=Paper= {{Paper |id=Vol-1726/paper-08 |storemode=property |title=Modeling and Decision Support for the Firms’ Pricing Policy under a Chaotic Dynamic of Market Prices |pdfUrl=https://ceur-ws.org/Vol-1726/paper-08.pdf |volume=Vol-1726 |authors=Ekaterina V. Orlova }} ==Modeling and Decision Support for the Firms’ Pricing Policy under a Chaotic Dynamic of Market Prices == https://ceur-ws.org/Vol-1726/paper-08.pdf

Modeling and Decision Support for the Firms’
Pricing Policy under a Chaotic Dynamic of
Market Prices

Ekaterina V. Orlova

Ufa State Aviation Technical University, Ufa, Russia
ekorl@mail.ru

Abstract. The article presents the results of the study of nonlinear
market prices dynamic, simulated using game theory model, maps and
theory of bifurcation. Market pricing is presented as a two-dimensional
map. Qualitative analyses of the firms’ pricing system properties using
the fixed points, analysis of the trajectories near these fixed points were
fulfilled Simulation of the market prices dynamic showed that the fixed
point of the map coincides with the local Nash equilibrium, so the analy-
sis of the stability of Nash equilibrium was done based on the maps’ fixed
points sustainability analysis. Numerical simulation results were visual-
ized, bifurcations of the fixed point were identified and transition from
periodic to chaotic mode was demonstrated. Sustainability analysis was
carried out with using Jacobian. The mechanism for the pricing decision
support, allowing under a chaotic market dynamics to ensure maximum
efficiency of the firms was proposed.

Keywords: nonlinear economic dynamics; visualization of chaos; pric-
ing decision making; fixed point stability criteria

1 Introduction

The last two decades the scientific literature has been widely discussing the
concept of deterministic chaos, chaotic dynamics occurring in different systems.
Different approaches and methods for chaos control in theoretical and applied
problem solution are suggested. At the same time a lot of attention is given to
managing the impact of low power, meaning that the system has a number of
characteristics, inherent properties and laws which allow to achieve expected re-
sults using weak management actions (without spending of significant resources).
In a number of studies of nonlinear systems dynamic (physical, chemical,
biological, economic), it was found that the dynamic chaos mode is a typical
phenomenon. Chaotic properties are manifested in a variety of systems, and if
the chaos is not found, the reasons for this may be either the existence of chaos
in a small area of the parameter space, or it is out of range of parameters.
Research associated with the problems of predictability of chaotic systems,
control of system dynamics and stabilization of chaos is rapidly developing in
different fields. Theoretical and applied studies in these fields have revealed an
unexpected property of chaotic dynamical systems: they are controlled by exter-
nal actions [3,13,18,21]. That is, by smaller impacts it is possible to significantly
affect the dynamics of chaotic systems, to stabilize their dynamics, transferring
it from the chaotic mode to the required periodic mode.
In relation to economic systems the following studies in the field of chaos
control were conducted: [1, 2, 5–8, 11, 12, 14, 15, 19, 20]. Chaos control is proposed
on the basis of production cost reduction [2], an increase in investment activ-
ity [11, 19], or control impacts are confirmed on the basis of revealed connection
between the previous and current variables values, i. e. by effective use of feed-
back [2, 19, 22].

2 Phenomenon of Chaotic Market Dynamic: the Model

Chaotic systems are a class of uncertain models that differs from deterministic
and stochastic systems in their properties [4]. In deterministic systems it is pos-
sible to build the future system trajectory from the initial state to infinite time
interval. In stochastic systems it is possible to estimate the future system state
into short time interval,determined by the prediction accuracy.
We consider a duopoly as a market model, when two firms-producers co-
operate on the market, and the pricing process is controlled by price (which
firms offer there differential products to consumers) change. Further it will be
illustrated that the pricing system is chaotic and then we will offer the decision
support model for stabilization market prices dynamic.
The main precondition of the model:

– The model of duopoly (model of price competition) with a differentiated
product in which the consumer demand function is given a utility function
with constant elasticity of substitution is used;
– Pricing dynamics is modeled as a map (recurrence relations);
– Modeling of pricing decisions is realized with using the game theory and
methods of nonlinear dynamic.

