.............. 1
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Chapter 3 MICROscopic Simulation: Many Interacting Agents
Section 3.1 Generic Framework
In the preceding chapter, we studied the single agent dynamics. Even in the stochastic case, where we considered large collections of N agents, in order to make statistical estimations of the probability distribution functions, the agents were totally independent. They did not "know" one of the other; they were copies of the same agents, living in parallel worlds which constituted different realizations of the stochastic dynamics, resulting from different realizations of the random numbers involved in the dynamics e.g. 2.17.
In this chapter, we start thinking about various agents living in the same world and even interacting. Such a multi-agent dynamics involves new results and new technical challenges.
Consider, say N=100 investors indexed by an index i=1,2, . . . .,N=100, whose wealths are governed each by the recursive equations of the type
2.17:
3.1 w i
(t) = w i
(t-1) + r i
(t-1)
If the r
i
's depend only on t or on their own wealth w
i
(with the same index i) then we just have a set of 100 independent equations of the type 2.3, 2.17 already studied in Chap. 2. However, if the q i
's are generic functions on the agents. In the "continuum limit" one has a highly interactive nonlinear differential system that is in general intractable:
3.2 dw i
/ dt = w i
+ r i
(w
1
, w
2
, . . . . , w
N
, t)
The simulation method 2.6, 2.17 however, goes through without problems for
3.2. In fact, it even holds if the functions r i
depend arbitrarily on the entire previous history of all the w i
's:
3.3 w
1
(t-1), w
1
(t-2), w
1
(t-3), . . . . , w
1
(0), w
2
(t-1), w
2
(t-2), w2(t-3), . . . . , w
2
(0),
. . . . . . . . . . . . . . . . . . . . w
N
(t-1), w
N
(t-2), w
N
(t-3), . . . . , w
N
(0),

Indeed, at each time t, knowing the previous values will allow us to compute the functions q
i
and substitute them in the recursion relations:
3.4 w i
(t) = w i
(t-1) + r i
(w
1
(t-1), w
1
(t-2), w
1
(t-3), . . . . , w
1
(0),
w
2
(t-1), w
2
(t-2), w
2
(t-3), . . . . , w
2
(0),
. . . . . . . . . . . . . . . . . . . .
w
N
(t-1), w
N
(t-2), w
N
(t-3), . . . . , w
N
(0),t) exactly like in 2.6, 2.16.
By contrast, the continuum approximation of such a system which involve values arbitrary in the past, would consist of higher order, non-linear, partial integro-differential equations with time dependent coefficients. Such systems not only do not admit in general analytic solutions but even their qualitative behaviour is in general unknown.
We can now be more explicit about what we mean by microscopic representation and microscopic simulation.
We have a system of many investors in which: the evolution of each wealth w i
(t) depends recursively on all the other w j
's through (in general non-linear stochastic) functions r i
(w
1
, w
2
. . . . (. . . .), t). This is a Microscopic
Representation.
In order to compute their time (co-)evolution we have in general to enact this recursion explicitly on the computer , by computing the "configuration" of the system [w
1
(t), w
2
(t), w
3
(t), . . . . , w
N
(t)], iteratively one time after the other.
This is Microscopic Simulation.
The point we are about to make in this book is not just that the continuum
Partial Differential Equations (PDE) approach is not practicable where the
Microscopic Representation approach is (due to the option of applying the
Microscopic Simulation). We are going to argue, that even in the special cases when both approaches PDE and Microscopic Representation admit analytic treatment, the continuum representation/ approximation leads occasionally to totally erroneous results. Moreover, we are going to characterize those effects and show that they are crucial in a very wide range of systems, phenomena and disciplines.
To this effect, we prepare below the analytic tools to treat microscopic representations of stochastic multi-agent systems. These techniques, originate in Statistical mechanics and in particular have been used to obtain the fundamental laws of systems in thermodynamics/ stochastic/ thermal equilibrium. We will show that the same very techniques can be used to uncover the fundamental laws in other domains even though the systems there are not in "thermal equilibrium".
Section 3.2 The Boltzmann Distribution; A naive model of money dynamics
In order to introduce the techniques relevant to the dynamics of stochastic probability distributions of ensembles of many discrete interacting elements, we describe here a classical statistical mechanics system. The example

