ON COMPETITION IN THE TELECOMMUNICATIONS MARKET Текст научной статьи по специальности «Математика»

Contributions to Game Theory and Management, XI, 7-21

On Competition in the Telecommunications Market*

Petr Ageev, Yaroslavna Pankratova and Svetlana Tarashnina

St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia E-mail: ageev. petr. vladimirovich@gmail. com E-mail: [email protected] E-mail: s. tarashnina@spbu. ru

Abstract The paper investigates the process of competition in the market of telecommunication services between three firms: the leader, the challenger and the follower. In this work we construct a model of competition between three players in the form of a multistage non-zero sum game. As a solution of the game we find a subgame perfect equilibrium. We illustrate the results with an example for three companies working on the Saint-Petersburg telecommunications market.

Keywords: telecommunications market, non-zero sum game, multistage game, subgame perfect equilibrium

1. Introduction

In the paper competition between three companies in the telecommunications market is investigated. All firms have different types: the leader, the challenger and the follower. The leader is a company that prevails in the market and acts in three main directions:

• expansion of the market by attracting new customers and finding new areas;

The challenger firm is a company that does not lag far behind the leader of the market and tries to become the leader by using attacking strategies. This firm uses strategies aimed to expand its market share, but those that do not cause active opposition to competitors.

In Stackelberg's monograph (H. Von Stackelberg, 1952) the competition on the market is presented by a multistage decision-making model. At the first stage, the decision is made by the leader, and at the next stage, taking into account the decision of the leader, the decision is made by the company-follower. At the same time, when making decisions, each of the firms pursues its own goal. In paper (Beresnev and Suslov, 2010) authors propose an algorithm for constructing an approximate solution of such a problem.

In this paper, we consider a more complicated problem. At the first stage, the leader and the challenger make decisions about which telecommunication services and at what prices to offer subscribers. At this stage, some customers make a choice in favor of the first or second company. At the next stage, the follower, taking into account the choice of competitors, decides on what services it would be better to offer to potential customers. At the same time, the follower tries to keep its customers

* This work was supported by the Russian Foundation for Basic Research (grant 17-5153030 ).

and, if possible, to attract a part of the competitors customers. At this stage, the remaining customers make their own choices.

We assume that each customer must choose one of the telecommunication services (tariff). If a customer decides to stay at his tariff with his company, we believe that he chooses the appropriate service from the relevant company. The leader and the challenger aim to maximize their profits by attracting some of the competitor's customers. The purpose of the follower is to maximize its profit and save the customers.

We formalize this problem of competition in the telecommunications market between the leader and the challenger as a nonzero-sum game in normal form. Then we consider a multistage game where this nonzero-sum game is realized as the first stage. The second stage is made by the follower. As a solution of this game we consider a subgame perfect equilibrium (SPE) (Hellwig and Leininger, 1987, D.W.K. Yeung, L.A. Petrosyan and N.A. Zenkevich, 2009, Nessah and Tian, 2014). Some examples are given and discussed in the paper.

The obtained solution allows each company to develop a long-term strategy to maximize its summing payoff. In the future, it would be interesting to study a strongly time consistency (Petrosyan, 1993) of the solution.

2. The game

To formalize the problem of competition in the telecommunication market we introduce the following assumptions:

• firms are informed on subscriber preferences which are created taking into account service prices;

•

service and the unit costs for it; the profit can only be positive; subscribers who have decided to use this service, and its price; subscriber should pay per month;

•

of a package of services, for example, a tariff consisting of v minutes for all outgoing calls, b gigabytes of Internet and z outgoing SMS messages. Further, the number of outgoing SMS messages is omitted from consideration, since to date SMS messages have been replaced by so-called messengers;

•

tariffs, in which the main emphasis is on the volume of Internet traffic in comparison with tariffs, in which the main emphasis is on the number of minutes for outgoing calls, is much less. Since quite often subscribers use the Internet to make calls, and the demand for such tariffs is higher, telecommunication operators set the price for Internet tariffs higher;

•

within the same operator are equal; tariffs;

costs are lower;

vices the players can offer.

We denote Fi by the leader, F2 is the challenger and F3 is the follower. Let N = {Fi;F2,F3} be the set of players - telecommunication companies, which provide services on the market.

Let I = {1,..., m} be the set of services (tariffs) that are offered on the telecommunications market. Each element of ir € I is some specific type of service. This service will be called the service type ir, offered by some firm.

