A computational method for solving n-person game Текст научной статьи по специальности «Математика»

Серия «Математика» 2017. Т. 20. С. 109-121

Онлайн-доступ к журналу: http://isu.ru/izvest.ia.

ИЗВЕСТИЯ

Иркутского государственного ■университета

УДК 519.853 MSG 91А06

DOI https://doi.org/10.26516/1997-7670.2017.20.109 A Computational Method for Solving TV-Person Game

R. Enkhbat, S. Batbileg

Institute of Mathematics, National University of Mongolia

N. Tungalag

The school of business, National University of Mongolia

Anton Anikin, Alexander Gornov

Matrosov Institute for System Dynamics and Control Theory, SB of RAS

Abstract. The nonzero sum «-person game has been considered. It is well known that the game can be reduced to a global optimization problem [5; 7; 14]. By extending Mills' result [5], we derive global optimality conditions for a Nash equilibrium. In order to solve the problem numerically, we apply the Curvilinear Multista.rt Algorithm [2; 3] developed for finding global solutions in nonconvex optimization problems. The proposed algorithm was tested on three and four person games. Also, for the test purpose, we have considered competitions of 3 companies at the bread market of Ulaanbaatar as the three person game and solved numerically.

Keywords: Nash equilibrium, nonzero sum game, mixed strategies, curvilinear multistart algorithm.

1. Introduction

Game theory plays an important, role in applied mathematics, mathematical modeling, economics and decision theory. There are many works devoted to game theory [6; 8; 9; 10; 11; 12; 4]. Most, of them deals with zero sum two person games or nonzero sum two person games. Also, two person non zero sum game was studied in [10; 15; 16] by reducing it. to D.C programming[l]. The three person game was examined in [2] by global optimization techniques. So far, less attention has been paid to computational aspects of game theory, specially A-person game. Aim of this paper to fulfill

this gap. This paper considers nonzero sum n person game. The paper is organized as follows. In Section 2, we formulate non zero sum n person game and show that it can be formulated as a global optimization problem with polynomial constraints. We formulate the problem of finding a Nash equilibrium for non zero sum n-person games as a nonlinear programming problem. A Global search algorithm has been propose in Section 3. Section 4 is devoted to computational experiments.

2. Nonzero Sum n-person Game

Consider the n-person game in mixed strategies with matrices (Aq, q = 1,2,... ,n) for players 1,2,... ,n.

Ai = (a9ili^in),q = l,2,...,n

i\ — 1,2,... ,k\, ... ,in — 1,2,..., kn,

Denote by Dp the set

p

Dp = {ue Rq | ^2/Ui = 1, Ui> 0, i = 1,... ,p}, p = ki,k2, ...,kn

i= 1

A mixed strategy for player 1 is a vector xl = (x\, x\, ■ ■ ■, xlki) € where x\ represents the probability that player 1 uses a strategy i. Similarly, the mixed strategies for q-th player is xq = (x\, x\,..., xl) € Dkq, q = 1,2,... ,n. Their expected payoffs are given by for 1-th person :

fci fc2 k„

fl(x ,X , . . . , X ) = 'y ^ 'y ^ . . . y ^ ai1i2...i„xiixi2 ■ ■ ■ Xi„ ■

¿1 = 1 ¿2 = 1 ¿n = l

and for q-th person

fci kn

fq{x ,X , . . . , X ) = ^^ ^^ . . . ^^ ai1i2...inXhXi2 • • • Xin '

¿1 = 1 ¿2 = 1 ¿n = l

q = 1,2 ,...,n.

Definition 1. A vector of mixed strategies xq € Dkq, q = 1,2,... ,n is a Nash equilibrium if

' f1(x1,x2,...,xn)>f1(x1,x2,...,xn), VxleDkl

< f (r1 t2 rn\ > f (r1 1 rl r1+1 rn\ \/r1 a D,

Jqy-1' ) , ■ ■ ■ ,-ij ) J , . . . , J/ , -Ij , -Ij , . . . ,.Xj j, »a c J^kq

k fn{xl,x2,...,xn) >fn{xl,x2,...,xn), VxneDkn.

