=Paper=
{{Paper
|id=Vol-2105/10000061
|storemode=property
|title=Universal Properties of the General Agent-Based Market Model
through Computational Experiments
|pdfUrl=https://ceur-ws.org/Vol-2105/10000061.pdf
|volume=Vol-2105
|authors=Alexander Weissblut,Ruslan Halutskyi
|dblpUrl=https://dblp.org/rec/conf/icteri/WeissblutH18
}}
==Universal Properties of the General Agent-Based Market Model
through Computational Experiments==
https://ceur-ws.org/Vol-2105/10000061.pdf
Universal Properties of the General Agent-Based Market
Model through Computational Experiments
Alexander Weissblut1 and Ruslan Halutskyi1
1Kherson State University, 27, Universitetska st., Kherson, 73000 Ukraine
veitsblit@gmail.com, nlahik2@gmail.com
Abstract. Research goals: to synthesize the general view market mathematical
model in accordance with new dynamic paradigm of economics, to reveal the
universal properties of general view markets.
During our investigation we developed and continuously improved a desktop
C# application Model for support the research process using computing
experiments. Here our task is revelation of the universal properties of general
view market as a result of simulation experiments using this software module.
Results of the research: the crucial factors which ensure the market stability are
the level of agreement in adaptive expectations and the share of planning with
adaptive expectations in a market. The increase of naive expectations leads to
stability loss, to bifurcations and finally to chaos in general view market. The
increase of number of firms also leads to stability loss and finally to chaos in
the general view market at appreciable naive expectations. We revealed that the
profits ratio and quantity outputs ratio of firms remains almost unchanged in
short-run period in general view markets. It seems an important stability factor
of many important real markets for which chaotic dynamics is usual.
Keywords: agent-based model, heterogeneous type, bifurcation, adaptive
expectations.
1. Introduction
Information technology in the economy made it possible to model artificial societies
and study economic models through the computer simulation. Economics has entered
the stage of deep transformation of its bases. In recent years the researchers are
renouncing the assumption of perfect rationality as unconditional basis of economic
agents’ behavior [1]. The neoclassical ‘rational man’ does not exist in reality:
economic agents act according to established rules, without being fully informed and
maximizing their own utility [2].
The real economic processes make a clear demonstration that neoclassical "rational
man" is not their subject. In real economy "optimal imperfect decisions" are taken by
simple and non-expensive calculations, well adapted to frequent repetitions, to
evolution: it is more efficient for perfectly rational firm to perform multiple
experiments with quantity to estimate the demand function rather than search for
nonrecurrent, instantaneous achieving of equilibrium [3]. All it means that the real
economy is dynamical system, and real processes of economy are iterative processes
of this system.
Now institutional school of economics analyzes economic systems as a result of
evolutionary process of participants’ interaction [4]. New paradigm of economics is a
mix of the nonlinear dynamical system theory and mathematical programming,
including game theory and optimal control theory [5]. And the main tool of new
economics is simulation modeling grounded on the basis of 3 computer paradigms
(object-oriented, dynamic and multi-agent system) [6].
This new economics allows explaining the phenomena which were not keeping
within traditional schemes. The evolutionary approach and analysis of the dynamics
allow to explain why one type of firm ousts another from the market, why sometimes
the economic system is stable, but in other cases is unstable [3, 7]. If the system has
multiple equilibriums, the dynamics and evolution is the selection mechanism of best
equilibrium according to certain criteria [8]. Traditional static models of competition
(e.g., Cournot, Bertrand and Stackelberg) were converted in dynamic models which
were investigated on existence, stability and local bifurcations of the equilibrium
points [9, 10, 11].
Within the limits of new economics it is natural to study reciprocity relations
[12]. Reciprocity or social responsibility implies that the firms not only pursue their
selfish goal of increasing profits, but are also ready to sacrifice some of their profits
for the benefit of consumers without direct compensation for it by the state [13]. Such
targets can be stipulated by the firms’ desire to get stable profits in the long run rather
than maximal short-run profits [14, 15]. Such forward-thinking firms-reciprocators
are considered in this paper. Their objective function is a weighted average of the
profits and consumer surplus of their market segment.