Denote by Ii5 I^d I3 subsets of I, which contain the offered services, respectively, by the leading firm, the challenger firm and the follower firm. Assume that Ii U I2 U I3 = ^d Ii n I2 n I3 = 0.

Let the following quantities are known:

• ck is the price of service i for the player Fk, where i € Ik and k € {1, 2,3};

• a,j is the unit costs of service i;

• f is the fixed costs (i.e. costs that are not depending on the volume of services) for the player's service Fk, k € {1,2, 3}. At the same time, the fixed costs are constant.

We denote by J = {1,..., n} the set of subscribers using the services offered on the market. Each element j € J is a certain subscriber (customer). We assume, that the subscriber chooses an offered by one of the firms service based on his internal preferences, which are specified by splitting the set J into two subsets JT and JP. JT includes subscribers, who mainly think about low price, when choosing an operator. In turn, JT is divided into JTl, which consists of subscribers, for whom the most important thing along with the price is the number of minutes for outgoing calls within the tariff, and JT2, consisting of those who, along with the price pays attention to the volume of Internet traffic provided within the tariff. The subset JP contains "conservative" subscribers. They are subscribers which can not change an operator, because it is a problem for various reasons, for example, corporate users.

We suppose that J = JT U JP, JT n JP = 0. We have

J0 = J = J? U J20 n J30. (1)

Expression (1) describes the distribution of the set of subscribers between the players at the initial stage of the game. Let the following relations hold:

|J? n Jt| > | J20 n Jt| > |J0 n Jt |J? n Jp| > |J20 n Jp| > |J30 n Jp|.

We assume that subscribers from the set JP n J° always choose player k, where k € {1, 2,3}, and the service that the operator offers at the moment, regardless of the offered tariff.

By the strategy of player Fk, where k € {1,2, 3} we define the pair s^T = (cir, ir), ir € Ik. We denote the set of strategies of Fk by Sk = {s^T : ir € Ik}. We assume that the firm's strategies are developed, based on the results of SWOT analysis, and are aimed at minimizing the risks associated with the identified weaknesses of each firms, and aimed to use the identified opportunities.

(c01 , V 0| . i;) ^ (c2 ,iP)' j J20 n JT2 and (4 : , i;) ^

(c1 , i;) j J20 n JT2 and (4 : , i;) ^

(c1 , i;) ^ (c2 ,iP)' j J20 n JT2 and (c0| : , i;) ^

We introduce the preference relationships for the subscriber j e J of services offered by firms F^d F2. Obviously, the player's strategy is characterized by the following indicators: the price ck, the number of minutes the number of gigabytes of mobile Internet bk.

For subscriber j e J e JT2 we say that the pair (cl, i;) is preferable to the pair (c?p , ip), i.e. (cli, i;) >- (c2p, ip), if at least one of the following conditions holds:

L 4 < 4 and 4 > 4;

2- 4 = 4 and 4 > 4;

3- 4 < 4 and 4 = 4-

For other cases:

1. if c1 < c2 and b1 < b2 , tl

0l Op 0| Op '

(c2p ,ip), if j e J0 n JT2-

2. if c1 > c2 and b1 > b2 , tl

0i 0p 0i 0p '

(c2p ,ip), if j e J0 n JT2-

3. if c1 = c2 and b1 = b2 , tl

0p 0i 0p '

(c2p,ip), if j e J0 n JT2-

4. if one of the previous three conditions is fulfilled, but j e J0 n j e J0 n JT2, then j e Jnd, where Jnd is a set of subscribers who can equally choose the services of both compared firms.

For subscriber j e J n JTi we say that the pair (c1, i;) is preferable to the pair (c2, ip), i.e. (c*, i;) >- (c2p, ip), if at least one of the following conditions holds:

1- 4 < clp and v1 > v0;;

2- 4 = cl and 4 > vi;

3- 4 < ct and 4 =

For other cases:

1- if 4 < c2p and v1 < v2;, tl (c2p ,ip), if j e J1 n JTi.

2. if c1 > c2 and v1 > v2 , tl

0l 0p 0l 0p '

(c2p,ip), if j e J1 n JTi •

c01 = c02 v01 = v02 ( c

0l 0p 0i 0p' v

(c2p,ip), if j e J1 n JTi •

4. if one of the previous three conditions is fulfilled, but j e J1 n j e

J0 n JTi, then j e Jnd, where Jnd is a set of subscribers who can equally choose the services of both compared firms.