It is clear that

fi(x1,x2,..., xn) = max^igj)^ f\{xx ,x2,......, xn),

-p ( rf> 1 /y>2 r^n \ - yY"| Q v /1 7"~\ f ( fy rpQ 1 rpQ

J > ^ j ... j y — J QK^ i ^ , ... , J/ , J/ , J/ , ... , «X/

fn{xl,x2, ...,xn) = maxxn&Dkn fn(xl,x2,.. .,xn~1,xn). Denote by

fcl fc2 fcq-1 fcq+1 fcrj

EE--- E E •••EoL- =

¿1 = 1 ¿2 = 1 ¿q_l = l ¿q+l = l ¿„ = 1

4 <piq(x\ x2,..., x«-\x«+\ ...,xn) = <piq(x\x9)>

iq = 1,2,... ,kq, q = 1, 2,..., n.

For further purpose, it is useful to formulate the following statement.

Theorem 1. A vector strategy (xl, x2,..., xn) is a Nash equilibrium if and only if

fq{x)><piq{x\&) (2.1)

ry - f/Y> fy rp ^ \

Jb — I Jb j Jb j . . . 2 J

iq = 1,2,..., kq,

q = 1,2 ,...,n.

Proof. Necessity: Assume that a: is a Nash equilibrium. Then by the definition, we have

fcl fc2 fc„

¿1 = 1 ¿2 = 1 ¿n = l

fcl fc2

kq—1 kq kq-\-\ kn

- """ X] X] X] •••X] aiii2-in^h ■ ■ ■

¿1 = 1 ¿2 = 1 ¿q-l = l ¿q=l ¿q+l = l ¿n = l

— f f'T1 >r2 'r9~1 >rn) (9 9">

— Jqy^ , Jj , ■ ■ ■ , JJ , .JJ , .JJ , . . . , jj j,

q = 1,2 ,...,n.

In the inequality 2.2, successively choose xj =1, iq = 1,2,... ,kq. We can easily see that fq{x) = tpiq(x\xq), for iq = 1,2,..., kq; q = 1,2,..., n. Sufficiency: Suppose that for a vector x € D^ x Dk2 x ... x Dkn,

conditions 2.1 are satisfied. We choose xq € D^ , q = 1,2,...,n and multiply 2.1 by xj respectively. We obtain

kg k-l kg kn

ErpQ -p (ry \ \ \ \ fJi /y> 1 rfQ rfQ

XiqJq{X) ^ • • • " ' ahi2...in-Lii ■ ■ ■ 1 • •

¿q = l ¿1 = 1 ¿q = l ¿„ = 1

q = l,2,...,n.

Taking into account that i = 1, q = 1,2,..., n, we have

z^1^2,...,^) VxqeDkq, q = l,2,...,n

which shows that a: is a Nash equilibrium. The proof is complete.□

Theorem 2. A mixed strategy x is a Nash equilibrium for the nonzero sum n-person game if and only if there exists vector p € Rn such that vector (x,p) is a solution to the following bilinear programming problem :

n n

maxF(x,p) = fq(xl, x2,..., xn) - ^pq (2.3)

(iC'P) q= 1 q= 1

subject to :

Vig(x\xq) <Pq, ig = 1,2 ,...,kg, (2.4)

Proof. Necessity: Now suppose that a; is a Nash point. Choose vector p as : pq = fq(x), q = 1,2,...,n. We show that (x,p) is a solution to problem 2.3-2.4. First, we show that (x,p) is a feasible point for the problem. By theorem 1, the equivalent characterization of a Nash point, we have

<Piq(x\xq) > fq(xl,...,xn), q = l,2,...,n.

The rest of the constraints are satisfied because xq € Dkq, q = 1,2,..., n. It meant that (x,p) is a feasible point. Choose any xq € Dkq, q = 1,2,... ,n. Multiply 2.4 by xqq, q = 1,2,... ,n. respectively. If we have sum up these inequalities, we obtain

fg(x)<pq, q = l,2,...,n.

Hence, we get

fci k2 k„ / n \ n

F{X,p) = I]xlxl• o

¿1 = 1 ¿2 = 1 ¿n = l \9=1 / 9=1

for all xq € Dq, q = 1,2,... ,n.

But with pq = fq(x),we have F(x,p) = 0 Hence, the point (x,p) is a

solution to the problem 2.3-2.4.