shows in particular the universality and robustness of certain results and their independence on the details of the microscopic dynamics.
This universality is very important: usually it is impossible to characterize exactly the microscopic interactions and circumstances. Therefore, if the
Microscopic Representation is going to be of any use, we have to show that many of the results are numerically and dynamically stable to the various changes in the unknown details.
We will formulate the system as a money-dynamics system but in fact it is formally equivalent with the emergence of the Boltzmann-Maxwell probability distribution of the energy of the molecules in any physical system with local interactions at thermal equilibrium (these terms will become clear below)..
It is a very striking and important result in statistical mechanics that whatever are the microscopic interaction rules among the molecules, the probability
P(E) of a molecule i to have a certain energy E(i) is given by an exponential law:
3.5 P(E) ~ exp (-E/kT) where k is an universal constant (called the Boltzmann constant) and T is the temperature.
We will find later that in typical market conditions (where the individual gains/ losses are proportional with the individual investment itself), it is the logarithm of individual wealth which takes the place of energy. The temperature T will be related to the amplitude of gain/loss fluctuations in the system (market price volatility) and/or to the subsidies policies in the system (social security policies, poverty bound, etc.).
In the next chapters we will explain the robustness of the Pareto wealth distribution law and of the Levy market price fluctuations in the nonequilibrium financial context via the robustness of the Boltzmann law in equilibrium statistical mechanics.
However, for the beginning we describe a less natural financial system in which the analog of Energy is the wealth itself rather than its logarithm. Gains and losses are assumed to be additive random quantities and the temperature will correspond to the average wealth itself.
The "financial" system is described as follows.
Consider a system of N people/ agents/ traders each labeled by an index i.
Each person i, holds at time t=0 a number w(i,t=0) >> 1 of one-dollar bills.
Time is discrete t= 0,1, ..., infinity.
At each time t, a pair of agents (i,j) is selected randomly. The interacting rule is simple: the first one, i gives to j one dollar if he has it and nothing if w(i) =
0.
We will show in the present Section that in this conditions, the system approaches very fast a stochastic equilibrium state. I.e. a state in which the probability distribution of the individual wealth is time independent. More precisely, the equilibrium probability distribution is:
3.6 P(w, t= infinity) ~ exp(-w/T)

where T ~ 1/P(0).
To avoid complications, we are going to choose the simplest circumstances, but they can be widely relaxed without affecting the result 3.6. For instance, one can start with an arbitrary initial distribution w(i,0); one can endow the agents with spatial dynamics and let them exchange bills only upon meeting one another at a common location. One can introduce 3 body interactions, i.e. i gives a bill to j only a third person k is present at the same location.
One can even allow exchange only through a subset (e.g. S= {i= 1, 7, 27, 44, etc}) of dealers: i.e. one person can give or take bills only if i or j is a dealer.
In particular, there might be just one dealer which then would act more like a
God, arbitrarily giving and taking from the persons which he arbitrarily selects
(except in our case, God might have w = 0 and default in those instances his/her supplier role).
The only important thing for 3.6 to hold is the equality of the probability for each person to be an i (giver) or a j (taker) if it is selected in the currently
"active" giver-receiver pair. Another trivial requirement is that the system does not split in disjoint non-interacting sub-sets. The probabilities to engage in an encounter may vary from one person to another.
In order to estimate the stochastic equilibrium (stationary) wealth distribution
P(w,t=infinity) one writes the flow of people leaving the wealth station w and will require that it equals the flow of people incoming the wealth station w
(one considers both types of "incoming" agents: the ones which had higher wealth and got impoverished and agents that were poorer and gained). We will use the techniques and results of Section 2.3 to write and solve the
Master Equation for the probability distribution P(w,t).
The expressions for the flows above are estimated as following:
- Assume there are N people in the system and each pair (i,j) has equal probability 1/N 2 to meet. Each agent has probability 1/N to be the donor and 1/N to be the receiver. We neglect the case i=j (which is higher order in 1/N).
- Assume the number of people with wealth w at time t is N P(w).
Then, the probability that one of them will be the giver (i from the (i,j) pair) is
3.7 P(w).
If this happen, he will leave the wealth station w for the wealth station w-1.
By analogy, the probability that i will be the receiver is P(w) but he will really receive a dollar only if the giver of the pair is not wealth-less (this happens with probability 1-P(0)). The resulting probability for i to leave the wealth station w for the higher wealth station w+1 is therefore:
3.8 P(w) [1- P(0)].
The probabilities 3.7-3.8 mean that in the Master Equation , one is reduced to the case of the equation 2.22 with
3.9 R(+) = P(w) [1- P(0)]/(2N) and
3.10 R(-) = P(w)/(2N) leading according to 2.29-2.31 to a distribution
3.11 P(w) ~ exp(w ln [1-P(0)])