Next, we look, if | Jnd| = 2k, where k e Z, then a half of the subscribers from Jnd choose the firm F1; and the other half choose the firm F2. If | Jnd| = 2k +1, where k e Z, then the firm F1 (firm with higher market position) chooses one subscriber j e J„d more.

To determine the preference of services offered by firms F2 and F3, we proceed similarly. To determine the preference of services offered by the firms F1 and F3, we proceed similarly.

(c1| : i;) ^ (clp ,iP) j J20 n JTi and (4 : , i;) ^

(4 : i;) ^ (4p ,iP) j J20 n JTi and (4 : , i;) ^

(4 : i;) ^ (4p ,iP) j J20 n JTi and (4 : , i;) ^

First, the subscriber chooses two of the three firms to compare their services for preference based on the value for subscribers j € J n JTl and the value

jk for subscrib ers j € J n JT2. He chooses those two firms for which this value is less. It should be noted that when the subscriber compares the leader company and the challenger company, he compares the corresponding values for one strategy chosen by each player, since at the first step of the game, a bi-matrix game is played between the leader firm and the challenger firm. For the third company, the subscriber compares all possible strategies, since the follower firm can choose any of them, and if for all of the theoretically possible strategies it turns out that the value vk for subscrib ers j € J n JTl and the value ^ for subscrib ers j € J n JT2,

respectively, is not the greatest one, then the follower firm becomes one of the two

j

We introduce the switching function Vj (s^ )•

y (sir) = i 1, if ir is the preferred service for subscriber j; j k ) [ 0, otherwise,

i.e., the function characterises the preference for the subscriber j € J of the service ir € Ik offered by the player Fk, compared to all other types of services that are offered on the market. For regular subscribers, i.e. for j € J n JP:

Vj(sj.) = 1 for all i € Ik, k € {1, 2,3}.

We assume that the services are selected by the subscriber for a month in advance. Let introduce the value gj(s^T) = (ckr — air), which characterises the profit of a company Fk from the subscriber j when the firm uses strategy s^T. The ir service, which offers a larger amount of Internet traffic is designated for ig6, and the service, which offers a greater number of minutes for outgoing calls for i"ni. Taking account of assumption 6 6, we obtain the following inequality

gj(skr) > gj(skr ) > ° for k € {1, 2, 3} ir € Ifc, j € J.

Denote by (s^T) the total profit of firm k from customers j € Jp n J°, which

ir

(sfc ) = gj (sfcT ), jeJp nJ0

where k € {1, 2,3} Mid ir € Ik. Then the payoff function of the leader firm we can define by the following way:

Hi(s11 ,s2P,J0) = —fi + £ gj(s11) x yj(s1i) x (1 — Vj(s2p)) + Gi(s11),

jeJrnJ0

where i; € Iu ip € I2.

The payoff function of the leader firm expresses the amount of profit of this player taking into account changes in income due to the loss and acquisition of subscribers. The payoff function for the challenger firm is the following

#2(4*, 4p, J0) = /2 + E gj(s2p) x Vj(s2p) x (1 - Vj(si1)) + G2(s2p),

j£jT nJ 0

where i; e I^ ip e I2.

The payoff function of the challenger firm expresses the amount of profit of this player taking into account changes in income due to the loss and acquisition of subscribers. The payoff function for the follower firm is

H3(si ,s2p ,s3s ,J 0) = -/3+

+ E gj(s0s) x Vj(s0s) x (1 - Vj(s1l)) x (1 - Vj(s2p))+ G3(s0s),

jeJTnJ0

where i; e I1, ip e /2, is e /3-

The payoff function of the follower firm expresses the total profit of this player taking into account changes in income due to the loss and acquisition of subscribers.

Vj(si) + Vj(s2p) + Vj (s3s) < 1, i; e I1, ip e I2, is e I3, j e Jnd- (2)

Inequality (2) shows that it might not be that for the subscriber j e Jnd two services are the most preferable at the same time compared to each other. For subscribers j e Jnd, in general, this situation can not be either, since, in fact, according to condition 4 from the block "other cases" when determining the preference of services, the subscriber j e Jnd chooses only one company.