Sufficiency: Now we have to show reverse, namely, that any solution of problem 2.3-2.4 must be a Nash point. Let (x,p) be any solution of problem 2.3-2.4. Let i be a Nash point for the game, and set pq = fq(x). We will show that x must be a Nash equilibrium of the game. Since (x,p) is a feasible point, we have

<Piq(x\xq) <pq, iq = 1,2,...,kg, q = 1,2,...,71. (2.5)

Hence, we have

fg(x) <pg, q = 1,2,..., 71. Adding these inequalities, we obtain

fcl fcn / n \ n

/•■:•'••/') = ££•••£ E<n..,n •<-'-<;iE/v " (2-6)

¿1 = 1 ¿2 = 1 ¿n = l \9=1 / 9=1

We know that at a Nash equilibrium F(x,p) = 0. Since (x,p) is also a solution, F(x,p) be equal to zero:

ki k2 k„ I n \ ra

= ££•••£ • • " E^ =0 (2J)

¿1 = 1 ¿2 = 1 ¿„ = 1 \9=1 / 9=1

Consequently,

fq(x)=Pq, 9= 1,2,..., 71. Since a point feasible, we can write the constraints (2.5) as follows:

fcl fc2 fcn

E E ■ ■ ■ E aiii2...inxiixi2 ■ ■ ■ xin ^ ¿1 = 1 ¿2 = 1 ¿„ = 1

fcl k2 kq-l kq+l fc„

>VV V V V </'.'. ./•'.r2 1 y/ •1 ^

¿1 = 1 ¿2 = 1 ¿q-l = l ¿q+l = l ¿n = l

for

iq = 1,2,... ,kq, q = 1,2,... ,n.

Now taking into account the above results, by theorem 1 we conclude that a: is a Nash point which a completes the proof. □

3. The Curvilinear Multistart Algorithm

In order to solve considered problems, we use curvilinear multistart algorithm. The algorithm was originally developed for solving box-constrained

optimization problems, therefore, we convert our problem from the constrained to unconstrained form using penalty function techniques. For each equality constraint g(x) = 0, we construct a simple penalty function g{x) = g2{x). For each inequality constraint q(x) < 0, we also construct the corresponding penalty function as follows:

a(x)-i if^)<0,

q[X) ~ \q2(x), ifq(x)>0.

Thus, we have the following box-constrained optimization problem: f(x) = f(x) + ^ J2gi(x) + |

i 3

X = {x € < Xi < xl, i = 1,..., n} .

where 7 is a penalty parameter, x and x - are lower and upper bounds. For original ^-variables the constraint is the box [0,1]; for p-variables box constraints are [0Values of p^ are chosen from some intervals. An initial value of a penalty parameter 7 is chosen not too large (something about 1000) and after finding some local minimums we increase it for searching another local minimum.

The proposed algorithm starts from some initial point xl € X. At each k—th iteration the algorithm performs randomly "drop" of two auxiliary points x1 and x2 and generating a curve (parabola) which passes through all three points xk, x1 and x2. Then we generate some random grid along this curve and try to found all convex triples inside the grid. For each founded triple we perform refining the triple minima value with using golden section method. The best triple became a start point for local optimization algorithm, the final point of which will be a start point for the next iteration of global method. Details are presented in Algorithms 1 and 2.

4. Computational Experiments

The proposed method was implemented in C language and tested on compatibility with using the GNU Compiler Collection (GCC, versions: 4.8.5, 4.9.3, 5.4.0), clang (versions: 3.5.2, 3.6.2, 3.7.1, 3.8) and Intel C Compiler (ICC, version 15.0.6) on both GNU/Linux, Microsoft Windows and Mac OS X operating systems.

The results of numerical experiments presented below were obtained on a personal computer with the following characteristics:

— Ubuntu server 16.04, x86_64

— Intel Core i5-2500K, 16 Gb RAM

— used compiler — gcc-5.4.0, build flags: -02 -DNDEBUG

Algorithm 1 The Curvilinear Multistart Algorithm

Input: xl € X - initial (start) point; K > 0 - iterations count; 5 > 0; N > 0; ea > 0 — algorithm parameters. Output: Global minimum point x* and /* = f(x*) l: for k <- 1,K do fk <- f{xk) 2: generate stochastic point i'el 3: generate stochastic point x2 € X 4: generate stochastic a-grid:

-1 = ai < .... <ai<-5<0<5< ai+1 < ... < aN = 1

5: Let x(a) = Projx {a2 ((xl + x2)/2 - xk) + a/2 {x2 - xl) + xk)

where Projx(z) - projection of point z onto set X. //note that x(—l) = xl, x(l) = x2, x(0) = xk. 6: /* <~ fk 7: ak^~ 0