-
-
-
In the limit N-> infinity, P(0) <<1 and with the notation
3.12 T= 1/P(0)
One gets
3.13 P(w) ~ exp (-w/T)
This is the famous Boltzmann-Maxwell distribution that governs the behavior of classical systems in stochastic equilibrium (in which case w i
. is the energy of the molecule i).
The fact that the exponent -1/T is related to the information on the lowest allowed value w=0 is quite general and very relevant for financial applications. We will see repeatedly that the social security policy towards the lowest income citizens is going to affect the Boltzmann / exponential wealth distribution 3.13 throughout the entire society.
The main ingredients insuring the emergence of this distribution are the additive character of the dynamics (one adds a random number to the wealth rather than e.g. multiplying by a random factor) the conserving character of the interactions (the total amount of dollars in the system is conserved) the economic fairness / "market efficiency": (the equality between the a priori probability for a participant to be elected as a donor or as a receiver: this is called by physicists "detailed balance" ). The concept of
"efficient market" is central to classical economics. By definition, in such a market, there is no way for any investor to make systematically more profit than the others.
Our relation in 2.4 between additive stochastic systems governed by the
Boltzmann law and the multiplicative stochastic systems governed by the
Pareto power law implies also a series of other correspondences: between the statistical mechanics of physical systems in thermal equilibrium and the
(not necessaryly equilibrium) financial market dynamics. In particular the detailed balance is related to the market efficiency (all investors have statistically equal chances to gain and loose fractions of their capital). In a slogan way: Pareto laws are logarithmic Boltzmann laws in disguise. We will develop this line of thought in the following chapters.
As emphasized at the ends of Sections 2.3, 24 (Eq 2.51), there is another crucial condition for obtaining 3.13: the very existence of a stationary distribution. This is a quite nontrivial effect: the probability distribution at a certain wealth station w (and its very existence) depends on the way the microscopic dynamics takes care of the extremely low w values. In particular,
Eq. 2.51 (and Eq. 2.31 and following text) requires that very low w values are disallowed. Physicists say that "energy has to be bounded from below". This situation, in which the wealth distribution of the richest depends crucially on the social security as applied to the poorest will be even more evident in the stochastic multiplicative/ Pareto law case.
Without a lower bound for the w, the dynamics becomes a free random walk for each individual wealth as described in the section 2.5. We avoid systematically for simplicity in this book to state and prove the Central Limit