The leader in the game is determined by the number of subscribers available to the company at the beginning of the game. At the end of the game, in the case of equality of the subscribers number in several companies, the leader is determined by the amount of total profit. Since, ceteris paribus, follower firm can both play-along with the leader firm and play along with the challenger firm, for definiteness, F3 F1

Thus, the competition on the telecommunications market can be formalised in the form of non-zero sum game T:

r =< N, S1, S2, S3, #1, H2, H3 > . How is the game going?

F1 F2

(services) in the first step of the game, i.e. they choose the strategies. Thus, previously two players play a non-antagonistic bi-matrix game.

F1 F2

F3

3. Each player seeks to maximise their win function.

Schematically, the game in expanded form for a one-stage case, when the set of strategies of each player contains two strategies, is shown in figure 1.

1 F1

F2

and independently of each other, playing a bi-matrix game. In positions 2, 3, 4, 5,

F1 F2 F3

H\ fi i Hj1

Hi Hi Hi H] Hi Hi Hi Hi

Wj Hj /f| Wj Wj /ij h]

Fig. 1.

choosing between s3 or s3 strategies. At the same time, if both of these strategies give him the same gain, when choosing his strategy, this player plays along with the firm-leader.

3. Nash equilibrium

We assume that for the same strategy for the same player the value gj (s^T) will be greater for the strategy s^T, if it offers a greater volume of the services, that is, if ii and i2 are "Internet" tariffs, than the value of gj(s].1) will be greater if the service i1

the volume of the service increases, the price for it increases, while the unit costs according to the assumptions for services of the same type are the same.

Let = {s1,...,s"}, S2 = {s2,...,s"}, S3 = {s3,...,s"} be the sets of strategies of Fi; F2, F3, correspondingly .

Suppose there is a payoff matrix (A, B)mxn for firms F^d F2. We search for a subgame perfect equilibrium from the end of the game. Suppose that players F^d F2 have chosen their strategies (s^, s2p) and have already played the first stage of the game. Let us compare two arbitrary strategies s31 and s32 of the company F3 and the payoff functions for these strategies:

H3(s1;, s2p, s3, J0) = — fs+

E gj(s^1) x V(s£) x (1 — Vj(s f)) x (1 — Vj(s )) + G3(s3.), (3)

jeJr nJ0

H3(s1;,s2p,s32, J0) = — f3+

(4)

E gj(s32) x V(s32) x (1 — Vj(s f)) x (1 — Vj(s )) + G3(s32),

je Jt nJ0 where:

G3 (s31 )= E gj (s31),

jeJp nJ"

G3 (s32 )= E gj (s32).

jeJp nJ"

Suppose that the set JP n J3 contains w3 subscribers, then we can rewrite the previous conditions in the form of:

G3(s3) = w3 X gj£jpnjo(s^1), G3(%2) = w3 X gjejpnJ0(s3

where w3 > 0.

Then, we get that in order for the strategy S31 to be no worse for the player F3 than the strategy s32, it is required that the following inequality be executed:

E gj(S31) X Vj(S31) X (1 - Vj(si')) X (1 - Vj(s;p))+

jgJtnJ0

w3 X gjejpnJ0(s31 )-

E gj(s32) X Vj(s32) X (1 - Vj(si')) X (1 - Vj(sip))-jeJT nJ0

w3 X gj'GJpnJ0 (s32 ) > ° Converting expression (5), we have

gJ nJ 0 (s31) ( E Vj (s31) X (1 - Vj (si')) X (1 - Vj (sip))

\j'£JT nJ 0 +w3 X gj£JpnJ0 (s31 )-

gje Jt n J 0 (s32) X I E Vj (s32) X (1 - Vj(si')) X (1 - Vj (sip))| -Vj'e Jt nJ0

(5)

(6)

-w3 X gjJpnJ0 (s32) > °

Note that gjJnJ0(s31) = gjJnJ0(s31 ) and gjJnJ0(s32) = gjeJpnJ0(s32)> gj

on the player's strategy (choosing service). We introduce the following notation:

gje Jt nJ0 (s31) = g(s31 gje Jt nJ0 (s32) = g(s32 )-

Then, expression (6) can be written as:

g(s31) X IE Vj (s31) X (1 - Vj (si')) X (1 - Vj (sip))+ W3 ) >

UeJT nJ 0

g(s32) X I E Vj(s32) X (1 - Vj(si')) X (1 - Vj(sip)) + W3 J . ue Jt nJ0

Given that g(s32 ) > w > Oand E V (s^ )x(1-Vj (s Î1 )x(1-Vj (s2p )) > 0, we

j£JTnJ0

obtain that the strategy s31 is more profitable for the player F3 when the following condition is fulfilled:

. E V (s32 ) x (1 - Vj (s Î1 )) x (1 - Vj (s2p)) + w3 g(s3 ) > J'ëJtnJ0_

g(s32)_ ]T Vj (s31 ) x (1 - Vj (sf )) x (1 - Vj (s ;p)) + W3 '

j£JTnJ0

Next, consider two arbitrary strategies si1 and si2 of player and the payoff function for them. Let the player F2 use a fixed strategy Write the payoff function of the first player for strategies si1 and si2.

H1(sl1 J0) = -/ + E gj (si1 ) x Vj(si1 ) x (1 - Vj (s )) + G1(si1), (8)

JgJtnJ0

Hi(si2 J0) = -/1 + E gj (si2) x Vj (si2 ) x (1 - Vj (s )) + Gi(si2), (9)

j£JTnJ0

where

Gi(s11 )= E gj (si1 )

jeJp nJ°

Gi (si2 )= E gj (si2 )•

jeJp nJ"

Let set JP n J0 contains w1 subscribers, then the previous two expressions can be written as:

G1(s11 ) = w1 x gjeJpnJ0(sï), G1(s12) = w1 x gjeJpnJ0(s12),

where w1 > 0.

Thus, in order for the strategy s^ to be no worse for player F1 in comparison

si12

E gj(s11 ) x Vj (sÎ1 ) x (1 - Vj (sf )) + W1 x gjeJpnJ? (s? )-jeJTnJ0

E gj (si2 ) x Vj (si2 ) x (1 - Vj (sp* )) (10)

jeJTnJ0

-w1 x gjeJpnJ» (s'i2 ) > ° Transforming inequality (10), we get

g-eJTnJ0(si1) X I E Vj(si1) x (1 - Vj(sf)) I + wi x gJnJo(si1 )-VjGJt nJ0 J

\ (11)

ffjeJtnjo (si2) x | E Vj(si2) x (1 - Vj (sf ))| -yjeJT nJ0 J

wi x gjeJpnJ0 (s12) > o, where

SjeJTnJ0 (si1) = gjeJpnJ0 (s11 )>

gjeJT nJ0 (s12) = gjeJp nJ0 (s12), since gj does not depend on the set to which the subscribers j belongs, but depends only on the strategy. We introduce the following notation:

gjeJT nJ0 (•si1) = g(s11), gjeJT nJ 0 (s12) = g(s12), then, expression (17) can be written as:

g(s11) x I E Vj (si1) x (1 - Vj (s2p))+ wi I > \j£JT nJ0 J

>g(si2) x I E Vj(si2) x (1 - Vj(s;p))^i

\j£JT nJ 0

Given that g(si2) > 0 wi > 0 and £ Vj(s®1) x (1 - vj(s |l)) > 0, we obtain

J£JtnJ0

that the strategy si1 is more preferable for the player Fi than the strategy si2 when the following condition is fulfilled:

. E Vj (si2) x (1 - Vj (s®p)) + wi

g(si1) > j£JtnJ0_

g(si2 )_ E Vj (si1) x (1 - Vj (s2p)) + wi

J£JtnJ0

Next, consider two arbitrary strategies s®1 and s®2 of player and the payoff function for these strategies. Let the player Fi use his fixed strategy sil. Carrying out reflections analogous to the case for firm we obtain a condition, when the strategy s®1 is more profitable for the player F2 than the strategy s®2, i.e.

