MATRITSALI O'YINNI CHIZIQLI DASTURLASH YORDAMIDA YECHISH SOLVING A MATRIX GAME USING LINEAR PROGRAMMING
Keywords:
Kalit so'zlar: matritsali o'yin, chiziqli dasturlash, simpleks usul, antagonistik o'yin, to'lov matritsasi, aralash strategiya, o'yin qiymati, dual masala., Keywords: matrix game, linear programming, simplex method, antagonistic game, payoff matrix, mixed strategy, game value, dual problem.Abstract
ANNOTATSIYA
Ushbu maqolada ikki o'yinchili antagonistik matritsali o'yinlarni chiziqli dasturlash masalalariga keltirish metodologiyasi batafsil bayon etilgan. Matritsali o'yinning asosiy matematik tushunchalari — to'lov matritsasi, sof va aralash strategiyalar, o'yin qiymati va optimal strategiyalar — izchil ravishda tushuntirilgan. Har ikki o'yinchi uchun dual chiziqli dasturlash masalalari tuzilish algoritmi keltirilgan. Amaliy qism sifatida 3×3 o'lchamli to'lov matritsa asosida muayyan o'yin masalasi qo'yilgan va simpleks usuli yordamida qadam-baqadam to'liq yechilgan: optimal aralash strategiyalar hamda o'yin qiymati aniqlanган. Tadqiqot natijalari qaror qabul qilish, iqtisodiy raqobat tahlili va muhandislik loyihalarida keng qo'llanilishi mumkin.
ABSTRACT
This paper presents a detailed methodology for reducing two-player antagonistic matrix games to linear programming problems. Core mathematical concepts of matrix games — payoff matrix, pure and mixed strategies, game value, and optimal strategies — are systematically explained. An algorithm for formulating dual linear programming problems for both players is provided. As a practical illustration, a specific 3×3 payoff matrix game is constructed and fully solved step-by-step via the simplex method, yielding optimal mixed strategies and the game value. The findings are applicable to decision-making, economic competition analysis, and engineering design.
