CC BY
6Margarita A. Gladkova and Nikolay A. Zenkevich
St.Petersburg State University,
Graduate School of Management,
Volkhovsky Per. 3, St.Petersburg, 199004, Russia E-mail: [email protected] [email protected]
Abstract In the paper a game-theoretical model of quality choice under competition is suggested. The game-theoretical model is presented as a two-stage game where production companies compete on an industrial market and consumer’s taste to quality in non-uniformly distributed. The strong Nash equilibrium in the investigated game was obtained in explicit form which allowed us to evaluate prices, companies market shares and revenues in the equilibrium. A case study for Internet-trading systems was used to approve the suggested quality choice mechanism.
Keywords: quality evaluation, quality measurement, consumer’s taste to quality, quality choice, two-stage game, Nash equilibrium, Stakelberg equilibrium, Pareto-optimal solution, optimal quality differentiation, index of consumers satisfaction.
1. Introduction
The problem of quantitative estimation of quality and the development of quality choice mechanism in case of competition are considered. Quality choice is an action that is based on changing of quantitative quality estimation.
The main theoretical goal of the research is to develop a quality choice mechanism which is based on construction and solution of the appropriate game-theoretical model of competition taking into account the information on consumers preferences. From practical point of view we are interested in quantitative quality estimation methods.
In this paper quality of a product (or service) is considered as quantitative estimation of its value expressed in monetary terms which an average consumer gets when buying this product (or service). Therefore when consumer is absolutely satisfied with the product (or service) its quality is equal to price he or she paid.
In order to estimate quality of an object basing on consumers’ opinions about object’s characteristics to the questionnaire we will consider an object as a system of these characteristics. Therefore, quality is calculated as some composite index, which allows to estimate the level of consumers satisfaction with the product. To realize this approach the technique presented in Hovanov et al., 1996 is used.
To define the preferable product quality under competition a game-theoretical model was built. This model let us analyze the companies’ decision-making process about goods production of demanded quality under competition.
* This work was supported by the Russian Foundation for Basic Research under grant No. 08-01-00301-a and and up to subject-matter plan of Graduate school of management, SPbU (project 16.0.116.2009).
The suggested game-theoretical model is an extension of the models presented in Benassi and Motta papers. They considered duopoly models under condition of vertical product differentiation.
Motta (Motta, 1993) analyzes two types of models of vertical differentiation in order to study the influence of price and quantity competition on the Nash equilibrium solution. The author shows that optimal product differentiation is higher in Bertrand competition rather than in Cournot. This model is upgraded to the case when market is uncovered and inclination to quality parameter is non-uniformly distributed (triangular distribution is analyzed). Another modification of this model is considered in Gladkova and Zenkevich, 2007.
In his paper Benassi (Benassi et al., 2006) considers duopoly model under condition of vertical product differentiation when market is uncovered. Author examines the influence of consumer concentration according to their willingness to pay for quality on companies behavior and decisions.
Noh’s paper (Noh and Moschini, 1993) is focused on sequential quality choice in the game-theoretical model of duopoly and vertical product differentiation (Stack-elberg model). The study is limited by the case of covered market. As well, similar problem of simultaneous and sequential quality choice is considered in Aoki and Pursa’s paper (Aoki and Pursa, 1996).
Theoretically the main goal of this paper is to find quality Nash equilibrium and optimal product differentiation in case of competition. To do that the game-theoretical model of duopoly was constructed, which is based on Tirole, 1988 and Gladkova and Zenkevich, 2009.
2. Game-theoretical model of quality competition
Two-stage game-theoretical model of duopoly under vertical product differentiation is investigated. It is assumed, that there are two firms - 1 and 2 - on some industrial market which produce homogeneous product differentiated by quality. The game consists of two stages, when at the first stage firms set its product quality level and on the second stage they compete in prices knowing the qualities. Suppose, that on each stage firms make their decisions simultaneously.
Suppose that each consumer has unit demand and has different inclination to quality. Assume that a customer is indicated by the parameter 0 G [0, 0] - ” inclination to quality” which defines a customer’s willingness to pay for quality.