Modern development of dynamic paradigm in economics is a wide stream of
researches. However it is a stream of examples which are not developing in the
general theory; their relations with real markets are often problematic [16]. The
traditional method of constructing a scientific theory is first to synthesize and
investigate the simplest possible mathematical model. And then we can study
complex real systems which are grounded on this basis. This traditional approach is
taken as a principle of our research.
This paper is a continuation of our previous works [17], [18]. There the elementary
market model corresponding to the new paradigm of economics has been synthesized
and investigated. That model describes a simplest market where firms have only one
difference in their type when some firms (egoists) are focused exclusively on short-
run profits, while others (reciprocators) take into account long-run factors. However
it`s not any special, specific market; actually any global market contains such
elementary local markets and consists of them. As suggests common sense then
dynamics of the global market is stratified on dynamics of such local markets.
Therefore it was naturally to state a hypothesis that the derived in [17], [18] properties
of the elementary model are universal, i.e. these properties are the properties of
general view market including real markets as a special case. Check of this hypothesis
makes the project of this work.
The paper goal is to synthesize the general view market mathematical model
according to the new dynamic paradigm of economics, to reveal the universal
properties of general view markets including real markets, to check up the hypothesis
about universality of properties of the model [17].
During our investigation we developed and continuously improved a desktop C#
application Model for support the research process using computational experiments.
Our next task is revelation of universal properties of general view market as a result
of simulation experiments using this software module.
The paper is organized as follows: in part 2 we synthesize the general view market
model; part 3 demonstrates desktop application Model for computing experiments;
sections 4.1 – 4.3 describe our market model researches using this application, section
4.4 formulates their results; part 5 concludes.
2. Agent-Based General View Market Model
In general, almost any microeconomic market model is constructed as follows: 1) n
firms operate in the market (to simplify the notation suppose n 2 ); 2) these firms
produce homogeneous products in quantities x1 (t ) and x2 (t ) in time period t ;
3) they use adaptive approach, i.e. they try to predict the quantity of their competitor
in the next time period; 4) let x ej (t 1) is the expected quantity of rival j by a firm i
in next period t 1 ( i, j 1, 2 ). Then under planning of their quantity xi (t 1) in the
next period the firms solve the following optimization problem:
MaxП1 ( x1 (t 1); x2e (t 1)) , MaxП2 ( x1e (t 1); x2 (t 1)) , (1)
where Пi , i 1,2 is a profit function of firm . The assumption about unchangeable
i
quantity of the competitor (i.e. firm i will use x j (t ) instead of x ej (t 1) when it
solves the optimization problem) is an example of imperfect, bounded rationality in
firm’s strategies; it is called naive expectations. As a rule these two approaches
(adaptive and naive) coexist in the market with a certain probability. Our model is
based on these assumptions.
We consider a market of homogeneous product, where exogenous parameter n(t )
indicates how many firms operate at time t . Each firm produces output xi (t ) , where
n
i 1,..., n(t ) . Thus the industry output of the market is Q(t ) x(t ) at time t .
i 1
Product price P is given by isoelastic demand function P P(Q) b(t ) / Q
( b(t ) 0 ). Such kind of demand function as a matter of fact is not a restriction.
Really, in a small neighborhood of a market state during the moment t any demand
function with elasticity b(t ) differs from the isoelastic one a little. Then in short-run
period dynamics of a market with such demand function differs a little also. And at a
structural stability they are qualitatively (i.e. orbitally) equivalent.
