p < λ
Tf (p) =
.
p
if λ ≤ p ≤ 1
Under the assumption that q > 1/2 , it follows immediately that the minimum
value of Tf (p) occurs when p = λ.
The continuum economy model described above can be summarized by saying
that if individuals pay the tax with probability p = λ, then the state will collect
exactly the fraction λ of the owed tax. Furthermore, any other value of p would
result in a higher payment of tax by the individuals in the population.
Motivation for the continuum model depends heavily upon the assumption of an
“exact law of large numbers”. In particular, this assumption is used to determine
the initial fraction of tax paid to tax owed, and it is used to determine the final
fraction of tax paid to tax owed (after implementation of the state’s audit policy).
3.2. The stochastic finite population model. We now describe the corresponding finite population model with n agents. In keeping with the main point of the
paper, we start with the degenerate distribution of strategies - all agents pay the
tax with probability p . However, in order to better justify this assumption we shall
also analyze the example as an n- person stochastic game and show that there are
precisely two types of Nash equilibria.
Assume each agent pays the owed tax of 1 dollar with probability p. Then the
amount each agent actually pays is a random variable, Xk , taking values 0 or 1.
In addition the initial total fraction of actual tax paid to tax owed is a random
variable that we denote by To (p). In terms of the random variables Xk ,
n
1X
Xk .
To (p) =
n
k=1
Since
Sn =
n
X
Xk
k=1
is a binomial random variable with parameters n and p, we know the distribution
of To (p). Note that for any value of p the strong LLN says that
To (p) ⇒ p a.s.
as the population size n tends to infinity.
7
Now we compute the final (after implementation of the government audit policy)
distribution of total tax paid to total tax owed. We again denote by Tf (p) the random variable that represents this fraction. Let B(m, p) denote a binomial random
variable with parameters m and p. Let Y` denote the final tax paid by agent `. It
is easy to compute the distribution of Y` .
Y`
0
1
2
Probability
(1 − p) [Prob (B(n − 1, p) ≥ λn) + (1 − q)Prob (B(n − 1, p) < λn)]
p
(1 − p)qProb (B(n − 1, p) < λn)
Since
n
Tf (p) =
1X
Y` ,
n
`=1
one could write down an explicit expression for the distribution of Tf (p) although
the expression would not be particularly useful. What we really want to know is
whether the expression for Tf (p) in the discrete case converges to the expression
for Tf (p) we obtained in the continuum case.
One would like to apply a law of large numbers to determine the limiting value
of Tf (p) as n tends to infinity. A bit of care is needed here since the distribution
of the random variables Y` depend upon n and the Y` are not independent. Since
we provide an alternative method analyzing this model later in the paper we give
only a rough outline of the computations needed to find the limiting value of Tf (p)
here. We consider two separate cases. First suppose p 6= λ. If p < λ , then
Prob (B(n − 1, p) < λn) → 1
and
Tf (p) ⇒ 2q − (2q − 1)p a.s.
Similarly, if p > λ,
Prob (B(n − 1, p) < λn) → 0
and
Tf (p) ⇒ p a.s.
Now suppose p = λ. In this case
Prob (B(n − 1, λ) < λn) →
1
2
and
E [Tf (p)] → q + p(1 − q).
In particular, Tf (p) does not converge in distribution to 2q + (2q − 1)p which is the
value of Tf (p) of the continuum model. We can summarize the comparison between
the finite population model and the continuum model as follows.
• If p 6= λ then the finite population model converges to the continuum model.
• If p = λ then the finite population model does not converge to the continuum model. In particular, the finite population model does not converge
to the continuum model at the value of p that is optimal in the continuum
model.
8
Note that for all values of p, there is no difficulty in the convergence of the initial
amount of tax paid. The difficulty occurs at the optimal value of p after implementation of the audit policy.
Finally in this subsection we state the result that justifies considering the symmetric case in which all agents select equal probabilities with which to pay the tax.
The proof is straightforward and omitted.