Then the utility function of the consumer with inclination to quality 0 (from now on we will simply call him/her ”the consumer 0”) when buying the product of quality s for price p is:
where 0 G [0, 0\ - inclination to quality of this customer. Here 0 s is maximum price that the consumer 0 is ready to pay for the product of quality s, i.e. the worth of the product for the consumer 0.
It is clear that the consumer 0 will purchase the product of quality s for price p if Uq (p, s) > 0 and won’t buy a product otherwise.
In the model assume that the parameter of inclination to quality 0 is a random variable and has triangular distribution with the following density function f (0):
(1)
f (0) =
for 0 < 0
for 0eA=(O, |]
for 0gB = (|,6]
0, for 0 > 6
Then distribution function can be presented as follows:
0, for 0 0
p02, for0GA=(O, |]
f0-^02-1. for0GB = (|,6]
1,
for 0 > 6
Here the parameter b G [0, 0] is an endpoint of the distribution support. Note that distribution function is continuous, differentiated and increasing on the interval [0, 6]. Figure 1 represents the view of the density function f (0).
rm
2Jb
▲
/ A B \
b/2
e
Figurel. The density function f (0).
Customer 0 is indifferent between buying a product of quality si for price pi d not buying anything, if characterize such customer.
and not buying anything, if 0s\ — p\ = 0. Therefore a value 0\ = Q\(p\, s\) = ^
0
4
Let firm i produces goods of quality si and the production costs for the product of quality si are ci. Lets, for instance, s2 > s1 and this values are known to both firms and customers. Assume that there is Bertrand competition in price. Let us denote by pi the price of the firm i for the product of quality si.
Customer 0 is indifferent between buying a products of quality s1,s2 for prices p1,p2 correspondingly, if 0s1-p1 = 0s2-p2. Therefore a value 02 = 02(p1,p2,s1,s2) =
-n-—221 characterize such customer.
s2 - s1
Define the demand functions Di(p1 ,p2, s^ s2) of the firm 1 and 2 correspondingly
as:
f02(P1,P2,S1,S2)
D1(p1,p2, s1, s2) = f (0)d0 = F(02(p1,p2, s1, s2)) - F(01 (p1, s1));
Jd l(pi,Sl)
D2(p1,p2,s1,s2) = f f (0)d0 = 1 - F(02(p1,p2,s1,s2)).
^2(pi,P2,Sl,S2)
Firm i’s payoff when producing a product of quality s* , where s* G [s, s] , will be defined as following function:
Ri(p1,p2, s1, s2) = pi ■ Di(p1,p2, s1, s2), (2)
where pi is the price of the firm i for the product of quality si.
The game-theoretical model of quality choice is defined as a two-stage model of duopoly, when :
— at the first stage firms i simultaneously choose a quality levels si;
— at the second stage assuming that the quality levels si are known both to competitors and customers, firms compete in product price, making their choices simultaneously.
To solve this game we use the backward induction. In this case Nash equilibrium is constructed in two steps. On the first step, assuming that product qualities si are known, we find equilibrium prices p*(s1,s2). Knowing p*(s1, s2), on the second step we find qualities s*, s2 of products of firm 1 and 2 correspondingly in Nash equilibrium.
As the density function of the parameter 0 is considered to be triangular, the explicit form of the demand functions will differ depending on the location of consumers 01 and 02 across the interval [0,b]. Theoretically, there are three possible cases:
1. 01,02 e A,
2. 01,02 e B,
3. 01 e A,02 e B,
where A = [0, 6/2] , B = (b/2, 6] are illustrated on the Figure 1.
To find the equilibrium prices prove the following theorem.
Theorem 1. Consider any concave density function f (0) defined over [0,6], where b > 0, 5, which is symmetric about the median of the distribution 6/2 and f (0) = f (1) = 0 and f (6/2) > 2. If 02* > 0* identifies the consumers at the perfect price Nash equilibrium in the game in case of vertical differentiation, then 0* is unique and 02* < 6/2.
Proof. Players’ payoff functions are:
R1 (p1,p2, s1, s2) = p1(F(02) - F(01)),
R2(p1, p2, s1, s2) = p2 (1 - F(02)),
where el = fi,02 = Ps\ I g.