Formally the firm is defined by its objective function. Firm maximizes both its
own profit X (P v) x fc (where v is the firm’s cost per unit in the market, fc
is fixed cost) and consumer surplus CS (difference between maximum price which
Q
consumer can pay and real price) CS P(q)dq P Q , where parameter
specifies the segment of the market, which the firm believes its own and optimizes;
is the minimal technologically possible product quantity. Then
Q b Q Q
CS b ln( ) Q b ln( ) 1 b ln , where ˆ e (specific
Q ˆ
choice of does not affect the model dynamics and so we suppose 1 ). Then
general profit function П Пi (t ) of firm is:
Q
П (1 ) CS (( P v) x fc) (1 ) b ln , (2)
where i (t ) is share of short-run own profit i (t ) in the objective function,
1 is share of consumer surplus CS , fc fci (t ) is a fixed cost. As a matter of fact
П is a weighted average of short-run profit and expected stable long-run profit.
The model of paper [17] is the elementary special case of this general model. There
we consider a market of homogeneous product, where n firms operate, among them
are k identical reciprocator firms with the same output x and n k identical selfish
firms with the same output y .
Dynamic of the model is considered for discrete time t 1,2,... . Our model is
uniquely defined by firms’ objective functions and their expectations types. It does
not use any additional assumptions or restrictions.
2.1 Dynamics Model Equations
In real life both decision making approaches (adaptive and naive) coexist in the
market with a certain probability. Let's obtain now the equations of a general market
model with the minimum account of adaptive expectations dictated by common sense.
According to such expectations firm i suggests that production quantities of its rival
j will be equal to x ej (t 1) ij (t ) xi (t 1) ij (t ) x j (t ) . Here ij (t ) 0 and ij (t ) 0
are parameters, defining shares of naive and adaptive expectations at this planning.
n (t ) n(t )
Let zi xi (t 1) x (t ) (t ) x (t 1) (t ) x (t ) is prospective industry
j i
e
j
j 1
ij i
j 1
ij j
output of the market, where ii (t ) 1 , ii (t ) 0 . Then the objective function for the firm
b
i has the form Пi i (( v) xi (t 1) v0 ) (1 )bi ln(zi ) in accordance
zi
with (2) (here 1, i i (t ) ). Then according (1) the point xi (t 1) of maximum
objective function Пi is found from the condition
n(t ) n (t )
bzi ij (t ) bxi (t 1) (t )
Пi
ij
j 1 j 1
i ( v) (1 i )bi 0 . Then
xi (t 1) zi2 zi
b n (t ) b i 1 i n (t )
zi2 ij j
v j 1
(t ) x (t )
v i j 1
ij (t ) zi . (3)
1 bi 1 i n (t )
Hence suppose that di
2 v i j 1
ij (t ) we obtain
b n (t ) b n (t )
( zi d i ) 2
v j 1
ij (t ) x j (t ) d i2 ; zi
v j 1
ij (t ) x j (t ) d i 2 d i .
Thus we obtain the dynamics equations of general view market model
b
i xi (t 1) wi (t ) di2 di wi (t ) (i 1,..., n(t ) ), (4)
v
1 bi 1 i
n(t ) n (t )
where i i (t ) (t ), w (t ) (t ) x (t ), d d (t) 2 v (t).
j 1
ij i
j 1
ij j i i i
i
In this paper we consider all actions, expectations and strategies of firms in short-
run period, therefore the equations parameters i and d i are assumed further as
constants which are independent of time.
Let the market of homogeneous product consists of m firms’ types, each type l
m
including kl identical firms: l 1,..., m, k1 ... km n. Then i k , ,
l 1
l il
m
wi (t ) kl il xl (t ), where il ij , il ij , xl (t ) x j (t ) at all j from type l .
l 1
Then owing to (4) xi (t 1) xl (t 1) at all i from type l . As a result dynamics in the
equations (4) has dimension m :
b
i xi (t 1) wi (t ) di2 di wi (t ) (i 1,..., m ), (5)
v
m
1 bi 1 i .
i kl il , wi (t ) kl il xl (t ) , di
m
where i
l 1 l 1 2 v i
The equations (5) are a special case of (4) and simultaneously their generalization,
i.e. they are equivalent to (4) in short-run period.