8: for i 1, (N - 2) do //Convex triplet

9: if f(x(a>i)) > f{x{ai+1)) and f{x{ai+1)) < f(x(ai+2)) then

//Refining the value of minima using //Golden-Section search method with accuracy ea 10: ak GoldenSectionSearch(/, oij, «¿+1, «¿+2, £<*)

11: if f(x(ak)) < fk then

12: ft <r- f(x(ak))

13. ^

14: end if

15: end if

16: end for

//Start local optimization algorithm 17: xk+1 LOptim(£(o;^)) 18: end for 19: X* Xk 20: /* f{xk)

The proposed algorithm was applied for numerically solving number of problems with 3 and 4 players. In all cases, Nash equilibrium points were found successfully. Problems 3.1-3.3 are of type (2.3) —(??) have been solved numerically for dimensions 2x2x2.

Algorithm 2 The Local Optimization Algorithm

Input: xl € X - initial (start) point; ex > 0 — accuracy parameter. Output: Local minimum point x* and /* = f(x*) l: repeat

2: dk = Xk - Projx(xk - Vf(xk))

//Perform local relaxation step, for example, with using standard

convex interval capture technique.

3: xk+1 = argmin f(xk + adk)

a> o

4: until ||a;fc+1 — x\\2 < £x

Problem 3.1

Let Ax is a\n = 2, a\l2 = 3, a\2l = -1, a\22 = 0, al2ll = 1, a\l2 = -2, a221 = a222 = iS alll = 1> a112 = a121 = 0>\22 = ~1> a2ii = -1. a2i2 = a22i = 2> a222 = and ^з is a?n = 3, a?12 = 2, a121 = a122 = —a211 = 0) 0-212 = a221 = — 1> a222 =

The optimization problem is :

/123 л 123 123 123 123

FyX j X j X ^ Pi ^ P2) Ps) = H- — Зж^^2

+ Qx\x\x\ — pi — p2 — Рз —> max

f 0/у>2/у»3 1 0^,2^3 ¿dJU-^JU-^ О -А/ -у-А/ <2 9 Ч 2 1 -Pi < о

/у.2^3 о ™ 2 ™ 3 л ^2^3 о23 11 ^¿.л*2^1 2 2 -Pi < о

/у>1/у>3 I 0/у»1/у»3 11 ' 12 1 ч /у> J. /у>"-> 2 1 Р2 < 0

/у>1/у>3 I 0/у»1/у»3 12 ' 2 1 I /у> 1 /у>3 -+- Х2Х2 -Р2 < 0

Q/y>l/V>2 I /у»1/у»2 Utl/^l п^ .X/ .X/2 1 2 < t • 1 < t • ^ х2х2 Рз < 0

j О /у>1/у>2 Q/y»l/y»2 \ ^.х- О/ оо/1^2 + + 2х\У2 -Рз < 0

ry 1 I /у> 1 JU \ Ju 2 = 1

/у>2 1 О/ \ -jb 2 = 1

о о /у><-) I /у><-) О/ \ -jb 2 = 1

ж} > 0 > 0 ,»1 > 0 ,4> о

х\ > 0 , х% > 0 >0 ,Р2 >0 рз>0

4 Nash equilibrium points have been found:

points Player 7* /y»*- Oy 1 7* /y»*- X 2 Pi p*

1 0 1 3

1. 2 0 1 1 0.0

3 0 1 2

1 1 0 2

2. 2 1 0 1 0.0

3 1 0 3

1 0.5191 0.4809 1.2281

3. 2 0.5888 0.4112 0.5 2.08 10"8

3 0.5382 0.4618 0.9327

1 0.75 0.25 1.5

4. 2 0.8333 0.1667 0.5 3.37 10"8

3 1 0 1.9583

Problem 3.2

Let Ai is a\n = 5, a\12 = 3, a\21 = 6, a\22 = 7, = 0, a\l2 a22l = a222 = h IS O-iii = a112 = 4, a121 = —h a122 = 0-211 O212 = 5) a22i = 4, 0222 = 9> and is afn = 2, af12 = 0, a?21 = a122 = — h a211 = — a212 = 022i = 8, a222 = 9.

