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ТЕХНИЧЕСКИЕ НАУКИ
СРАВНИТЕЛЬНЫЙ АНАЛИЗ ДЛЯ АЛГОРИТМА РЕШЕНИЯ ДВУХПАРАМЕТРИЧЕСКИХ ИГРОВЫХ МОДЕЛЕЙ Багинян М.К. Email: Baghinyan17113@scientifictext.ru
Багинян Мгер Каренович - аспирант, отдел компьютерных систем и информатики, Национальный политехнический университет Армении, г. Ереван, Республика Армения
Аннотация: в этой статье выполнен сравнительный анализ для алгоритма решения двухпараметрических игровых моделей. В первой части исследования представлены сравнения функциональных значений цены игры и вероятностей стратегий, с численными решениями задачи в разных точках d, t параметров. Такой же анализ выполнен для разных количеств дискретов. Точность и надежность результатов проанализированы с помощью числового примера. Все представленные данные получены с помощью пакета прикладных программ. Окончательные данные представлены в форме таблиц.
Ключевые слова: параметрическая игровая модель, дифференциальные преобразования, сравнительный анализ, задача математического линейного программирования, алгоритм решения двухпараметрических игровых моделей.
COMPARATIVE ANALYSIS FOR TWO-PARAMETRIC GAME MODEL SOLVER ALGORITHM Baghinyan M.K.
Baghinyan Mher Karen - PhD Graduate, DEPARTMENT OF COMPUTER SCIENCE AND INFORMATICS, NATIONAL POLYTECHNIC UNIVERSITY OF ARMENIA, YEREVAN, REPUBLIC OF ARMENIA
Abstract: in this paper the comparative analysis was performed for two-parametric game model solver algorithm by means of a numeric example. In the first part of this paper the functional meanings of the value of the game and the probabilities of the given strategies were compared with the numerical solutions for the given points of d,t parameters. The same analysis was performed for the different numbers of discrets. The accuracy and reliability of the outcome values was analyzed. All the data shown in this paper were obtained with the help of an applied software package. The final results are given in the form of tables.
Keywords: parametric game model, differential transform, comparative analysis, parametric linear programming problem, two-parametric game model solver algorithm.
UDC 004.052.42
Experimental details
To check the performance and accuracy of the two-parametric game model solver algorithm [1, 2], we consider an example:
The following game model payoff matrix with d, t functional coefficients is presented below:
"1 + t 2d 1"
P =
3 - 2t d 2
(1)
1 + 3t 4 d
The parametric linear programming problem of the game will be the following:
(3)
F(d,t) = x/d,t) + x2(d,t) + x3(d,t) ^ max
xl ( d,t )x2 ( d,t ).x3( d,t). (1 + t)xl(d,t) + 2dx2(d,t) + x3(d,t) < 1, (3 - 2t)xx(d,t) + dx2(d,t) + 2x3(d,t) < 1,
(2)
(1 + 3t)xl(d,t) + 4x2(d,t) + dx3(d,t) < 1, xr(d,t),x2(d,t ),x3 (d,t) > 0 : For solving the game we choose the following parameters: KY = K2 = 5,
H = H 2 = 1 (the parameters needed to apply the method of differential transforms [4, 5]), d=10, t=10. With the help of the applied package of program [3] the following solutions are obtained:
The parametric function of the value of the game:
V(d,t) = 1 /(0.127(d -10/(t -10/ -0.01(d - 10/(t -10/ + 0.0008(d -10/(t-10/ -
- 7.16* 10-5 (d -10/ (t -10 / + 5.56* 10-6 (d -10/ (t -10)° - 4.04* 10-7 (d -10/ (t -10)° + + 2.3* 10-19( d -10)° (t -10/ -1.56* 10-20(d - 10/(t -10 / + 2.96* 10-21(d -10/ (t -10/ + + 9.93* 10-23(d -10/(t -10/ + 6.6 * 10-24(d -10)4(t -10/ - 2.11* 10-24(d -10/(t -10/ + +1.86* 10-20(d -10 J0 (t -10 / -1.4 * 10-20(d -10)1 (t -10)2 +1.75* 10-21(d -10)2 (t -10 / -