In particular, in two-dimensional model [17] ( m 2 ) for firm i
n n
i ij 1 p(k 1) x , wi (t ) ij x j (t ) q(k 1) x(t ) (n k ) y(t ) wx (t ),
j 1 j 1
1 b 1 1 b 1
n
di
2 v j 1
ij 2 vk (1 p(k 1)) d . (6)
So for two-dimensional model [17] equations (5) have the form
b
x x(t 1) wx d 2 d wx
v , (7)
y (t 1) b w w
y v
y y
where x 1 p(k 1), wx q(k 1) x(t ) (n k ) y(t ), y 1 p(n k 1),
1 1 b
wy kx(t ) q(n k 1) y(t ), d (1 p(k 1)).
2 vk
We usually use further the following simplest after two-dimensional version of (5)
for the illustrations of results of computational experiments. In this version we
consider a market of three firms’ types: k 1 and correspondingly k 2 reciprocator firms,
( k k 1 k2 ) and dn n k identical selfish firms. Here as well as above 1 2 ,
3 1, 13 23 31 32 0, ij 1 ij . Then (5) has the form
b
i xi (t 1) wi (t ) di2 di wi (t ) (i 1,2,3 ), (8)
v
where 1 1 11(k1 1) 12k2 , 2 1 22 (k2 1) 21k1, 3 1 33dn,
w1 (t ) 11(k1 1) x1 (t ) 12k2 x2 (t ) dnx3 (t ), w2 (t ) 22 (k1 1) x2 (t ) dnx3 (t ),
1 bi 1 i
w3 (t ) 33 (dn 1) x3 (t ) k1x1 (t ) k2 x2 (t ), di i (i 1,2), d3 0.
2 v i
2.2 Equilibrium Conditions
In a Nash equilibrium point we have xi (t 1) xi (t ) xi at all t 1,2,... and
i 1,..., m . Hence x (t 1) xi at all i and t .
e
i
Proposition 1. There is unique Nash equilibrium point in a general market model (5).
m
Proof. In an equilibrium point zi xi (t 1) xej (t ) x j z at all i 1,..., m .
j i j 1
Therefore owing to (3)
b n bi 1 i n b m b 1 i
z2
v j 1
ij (t ) x j
v i j 1
ij z
v l 1
kl il xl i
v i
i z (i 1,..., m),
m
where i k , x x at all i from type l , l 1,..., m . Hence
l 1
l il l i
m
k x a
l 1
l il l i (i 1,..., m ), (9)
v 2 1 i
where ai z i i zi . Since matrix of types parameters ( kl il ) is
b i
nonsingular m m matrix on construction then the system of linear equations (9) has
one and only one solution, Q.E.D. .
For two-dimensional system (7) this Nash equilibrium point is the same, as in [17]
and also is set by the same formula.
Proposition 2. There is unique Nash equilibrium point in a dynamical system (7):
b b
( kG q ( n k 1)) (k q (1 / G )( n k 1))
y v
*
x Gy v
* * , (10)
( kG ( n k )) 2 (k (1 / G )( n k )) 2
nk
p (n k ) q ( (1 ) )
where function G G ( p, q, n, k , ) k .
(2 1)(1 p(k 1))
Proof. Since (6) equation (3) has the following form for any reciprocator firm i
b 1 1 b
zi2 (q(k 1) x(t ) (n k ) y(t )) (1 p(k 1))zi . (11)
v 2 vk
b
For any selfish firm equation (3) takes the form zi2 (kx (t ) q(n k 1) y(t )) . But in the
v
Nash equilibrium point x(t 1) x(t ) xi (t ) x , y(t 1) y(t ) y j (t ) y at all i , j and
t 0,1,... . Then since (11) we get
b
(kx (n k ) y)2 (k x q (n k 1) y)
v
b 1 1 b
(q(k 1) x (n k ) y) (1 p(k 1))zi . (12)
v 2 vk
From second equation (12) we obtain the response function
nk
p(n k ) q ( (1 ) )
x
k G .
y (2 1)(1 p (k 1))
To calculate the coordinates of a fixed point, we substitute the expression of y
through x in the first equation (12), Q.E.D. .