For this problem Nash equilibrium points are:

points Player * O/ ^ ryii * X 2 Pi p*

1 1 0 5

1. 2 1 0 2 0.0

3 1 0 2

1 0.5 0.5 4.8181

2. 2 0.5454 0.4545 4.5 2.6 10"8

3 0 1 3.4545

1 0.8 0.2 4.0

3. 2 1 0 3.2 9.9 10"8

3 0.5 0.5 1.2

Problem 3.3 Let A\ is a\u = 3, a}i2 = 2, a}2i = 1, 0122 = 5, 0211 0212 = 4, a22i = 1, a222 = 3, A2 is a\u = 3, a\ 12 = 2, a\2l = 4, a\22

a2ll = 1. a212 = 8. a221 = 6) a222 = 6) and ÍS 0?n = 3, flfl2

ai2i = 9) ai22 = a2ii = 4, o2i2 = 7, a22i = 2, a222 = 3.

Also, we found 4 Nash equilibrium points are:

points Player 7* /у»*-•Jb 1 7* /у»*- X 2 Р*г р*

1 1 0 1

1. 2 0 1 4 0.0

3 1 0 9

1 0 1 4

2. 2 1 0 8 0.0

3 0 1 7

1 0.5 0.5 1

3. 2 0.0 1.0 5 0.0

3 1.0 0.0 5.5

1 0.7 0.3 1

4. 2 0 1 4.6 -1.33 • 10"15

3 1.0 0.0 6.9

Problem 3.4 We have considered competitions of 3 companies sharing the bread market of city Ulaanbataar where each company maximizes own profit depending on its manufacturing strategies. The problem was formulated as the three-person game with profit matrices A = {(%•&}, B = {bijk}, C = {cijk}, i = l,5,j = l,6,k = 1,4. The matrix data can be downloaded from [17]. In this case the problem had 18 variables with 18 constraints. The solution of the problem found by the curvilinear multistart algorithm was:

Player /у>* Ju Y ■l2 х3 /у>* Jj 4 х5 х6 р* р*

1 0 0 0 0 1 65

2 0 0 0 0 0 1 160 0.0

3 1 0 0 0 53

It means that first and second companies must follow their 5-th and 6-th production strategies while third company applies for its 1-st production strategy. Companies's maximum profits were 65, 160 and 53 respectively. Problem 4.1 Let A\ is a}m = 1 , a}211 = 0, a}121 = 0, a\112 = 0, a2111 = l)a2112 = 0,0-2121 = 0) 0-2211 = 0,0-1122 = h a1212 = 0, o}22l = 0, a2221 = 0) a2212 = 0, 02i22 = 0, 0-1222 = 0) a2222 = 1,

A2 is Onu = 0, Oi2n = l,On2i = 0, Om2 = 0,a2m = 0, a2112 = l)02i2i =

0, o22n = 0, On22 = 0, oi2i2 = 0, oi22i = l,o222i = 0, o22i2 = 1, o2i22 = 1) ai222 = 0, o2222 = 1.

and is of m = 0,of2n = l,o?i2i = 0,a?112 = 0 ,o^m = 0,a^112 =

l)02i2i = l)022n = l,On22 = 0, oi2i2 = l,Oi22i = 0, o222i = 1, o22i2 = 0) ®2122 = ®1222 = h ®2222 = 0)

A4 is aim = 0,0^211 = 0,0^121 = 0,aj112 = l,a42111 = 0,a42112 =

o, 02121 = 1) O22H = 0, O1122 = I)ai2i2 = 2, O1221 = 0, O2221 = l)a2212 =

1) a2122 = a1222 = — 1) 0-2222 = 0-

Solution of this problem is also not unique and consist of several sets, such as:

1) F* = 0, a;1* = (0,1)T, a;2* = (0, l)T,x3* = (t, 1 - t)T, x4* = (1,0)T, Pi = 0) P*2 = 0, P% = 1, and pi = 1 - t, where t € [0, 0.5].

2) F* = 0, a;1* = (0,1)T, a;2* = (u, 1 - u)T, x3* = (0,1)T, x4* = (1,0)T, Pi = 0, P*2 = 0, p% = 1, and p\ = 1, where u € [0,1].

3) F* = 0, a;1* = (0,1)T, a;2* = (0, l)T,xs* = (v, 1 - v)T, x4* = (0,1)T, p\ = 1 - z\, p*2 = 1, pi = 0, and pi = v, where v € [0.5,1].