E Vj(s22) x (1 - Vj(sil)) + w®

g(s21 ) > j'eJtnJ0_

g(s®2)_ E Vj (s®1) x (1 - Vj (sf)) + w®'

j£JTnJ0

where g( si,2) > 0 > 0 and £ Vj(si,1) x (1 - Vj(s |1)) > 0.

jeJTnJ0

Thus, in a two-stage non-zero sum game r =< N, Si, S2, S3, Hi, H2, H3 >, the strategies of the players s s i , s3 lead to a subgame perfect equilibrium if the following conditions are satisfied:

g(si^ > jeJtnJ0

E Vj(si2) x (1 - Vj(si)) + wi

g(sl2) E Vj(si) x (1 - Vj(s2))+ wi'

jeJTnJ0

E Vj(si2) x (1 - Vj(s i)) + W2

g(s2^ jeJT nJ0

>

g(s22) E Vj(s2) X (1 - Vj(si))+ W2

jeJTnJ0

E Vj(s32) x (1 - Vj(si)) x (1 - Vj(s2)) +

g(s3) > jeJT nJ0

g(s32) E Vj(si) x (1 - Vj(si)) x (1 - Vj(si)) + w3 '

jeJTnJ0

for Vsi2 G {Si} Vsi,2 G {S2} Vs32 G {S3} where wfc = | Jp n Jfc0|, k G {1, 2,3}. These inequalities can be rewritten as:

E V(si2) x (1 - Vj(s2))+ wi

g(si) > jJJ- x g(si2),

— Vj (si) xn v.^v

E Vj (si) x (1 - Vj (si)) + wi

jeJTnJ0

E Vj(s22) x (1 - Vj(s i))+ w2

g(s2) > JJ0- x g(s22), (14)

2 E Vj(s2) x (1 - Vj(si)) + w2 2 )J 1 '

jeJTnJ0

E Vj(s32) x (1 - Vj(s i)) x (1 - Vj(s2)) + w3

g(s3) > JJ- x g(s32),

E Vj (s3) x (1 - Vj (si)) x (1 - Vj (si)) + w3

jeJTnJ0

for Vsi2 G {Si} Vs22 G {S2} Vs32 G {S3}, where wfc = | Jp n Jfc0|, k G {1, 2,3}. So, we have proved the following theorem.

Theorem 1. In a non-zero sum two-stage game r =< N, Si; S2, S3, Hi; H2, H3 > the strategies s i, si, s3 lead to a subgame perfect equilibrium if inequalities (15) are fulfilled.

Proof. The proof follows from the construction.

It can be noted that the ratio of payoffs from the use of the strategy (service), leading to the subgame perfect equilibrium in the game for player , to the amount of a payoff from the use of any other strategy should be not less than the ratio of the number of all subscribers who have chosen the services of player when using any other strategy, to the number of all subscribers who have chosen the service leading to the subgame perfect equilibrium.

4. Example

We determine the strategies of the players taking into account the results of the SWOT-analysis (Bogomolova, 2004), that has been conducted with using a real data set for three companies working on the Saint-Petersburg telecommunications market.

First, we assume that Ii = {1, 2} I® = {3,4} 13 = {5, 6}.

• Tariff 1 contains 200 minutes of outgoing calls, 2 Gigabytes of Internet traffic.

Fixed costs / are equal to 70, the unit cost ai is equal to 60.

•

Fixed costs / are 70, the unit cost a® is 50.

Fixed costs /3 are equal to 60, the un it cost a3 is equal to 70.

Fixed costs /4 equal to 60, the un it cost a4 equals to 60.

Fixed costs /5 equal to 50, the un it cost a5 is 70.

Fixed costs /3 equal to 50, the un it cost a6 equals to 60.

That is, the size of the fixed costs takes into account that for leading companies the fixed costs are relatively less per. They depend on a number of staff.

Let J = {1,2, 3,4, 5,6, 7,8, 9,10,11,12,13,14,15,16,17} Divide JT into two sets JT1 and JT2 • JT1 includes the customers for whom the determining factor when choosing a service is the number of minutes per month for outgoing phone calls. JT2 includes the customers, for whom the volume of monthly Internet traffic is important along with the price. Thus, taking into account tariffs 1-6, we have JT1 = {1, 2, 3,4, 5} JT2 = {6, 7, 8, 9}.

The set JP includes customers 10,11,12,1314,15,16,17.

Let J0 n JP = {10,11,12,13} J0 n JP = {14,15,16} J30 n JP = {17}.

Assume that

J0 n JT = {1, 4, 6, 9}, J0 n JT = {2, 5, 7}, J30 n JT = {3, 8}.

Let us move on to the strategy sets: Si = {s} s®} S® = {s2, s®} S3 = {s3, s3},

si = (300,1), si = (330, 2), s® = (310, 3), s® = (320, 4), s3 = (320, 5), s3 = (340, 6).