Let’s first calculate the derivative of R2 with respect to p2 and equal it to zero:
= 1 _ F{e2) - _P2_/(02) = 0.
dp2 s2 - s1
Thus we get the following equation:
z(02) = 1 - F(02), (3)
where z(02) = (t + 92)f(92), t = S2PlSi > 0.
The following inequality correct:
Besides, z(0) = tf (0) = 0 < 1 - F(0) = 1. Thus
Z{1) Z(0)<1~F(0)-Hence, the solution of the equation (3) is 9* < ^-
Note that on the interval [0, 6/2] the function R2 is strictly concave with respect to p2 (or 02). In particular, for the triangular distribution (see Figure 1) we have:
92Rl -______„__6_____ < 0
dp2 b (s2 - si)
Hence in the critical point 9*, where the equation (3) is satisfied, there is a
maximum point of the payoff function R2 on the interval [0,b/2j.
Lets prove that in 92* the largest value of the function R2 on the interval [0, b] is achieved and this point is unique.
Lets analyze the equation (3). As the distribution function F(9) is strictly increasing on [0,b], then right part of the equation 1 — F(92) is strictly decreasing on [0,b]. ;
Left part of the equation (3) z(92) is strictly increasing until f (92) > 0. This is
due to the view of the derivative: z (92) = f (92) + ( + 92)f (92).
The function z(92) is continuous and z(b) = z(0) = 0. Then the largest value of the function z(92) on [0,b] is achieved in the point 92 = 92. The inequality
f (62) > 0 is true for any Besides, on this interval z ($2) > 0. Hence O2 > ^
and 02* e [0, 02].
Consider now an interval 02 e [02, 6] and show that on this interval there is no
point where the largest value of the function R2 is achieved.
To do so, introduce a function p(02) = 1 - F(02) - z(02) and calculate its
derivative p (02) = -2f (02) - (t + 02) f' (02).
As the density function f (02) is decreasing and concave on the interval [02,6], then f (02) is decreasing or p (02) is increasing functions.
Calculate the value of the function derivative p(02) in points 02 and 6. Then:
p' (£2) = -2f (02) - (t + 02) f' (&) = -f (%) - z' ($2) =
= -f (O2) < 0, z' (02) =0.
The derivative p (6) = -2f (6) - (t + 6) f (6) = - (t + 6) f (6) > 0, as f (6) = 0 and f' (6) < 0.
In sum, p (02) is increasing, p (02) < 0 and p (6) > 0. Thus there is a point
02 = 02, where p (02) =0, and the minimum is achieved there.
As far as p (02) < 0 and p(6) = 0, then on the whole interval [02, 6] the function
92,b
and then on the interval
(92) < 0. Then 1 — F(92) < z(92) for any 92
[92, b] there are no points where the equation (2.3) is true.
Then there is unique parameter value 9* in price equilibrium where 92* > 9* and 92* < b/2.
It is implied from theorem 1 that in order to find price in equilibrium it is sufficient to consider only one case when the parameters 01,02 e A (see Figure 1). Then the demand functions of firms 1 and 2 can be presented as following:
2
Dl(pi,p2, Si, S2) = To b2
P 2 - p 1 «2 - «1
2
¥
n f ^ 1 2 fP2~ Pi
U2{Pl,P2, Si, S2j = 1 - 72 --------
b2 \ S2 — Si
and the payoff functions can be written in the explicit form as:
2
2
2
R2 (pi, P2, Si, S2) = P2
P 2 ~P 1 S2 ~ «1 ,
Lets find prices in equilibrium p*,p2* when qualities of goods are s1 and s2.
The price values p1,p* can be found as a solution of the following system of equations:
( dRi _ 2 f(P2-Pl\2 o (Pl\2 ov (P2 -Pi) A n
“3W ~2p\s2-Slf J
2 = 1 _ 6 (P2 -Pl\ _ 4Pl_ = 0
2 FV«2 -Si) W^~~) ■
To solve the system of equations we make a substitution p2 = mp1, where a coefficient m > 1. Then first equation can be rewritten as the following quadratic equation in m:
to2 - 4m + 3 - 3^-5^- = 0.