Theorem which governs this case, but will just explain in simple words what actually happens.
As explained in Section 2.5, according to the Central Limit Theorem, in the absence of a lower bound, the dynamics 3.9-3.10 would lead to an ever expanding and flattening Gaussian distribution. In the presence of the lower bound social security policy, the results differ dramatically: one reaches eventually a stationary normalizable distribution.
The dynamical mechanism is that even if one starts with a Gaussian distribution P(w) around a finite value w, this will eventually expand to (upper and) lower wealth values until a significant number of individuals reach wealth 0.
There, rather than continuing the road towards worse poverty, they are stopped by the "social security" rule that an individual with wealth 0 is exempt from giving a dollar even if he is the "donor" of a pair.
Moreover during the next encounters, he will have only the occasion to get richer (w=1) but not poorer (w=-1).
This effective subsidy of the poor is of course on the expense on the rest of the population. Indeed, while the probability of the agents with w> 0 to get poorer in an encounter is not affected by the social security rule, their probability of getting richer is diminished by the probability of encounter a wealth-less "donor".
As a result, the expansion of the distribution towards upper wealth is affected too: on top of the diffusion which in principle would have assured the expansion of the probability distribution to arbitrary large values of w, one has now a drift of the agents towards lower wealth values. This drift is due to the money they loose in the encounters with wealth-less partners and it is therefore proportional to < R> t = -P(0) t.
This drift is in effect a flat tax for all the w > 0 persons.
The volume of wealth carried by the "downward" drift is exactly equal to the amount of "social security" subsidies: the amount which the w=0 individuals is exempt from paying during their encounters as donors.
This results into a very large population at w=0 and a sharp exponential decay as w increases.
Section 3.3 A less naive model of money dynamics
Certainly Eq. 3.13 is a quite equalizing social wealth distribution (no person has significantly more than the average wealth). However it is not a very efficient one.
It is very inconvenient to have every one in the poverty range and likely to need in the near future subsidies (especially if this is in reality not arranged at the personal encounter level as in our model, but by the state intervention).

-
Moreover, the uncertainty of whether one meets a real donor or a w=0 one is not contributing to a reliable social and economic life.
The resulting economy reminds one of the old characterization of the communist society vs the capitalist one: "in capitalist societies there are very rich people and very poor people while in socialist ones there are only poor people".
In conclusion, what this model misses is the existence of a large middle class. In terms of emergence jargon: a society that wishes to accomplish macroscopic objectives has to insure the emergence of macroscopic economic features beyond the average individual wealth.
In particular, in the Boltzmann system the typical departures (thermal fluctuations) from equilibrium of the intensive (average) quantities (analog to the market price) are Gaussian and microscopic (order 1/sqrt(N)). While it is reassuring that the pressure in your part of the room is not going to have a significant fluctuation during your life-time, this is not compatible with the daily market dynamics which relies on frequent significant gain opportunities arising stochastically with high probability.
Fortunately we will find that in fact the free society is taking care of these problems through the auto-catalytic property of the capital: the dynamics of the individual w's is not additive but (roughly, stochastically) multiplicative.
Rather than producing during the economic activity fixed amounts per agent, the capital produces returns (positive or negative depending on the "luck") proportional to the investor's wealth).
The implications of this dynamics is very different from the dynamics studied above.
Yet we can find a connection to it similarly to the connection we realized in
2.4 between the multiplicative and additive stochastic dynamics of single agents.
In order to study a multiplicative multi-agent system one has to make more effort but the results will be stronger. In particular, we will find that the one can find a stable power law with constant exponent for a multiplicative dynamics with a probability distribution of the multiplicative factor q i
(t,w) that depends explicitly on time t and on the average wealth w(t). Therefore the global dynamics can be very diverse (including booms, crises and crashes) while the relative probability distribution of the relative individual wealth remains fixed in time.
Let us indeed consider the multiplicative stochastic dynamics which consist in the following steps: select randomly at each time t one of the agents i, in order to update its wealth w i
(t) -> w i
(t+1). All the other wealths are left unchanged.