Solution (1) and (2) meets in point xl* = (0, l)T,a;2* = (0,1)T,x3* = (0,l)T,x4* = (l,0)T.

Some single points of Nash equilibrium also are:

p* 1 * X O* xz q * xA 4* X Pi P*2 Pt Pt

0 (1, 0) (0,1) (1, o) (0,1) 0 0 1 2

0 (1, 0) (1,0) (1, o) (0,1) 0 0 0 1

5. Acknowledgements

This work was partially supported by the research grants P2016-1228 of National University of Mongolia and by the research grant 15-07-03827 of Russian Foundation for Basic Research.

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Enkhbat Rentsen , Dr. Sc., Professor, National University of Mongolia, Baga toiruu 4, Sukhbaatar district, Ulaanbaatar, Mongolia, tel.: 97699278403 (e-mail: renkhbat46@yahoo.com)

Sukhee Batbileg, Dr. Ph., Lecturer, National University of Mongolia, Baga toiruu 4, Sukhbaatar district, Ulaanbaatar, Mongolia, tel.: 97699182806 (e-mail: batbileg@seas.num.edu.mn)

Natsagdorj Tungalag, Dr. Ph., Professor, National University of Mongolia, Baga toiruu 4, Sukhbaatar district, Ulaanbaatar, Mongolia, tel.: 97699182806 (e-mail: tungalag88@yahoo.com)

Anikin Anton, Matrosov Institute for System Dynamics and Control Theory SB RAS, 134, Lermontov st., Irkutsk, 664033, Russia, tel.: (3952)453082

(e-mail: anton.anikin@htower.ru)

Gornov Alexander, Doctor of Sciences (Technics) Professor, Matrosov Institute for System Dynamics and Control Theory SB RAS, 134, Lermontov st., Irkutsk, 664033 Russia, tel.: (3952)453082 (e-mail: gornov@icc.ru)

Р. Энхбат, С. Батбилэг, Н. Тунгалаг, А. Аникин, А. Горнов Вычислительный метод для игр с ненулевой суммой для IV-лиц

Аннотация. Рассматривается игра с ненулевой суммой для ^игроков. Хорошо известно, что игра может быть сведена к глобальной задаче оптимизации [5; 7; 14]. Обобщая результаты, полученные Миллсом [5], мы имеем условия гло-

бальной оптимальности для равновесия по Нэшу. Для отыскания равновесий по Нэшу в построенной игре используется подход, базирующийся на ее редукции к невыпуклой задаче оптимизации; для решения последней применяется алгоритм глобального поиска, мы применяем Curvilinear Multistart Algorithm [2; 3], специально модифицированный для нашей редуцированной задачи невыпуклой оптимизации. Предложенный алгоритм протестирован на играх с тремя и четырьмя игроками. Кроме того, мы рассматривали маркетинговую задачу соревнования по ценам трех компаний на хлебном рынке Улан-Батора. Приводятся и анализируются результаты вычислительного эксперимента.

Энхбат Рэнцэн , доктор физико-математических наук, профессор, Национальный университет Монголии, 4, ул. Бага Тойру, Округ Сухэ-Батора, г. Улан-Батор, Монголия, тел.: 976-99278403 (e-mail: renkhbat46@yaboo.com)

Батбилэг Сухэ , кандидат физико-математических наук, преподаватель, Национальный университет Монголии, 4, ул. Бага Тойру, Округ Сухэ-Батора, г. Улан-Батор, Монголия, тел.: 976-99182806 (e-mail: batbileg@seas .num. edu.mn)

Тунгалаг Нацагдорж , кандидат экономических наук, профессор, Национальный университет Монголии, 4, ул. Бага Тойру, Округ Сухэ-Батора, г. Улан-Батор, Монголия, тел.: 976-99146101 (e-mail: tungalag88@yaboo.com)

Аникин Антон, Институт динамики систем и теории управления им. В. М. Матросова СО РАН, 664033, г. Иркутск, ул. Лермонтова, 134, тел.: (3952)453082 (e-mail: anton.anikin@htower.ru)

Горнов Александр, доктор технических наук, профессор, Институт динамики систем и теории управления им. В. М. Матросова СО РАН, 664033, г. Иркутск, ул. Лермонтова, 134, тел.: (3952)453082 (e-mail: gornov@icc.ru)