Proposition 2. For the stochastic tax game the only Nash equilibria are those with
all probabilities equal to 0 or 1 and the symmetric Nash equilibrium with all probabilities equal. The value of p that corresponds to the symmetric Nash equilibrium
is given as the solution of the equation (in the unknown p)
1 − 2qProb (B(n − 1, p) < bλnc) = 0,
where bλnc denotes the smallest integer greater than or equal to λn.
It is now easy to compute the total tax collected by the state if the agents select
one or the other of these strategies.
Proposition 3. Under the symmetric Nash equilibrium, the total tax paid is a
random variable and the expected value is equal to n. Under the Nash equilibrium
with all probabilities equal to 0 or 1, the total tax paid is deterministic and equal
to bλnc. The fraction of the collected tax to the owed tax is asymptotic to λ as n
tends to infinity.
This result emphasizes the extent to which the continuum model fails to approximate the finite agent model. Furthermore, if the agents act in a homogeneous
fashion and initially pay the tax with probability λ then the expected amount
collected by the state equals the tax owed and the final (after a possible audit)
expected amount paid by each agent equals 1. This is in stark contrast to the
result obtained by the continuum of agents model. In that case each agent pays an
expected amount equal to λ and the fraction of the owed tax collected by the state
equals λ. This contrast between the two cases holds for all n arbitrarily large.
In Section 5 below we show how to analyze the tax example by using the central
limit theorem which will give a continuum model that is a good approximation to
the finite agent model.
4. A Random Matching Example
For our second example we consider an inherently random process that is very
often approximated by a deterministic process through an application of a “law
of large numbers”. Models in which random matching serves as random shock are
perhaps more often approximated by deterministic process because the probability
distribution defined by a random matching process is considerably more difficult to
work with than the binomial distribution that appears in the tax example above.
However a difficulty similar to the one presented in the tax example can crop up in
the matching models. Here is an example.
Consider a population of agents of two types, a and b. These types posses some
complementary resource so that when an a type is paired with a b type, production
or beneficial trade occurs. Suppose that some of the benefit of the production will
be retained for consumption by the paired agents and some of it will be contributed
to a public resource project. If there is sufficient total contribution to the public
project, it will be completed and return a benefit to each member of the entire
9
population which is then consumed. We assume that the nature of the project is
such that the resources needed for completion are proportional to the size of the
population it will serve. The agents must decide how much to consume and how
much to invest in the public project to maximize individual consumption.
We model this simple example more precisely by assuming that there are n type
a agents and n type b agents in the population1. Suppose agents are simultaneously
randomly matched with each match equally likely. If a type a is paired with a type
b then both agents realize a benefit of 1 unit of consumable good, and we say the
agent is favorably matched. If two type a agents or two type b agents are paired
neither realizes a benefit. Before matching, agents commit to an investment of x
in a public works project in case they are favorably matched. They consume the
remainder, 1−x, of the benefit of a favorable match. If the total amount contributed
to the project exceeds an amount P multiplied by the population size, the project
is completed and returns an amount r to each agent in the population. The agents
wish to optimize expected consumption.
Let Nab denote the random variable equal to the number of {a, b} pairs in a
random match of the 2n agents, and assume that each agent invests x if favorably
matched. The amount of good available to an agent for consumption , C(x), is
given in terms of Nab as follows.