Si
Taking into consideration that m > 1, we receive:
m = 2+yi + 3(S2 ^ . (4)
From the system of equation we get the price Nash equilibrium in explicit form:
pl(sus2) = ^l
bsi m — 3
a/6 \J (3m — l)(m — 3) ’
(5)
P*{su S2) = h*L. , m(m 3) ;
a/6 \J (3m — l)(m — 3)
where m is given as (5).
Lets now calculate in the explicit form the demand of the firm 1 and 2 and its payoffs in equilibrium as the functions of qualities:
Dt(si, s2) = DKpKs!, s2),p*(si, s2), si, s2) =
(6)
D2(s1,s2) = D*(p*1(s1, s2),p*(s1, s2), si, s2) =
D* í \ D* í * í \ * Í \ \ 26SI 'm('m 3)
R^si, s2) = i2i(pi(si, s2),p2{si, s2), si, s2) = —7= * ,,0 v ^3, y 0, ;
^6 V (3m — 1)°(m — 3)
fi2(si, S2) = R2(p\(si, s2),_p^(si, s2), Si, S2) = --------m ~
(7)
a/6 (3m — l)3(m — 3)
At the second stage of the game we find qualities in Nash equilibrium s1,s2 e [s, s] according to the payoff functions i?*, R2, where s <s are given.
The partial derivative of the firm 2’s payoff R2 with respect to s2 is equal to:
<9i?2(si, s2) _ b(s2 - si)a/6 m(3m2 - 7m + 6)
ds
2 \Js\ + 3(s2 - si)2 y(3m - l)5(m - 3)
It is easy to check that taking into account s2 > si the partial derivative ’ S2^ > 0 , i.e. the function R\(si, s2) is strictly increasing with respect to s2 . Thus, the firm 2’s equilibrium strategy will be the choice of maximum possible quality value, i.e. s2 = s .
To find the equilibrium value s1 of the firm 1 make the following substitution of variables s* = ks, where 0 < k < 1 is an unknown parameter. Then the parameter k can be found using the following condition:
dRl(ks,s)
dk
0.
(8)
The explicit view of this equation solution is too lengthy but for any given quantitative value of the parameter b, it is possible to get the quantitative value of the parameter k, using computer algebra system Maple.
For instance, when the parameter of consumer’s willingness to pay for quality
0 defined in the interval [0; 0, 5] , i.e. b = 0, 5, the solution of the equation (9) is k = 0, 6543. In this case Nash equilibrium will be the following:
s* = 0, 6543s,
(9)
Then substituting this solution in (5), we get the value of the parameter m = 3, 3555.
According to the expressions for the equilibrium prices (6), demands (7) and payoffs (8), we get final expressions for equilibrium prices p*,p2, demands D*,D* and payoffs R*, R* in equilibrium with respect to the parameters b and k as following:
Pi =
kbs
^Jk2 + 3(1 - k)2 -k _
a/6 Y 3^/k2 + 3(1 - k)2 + 5k ’
p* = ^.(2k+ y/k2 + 3(l-k)2) •
I yjk2 + 3(1 - k)2 -k 3\Jk2 + 3(1 — k)2 + 5k'
D*
2\/k2 + 3(1 — k)2 + Ak .
3(3 v^2 + 3(1 - k)2 + 5k) ’
p* _ 2\Jk2 + 3(1 - k)2 + Ak
2 _ 3y/k2 +3(1 -k)2 + 5k'
R*
2kbs
3a/6
• (2k + \Jk2 + 3(1 - k)2^j ■
R* = 2bg.(2k+ ^Jk2 + 3(1 - k)2
\
\
\/k2 + 3(1 - k)2 -k 3^/k2 + 3(1 - k)2 + 5k)
3 ’
\/k2 + 3(1 - k)2 -k 3^/k2 + 3(1 - k)2 + 5kJ
So = s.
2
Note that firm 2 which produces the product of the higher quality s2 gets higher payoff in equilibrium than firm 1, as far as:
ñ* 7~>*
2 R1
• (2*+v*2 + 3(1 - *o2)
V^2 + 3(1 - k)2 -k S^k2 + 3(1 - k)2 + hkj
x
2J > 0.