3.14
-
extract a random number q i
(t,w) from a probability distribution R(t,w) which may depend on t and the current average wealth w(t) (but the same for all the agents i). update the wealth of the agent i according to the equation: w i
(t+1) = q i
(t,w) w i
(t) w k
(t+1) = w k
(t) for all other k's
If this following the operation 3.14 (i.e. the updating of w i
and w), w i
(t+1) or any other individual wealth w k
(t+1) = w k
(t) becomes less than a certain fixed fraction of the new average wealth s w(t+1), then their value is updated to w i
(t+1) = s w (t+1).
The last operation can be thought as a social security policy: individuals which fall below a certain fixed ratio s of the average current wealth w(t) are subsidized (they receive an amount which bring their wealth back to s w(t).
I.e.
3.15 w i
(t) > s w (t)
It can be shown (and it is verified by the consistency with the end result) that the influence on w(t+1) of the updating of each w i
(t) vanishes for N ->infinity, therefore in practice it is not so important if one uses actually the easier to enforce condition w i
(t+1) > s w (t). However, over longer time scales (O(N) and longer), of course w(t) evolves significantly.
In fact the dynamics of this system is non-stationary, and the time evolution of w(t) depends on c and on the distribution of r i
(t,w):
For times when < ln r i
(t,w)> > 0, one has a growth economy while for times when < ln r i
(t,w)> < 0, the total wealth shrinks. For times for which
< [ln r i
(t,w)] 2 > - < ln r i
(t,w)> 2 is large, the market is "nervous" and has large volatility, while for times with small < [ln r i
(t,w)] 2 > - < ln r i
(t,w)> 2 the market is quiet and the fluctuations small.
The surprising thing is that during all these periods of growth, decay, febrile activity, quiescence, the probability distribution Q(x) of the relative individuals wealth remains unchanged.
By relative wealth of the individual i, we mean its wealth normalized to the average wealth.
3.18 x i
(t) = w i
(t) / w (t)
In terms of 3.18, the equations 3.14-3.15 become:
3.19 x i
(t+1) = [ q i
(t,w) w (t)/ w (t+1) ] x i
(t)
x k
(t+1) = [ w (t)/ w (t+1) ] x k
(t) and respectively

3.20 x i
(t) > s
x k
(t) > s
These equations (one for each i) are the same as 2.41 and 2.52 with an effective random multiplicative coefficient q which is frequently renormalized by the factor w (t)/ w (t+1). This insures that the condition < ln q > < 0 Eq.
2.52 is fulfilled by the dynamics of x i
.
Thus, all the conditions are fulfilled for the emergence of the power law 2.49:
3.21 Q(x) = 1/Z x -1-

3.22
Where:
Z =
 s
∞
x -1-
 d x = s -

/
 is the normalization factor which insures that the integral of Q over x is 1.
For x < s, Q(x) =0 according to 3.20.
The detailed expression in 2.49 for the exponent
 is not very useful for the present case because the probability distribution of the factor in 3.19 and in particular its average and variance are not explicitly known (and on top of it, from time to time the x i
are updated to the value s according 3.20).
It turns out that the best way is to estimate the exponent
 is to use the fact that one knows that the average value of x is by the definition 3.18 equal to1.
3.23
Therefore, one can fix the parameters in P(x) by using:
< x > =1/Z

∞ s
x P(x) = 1
Substituting the distribution 3.21 one gets an implicit equation for

3.25
3.26
3.24 =1/Z

∞ s
x x -1-

dx = [s 1-

/ (

]/ [s -

/

Consequently one gets an algebraic relation between s and

:
1 = s

/ (

] which finally gives
 as a function of s only:

s

This is a quite surprising result: one would have expected (as suggested by the formula 2.49 that the properties of the probability distribution of q i
(t,w)
(its average, variance) should be the important elements.
The intuitive explanation is related with the fact that the lower bound 3.15 is more like a "running wall" which is coupled to the average wealth: as the individual wealths "run to infinity" through the multiplicative stochastic dynamics, the wall runs after them pressing the lower values. Therefore even if one has a large < ln r i
(t,w)> this will not lead because of 3.14 to larger relative wealths x i
(t) = w i
(t) / w (t) values, since w(t) will increase with exactly the same speed.
So, in terms of the relative wealth of each agent it is as if the lower bound is fixed as in section 2.5 Eq. 2.51. The "only" role of coupling each individual wealth to the average wealth through the social security poverty lower bound is that now the relevant quantities ("degrees of freedom") for the probability distribution are the individual relative wealths rather than the wealth itself.