1 − x + r if Nab x ≥ nP
and the agent is favorably matched



r
if Nab x ≥ nP and the agent is not favorable matched
.
C(x) =
1−x
if Nab x < nP
and the agent is favorably matched



0
if Nab x < nP and the agent is not favorably matched
4.1. Continuum Economy. We first consider how one would model the consumption - investment problem in a continuum economy. Since the number of matches
of type {a, b} does not make sense in a continuum model, we describe the matching
process in terms of Xab , the fraction of pairs of type {a, b} in a simultaneous bilateral match of the population. The “Law of Large Numbers” implies that with equal
fractions of type a and b agents in the population, we should take Xab = 1/2 in
the continuum model. One-half of the matched pairs will be type {a, b}, one-fourth
will be type {a, a}, and one-fourth type {b, b}. 2. We shall assume that
r > P,
and find the optimal investment amount. As a function of x, the expected consumption, E[C(x)], of an agent is given by
(1/2) (1 − x)
if 0 ≤ x < 2P
E[C(x)] =
.
(1/2) (1 − x) + r if
x ≥ 2P
Under the assumption that r > P it follows immediately that the optimal value of
x is 2P , and the maximum expected consumption is
E [C(2R)] =
1
− P + r.
2
1One could allow a fraction α of type a agents and a fraction β of type b agents. The entire
example goes through with very minor modifications and the same conclusions can be drawn.
2If the fraction of type a agents in the population is α and the fraction of type b agents is β,
then the law of large numbers tells us that the (limit) fraction of pairs of type {a, b} equals 2αβ.
10
4.2. Finite Economy. Now consider what actually occurs when each agent uses
an investment amount x = 2P in a random match of the n type a agents and n type
b agents. If n is sufficiently large, then with probability close to 1/2, the realized
value of Xab = Nab /n will be less than 1/2 and with probability approximately
1/2 the realized value of Xab will be greater than 1/2. Since the project will be
completed only if Xab x ≥ P , it follows that the expected payoff to an agent when
x = 2P will be approximately
1
E[C(2RP )] u (1/2)( − P + r) + (1/2)
2
1
−P
2
=
1
1
− P + r.
2
2
which differs from the expected consumption by an agent in the continuum model
by r/2. Note that this discrepancy holds for arbitrarily large population sizes. In
other words, suppose we analyze the consumption-investment problem as a stochastic optimization problem and look only for symmetric strategies (all select the
same contribution value x). If we use the solution suggested by the deterministic
approximation (obtained by assuming an exact law of large numbers), the limiting
value, as population size tends to infinity, of optimal expected consumption will
not approach the value obtained via the deterministic model.
In fact the discrepancy between the deterministic continuum model and the
stochastic model for arbitrarily large populations is even more serious when r > P
but r is close to P . In that case the agents will be worse off in the stochastic model
if they use a value of x = 2P than they would be if the used x = 0 (that is, no
investment).
5. Discontinuities in Payoffs
It should be clear from the two examples that the difficulty encountered in passing from the finite population model to the continuum model is connected with the
discontinuity in the cost (or reward) to agents as a policy shift occurs. We now
formulate a general finite agent model and show exactly when a discontinuity in
costs or rewards will be a problem.
We consider finite agent economies that can be completely described in terms of
a multivariate random variable together with a payoff function that assigns a payoff
to values of the random variable for all agents. Hence the payoff to an individual
will not only depend upon the individual realization of the random variable, but
also on the aggregate realization for the entire population. This is a general setting
that includes the tax example we presented earlier as well as many cases found in
the literature.
We assume X is a random variable with values in RM with distribution function
F . We do not exclude the possibility that some components of X may describe
deterministic properties of agents. Assume a population of n agents and let Xk for
k = 1, ..., n be independent realizations of the random variable. We shall refer to
the realization, Xk of the distribution F for an individual as an individual shock,
although some of the components of Xk might describe deterministic types or attributes. Let τn denote a symmetric function on the n-fold product RM ×· · · × RM
with values in RL for some L. Hence each component of τn is a real valued symmetric function. The two most important examples are the empirical distribution
11
and the empirical mean. The empirical distribution is defined by
n
(5.1)
1X
IA(x) (Xk )
n
Fn (x1 , ..., xM ) =
k=1
where
A(x) =
M
Y
(−∞, xi ].
i=1
For a fixed value of x = (x1 , ..., xM ) , Fn (x) is a random variable with values in R.
The empirical mean is defined by
n
µn =
1X
Xk .
n
k=1
Hence µn is random variable with values in RM .
We shall assume that the function τn is chosen in such a way that as n tends to
infinity, τn converges in distribution to a random variable τ . So, for example, the
strong law of large numbers tells us that µn converges a.s. to µ, the multivariate
mean of F . The ith component of µ is just computed from the ith marginal
distribution of F . Similarly, a generalization of the Glivenko-Cantelli theorem (see
[9]) states that
Fn (x1 , ..., xM ) ⇒ F (x1 , ..., xM ) a.s.
where the convergence is uniform in x = (x1 , ..., xM ).
Now suppose individual agents receive a “payoff” that depends on individual
realization of the distribution F and the empirical distribution defined by the realizations of each agent in the economy through a function τn as defined above. In