According to the model construction there two asymmetric Nash equilibriums in the model (A:s, s) and (s, A:s) , which are beneficial for 2 and 1 players correspondingly. It is easy to note that both equilibriums are Pareto optimal, which means that they are strong equilibriums (Petrosyan et al., 1998).
Therefore when choosing optimal strategies under condition of competition the players face the problem of so called fight for leadership, like in the game ’’Battery of sexes”(Petrosyan et al., 1998). Thus, each firm will try to become a leader, i.e. to start the production of the higher quality goods, which will give the firm the more profitable position in equilibrium.
Note as well that if we consider the Stackelberg model (firm 2 is a leader, firm 1 is a follower) the result will be similar but there will be only one equilibrium (ks,s). Besides, the leader (firm 2) use its right to act first and will take up the more benefit position in equilibrium.
x fy + V*2 + 3(1 - A)
3. Numerical example
In this section the numerical example of application of the suggested game-theoretical model of competition and vertical differentiation for the market of Internet-trading systems, which are used for exchange auction.
Internet-trading (or electronic-trading) is new, simple in use and highly effective software package(e-trading platforms). Stock markets are not physical locations where buyers and sellers meet and negotiate, Internet-trading allows traders to transact from remote locations. As well, it gives access to a huge amount of analytical information.
There are more than 20 Internet-trading systems now which are used in Russia. Some brokers develop their own systems, others use systems that were developed by specialist software providers. In Russia specialist software is dominated. Here first of all QUIK, NetInvestor, TRANSAQ and ”ITS-Broker” can be called. The Internet-trading system developed by QUIK is the most popular and is used by more than 60 brokers (more than 3500 users).
There are more than 100 exchange brokers, which uses the Internet-trading systems. The list of such organizations is presented on the web-sites of the biggest Russian stock exchanges - ’Moscow Interbank Currency Exchange” (MICEX Group) and ’’Russian Trading System” (RTS).
The main purpose of the Internet-trading system is an on-line access to trading systems. It lets the users to get the stock information and to make transactions. As well it gives information about current state of the investment portfolio (the quantity of bought/sold stock certificates),position with regard to monetary resources and usually there is an option of price charts review and other additional features.
When doing empirical investigation of the Internet-trading system quality we distinguished eight main characteristics of such system, namely:
— quantity of available exchanging markets;
— operation speed, i.e. the speed of referring a request and getting information;
— system functionality (availability of price quotations, time-series and charts construction), i.e. availability of integrated analytic;
— technical support;
— ability of data export;
— possibility of system upgrading by a user;
— price for a system and its maintenance;
— guarantee and durability, i.e. responsibilities of development company for any possible errors, its elimination and compensation for losses.
3.1. Sample description
Data accumulation was organized using an experts questionnaire survey. Based on the results of the survey we have a sample of 29 respondents. In our research we were interested in opinions of the direct users of Internet-trading systems, namely, members of the department of trading systems management and economists from brokers companies (commission houses), who works with such systems. By geographic location we chose users of Internet-trading systems from such biggest Russian cities as Moscow, Saint-Petersburg and Ekaterinburg.
Broker companies often works with several Internet-trading systems, which allows them to satisfy different groups of investors. As the QUIK system is the dominant Internet-trading system on Russian market, we distinguish two types of systems
- the QUIK system and the OTHER system (which includes all other systems). According to this suggestion, we get that 22 respondents are the users of the QUIK system and 20 respondents work with the OTHER system.
3.2. Analysis and evaluation of the quality of Internet-trading system
The definitions of quality that are given in ISO 9000 (2005) documents distinguish the systematic formation of all object characteristics. Thus, we can accept quality of an Internet-trading system as some generalized quantitative estimation of quality or composite quality index.
In this research respondents are consumers and experts at one time. On the basis of their opinions about each characteristic of Internet-trading system we evaluated this system quality in general. To realize this approach the composite index method is used. This method is realized in ASPID-3W (Hovanov et al., 1996). Some specifications of this method use and application for the evaluation of a quality of any complex technical system under condition of information deficiency are presented, for example, in Hovanov et al., 2009.