- 2.14* 10-22( d -10/ (t -10)2 + 2.89* 10-23 (d -10)4 (t -10) 2 - 3.25* 10-24(d -10)5 (t -10)2 -
- 8.79* 10-21(d -10)° (t -10)3 + 8.54* 10-22(d - 10)1(t -10)3 - 9.6 * 10-23(d -10)2 (t -10)3 + + 9.46* 10-24(d -10)3 (t -10)3 - 3.03* 10-25(d -10)4 (t -10)3 + 2.3* 10-26(d -10)5 (t -10)3 + + 3.44* 10-22(d -10)°(t -10)4 -8.68* 10-23(d -10)l(t -10)4 +17* 10-23(d -10)2(t -10)4 --1.68* 10-24(d -10)3 (t -10)4 +1.96* 10-25(d -10)4 (t -10)4 - 2.16* 10-26(d -10)5 (t -10)4 + + 4.96* 10-23(d -10(t -10)5 - 7.83* 10-24(d - 10)1(t -10)5 +1.03* 10-24(d -10)2 (t -10)5 -
- 7.43* 10-26(d -10)3 (t -10)5 + 1.26* 10-26(d -10)4 (t -10)5 - 2.5* 10-27(d -10)5 (t -10)5) The functions of the probabilities of the given strategies:
X (d, t) = 0 (4) X2(d,t) = (-0.004d + 2.3 * 10-19t + (d -10/ (-6.97 * 10-24t + 6.97 * 10-23) -
- 7.25* 10-28(d -10/(t -10/ - 2.67* 10-27(d -10/(t -10/ --1.49* 10-26(d -10/(t -10/ - 5.2 * 10-25(d -10/(t -10)2 -
- 3.75* 10-7 (d -10/ + (d -10/ (4.54* 10-23t + 4.54* 10-22) + + 5.55* 10-27(d -10/(t -10/ + 4.44* 10-26(d -10/(t -10/ + + 3.03* 10-25(d -10/(t -10/ + 2.22* 10-24(d -10/(t -10)2 + + 3.77* 10-6 (d -10/ + (d -10/(-2.11* 10-22t + 2.11* 10-21) -
-3.63* 10-26(d -10/(t -10/ - 4.65* 10-25(d - 10/(i -10)4 - (5)
- 2.68* 10-24(d -10/ (t -10/ -1.98* 10-23(d - 10/(i -10)2 -
- 3.86* 10-5 (d -10)3 + (d -10)2 (2.96* 10-21t - 2.96* 10-20) + + 5.81* 10-25(d -10)2(t -10/ + 4.85* 10-24(d -10)2(t -10)2 + + 2.06* 10-23(d -10)2(t -10/ + 3.57* 10-22(d -10)2(t -10)2 + + 0.0004(d -10)2 + (d -10)(-3.55* 10-20t + 3.55* 10-19) -
- 6.61* 10-24(d - 10)(t -10/ - 4.79* 10-23(d - 10)(t -10)4 -
- 7.94* 10-23( d - 10)(t -10 )3 - 4.12* 10-21(d - 10)(t -10)2 + 4.96* 10-23(d -10 / + + 3.44* 10-22(d -10/ +1.16* 10-21(t -10/ +186* 10-20(t -10)2 + 0.886)/V(d,t)
Xi(d,t) = (-0.006d + (d-10)5(4.86* 10-24t-4.8* 10-23)-1.78* 10-27)(d-10)5(t-10)5 --1. 86* 10-26(d-10)5(t-10/ + 3. 79* 10-26(d-10)5(t-10)3 -2. 73* 10-24(d-10)5(t-10)2 -- 2.88* 10-8 (d -10)5 + (d -10 )4 (-3.8* 10-23t + 3.88* 10-22) + 7.12* 10-27 (d -10 )4 (t -10 )5 + +1.51* 10-25(d-10)4(t-10)4 -6.07* 10-25(d-10)4(t-10)3 + 2.67* 10-23(d-10)4(t-10)2 + +1.7* 10-6(d-10)4 + (d-10)3(3.11* 10-22t-3.11* 10-21)-3.79* 10-26(d-10)3(t-10)5 --1.21* 10-24(d-10)3(t-10)4 +1.21* 10-23(d-10)3(t-10)3 -1.94* 10-22(d-10)3(t-10)2 - (6) -3.3* 10-5(d-10)3 + 4.55* 10-25(d-10)2(t-10)5 +1.21* 10-23(d-10)2(t-10)4 --1.16* 10-22(d -10)2 (t -10)3 +1.4* 10-21(d -10)2 (t -10)2 + 0.0004^d -10)2 + + (d-10)(1.99* 10-20t-1.99* 10-19)-1.21* 10-24(d- 10)(t-10)5 --3.88* 10-23(d-10)(t-10)4 + 9.33* 10-22(d- 10)(t-10)3 --9.95* 10-21(d-10)(t-10)2 -9.95* 10-21(t-10)3 + 0.14/V(d,t) Results and discussion
In order to check the accuracy of the obtained results, we perform comparative analysis between the values of the Xx(d,t), X2 (d,t), X3(d,t), V(d,t) functions in selected points of the d,t parameters and the numerical results of the same problem for the 10 <= d < 20.4 , 10 <= t <= 50 ranges of the given d, t parameters. As we see from table 1 the results obtained by both ways of solving the model are equivalent with the exception of the last points for d parameter, which is explained by the level of bias while determining the bounds of optimality of d,t parameters.