The model designations: i ∈ I – producer’s number; pi – price of the i-th
producer (firm); qi – the volume of sales of i-th producer; qi (p̄) – demand for
the product of i-th producer; ci – unit costs for the product of i-th producer;
Πi (p̄) = (pi − ci ) qi (p̄) – profit of i-th producer.
Prices vector p̄∗ = (p∗i , i ∈ I) is a local Nash equilibrium when it satisfies the
following conditions of local maxima conditional of the profit function:

∂Πi ∂ 2 Πi
= 0, < 0.
∂pi ∂(pi )2

The strategy of i-th producer is a price changing, which is proportional to
the change of its profits with some constant ki > 0 in order to maximize their
effectiveness, i. e. to achieve the global sustainable Nash equilibrium. The model
of competitive interaction of firms described by two-dimensional system of dif-
ferential equations was offered in [16, 17] and represented as:

(−p21 (t)p2 (t)+2c1 p1 (t)p2 (t)+c1 p22 (t))

 p1 (t + 1) = p1 (t) + k1
 2 ,
(p21 (t)+p1 (t)p2 (t))
−p (t)p2 (t)+2c2 p1 (t)p2 (t)+c2 p21 (t))
(1)
 p2 (t + 1) = p2 (t) + k2 ( 1 2 2

2 ,
(p2 (t)+p1 (t)p2 (t))

where p1 (t), p2 (t) are product prices of first and second producers at discrete
time t; second terms in both equations show the prices changing in period t, and
how this changing will affect the price in the next period (t + 1). Parameters k1
and k2 characterizes the prices increasing due to changes in the firms’ pricing
policy; c1 and c2 represents a production cost of the first and second producer
respectively.
Nash equilibrium, characterized by a pair of prices p∗1 and p∗2 , is the solution
of the following equations:
 q
 p∗ = c1 + c1 1 + p∗2 ,
1 A1
q (2)
 p∗ = c + c 1 + p∗1 .
2 2 2 A2

Analyze the evolution of the system (1) according to the parameter values k1 ,
k2 , A1 , A2 identify the area of stability, bifurcations and chaos. Such analysis has
been made based on the known analytical and graphical criteria of dynamic chaos
[10, 13]: the Lyapunov exponents, bifurcation diagrams, attractors of the system
by varying the system parameters. Then, in order to control the system dynamic
on the basis of the revealed laws, it is necessary to find the method for parameters
changing in order to provide an expected mode of the system dynamic. For each
firm, this means the monitoring and controlling of their costs Ai and changing
in profit for the period and the selection of appropriate control ki . In practice
this means the variation of price, which causes both firms to balance interests,
i. e. the Nash equilibrium.
From the economic point of view, this means that firms choose the mode (and
parameters), which will lead to a change in the market, resulting in price levels
will evolve predictable dynamic. If such control is permissible, then a transition
to the market equilibrium will proceed, that ensure for each firm the maximum
effectiveness.
The fixed point (p10 , p20 ) of the system (1) is a point which goes into itself
under a single iteration of the map and is determined on the basis of equations:
(
p10 = f (p10 , p20 ),
p20 = g(p10 , p20 ),

where f – the function p1 (t + 1) of p1 (t) and p2 (t), g – the function p2 (t + 1)
of p1 (t) and p2 (t) of the map (1).
The decision of the latter system of equations will obviously be the same as
the solution of the system (2). Therefore, the fixed point of map (1) coincides
with the Nash equilibrium, for a certain competitive interaction between firms.
The nature of the stability of a fixed point is defined by its multipliers, which are
the eigenvalues of the perturbation matrix (Jacobian) and their number is equal
to the dimension of the display. The bifurcation analysis and stability analysis
of two-dimensional maps is carried out on the basis of parameters – invariants
of Jacobi matrix. For the two-dimensional map there are track and Jacobian
of Jacobi matrix. Jacobi matrix of the dynamic system (1) at a fixed point is
as follows: 0 0
Mc = fp0 1 fp0 2 (3)
gp1 gp2 (p ,p ).
10 20

Eigenvalues of this matrix are multiples of the map (1) µ1 and µ2 , for which
the relation is performed: µ2 − Sµ + J = 0, where S and J – two invariants of
Jacobi matrix – track and Jacobian, also S = µ1 + µ2 , J = µ1 · µ2 . In accordance
with a triangle of stability [9], the conditions of stability of a fixed point are
presented as: 
1 − S + J > 0,

1 + S + J > 0, (4)

J < 1.


3 Pricing Decision Making: Visualization and
Justification

We form the prices Nash equilibrium p∗1 and p∗2 , and find the conditions to achieve
and maintain this balance for given costs Ai . If we fix a cost A1 ,vA2 at the
level 0.07 and 0.12 respectively, the Nash equilibrium prices (2) are: p∗1 = 0.15;
p∗2 = 0.23. Jacobi matrix (3) at this point would be:

c = 1 − 30.5k1
M
4k1
2.9k2 1 − 9.9k2 (p∗ ,p∗ )
1 2

for which the characteristic polynomial µ2 − Sµ + J = 0 has a trace and the
Jacobian:

S = 2 − 9.9k2 − 30.5k1 ,
J = (9.9k2 − 1)(30.5k1 − 1) − 11.7k1 k2 = 290.25k1 k2 − 30.5k1 − 9.9k2 + 1.