Consequently one can fulfill the condition 2.52 without assuming that everybody is a loser (has negative returns < ln q > < 0). The relative wealth has this property from its definition: if evrybody becomes richer, the average wealth increases and the average relative wealth of most of the agents decreases (with the notable exception of the social security cases, which benefit from the subsidies).
It is important to remark that we have here a probability distribution 3.21 which, while obtained with techniques similar to the Boltzmann distribution, differs from it in very significant ways. First, the system is not in equilibrium, in fact for q i
(t) distribution with average significantly larger than 1, the values of the w i
's will increase continuously. Second the power law 3.21 implies that, as opposed to 3.13, in the multiplicative system there is a very significant probability for values of w i
that are orders of magnitude larger than the mean w(t). In fact, in real economic systems, the formula 3.21 has been repeatedly obtained by measurements over the last 100 years in all the capitalistic economies. The measured value for exponent of the power law was roughly constant around 2.4 - 2.6.
So, in fact there will be a very significant middle class, significant, nongaussian fluctuations and in fact also a few very (macroscopically) rich individuals. Of course this opens also the way to unpleasant features: the diffusion of wealths to infinity (inflation) and very large, unhealthy fluctuations.
We will see that the introduction of incremental tax (or finite resources) limits both these unwanted effects.
Far from depicting an idealized picture of the capitalistic society, we are trying to use the generic universal features of the different dynamics in order to characterize the generic mechanisms of the market. It is clear that upon exhausting the generic information one will have to introduce (as we do in later chapters) more details on the actual behavior of the individuals and of the market.
In conclusion, the exponential probability is definitely very different from the realistic power one and in fact it implies that all the values of w are concentrated within a very limited range.
In nature, this is usually not the case in interesting systems.
In fact, it is the autocatalytic dynamics of certain systems which allows them to escape the additive dynamics and the narrow dynamical range.
The reason for treating in such a detail the Boltzmann systems is not only their analytic solvability but mainly the fact that the auto-catalytic
(multiplicative) dynamics of highly interactive nonlinear complex systems reduces to it upon appropriate transformations which we detail in the rest of this chapter.
-
-
Section 3.4 A Simple Financial Microscopic Simulation Model : returns, taxes and subsidies
To make the microscopic simulation procedure 3.3 clearer let us take the example in which the variation of the wealth w i
(t) of the individual i at time t is composed by a random gain factor r i
, proportional taxation (of a fraction f) of the wealth and