other words the payoff to an individual in an n agent economy is given by
Y` (n) = f (X` , τn (X1 , ..., Xn ) .
For simplicity of the exposition we assume that f is real valued although one could
consider RK valued payoff functions as well.
Example 4. We can expand upon our earlier Example 1 in this notation by assuming individuals receive a random endowment X from a distribution F and as
strategy “report” an amount X̃. We then let
!
n
n
X
X
(5.2)
Y` (n) = µn I{0}
Xi −
X̃i
i=1
i=1
where I denotes the indicator function. Of course, the actual “payoff” to agent `
will depend upon exactly how X̃ is defined, but in any case we see that Y` (n) will
equal 0 unless the total reported endowments equals the total actual endowments.
If, for example, X̃` is defined (through some mechanism) to equal X` then we simply
have Y` (n) = µn , the empirical mean.
We can now state the first convergence result that justifies a continuum model
as a good approximation to a finite agent model.
Theorem 5. Let Df denote the set of discontinuities of the payoff function f in
case individual payoff depends on the individual shock and a symmetric function
τn of the population shocks. Assume individual shocks, X` , are realizations from
12
a common distribution F and τn converges in distribution to a distribution τ as n
tends to infinity. If
Prob ((X, τ ) ∈ Df ) = 0
then for each agent
Y` (n) = f (X` , τn (X1 , ..., Xn )) ⇒ f (X, τ )
in distribution as n tends to infinity.
Proof. Note that by our assumptions on the domain of f , Df ⊂ RM × RL . The
continuous mapping theorem of probability (see [5] ) says that if g maps the metric
space M into the metric space M̃ and Prob (X ∈ Dg ) = 0, then Xn ⇒ X in
distribution implies that g (Xn ) ⇒ g (X) in distribution . In our case, we assume
the (multivariate) random variables τn converge in distribution to τ , hence (X, τn )
converges in distribution to (X, τ ). From the assumption on f , the result follows
immediately.
Many, and perhaps the majority, of continuum models that rely on an appeal to
a law of large number take τn = µn , the empirical mean of random shocks. Both
Example 1 and the tax example are of this form. Even in the case of random matching models, the fraction of pairs of a particular type is usually the aggregate feature
of the model that plays the major role in policy selection or policy modification (see
for example [15, 18]).
With the above convergence result at hand we can now easily understand the
difficulties presented by our examples. We first consider the tax example in the
context of this result.
Example 6. We rewrite the tax example in the notation used above and explain
why the continuum model cannot be a good approximation to the finite agent
model. Let X denote the initial amount of tax paid by an individual. Hence X is
a random variable taking values 1 and 0 with probabilities p and 1 − p respectively.
We assume p = λ. Let Q be the “audit” random variable taking values 1 and 0
with probabilities q and 1 − q respectively. Define the function ϕ(t) by ϕ(t) = 1 if
t < 0 and ϕ(t) = 0 if t ≥ 0. Define
Y` (n) = f (X, µn ) = X` + 2(1 − X` )Qϕ (µn − λ)
where µn denotes the empirical mean as usual. By the strong LLN µn converges a.s. (hence in distribution) to the constant random variable λ. However,
Prob ((X, λ) ∈ Df ) = 1 since ϕ is discontinuous at 0. Hence we cannot use the
continuous mapping theorem to conclude that Y` (n) converges to f (X, µ) = X.
Essentially the same difficulty occurs in the random matching example. One
could rewrite the payoff in the random matching example in the same way that we
wrote the payoff to individuals in the tax example and observe that the probability
that the limiting (constant ) random variable lies in the discontinuity set of the
payoff function is 1 and not 0.
In the next section we present a simple trick that allows one to rewrite the payoff
function in such a way that one can apply the continuous mapping theorem to get
an appropriate continuum model.
13
6. Stochastic Continuum Models
In this section we show how to make simple modifications of finite agent models
that avoid the difficulties presented by discontinuities in payoff functions. As we
pointed out in the Introduction, the LLN essentially discards information about
variance and if the payoff function has discontinuities, variance can play a critical
role in realized payoff. In particular, if the payoff function is discontinuous, then
minute changes in aggregate or average shocks may drastically change payoff values.
We first show how a simple modification of the tax example will lead to a continuum model that accurately approximates the finite population model.
Example 7. We assume the same notation and parameters as in Example 6. We
assume that agents pay the tax of 1 with probability p. The payment by an agent
in the original model with n agents is
!
n
1 X
(
Xi − λn) .
(6.1)
Y` (n) = f (X` , µn ) = X` + 2(1 − X` )Qϕ
n i=1
p
Suppose we now replace the factor 1/n in the argument of ϕ by 1/( np(1 − p) so
that
!
Pn
i=1 Xi − λn
p
(6.2)
Y` (n) = f (X` , µn ) = X` + 2(1 − X` )Qϕ
.
np(1 − p)
Note that we have not changed the value of Y` (n) with this change in normalizing
factor, so the right side of equation (6.1) equals the right side of equation (6.2). If
p = λ and we now take a limit as n tends to infinity, the argument of ϕ converges in
distribution to the standard normal random variable Z by the central limit theorem.
If p > λ, the argument diverges to +∞ and if p < λ the argument diverges to −∞.
With this new normalization
Prob (Z ∈ Dϕ ) = 0,
and we can apply the continuous mapping theorem. Note that in the case p 6= λ,
the original normalization of 1/n would have sufficed to obtain a limit distribution
for Y . Hence, the final tax payment of an agent in the continuum model is