In (10) by the data we get x 0 , y 0 . In view of the following proposition 3 it
* *
also ensures nonsingularity of a matrix (9) in proposition 1.
3. Desktop Application Model for Computing Experiments
During our research we developed desktop application Model to support the research
process using computational experiments with dynamic systems. The main purpose of
the application is to provide the best service for research cycle: hypothesis
experiment hypothesis. It’s impossible to realize new idea with new device
immediately, at once after it appearance for natural experiments. However here we
can do it using application window with the appropriate tools. The results of new
experiment give rise to new ideas, which we can check immediately using new
windows and so on. Therefore intensive researches with multidimensional dynamical
systems during this work have demanded efforts for computational speedup of the
application. The goal of Model is the highest possible support for research process.
Model is a C# application created on the basis of the graphical interface of the
System.Drawing and System.Windows.Forms C# system libraries. All calculations
related to the model are localized in the calc method, which makes it easy to modify
the equations of the model or move to other models.
Model application additionally uses Open Maple to work with differential equations
and 3D graphs. Open Maple is access interface to Maple computational core from
various programming languages: C#, Java, Visual Basic etc. In addition to the above
standard namespaces is also used the System.Runtime.InteropServices namespace,
which allow us to make links to the Maple dynamic linking core library - maplec.dll.
The following figure demonstrates the main application window which
automatically appears when you open it.
Fig. 1. Main window of the Model application
In the center of the window is located a two-dimensional projection of Lorentz
system’s attractor. In fig. 1 above in the left corner are the application menu buttons.
From left to right: 1. Save button is used to save current model which is displayed on
the screen with all the given parameters’ values and settings under the chosen user
name. 2. Edit button is used to modify the current model. 3. Open button
demonstrates a list of saved models’ names with the date of their last modification,
which allows you to select and open a window of any of them. 4. Add button is served
to define new models. 5. Delete button gives possibility to delete the current model
(depicted on the screen) from the list.
The following fig. 2 shows the application window for market model of this paper.
Fig. 2. Model application window for general view market model
On the right are 5 types of graphs, which are used most often; their examples are
pointed out later in the paper. We can set model parameters and the initial values of
the model trajectory using counters on the left. After these settings the graph of given
model automatically appears in the center of the window. The number of iterations we
can be set on the scroll bar above the graph. In the center of the window is also
displayed the animation of the selected path when the button (near the scroll bar) is
pressed.
When you click Step button on the left, you can set step of changing for a list of
parameters. If you click Value button, you can obtain the table with coordinates of
model trajectory for given iterations.
But the main tool to support computational investigations in Model application is
easy modification of a current model after pressing of Edit button (fig. 3).
Modification window is located over the current model window, which allows using
both windows at the same time. After left click on the model equation in the field The
dynamical system will move to the field Equation, where it can be changed. After
pressing Add the modified equation will return back. Similar procedure can be done
with parameters. We can also add new equations and parameters and delete the
previous ones. In the field System name we can specify the name of the new model
modification. After clicking Save button, new model falls into the saved list. If you
click Change, the new modification will be saved under the name of the current
model, which is deleted. When you click Back, the modification is temporarily
suspended and we return to the current window. View button displays information
about the model (equations, parameters and settings).
Fig. 3. Model application window for modifying the current model
4. Investigation of General View Market Model via Computing
Experiments
4.1. Dependence of General View Market Model on Number of Firms
According to [18] with number of firms increase a market moves from stability to
chaos. Whether so it for the model of this paper? Let in system (8) k1 k 2 10 ,
b 200 , v 2 , 1 2 0.99 , 11 22 33 0.5 , 12 21 0.12 , 1 2 0.1 .