The first stage of empirical research was data analysis and processing in order to define the observed Internet-trading systems’ quality. For this purpose respondents were asked to evaluate their satisfaction level concerning each system characteristic that were mentioned at the beginning of this section. Besides, if a respondent uses several Internet-trading systems he or she should estimate his satisfaction levels for each system.
Then, using the composite index method, the Internet-trading systems qualities were received in 3 steps:
1. first the composite indexes of satisfaction level for each system characteristic for the QUIK system were defined; and for each system characteristic the composite indexes for the OTHER system were calculated;
2. on the basis of the composite indexes the generalized composite indexes of consumer satisfaction a2 and a1 for the QUIK system and for the OTHER system correspondingly were evaluated ;
3. on the basis of the generalized composite indexes a2 and a1 the qualitative estimation of system quality is received using the following formulas: s2 = a2p2 (for the system QUIK) and s1 = a1p1 (for the system OTHER), where p2,p1 are prices of the system QUIK and OTHER.
The formula we are going to use for quantitative estimation of quality needs to be explained. The questionnaire contained the following question: ”If you aren’t absolutely satisfied with your current Internet-trading systems, could you please evaluate how much are you ready to increase the maximum price for which you would buy a system that absolutely satisfies you (in percentage)?” Therefore if a respondent is absolutely satisfied with a system, then according to the understanding of quantitative estimation of quality the quality value s = p0 , where p0- is the price of the system. At the same time if consumer satisfaction level is 0 < a < 1, then s = apo .
Lets discuss now the relation of empirical and theoretical models. According to our model assumptions if a consumer 00 is absolutely satisfied with its current Internet-trading system then the maximum price he or she is ready to pay for it is equal to 00p0. On the other hand: 00p0 = p0 + Ap, where Ap is an increment of the current price for which a respondent is ready to buy the considered Internet-trading
Ap
system. Hence, Qo = 1 + > 1. Lets denote respondent’s willingness to pay for
quality for: 0 =
Therefore the utility function of the consumer whose willingness to pay for quality is 0 can be presented as:
where 0 G [0,6] , and right interval endpoint b is defined from the questionnaire answers of respondents.
According to the data processing algorithm and analysis it is clear that usage of the game-theoretical model of competition presented in section 2 is reasonable for our sample.
3.3. Results
This subsection presents the results of the realization of the algorithm described above using the data obtained from the consumers survey.
The data is collected by questionnaire survey. The questionnaire consists of twelve questions divided by three groups which helped us to analyze the consumers and Internet-trading systems specification.
First question group covered basic things such as what systems respondents use, what for and how much they are satisfied in general with what they have now.
Second question group is about Internet-trading systems’ characteristics. There a respondent is asked to put in order these characteristics according to its level
(10)
of importance. Then a respondent has to answer which systems’ characteristics he or she is satisfied with and evaluate the level of that satisfaction (using 5 pointed Lakert scale).
Third question group is about consumer’s preference about Internet-trading systems:
— which system is a key system for the organization;
— how much they are ready to increase the maximum price for which they would
buy a system that absolutely satisfies them;
— which system they would like to use in the future;
— which brand is more preferable.
The estimation of the quality of the system QUIK (product 2) and OTHER (product 1) is realized using the consumers’ answers on the second question group. Thus, with the help of the ASPID-3W, the composite indexes of consumers’ satisfaction for each Internet-trading system characteristic were obtained (see table 2). Weights coefficients were received from consumers range of the importance of each of 8 characteristic of the Internet-trading systems. They are presented in the table
1.
Then the composite indexes of consumers satisfaction a1 = 0, 572 and a2 =
0, 545 for system QUIK and OTHER correspondingly were calculated.