X + 2(1 − X)Q
if p < λ

X + 2(1 − X)Qϕ (Z) if p = λ .
Y =

X
if p > λ
We can now easily compute the expected tax
model.

 p + 2(1 − p)q
λ + (1 − λ)q
E [Y ] =

p
paid by agents in the continuum
if
if
if
p<λ
p=λ .
p>λ
The continuum model obtained as a limit with the new normalization is therefore
consistent with the heuristic explanation of the behavior of the finite agent model
for large populations that we provided in Section 5.
In a similar manner we can determine an appropriate continuum model for the
consumption-investment problem with random matching.
14
Example 8. We rewrite the consumption for individual agents in our random
matching example by introducing a random variable W which takes on the value 1
if an agent if favorably matched (matched with an agent of opposite type) and the
value 0 if the agent is not favorably matched. Let ϕ(t) = 0 if t < 0 and ϕ(t) = 1 if
t ≥ 0. Then for x > 0,
C(x) = (1 − x)W + rϕ (Xab x − P )
Nab −
xNab − P n
= (1 − x)W + rϕ
= (1 − x)W + rϕ
n
n
P
x
!
n
where the last equality follows since ϕ(kt) = ϕ(t) for any positive p
constant k. We
now replace the n in the denominator of the argument of ϕ with n(1/2)2 since
this will not change the value of C(x). We obtain
!
Nab − Px n
p
C (x) = (1 − x)W + rϕ
.
n(1/2)2
Now if we take x = 2P and we let n tend to infinity, the argument of ϕ converges in
distribution to the standard normal random variable by the central limit theorem
for random matching (see [13, 17]). If x > 2P the argument diverges to +∞, and
if 0 < x < 2P , the argument diverges to −∞. In any case, we may apply the
continuous mapping theorem since, as in the tax example, Prob (Z ∈ Dϕ ) = 0.
Therefore, the appropriate continuum model that approximates the finite agent
stochastic model is itself stochastic and defined by the random variable that specifies
consumption of agents for a given level of investment;

W
if
x=0



(1 − x)W
if 0 < x < 2P
.
C(x) =
(1 − 2P )W + rϕ(Z) if
x = 2P



(1 − x)W + r
if
x > 2P
It is again easy to compute the expected consumption of individual agents.