Fig. 4. The bifurcation diagram of dependence of quantity x1 on n .
Here the horizontal axis represents the number of firms n from 20 to 50; the
ordinate axis represents the quantity x1 (t ) of first reciprocator firm on attractor of the
trajectory. The path has the equilibrium stable state at n 20 . However as we can see
at n 21 bifurcation occurred and instead of equilibrium point there is a stable cycle.
There values of x1 are approaching the point x1* 40 for even t and the point x1* 10
for odd t . By doubling the lag between iterations only even or only odd iterations
will be considered, and thus either point x1* 40 , or x1* 10 respectively would be the
equilibrium stable state. Stable cycle has four points for n 25 (fig. 4). There was a
new cycle doubling (flip) bifurcation. Calculations show that with parameter n
increase doubling bifurcations continue following Sharkovskii’s order. At n 45
there is a state of dynamic chaos (fig. 4).
Process of division of stable equilibrium on some directions will clear up, if during
it we trace profit changes. Model tools allow us to demonstrate the dependence
between reciprocator firm’s profit and number of firms n for the same parameter
values that in bifurcation diagram 4 above.
Fig. 5. The bifurcation diagram of dependence of profit on the number of firms n .
It appears that the real choice here is unique and depends on quantity output. The
smaller quantity output the bigger the firm’s profit. Moreover, the profit for bigger
output direction varies around zero and often converts into a loss. But quite
unexpected is the effect well visible in a fig. 5: firm’s profit in chaotic state is on
average greater than in stable state. This example illustrates typical, many times
investigated via Model behavior of dynamics of the general view market model with
increasing number of firms.
Analysis of computing experiments for model (7) in [18] show, that such behavior
arises provided that firms in the market are not identical, reciprocators and egoists are
also presented enough there. How can we generalize such condition for the general
view market?
Let in system (8) 12 21 0.5 instead of 0.12 above saving all other parameters.
Then in (8) disappear difference between first and second types of reciprocators, they
unite in one type. Such system has stable equilibrium at all n . By 12 21 0.4 the
whole attractor is a cycle of an order 2 at all n . By 12 21 0.2 it is a cycle of an
order 4 at all n . By 12 21 0.14 a state of dynamic chaos arises at n 140 . At
12 21 0.12 we return to fig. 4, where chaos arises by n 45 .
But the less value of 12 21 the greater difference between types of reciprocators
and so the market is more heterogeneous. All our computing experiments lead to the
following conclusion. The more difference (segregation) between firms i.e. the more
types of firms are in a market, the faster this market directs to complex dynamics and
to chaos due to increase of firms’ number.
4.2. The Crucial Factors which Ensure Stability in General View Market
Apparently the main assumption of the traditional neoclassical economics is the idea
of automatic stabilization and market order due to increasing the number of
independent firms and achievement of perfect competition. This is realization of
Adam Smith's ‘invisible hand’ [19]. Then how stability is possible in real markets
with the effects revealed in the previous section?
We found [17] that adaptive behavior is the main tool that ensures the stability of
model (7). While increasing of number of firms directs a market to complex dynamics
and finally to chaos the increase of adaptive expectations acts in an opposite direction.
Due to increase of adaptive expectations predictability and stability of market
becomes stronger; due to increase of naive expectations the market loses stability and
chaos grows. Whether it is true for multidimensional model of this paper?
Let k1 k 2 10 , b 200 , v 2 , 1 2 0.99 , 12 21 0.12 , 1 2 0.1 ,
33 0.5 as above. But now n 35 and q 11 1 11 22 1 22 is a variable
parameter of following bifurcation diagram. Here q is the parameter of share in
output of a market planned under naive expectations.
Fig. 6. The bifurcation diagram of dependence of quantity x1 on q in the system (8).