Firm Strategy Tariff Fixed costs Unit costs Minutes Gigabytes Price

Leader S1 S? 1 2 70 70 60 50 200 100 2 6 300 330

Challenger s? 3 4 60 60 70 60 200 150 3 5 310 320

Follower S3 S32 5 6 50 50 70 60 150 200 4 7 320 340

We calculate the players payoffs for all situations in this game. Let us make clear how the players payoffs in an arbitrary situation are determined. Consider the situation in which player F} uses strategy s }, player F2 use s s }, pi ayer F3 use s S3, so we have situation (s}, s2, S3).

Determine what services the two firms customerss j G Jn j G Jn JT2 will

be compared to the preference. To do this, we calculate the values that characterize the ratio of the cost of services to their volume. For subscribers j G J n JTl :

• player F has

F J 1 200

• player F2 has tj^o^

• player F3 tos 320 or 340.

150

100

Thus, subscribers of j G J n JTl will compare the service preference of the first and second players. Similarly, for subscribers j inJ n JT2:

• player Fi has

, P 310

• player F2 —;

• player F3 has 320 or 340.

Thus, subscribers of j G Jn JT2 will compare the service preference of the second and third players.

Then Hi (si, si) = 2090, #2^1, si) = 1140, Hs(si,s2,s£) = 700, as the first player will be chosen by customers 1,2, 3,4, 5,10,11,12,13; second player will be chosen by customers 6, 8,14,15,16; the third player will be chosen by customers 7, 9, 17

depending on strategies:

Strategy profile Customers of firm Fi Customers of firm F2 Customers of firm F3 Payoffs (Hi, H2, H3)

sl> «2,«3 {1, 2, 3,4, 5,10,11,12,13} {6, 8,14,15,16} {7, 9,17} (2090,1140, 700)

{7, 8,10,11,12,13} {1, 2, 3,4, 5,14,15,16} {6, 9,17} (1610,1860, 700)

sLs! ,«3 {1, 2, 3,4, 5,10,11,12,13} {6, 7, 8, 9,14,15,16} {17} (2090,1760, 200)

ll1 s1j «2, «3 {7, 8,10,11,12,13} {1, 2, 3,4, 5,14,15,16} {17} (1610, 2540, 200)

«2 j «2 ,«3 {1, 2, 3,4, 5,10,11,12,13} {6, 8,14,15,16} {7, 9,17} (2090,1140, 790)

«2 «1 «2 s1j «2, «3 {7, 8,10,11,12,13} {1, 2, 3,4, 5,14,15,16} {6, 9,17} (1610,1860, 790)

12 2 a1> «2, «3 {1, 2, 3,4, 5,10,11,12,13} {6, 8,14,15,16} {7, 9,17} (2090,1240, 790)

222 «1,52 ,«3 {7, 8,10,11,12,13} {1, 2, 3,4, 5, 6, 9,14,15,16} {17} (1610, 2540, 230)

The game with the players payoffs is presented in figure 2. Now let us find a subgame perfect equilibrium in this two-stage game. To do this, we consider the game from the end. In nodes 2,3,4, 5, player F3 makes his decision. Accordingly, in all of these nodes, he chooseC< the strategy s§, because it gives to him the highest payoff. Depending on the strategy selected by the third players, in node 1 a bimatrix game is played between the first and the second players. The corresponding payoff matrix is presented in table:

s1 s2 s 2 s 2

s 1 (2090,1140) (2090,1240)

si (l610,1860) (l610, 2540)

In that game firms F1 and F2 choose their strategies simultaneously. The solution of this bimatrix game is the situation (si, s2) with payoffs (2090,1240). Note that in nodes 2, 3,4, 5 player F3 chooses the strategy s§. All this forms a subgame perfect equilibrium which can be written as [s} s2; s§, s3, s3, s§]. Figure 2 shows the constructed subgame perfect equilibrium.

5. Conclusion

In conclusion, we would like to note that using various intellectual response techniques for the process of competition is a large and interesting range of tasks, the

2090 2090 2090 2090 1610 1610 1610 1610 1140 1140 1760 1240 1860 1S60 2540 2540 700 790 200 790 700 790 200 230

ih)

04,B)2x2

(4)

(M

Fig. 2.

relevance of which does not decrease over time, but on the contrary, in the conditions of increasing globalisation, opening of new markets, growth of production capacities and production volumes is increasing, thereby increasing the scientific and practical value of the works devoted to this subject.

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