Table 1. Comparative analysis
t d X1 (d, t) X2 (d, t) X3 (d, t) V (d, t) X1 X2 X3 V
10 10 0 0.36 0.64 7.84 0 0.36 0.64 7.84
10 10.1 0 0.359684 0.640316 7.905929 0 0.359 0.6403 7.905
10 10.2 0 0.359375 0.640625 7.971875 0 0.359 0.6406 7.971
10 10.3 0 0.359073 0.640927 8.037838 0 0.359 0.6409 8.037
10 10.4 0 0.358779 0.641221 8.103817 0 0.358 0.641 8.103
10 10.5 0 0.358491 0.641509 8.169811 0 0.358 0.641 8.169
10.1 10 0 0.36 0.64 7.84 0 0.36 0.64 7.84
10.2 10 0 0.36 0.64 7.84 0 0.36 0.64 7.84
10.3 10 0 0.36 0.64 7.84 0 0.36 0.64 7.84
10.4 10 0 0.36 0.64 7.84 0 0.36 0.64 7.84
10.5 10 0 0.36 0.64 7.84 0 0.36 0.64 7.84
11 11 0 0.357143 0.642857 8.500002 0 0.357 0.642 8.5
11 12 0 0.354825 0.645175 9.161417 0 0.354 0.645 9.16
11 13 0 0.352789 0.647211 9.825106 0 0.352 0.647 9.82
11 14 0 0.350512 0.649488 10.49612 0 0.351 0.648 10.48
11 15 0 0.346841 0.653159 11.18979 0 0.35 0.65 11.15
12 12 0 0.354825 0.645175 9.161417 0 0.354 0.645 9.16
13 12 0 0.354825 0.645175 9.161417 0 0.354 0.645 9.16
14 12 0 0.354825 0.645175 9.161417 0 0.354 0.645 9.16
15 12 0 0.354825 0.645175 9.161417 0 0.354 0.645 9.16
20 20 0 0.0955 0.904 19.019 0 0.345 0.654 14.47
20 50 0 0.0955 0.904 19.019 0 0.345 0.654 14.47
20.4 50 0 0.0009 0.999 21.156 0 0.345 0.654 14.738
The values of the function V(d,t) in some d,t points for different K1rK2 values of discrets are portrayed in table 2 as well as numerical items of the game value of non parameter game for each d,t point. The research of the experiments shows that the accuracy of the results grows while increasing the numbers of K1rK2 discrets. In particular, for the given example the most precise results are achieved in case of the Kj=10,K2=10 counts of discrets.
Table 2. Accuracy of the results
t d V(d,t) V
KI=2,K2=2 Ki=3,K2=3 KI=4,K2=4 Ki=5,K2=5 Ki=10,K2=10
10 10 7.84 7.84 7.84 7.84 7.84 7.84
10 10.1 7.905924 7.905929 7.905929 7.905929 7.905929 7.905
10 10.2 7.971839 7.971876 7.971875 7.971875 7.971875 7.971
10 10.3 8.037716 8.037841 8.037838 8.037838 8.037838 8.037
10 10.4 8.103525 8.103826 8.103817 8.103817 8.103817 8.103
10 10.5 8.169236 8.169834 8.169811 8.169811 8.169811 8.169
10.1 10 7.84 7.84 7.84 7.84 7.84 7.84
10.2 10 7.84 7.84 7.84 7.84 7.84 7.84
10.3 10 7.84 7.84 7.84 7.84 7.84 7.84
10.4 10 7.84 7.84 7.84 7.84 7.84 7.84
10.5 10 7.84 7.84 7.84 7.84 7.84 7.84
11 11 8.495203 8.500375 8.499973 8.500002 8.5 8.5
11 12 9.119901 9.167806 9.16033 9.161417 9.16129 9.16
11 13 9.674784 9.859236 9.815624 9.825106 9.823535 9.82
11 14 10.11665 10.60858 10.45068 10.49612 10.48663 10.48
11 15 10.40509 11.4739 11.03368 11.18979 11.15179 11.15
12 12 9.119901 9.167806 9.16033 9.161417 9.16129 9.16
13 12 9.119901 9.167806 9.16033 9.161417 9.16129 9.16
14 12 9.119901 9.167806 9.16033 9.161417 9.16129 9.16
15 12 9.119901 9.167806 9.16033 9.161417 9.16129 9.16
Conclusion
Comparative analysis was performed between the acquired results and already known solutions. The obtained results was analyzed for the different values of K1, K2 parameters for the function of the game V(d,t) and for the Xl(d,t), X2(d,t), X3(d,t) functions in some of the d,t points. The precision of the presented values were substantiate.
References / Список литературы
1. Baghinyan M.K. A solution method for none cooperative multi-parametric game models based on differential transforms, Bulletin NPUA. Series of Technical Sciences. Yerevan, 2017. Volume 1. Р. 226-232.
2. Baghinyan M.K. Ап applied software package for solving parametric game models, proceedings of engineering academy of armenia, scientific and technological collected articles. Yerevan, 2017. Volume 14. № 2. Р. 303-308.
3. Baghinyan M.K. Solving parametric game models by applying differential transform method. Proceedings of NAS RA and NPUA, Series of technical sciences, 2017. Issue. 70. № 1. Р. 123-130.
4. Пухов Г.Е. Дифференциальные преобразования и математическое моделирование физических процессов. Киев: Наукова думка, 1986. 158 с.
5. Симонян С.О., Аветисян А.Г.Прикладная теория дифференциальных преобразований: Монография. Ереван: Издательство ГИУА «Чартарагет», 2010. 361 с.