Here the ordinate axis represents the quantity x1 (t ) of first reciprocator firm on
attractor of the trajectory; the horizontal axis represents the parameter value of q
multiplied by 10. This rescaling is done for the sake of clarity. In figure the same
behavior that in [18]. And common sense prompts too, that increase of naive
expectations conducts to chaos. However, in multidimensional model it is incorrectly
to estimate a share of planning with naive expectations by the use of parameter
q 11 22 of this example. Apparently we should estimate it by ratios of parameters
ij and ij on all i and j . Formal definition will be given in section 4.4.
Computing experiments and common sense also testify that in multidimensional
systems it is incorrectly to estimate adaptation only by the use of a share of planning
with naive expectations. Let's consider an example. Let k1 k 2 10 , b 200 , v 2 ,
1 2 0.99 , 11 22 33 0.5 , 12 21 0.2 , 1 2 0.1 . At such values of
parameters all trajectories of dynamical system (8) are drawn to stable equilibrium at
all n . Let's now move away values 11 and 22 from their average 0.5 on quantity
11 0.5 0.5 22 . Other parameters we save unchanged. Then at 0 0.2 the
attractor consists of stable cycles. At 0.2 there are cycles of an order 3.
Fig. 7. The bifurcation diagram of dependence of quantity product x1 on n at 0.2 .
Such order of a cycle means that there has already been passed all Sharkovskii’s order
of conditions and there is a dynamic chaos at 0.2 . We observe the similar trends if
average of values 11 and 22 move away from 33 or if 12 move away from 21 .
Numerous computing experiments and common sense testify that stability of the
market critically depends on agreement of adaptive expectations of firms at planning.
In particular, it depends on how close are all parameters ij and respectively all ij .
In addition we note that condition in the end of section 4.1 is only a special case of
this condition: the more types of firms in a market the lower there level of the
agreement of adaptive expectations.
4.3. The Stability Factor of Market in Chaotic State
This part reveals the factor that ensures the stability of the market in a complex and
even chaotic dynamics. If any type of firms increases their profit more quickly than
their rivals then these firms will survive and expand their type among all firms [20].
In model (5) the ratio of profit of firm i from type l at period t
l (t ) (P(t ) v) xi (t ) to profit of firm j from type k k (t ) ( P(t ) v) x j (t ) at
the same time period is:
l (t ) ( P(t ) v) xi (t ) xi (t )
lk (t ) .
k (t ) ( P(t ) v) x j (t ) x j (t )
This is the unexpected finding of our research [18] during computing experiments. In
model (7) lk (t ) is adiabatic invariant of a dynamical system, i.e. it is almost
independent on t at t 2 for all acceptable values of parameters. Direct generalization
of this fact on model (5) proves to be true by all already made computational
researches. For example consider the phase curve that corresponds to trajectory with
dynamic chaos in fig. 4.
Fig. 8. Projection of phase curve of trajectory from fig. 4 at n 45 to a plane x1x3.
The more chaotic dynamics, the more densely populated points on phase curve.
But anyway it almost coincide with line segment, whose slope is equal to lk (t ) . We
can suppose that rare small deviations from a straight line on fig. 8 are just technical
failures at calculations. But look now on next fig. 9 with phase curve of trajectory of
system (8) at parameters n 100 , k1 k2 10 , b 200 , v 2,1 2 0.99 , 11 0.68 ,
22 0.32, 33 0.5, 12 21 0.12, 1 2 0.1 .
Fig. 9. Projection of phase curve with less level of agreement.
Here deviations from a straight line are already indisputable. The cause of difference
from the previous example that here parameters 11 0.68 and 22 0.32
considerably deviate from their average 0.5 33 . As it is noted in the previous
section, it means reduction of level of agreement of adaptive expectations in the
market, the key factor of stability in a market. All computing experiments show that if
this level increases the value lk (t ) comes nearer to a constant.