Table 1. The weights coefficients for each characteristic
Characteristics Weights
quantity of available markets 6.103
operation speed 7,172
integrated analytic 3,552
technical support 5,517
data export 4,103
ability to self upgrading 2,828
price 4,379
guarantee 3,586
Table 2. Composite index of satisfaction for each characteristic
Characteristics Quik Other
quantity of available markets 0,639 0,774
operation speed 0,549 0,701
integrated analytic 0,492 0,498
technical support 0,699 0,612
data export 0,610 0,500
ability to self upgrading 0,394 0,407
price 0,507 0,636
guarantee 0,470 0,450
For each system the quality is evaluated according to the formula: si = aipi , where i =1 means the OTHER system and i = 2 - the QUIK system. The price for the Internet-trading system OTHER is: p1 = 119000 RUB. (this price is an arithmetic mean of prices for each system from the group OTHER, which are presented on companies web-sites) and the price for the Internet-trading system
QUIK is p2 = 140000 RUB. correspondingly. Therefore the estimations of Internet-trading systems’ qualities s1 = 68068 RUB. and s2 = 76300 RUB. correspondingly.
Let’s estimate now the range of Internet-trading systems qualities, i.e. estimate the parameters s and s. To do that evaluate first the composite indexes of consumers satisfaction using the ASPID-3W for a consumer who appraises his satisfaction with each system characteristic as ”1 - absolutely unsatisfied” and for a consumer who appraises his satisfaction with each system characteristic as ”5 - totally satisfied”. The results are a = 0,056 and a = 1,000 for these consumers correspondingly. Thus, the range of Internet-trading systems qualities are s = api = 6664 RUB. and s = ap2 = 140000 RUB.
The endpoint b of the parameter 0 is evaluated as b = max{max Ap1, max Ap2} = 0, 5. Therefore, 0 e [0; 0, 5].
Lets test a hypothesis about the triangular distribution of the parameter 0 of the consumer willingness to pay for quality over the interval [0;0, 5] .
As the sample is not big enough, consider the hypothesis of the triangular distribution of the average consumer willingness to pay for quality for the sampling group of consumers. Therefore, we have to test the hypothesis that parameter 0 has triangular distribution on the interval [0,15916; 0, 220852].
Table 3 presents the results of subsidiary calculations when testing distribution hypothesis using Kolmogorov test. Here xi and xi+1 - cell boundaries which are the result of the sample division, li - theoretical frequencies for cell i, F* - the value of empirical distribution function, F - the value of expected (theoretical) distribution function.
Using Kolmogorov test, a statistic is calculated according to the formula:
A* = sup | F*(xi) — F(xi) |= 1,124498.
Therefore, the triangular distribution hypothesis is accepted with significance value equal to 0,01.
The comparison of the empirical results and game-theoretical results described in the previous section is presented below.
Table 3. Triangular distribution test
Xi h p* F F* -F
0,15916 0,167973 1 0,01 0,034327 0,024327
0,167973 0,176786 9 0,1 0,152158 0,052158
0,176786 0,185599 15 0,25 0,353875 0,103875
0,185599 0,194412 27 0,52 0,63245 0,11245
0,194412 0,203226 25 0,77 0,834055 0,064055
0,203226 0,212039 14 0,91 0,951774 0,041774
0,212039 0,220852 9 1 0,985608 0,014392
The equilibrium estimation of the qualities of the Internet-trading systems QUIK and OTHER are:
i s* = 0, 6543s = 91602,
\ s*2=l= 140000.
Note that both values s*, s* are from the quality range, i.e. s* e [6664,140000]. Comparing this values with those we get in the experiment s1 = 68068 and s2 =
76300, it can be found that both Internet-trading systems’ developers should increase the system quality as well as quality differentiation.
To evaluate price, recall first that p = p0 — s (see (14)). Then the Internet-trading systems’ prices in equilibrium are:
Í P01 = P*i + sí = 95305,
1 PÍ2 = PÍ + sí = 152424.
This results means that when the Internet-trading systems are more differentiated in quality companies may differentiate more in prices. Indeed, the price difference for the Internet-trading systems is equal now to 21 000 RUB., and according to the game-theoretical model results it may be increased to more than 57 000 RUB. In this case companies’ market shares are equal correspondingly to:
( D\ = 0, 247,
\ D = 0, 740.
This result represents the situation on the market today, as the rate of consumers of the Internet-trading systems OTHER and QUIK is about 1 to 3.
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