The equilibrium stability conditions formulated by the system (4). Solving
this system with respect to k1 and k2 we obtain the range of the speed of ad-
justment k1 and k2 for Nash equilibrium (0.15; 0.23):

290.25k1 k2 > 0,

290.25k1 k2 − 61k1 − 19.8k2 + 4 > 0,

290.25k1 k2 − 30.5k1 − 9.9k2 + 1 < 1.

Fig. 1. Decision making area (gray) for the parameters k1 and k2

The solution of this system of inequalities that define the triangle of stability
for the fixed point of the two-dimensional map prices (1), is shown in Fig. 1.
Thus, the identified area of admissible values of adaptive price parameters k1
and k2 is highlighted in gray. To ensure the stability of the Nash price equilibrium
(0.15; 0.23) in the conditions of the given cost values of both firms at 0.07 and
0.12 will allow the use of such firms adaptation strategy, which is based on a set
of adaptation options combinations that are within the acceptable area.
As soon as the control parameters deviate from the permissible values, there
occurs the equilibrium stability loosing and the system goes to another unstable
mode – chaos. That is, in any initial price of firms if they use a pricing strat-
egy in accordance with the values determined above, the firm definitely reaches
the Nash equilibrium. These pricing values are not able to change the Nash
price equilibrium, so to increase the efficiency the firm should use the proposed
decisions.
Figure 2 shows the chaotic attractors of system (1) in terms of unit costs
A1 = 0.07; A2 = 0.12 and variations in the parameters k1 , k2 . For instance, when
the values of the chaotic attractor of price adaptation parameters are k1 = 1,
k2 = 1.15 it demonstrates the chaotic pricing system dynamic. This mode is de-
termined by nonlinear system properties and manifests itself in an exponentially
rapid divergence of initially close trajectories in a bounded phase space.
The chaotic nature of the prices system dynamics is due to the instability of
the phase trajectories, the growth of small initial perturbations in time, mixing
elements of the phase space and, as a consequence, leads to unpredictable system
dynamic over long term.
In making pricing decisions in addition to the criterion of economic efficiency
the firms must take into account the objective nature of the market dynamics
as a whole and take into account the possibility of a chaotic regime of market
dynamics.
Fig. 2. Chaotic attractors of market price system (1) under variations of parameters
k1 and k2 (a-i)

4 Results and conclusions

The described approach for firms pricing using the methods of game theory,
nonlinear dynamics and bifurcation theory provides a new perspective on the
dynamics of the process of competitive interaction of the firms and makes it
possible to conduct a qualitative and visual analysis of the system properties
with help of the singular points of the phase space (fixed points) and to analyze
the systems trajectories near these fixed points.
It is shown that the market pricing system has complex and diverse types of
dynamics, so that the structure of the phase space and its dependence on the
parameters of this structure are very complex. The phase space of the system is
heterogeneous and has two basic types of system dynamic – stability and chaos.
Prices dynamics is modeled using a two-dimensional map; coordination of
firms’ pricing decisions is based on monitoring the stability of the Nash equi-
librium. The analysis of the developed model shows that the Nash equilibrium
coincides with the map fixed point prices. Therefore, the analysis of Nash equi-
librium stability is carried out on the basis of the analysis of the map fixed
points sustainability. Numerical simulations are demonstrated the existence of
fixed point bifurcations. Chaos in market pricing model means that when one
firm change its price even slightly this can lead to unpredictable market prices
changing of another producers and total market in long term. Therefore, all
producers must have the tools of chaos control.
In order to avoid unexpected chaotic dynamics in market prices the mecha-
nism for decision making support is offered and is based on the price changing
proportional to marginal profit changing of each firm. These mechanism would
ensure the stability of the Nash equilibrium, and therefore would balance the
firms’ economic interests, would coordinate price decisions and maintain maxi-
mum firms efficiency.
For the complex research and analysis of firms competitive interaction we use
the author program for modeling and visualization of nonlinear pricing dynamics
and decision support in firms price strategies. The program is designed to simu-
late the strategic cooperation in the firms pricing process, use a four-parameter
map and form the optimal pricing policy in oligopoly, provided the effective con-
trol and decision making under prices chaotic dynamics. The program has the
following functions:

– Assessment of local Nash equilibrium of prices;
– Identification of bifurcations of fixed price points in the map;
– Identification of modes of stability and dynamical chaos in pricing;
– Identification of transition scenarios to dynamical chaos;
– Forming the pricing decisions, ensuring stable mode of market prices dy-
namic.

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