-
-
-
-
equal redistribution of the taxed amount.
The gain factors r i
are considered in this simple model externally given (by how lucky or unlucky the individual is in its investment). In principle r i
may depend on time and also have some causal component. We will start by assuming that they are given time independent random numbers produced by the computer with a (Gaussian) probability distribution of square standard deviation D centered around m=0. We will relax it later.
So the random gains contribute an amount
3.22 r i
w i
(t-1) to the new wealth w i
(t) at time t.
The taxation of each individual by a fraction f of its wealth results in a negative contribution to its wealth:
3.23 - f w i
(t-1)
but then the taxes are redistributed equally among the individuals and provide a positive contribution
3.24 + f w ( t-1) to each of them.
We noted by w (without subscript index) the average wealth:
3.25 w(t) = W(t) /N = (w
1
(t)+ w
2
(t)+ w
3
(t)+
. . . .
+ w
N
(t))/N
Adding the contributions 3.22-3.24 to w i
(t-1), one obtains the new wealth w i
(t) of each individual i at time t:
3.26 w i
(t) = (1+r i
) w i
(t-1) - f w i
(t-1) + f w ( t-1)
This system of many coupled finite difference equations defines a
Microscopic Simulation model .
The microscopic simulation of this system proceeds as follows.
Suppose the initial wealth of the all the individuals at time 0 is given:
3.27 w
1
(0), w
2
(0), w
3
(0),
. . . .
, w
N
(0), one then:
- 1) computes the average w(0) according 3.25
2) computes f w(0) and - f w
1
(0), - f w
2
(0),
. . . .
, - f w
100
(0),
3) computes the products r
1
w
1
(0), r
2
w
2
(0),
. . . .
, r
100
w
100
(0)
4) adds all the contributions 2) ,3) to w i
(0) for each i to obtain the new wealths at time 1:
3.28.a
3.28.b w
1
(1) = (1+r
1
) w
1
(0) - f w
1
(0) + f w ( 0) w
2
(1) = (1+r
2
) w
2
(0) - f w
2
(0) + f w ( 0)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.28.c w
100
(1) = (1+r
100
) w
100
(0) - f w
100
(0) + f w ( 0)
5) keeps iterating this procedure to obtain w
1
(2), w
2
(2), w
3
(2), . . .
,w
N
(2), from the w
1
(1), w
2
(1), w
3
(1), . . . , w
N
(1) obtained at 1.24 a-c:
3.28.d w
1
(2) = (1+r
1
) w
1
(1) - f w
1
(1) + f w ( 1)
3.28.e w
2
(2) = (1+r
2
) w
2
(1) - f w
2
(1) + f w ( 1)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.28.f w
100
(2) = (1+r
100
) w
100
(1) - f w
100
(1) + f w ( 1) and so on.
Note that the system is quite nontrivial because the next value of each w i
is influenced by all the others through the appearance of the average w in the term f w in the second members of 3.28.

Note the difference between this iterative simulation procedure and writing a closed analytic formula like 2.2, 2.9, 2.12, 2.15, 2.34, 3.13.
One may question the wisdom of going through all these pains of iterating repeatedly a large number of equations when in the end one is interested only in macroscopic average properties . Wouldn't it be better to consider the average equation and follow its time evolution and find its asymptotic behavior?
In fact this approach has been used for many years in a wide range of biological and social sciences and even in physics. We will show that this
"representative agent" approach is flawed: it is just not true that the solution of the average equation is the average over the solutions of the equations of a system.
We will describe this situation in detail in the following sections and chapters.
Just for a very metaphoric explanation consider the owner of many transoceanic vessels that would like to know the average number of vessels that would make the trip back. Should he consider an average ocean with all the wave crests and dips averaged, the result would be a gross over-estimation.
Section 3.5 What is wrong with the average agent? exp sum << sum exp
The answer to the question in the title requires a little complicated analysis, so we give here a 1-paragraph preview to this sub-section:
Autocatalytic processes lead to exponential time development of the microscopic elements undergoing them.
Therefore, even microscopic initial differences (implied by microscopic noise/ discreetness) are amplified to macroscopic differences.
In turn, such an extreme heterogeneity of the elementary processes invalidates their continuum averaging or representative agent methods.
One may hope to be able to characterize the dynamics of the system 3.28 by taking the average.
3.29 <w i
(t)> = <(1+r i
) w i
(t-1)> - f <w i
(t-1)> + f < w ( t-1)> i.e.
3.30 <w i
(t)> = < w i
(t-1)> + < r i
w i
(t-1)>
One might hope to write an recursion equation for w(t) = <w i
(t)>:
3.31 w(t) = w (t-1) + < r i
> w (t-1)
However, this can be done only if one assumes
3.32 < r i
(t) w i
(t-1) >= < r i
(t)> < w i
(t-1) >
This approximation is essentially an assumption that the r 's are random independent variables (and independent on the w's). The little one can say in order to defend this approximation is that this is the only way to force the average <w(t)> to appear as the only variable in the single equation 3.28. If the averaging approach fails as (we will see below) it happens, this is one of the un-warranted approximations to be blamed for it.
With this approximation, one gets by iterating 3.31:
3.32 w(t) = (1+ <r
i
>) w (t-1) = (1+ <r
i
>) 2 w (t2) = ….
Which ends up with
3.33 w (t) = (1+ <r
i
>) t w (0)