1/2
if
x=0



(1 − x)/2
if 0 < x < 2P
E [C(x)] =
(1/2) − P + (r/2) if
x = 2P



(1 − x)/2 + r
if
x > 2P
We see again that this continuum model correctly approximates the finite population model for large population sizes with each possible value x of investment.
7. Conclusion
The Law of Large Numbers is frequently used to justify deterministic continuum
models as approximations to stochastic finite agent models. We provided some simple examples to show that if payoffs or policies that define models are discontinuous
functions of random shocks, then the continuum model obtained by “averaging out”
random shocks may not be a good approximation to the finite agent model even
for arbitrarily large population sizes. However, a good continuum approximation
can often be obtained if one is able to write the payoff functions for the finite agent
models in a way that allows a renormalization that does not change the value of
the payoff, but has the property that the probability that the resulting limit distribution takes values in the discontinuity set of the payoff is equal to zero. The
15
main mathematical tools needed to obtain these results are the continuous mapping
theorem for random variables and central limit theorems.
We believe there is a significant additional advantage to the approach we have
outlined in Section 6. As noted in the Introduction, it is well recognized that one
must take care when using continuum models as approximations to finite agent
models if population sizes are only moderately large. But, as pointed out in [16],
the cases between the extremes of very small population sizes (two agents) and infinitely large population sizes are often the most interesting. By using a continuum
model obtained through an application of a central limit theorem and the continuous mapping theorem, one obtains good approximations even for moderately large
population sizes. Thus a model that is intractable because of the stochastic interaction of individual agents may have a tractable stochastic continuum approximation.
In this paper we have concentrated on the difficulties to approximation by continuum models caused by discontinuities in payoff functions. These discontinuities
can make a critical difference in payoff for very small variance in aggregate shock. A
similar problem occurs in infinite time horizon models with random matching. For
these models, very small variance in random shock can accumulate over the infinite
time horizon so that the “steady state” for a continuum model may be quite different from the limit of the steady state of a finite population model as population size
tends to infinity. In [18] the authors show that the behavior of an evolutionary game
with a deterministic matching process can be quite different from the behavior of
the same evolutionary game with a random matching process. As population size
increases, a random matching process approaches (in probability) a deterministic
matching process by a law of large numbers for random matching (see [6]). One
would like to know the extent to which an evolutionary game with a deterministic
matching process is a good approximation to an evolutionary game with a random
matching process if the population size is extremely large. We leave a discussion of
this problem for a future paper.
References
[1]
C. Alós-Ferrer, Random matching of several infinite populations, Annals of Operations Research, 114 (2002), 33-38.
[2] C. Alos-Ferrer, Dynamical systems with a continuum of randomly matched agents, Journal
of Economic Theory 86 (1999), 245-267.
[3] R. Aumann, Markets with a continuum of traders, Econometrica, 32 (1964), 39-50.
[4] P. Berti, M. Gori, and P. Rigo, A note on the law of large numbers in economics, IDEAS,
December 2009.
[5] Billingsley, Probability and Measure, John Wiley & Sons, 1995.
[6] R. Boylan, Laws of large numbers for dynamical systems with randomly matched individuals,
Journal of. Economic Theory 57 (1992), 473-504.
[7] D. Duffie and Y. Sun, Existence of independent random matching, The Annals of Applied
Probability 17 (2007) 386-419.
[8] J. Doob, Stochastic Processes, John Wiley & Sons, 1953.
[9] J. Elker, D. Pollard, and W. Stute, Glivenko-Cantelli Theorems for Classes of Convex Sets,
Advances in Applied Probability, 11 (1979), 820-833.
[10] M. Feldman and C. Giles, An Expository Note on individual Risk without Aggregate Uncertainty, Journal of Economic Theory 35, (1985), 26-32.
[11] M. Hohnisch, Hildenbrand distribution economies as limiting empirical distributions of random economies, Bonn Econ Discussion Papers.
[12] K. Judd, The law of large numbers with a continuum of iid random variables, Journal of
Economic Theory 35 (1985), 19-25.
16
[13] H. Levene, On a matching problem arising in genetics, Annals of Mathematical Statistics 20
(1949), 91-94.
[14] W. Hildenbrand, Core and Equilibria of a Large Economy, Princeton University Press, 1974.
[15] M. Kandori, G. Mailath and R. Rob, Learning, mutation and long run equilibria in games,
Econometrica 61 (1993), 29-56.
[16] J. Miller and S. Page, Complex Adoptive Systems, An Introduction to Computational Models
of Social Life, Princeton University Press, 2007.
[17] R. Molzon, Approximation of stochastic interaction of agents and a central limit theorem
(preprint).
[18] A. Robson and F. Vega-Redondo, Efficient equilibrium selection in evolutionary games with
random matching, Journal of Economic Theory 70 (1996), 65-92.
[19] H. Uhlig, A law of large numbers for large economies, Economic Theory 8 (1996), 41-50.
17