4.4. Universal Properties of General View Market Model
Let's formulate the formal statements which are clearing up derived results of
computing researches. First of all let’s formalize the key concept of level of
agreement in adaptive expectations in a market.
n
Let xike (t 1) is quantity of firm k expected by a firm i , Qie (t 1) x (t 1) is
k 1
e
ik
prospective industry output of a market expected by a firm i during next time period
Q e (t 1) Q ej (t 1)
t 1 . For firms i and j we put max i , where Q(t ) is industry
ij
t Q(t )
output of the market in period t . The value ij characterizes disagreement in adaptive
expectations of firms i and j . Value max ij we will call the level of
i, j
disagreement in adaptive expectations in the market. Thus value 1 we will call the
level of agreement in adaptive expectations in the market.
Proposition 3. The ratio of profits lk (t ) is equal to a constant with accuracy 3
at all t 2 for any fixed values of parameters of model (5).
Owing to this statement dynamics of a general view market model is stratified on
dynamics of the local markets (7) from [17] with accuracy of the order . That is why
all derived in [17], [18] and considered above properties of the local markets are
generalized on the general view market of this paper. This fact explains universality
of their properties. The formal reduction of following statements to results from [17],
[18] is also based on this statement.
Let firm i suggests that production quantities of its rival j will be equal to
x ej (t 1) ij (t ) xi (t 1) ij (t ) x j (t ) during next time period t 1 , where ij (t ) 0 and
1 n n ij
ij (t ) 0 , i, j 1,..., n . Then the value
n2 i 1 j 1 ij ij
we will call the share
of planning with naive expectations and the value 1 we will call the share of
planning with adaptive expectations in the market. Thus 0 if in the market there
are no naive expectations, and 1 at total using naive expectations for planning.
Proposition 4. At 0 the unique Nash equilibrium of proposition 1 is stable for
all possible values of parameters of a general view market model (5).
Proposition 5. At 1 the unique Nash equilibrium of proposition 1 is unstable
for sufficiently large number of firms n and all other acceptable values of parameters
kl k 3
of model (5) if 3 and l 3 for all types of firms l 1,..., m , where
n n 4
is the level of disagreement in adaptive expectations in the market.
Proposition 6. In a general view market model (5) flip bifurcations (cycle doubling
bifurcations) occur following all Sharkovskii’s order and finally chaos state occur
with an increase of from 0 to 1.
Proposition 7. In a general view market model (5) flip bifurcations occur and
finally chaos state occur with an increase of number of firms in the market provided
sufficiently large 1 .
As model (5) is equivalent to a general view market model (3) in short-run period,
so actually propositions 3 – 7 describe universal properties of general view markets,
including real markets as particular case.
5. Conclusion
Thus we have synthesized the heterogeneous agent-based model of general view
market according to new economics paradigm as intersection of dynamic system
theory, mathematical programming and game theory.
During our investigation we developed and continuously improved a desktop
application Model for support the research process using computing experiments. As
a result of simulation experiments via Model application we have revealed the
following universal properties of general view market, including real markets. They
are derived by generalization and specification of the basic properties of model [17].
The crucial factors which ensure the market stability are the level of agreement in
adaptive expectations and the share of planning with adaptive expectations in a
market. If no any firm use naive expectations in the market there is unique Nash
equilibrium which is stable for all acceptable values of parameters. The increase of
naive expectations leads to stability loss, to flip bifurcations and finally to chaos in
general view market.
The increase of number of firms also leads to stability loss, to bifurcations and
finally to chaos in the general view market at appreciable naive expectations. It
appears that really the choice of equilibrium at these bifurcations is unique.
We revealed that the profits ratio and quantity outputs ratio of firms remains
almost unchanged in short-run period in general view markets. It seems an important
stability factor of many important real markets for which chaotic dynamics is usual.
In the further researches we plan to trace demonstrations of these universal
properties on examples of real markets in details.
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