while the correct approximation will turn out below to be:
3.34 w (t) ~ (1+r
MAX
) t where MAX is the index i =MAX for which r is maximal: r
MAX
> r j
for all j not equal to i.
With a cheap play of words, one may claim that in this system, the exception
(largest value) rules.
To see how misleading this "representative investor" result 3.33 is, let us compare it to the exact result in the solvable case f=0 in which the equations
3.28 become un-coupled.
3.25.a
3.25.b w w
1
2
(t) =
(t) =
(1+r
(1+r
1
2
) w
) w
1
2
(t-1)
(t-1)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.25.c w
100
(t) = (1+r
100
) w
100
(t-1)
In fact the system 3.25 becomes a set of independent equations which can be solved exactly as in 3.32:
3.26.a w
1
(t) = (1+r
1
) t w
1
(0)
3.26.b w
2
(t) = (1+r
2
) t w
2
(0)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.26.c w
100
(t) = (1+r
100
) t w
100
(0)
Even if the initial wealths of the individuals are equal:
3.27 w
1
(0)= w
2
(0)= w
3
(0)= . . . . ,=w
N
(0) = w(0) this will not continue to be the case at all after enough time: the individual with the largest r i
will totally dominate the others:
3.28 r
MAX
> r i and t >> 1 => (1+ r
MAX
) t >> (1+ r i
) t which means that at asymptotic times, the ratio between the maximal w i and the others diverges:
3.29 w
MAX
(t) / w i
(t) = (1+r
MAX
) t
/(1+r i
) t ----> infinity
Consequently, w
MAX
(t) will dominate the average wealth:
3.30 < w (t) > ~ (1+ r
MAX
) t w(0)
In words: in a sum of terms depending exponentially on time, the term with the largest base will eventually dominate totally for large enough times.
It is now clear what is the difference between the representative investor solution and the correct one:
3.31 (1+ <r
i
> ) t < < < (1+r i
) t > i
One can see the same effect in the simplest example of just 2 individuals, one with r
1
= 0.1 and the other with r
2
= -0.1, the "average/ representative" investor between the 2 will have r= r
1
+r
2
)/2 = 0 gain rate and consequently, stationary wealth:
3.32 "w "(t) = w(0)[1+ (0.1-0.1)/2] t = w(0)
By contrast, the average wealth will increase exponentially:
3.33 w(t) = w(0) [ 0.9
t + 1.1
t ] /2 ~ 1.1
t w(0) /2
Another "slogan" way to put it is: the exponent of the average is much smaller than the average of the exponents.
3.33 e 0 x t < < (e 0.1t + e -01t )/2
One sees here the origin of the failure of the representative agent approaches: in systems with autocatalytic individual behavior, individuals grow exponentially and even small microscopic inequalities are amplified to macroscopic differences. Consequently, the population becomes nonhomogenous and the macroscopic dynamics stops being the average of the microscopic ones.

-
-
-
This establishes a triangle that will dominate much of our following examples:
Failure of representative agent in auto-catalytic systems
Emergence of Macroscopic heterogeneity
Necessity of Microscopic simulation.
The truth is that the microscopic representation [Solomon 95] in terms of the relevant discrete microscopic stochastic objects is all it is required: the choice of simulation over other methods once a discrete set of different objects is studied is usually forced upon us by the technical limitations of the analytical tools. However, whenever possible, the analytical treatment of microscopically represented systems is not spurned upon.